3.10.53 \(\int \frac {A+B x}{x^{3/2} (a+b x+c x^2)^3} \, dx\)

Optimal. Leaf size=664 \[ -\frac {-A \left (36 a^2 c^2-35 a b^2 c+5 b^4\right )+c x \left (a B \left (b^2-28 a c\right )-A \left (5 b^3-32 a b c\right )\right )+a b B \left (b^2-16 a c\right )}{4 a^2 \sqrt {x} \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {3 \left (a b B \left (b^2-8 a c\right )-A \left (60 a^2 c^2-37 a b^2 c+5 b^4\right )\right )}{4 a^3 \sqrt {x} \left (b^2-4 a c\right )^2}+\frac {3 \sqrt {c} \left (a B \left (56 a^2 c^2-10 a b^2 c-8 a b c \sqrt {b^2-4 a c}+b^3 \sqrt {b^2-4 a c}+b^4\right )-A \left (60 a^2 c^2 \sqrt {b^2-4 a c}+124 a^2 b c^2-47 a b^3 c-37 a b^2 c \sqrt {b^2-4 a c}+5 b^4 \sqrt {b^2-4 a c}+5 b^5\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} a^3 \left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {3 \sqrt {c} \left (a B \left (56 a^2 c^2-10 a b^2 c+8 a b c \sqrt {b^2-4 a c}-b^3 \sqrt {b^2-4 a c}+b^4\right )-A \left (-60 a^2 c^2 \sqrt {b^2-4 a c}+124 a^2 b c^2-47 a b^3 c+37 a b^2 c \sqrt {b^2-4 a c}-5 b^4 \sqrt {b^2-4 a c}+5 b^5\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{4 \sqrt {2} a^3 \left (b^2-4 a c\right )^{5/2} \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {c x (A b-2 a B)-2 a A c-a b B+A b^2}{2 a \sqrt {x} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]

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Rubi [A]  time = 1.71, antiderivative size = 664, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {822, 828, 826, 1166, 205} \begin {gather*} -\frac {-A \left (36 a^2 c^2-35 a b^2 c+5 b^4\right )+c x \left (a B \left (b^2-28 a c\right )-A \left (5 b^3-32 a b c\right )\right )+a b B \left (b^2-16 a c\right )}{4 a^2 \sqrt {x} \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {3 \left (a b B \left (b^2-8 a c\right )-A \left (60 a^2 c^2-37 a b^2 c+5 b^4\right )\right )}{4 a^3 \sqrt {x} \left (b^2-4 a c\right )^2}+\frac {3 \sqrt {c} \left (a B \left (56 a^2 c^2+b^3 \sqrt {b^2-4 a c}-10 a b^2 c-8 a b c \sqrt {b^2-4 a c}+b^4\right )-A \left (60 a^2 c^2 \sqrt {b^2-4 a c}+124 a^2 b c^2+5 b^4 \sqrt {b^2-4 a c}-47 a b^3 c-37 a b^2 c \sqrt {b^2-4 a c}+5 b^5\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} a^3 \left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {3 \sqrt {c} \left (a B \left (56 a^2 c^2-b^3 \sqrt {b^2-4 a c}-10 a b^2 c+8 a b c \sqrt {b^2-4 a c}+b^4\right )-A \left (-60 a^2 c^2 \sqrt {b^2-4 a c}+124 a^2 b c^2-5 b^4 \sqrt {b^2-4 a c}-47 a b^3 c+37 a b^2 c \sqrt {b^2-4 a c}+5 b^5\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{4 \sqrt {2} a^3 \left (b^2-4 a c\right )^{5/2} \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {c x (A b-2 a B)-2 a A c-a b B+A b^2}{2 a \sqrt {x} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x^(3/2)*(a + b*x + c*x^2)^3),x]

[Out]

(3*(a*b*B*(b^2 - 8*a*c) - A*(5*b^4 - 37*a*b^2*c + 60*a^2*c^2)))/(4*a^3*(b^2 - 4*a*c)^2*Sqrt[x]) + (A*b^2 - a*b
*B - 2*a*A*c + (A*b - 2*a*B)*c*x)/(2*a*(b^2 - 4*a*c)*Sqrt[x]*(a + b*x + c*x^2)^2) - (a*b*B*(b^2 - 16*a*c) - A*
(5*b^4 - 35*a*b^2*c + 36*a^2*c^2) + c*(a*B*(b^2 - 28*a*c) - A*(5*b^3 - 32*a*b*c))*x)/(4*a^2*(b^2 - 4*a*c)^2*Sq
rt[x]*(a + b*x + c*x^2)) + (3*Sqrt[c]*(a*B*(b^4 - 10*a*b^2*c + 56*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 8*a*b*c*Sq
rt[b^2 - 4*a*c]) - A*(5*b^5 - 47*a*b^3*c + 124*a^2*b*c^2 + 5*b^4*Sqrt[b^2 - 4*a*c] - 37*a*b^2*c*Sqrt[b^2 - 4*a
*c] + 60*a^2*c^2*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(4*Sqrt[2]
*a^3*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (3*Sqrt[c]*(a*B*(b^4 - 10*a*b^2*c + 56*a^2*c^2 - b^3*S
qrt[b^2 - 4*a*c] + 8*a*b*c*Sqrt[b^2 - 4*a*c]) - A*(5*b^5 - 47*a*b^3*c + 124*a^2*b*c^2 - 5*b^4*Sqrt[b^2 - 4*a*c
] + 37*a*b^2*c*Sqrt[b^2 - 4*a*c] - 60*a^2*c^2*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sq
rt[b^2 - 4*a*c]]])/(4*Sqrt[2]*a^3*(b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]])

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 822

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x
)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 826

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 828

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[((
e*f - d*g)*(d + e*x)^(m + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[((d
+ e*x)^(m + 1)*Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x])/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c,
d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && FractionQ[m] && LtQ[m, -1]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {A+B x}{x^{3/2} \left (a+b x+c x^2\right )^3} \, dx &=\frac {A b^2-a b B-2 a A c+(A b-2 a B) c x}{2 a \left (b^2-4 a c\right ) \sqrt {x} \left (a+b x+c x^2\right )^2}-\frac {\int \frac {\frac {1}{2} \left (-5 A b^2+a b B+18 a A c\right )-\frac {7}{2} (A b-2 a B) c x}{x^{3/2} \left (a+b x+c x^2\right )^2} \, dx}{2 a \left (b^2-4 a c\right )}\\ &=\frac {A b^2-a b B-2 a A c+(A b-2 a B) c x}{2 a \left (b^2-4 a c\right ) \sqrt {x} \left (a+b x+c x^2\right )^2}-\frac {a b B \left (b^2-16 a c\right )-A \left (5 b^4-35 a b^2 c+36 a^2 c^2\right )+c \left (a B \left (b^2-28 a c\right )-A \left (5 b^3-32 a b c\right )\right ) x}{4 a^2 \left (b^2-4 a c\right )^2 \sqrt {x} \left (a+b x+c x^2\right )}+\frac {\int \frac {-\frac {3}{4} \left (a b B \left (b^2-8 a c\right )-A \left (5 b^4-37 a b^2 c+60 a^2 c^2\right )\right )-\frac {3}{4} c \left (a B \left (b^2-28 a c\right )-A \left (5 b^3-32 a b c\right )\right ) x}{x^{3/2} \left (a+b x+c x^2\right )} \, dx}{2 a^2 \left (b^2-4 a c\right )^2}\\ &=\frac {3 \left (a b B \left (b^2-8 a c\right )-A \left (5 b^4-37 a b^2 c+60 a^2 c^2\right )\right )}{4 a^3 \left (b^2-4 a c\right )^2 \sqrt {x}}+\frac {A b^2-a b B-2 a A c+(A b-2 a B) c x}{2 a \left (b^2-4 a c\right ) \sqrt {x} \left (a+b x+c x^2\right )^2}-\frac {a b B \left (b^2-16 a c\right )-A \left (5 b^4-35 a b^2 c+36 a^2 c^2\right )+c \left (a B \left (b^2-28 a c\right )-A \left (5 b^3-32 a b c\right )\right ) x}{4 a^2 \left (b^2-4 a c\right )^2 \sqrt {x} \left (a+b x+c x^2\right )}+\frac {\int \frac {\frac {3}{4} \left (a B \left (b^4-9 a b^2 c+28 a^2 c^2\right )-A \left (5 b^5-42 a b^3 c+92 a^2 b c^2\right )\right )+\frac {3}{4} c \left (a b B \left (b^2-8 a c\right )-A \left (5 b^4-37 a b^2 c+60 a^2 c^2\right )\right ) x}{\sqrt {x} \left (a+b x+c x^2\right )} \, dx}{2 a^3 \left (b^2-4 a c\right )^2}\\ &=\frac {3 \left (a b B \left (b^2-8 a c\right )-A \left (5 b^4-37 a b^2 c+60 a^2 c^2\right )\right )}{4 a^3 \left (b^2-4 a c\right )^2 \sqrt {x}}+\frac {A b^2-a b B-2 a A c+(A b-2 a B) c x}{2 a \left (b^2-4 a c\right ) \sqrt {x} \left (a+b x+c x^2\right )^2}-\frac {a b B \left (b^2-16 a c\right )-A \left (5 b^4-35 a b^2 c+36 a^2 c^2\right )+c \left (a B \left (b^2-28 a c\right )-A \left (5 b^3-32 a b c\right )\right ) x}{4 a^2 \left (b^2-4 a c\right )^2 \sqrt {x} \left (a+b x+c x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {\frac {3}{4} \left (a B \left (b^4-9 a b^2 c+28 a^2 c^2\right )-A \left (5 b^5-42 a b^3 c+92 a^2 b c^2\right )\right )+\frac {3}{4} c \left (a b B \left (b^2-8 a c\right )-A \left (5 b^4-37 a b^2 c+60 a^2 c^2\right )\right ) x^2}{a+b x^2+c x^4} \, dx,x,\sqrt {x}\right )}{a^3 \left (b^2-4 a c\right )^2}\\ &=\frac {3 \left (a b B \left (b^2-8 a c\right )-A \left (5 b^4-37 a b^2 c+60 a^2 c^2\right )\right )}{4 a^3 \left (b^2-4 a c\right )^2 \sqrt {x}}+\frac {A b^2-a b B-2 a A c+(A b-2 a B) c x}{2 a \left (b^2-4 a c\right ) \sqrt {x} \left (a+b x+c x^2\right )^2}-\frac {a b B \left (b^2-16 a c\right )-A \left (5 b^4-35 a b^2 c+36 a^2 c^2\right )+c \left (a B \left (b^2-28 a c\right )-A \left (5 b^3-32 a b c\right )\right ) x}{4 a^2 \left (b^2-4 a c\right )^2 \sqrt {x} \left (a+b x+c x^2\right )}-\frac {\left (3 c \left (a B \left (b^4-10 a b^2 c+56 a^2 c^2-b^3 \sqrt {b^2-4 a c}+8 a b c \sqrt {b^2-4 a c}\right )-A \left (5 b^5-47 a b^3 c+124 a^2 b c^2-5 b^4 \sqrt {b^2-4 a c}+37 a b^2 c \sqrt {b^2-4 a c}-60 a^2 c^2 \sqrt {b^2-4 a c}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx,x,\sqrt {x}\right )}{8 a^3 \left (b^2-4 a c\right )^{5/2}}+\frac {\left (3 c \left (a B \left (b^4-10 a b^2 c+56 a^2 c^2+b^3 \sqrt {b^2-4 a c}-8 a b c \sqrt {b^2-4 a c}\right )-A \left (5 b^5-47 a b^3 c+124 a^2 b c^2+5 b^4 \sqrt {b^2-4 a c}-37 a b^2 c \sqrt {b^2-4 a c}+60 a^2 c^2 \sqrt {b^2-4 a c}\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx,x,\sqrt {x}\right )}{8 a^3 \left (b^2-4 a c\right )^{5/2}}\\ &=\frac {3 \left (a b B \left (b^2-8 a c\right )-A \left (5 b^4-37 a b^2 c+60 a^2 c^2\right )\right )}{4 a^3 \left (b^2-4 a c\right )^2 \sqrt {x}}+\frac {A b^2-a b B-2 a A c+(A b-2 a B) c x}{2 a \left (b^2-4 a c\right ) \sqrt {x} \left (a+b x+c x^2\right )^2}-\frac {a b B \left (b^2-16 a c\right )-A \left (5 b^4-35 a b^2 c+36 a^2 c^2\right )+c \left (a B \left (b^2-28 a c\right )-A \left (5 b^3-32 a b c\right )\right ) x}{4 a^2 \left (b^2-4 a c\right )^2 \sqrt {x} \left (a+b x+c x^2\right )}+\frac {3 \sqrt {c} \left (a B \left (b^4-10 a b^2 c+56 a^2 c^2+b^3 \sqrt {b^2-4 a c}-8 a b c \sqrt {b^2-4 a c}\right )-A \left (5 b^5-47 a b^3 c+124 a^2 b c^2+5 b^4 \sqrt {b^2-4 a c}-37 a b^2 c \sqrt {b^2-4 a c}+60 a^2 c^2 \sqrt {b^2-4 a c}\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} a^3 \left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {3 \sqrt {c} \left (a B \left (b^4-10 a b^2 c+56 a^2 c^2-b^3 \sqrt {b^2-4 a c}+8 a b c \sqrt {b^2-4 a c}\right )-A \left (5 b^5-47 a b^3 c+124 a^2 b c^2-5 b^4 \sqrt {b^2-4 a c}+37 a b^2 c \sqrt {b^2-4 a c}-60 a^2 c^2 \sqrt {b^2-4 a c}\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{4 \sqrt {2} a^3 \left (b^2-4 a c\right )^{5/2} \sqrt {b+\sqrt {b^2-4 a c}}}\\ \end {align*}

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Mathematica [A]  time = 2.31, size = 628, normalized size = 0.95 \begin {gather*} \frac {\frac {A \left (-36 a^2 c^2+35 a b^2 c+32 a b c^2 x-5 b^4-5 b^3 c x\right )+a B \left (-16 a b c-28 a c^2 x+b^3+b^2 c x\right )}{a \sqrt {x} \left (4 a c-b^2\right ) (a+x (b+c x))}+\frac {\frac {3 \left (A \left (-60 a^2 c^2+37 a b^2 c-5 b^4\right )+a b B \left (b^2-8 a c\right )\right )}{\sqrt {x}}+\frac {3 \sqrt {c} \left (-\frac {\left (A \left (60 a^2 c^2 \sqrt {b^2-4 a c}+124 a^2 b c^2-47 a b^3 c-37 a b^2 c \sqrt {b^2-4 a c}+5 b^4 \sqrt {b^2-4 a c}+5 b^5\right )-a B \left (56 a^2 c^2-10 a b^2 c-8 a b c \sqrt {b^2-4 a c}+b^3 \sqrt {b^2-4 a c}+b^4\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\left (A \left (60 a^2 c^2 \sqrt {b^2-4 a c}-124 a^2 b c^2+47 a b^3 c-37 a b^2 c \sqrt {b^2-4 a c}+5 b^4 \sqrt {b^2-4 a c}-5 b^5\right )+a B \left (56 a^2 c^2-10 a b^2 c+8 a b c \sqrt {b^2-4 a c}-b^3 \sqrt {b^2-4 a c}+b^4\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} \sqrt {b^2-4 a c}}}{a^2 \left (b^2-4 a c\right )}+\frac {2 \left (A \left (-2 a c+b^2+b c x\right )-a B (b+2 c x)\right )}{\sqrt {x} (a+x (b+c x))^2}}{4 a \left (b^2-4 a c\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(x^(3/2)*(a + b*x + c*x^2)^3),x]

[Out]

((2*(-(a*B*(b + 2*c*x)) + A*(b^2 - 2*a*c + b*c*x)))/(Sqrt[x]*(a + x*(b + c*x))^2) + (a*B*(b^3 - 16*a*b*c + b^2
*c*x - 28*a*c^2*x) + A*(-5*b^4 + 35*a*b^2*c - 36*a^2*c^2 - 5*b^3*c*x + 32*a*b*c^2*x))/(a*(-b^2 + 4*a*c)*Sqrt[x
]*(a + x*(b + c*x))) + ((3*(a*b*B*(b^2 - 8*a*c) + A*(-5*b^4 + 37*a*b^2*c - 60*a^2*c^2)))/Sqrt[x] + (3*Sqrt[c]*
(-(((-(a*B*(b^4 - 10*a*b^2*c + 56*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 8*a*b*c*Sqrt[b^2 - 4*a*c])) + A*(5*b^5 - 4
7*a*b^3*c + 124*a^2*b*c^2 + 5*b^4*Sqrt[b^2 - 4*a*c] - 37*a*b^2*c*Sqrt[b^2 - 4*a*c] + 60*a^2*c^2*Sqrt[b^2 - 4*a
*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/Sqrt[b - Sqrt[b^2 - 4*a*c]]) - ((a*B*(b^4
- 10*a*b^2*c + 56*a^2*c^2 - b^3*Sqrt[b^2 - 4*a*c] + 8*a*b*c*Sqrt[b^2 - 4*a*c]) + A*(-5*b^5 + 47*a*b^3*c - 124
*a^2*b*c^2 + 5*b^4*Sqrt[b^2 - 4*a*c] - 37*a*b^2*c*Sqrt[b^2 - 4*a*c] + 60*a^2*c^2*Sqrt[b^2 - 4*a*c]))*ArcTan[(S
qrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(Sqrt[2]*Sqrt[b^2 - 4*a*c]
))/(a^2*(b^2 - 4*a*c)))/(4*a*(b^2 - 4*a*c))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 15.14, size = 986, normalized size = 1.48 \begin {gather*} \frac {-15 A x^2 b^6-30 A c x^3 b^5+3 a B x^2 b^5-25 a A x b^5-15 A c^2 x^4 b^4+6 a B c x^3 b^4+91 a A c x^2 b^4-8 a^2 A b^4+5 a^2 B x b^4+3 a B c^2 x^4 b^3+227 a A c^2 x^3 b^3-20 a^2 B c x^2 b^3+194 a^2 A c x b^3+111 a A c^3 x^4 b^2-49 a^2 B c^2 x^3 b^2-25 a^2 A c^2 x^2 b^2+64 a^3 A c b^2-37 a^3 B c x b^2-24 a^2 B c^3 x^4 b-392 a^2 A c^3 x^3 b-4 a^3 B c^2 x^2 b-364 a^3 A c^2 x b-180 a^2 A c^4 x^4+28 a^3 B c^3 x^3-128 a^4 A c^2-324 a^3 A c^3 x^2+44 a^4 B c^2 x}{4 a^3 \left (4 a c-b^2\right )^2 \sqrt {x} \left (c x^2+b x+a\right )^2}-\frac {3 \left (5 \sqrt {2} A \sqrt {c} b^5-\sqrt {2} a B \sqrt {c} b^4+5 \sqrt {2} A \sqrt {c} \sqrt {b^2-4 a c} b^4-47 \sqrt {2} a A c^{3/2} b^3-\sqrt {2} a B \sqrt {c} \sqrt {b^2-4 a c} b^3+10 \sqrt {2} a^2 B c^{3/2} b^2-37 \sqrt {2} a A c^{3/2} \sqrt {b^2-4 a c} b^2+124 \sqrt {2} a^2 A c^{5/2} b+8 \sqrt {2} a^2 B c^{3/2} \sqrt {b^2-4 a c} b-56 \sqrt {2} a^3 B c^{5/2}+60 \sqrt {2} a^2 A c^{5/2} \sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{8 a^3 \left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {3 \left (-5 \sqrt {2} A \sqrt {c} b^5+\sqrt {2} a B \sqrt {c} b^4+5 \sqrt {2} A \sqrt {c} \sqrt {b^2-4 a c} b^4+47 \sqrt {2} a A c^{3/2} b^3-\sqrt {2} a B \sqrt {c} \sqrt {b^2-4 a c} b^3-10 \sqrt {2} a^2 B c^{3/2} b^2-37 \sqrt {2} a A c^{3/2} \sqrt {b^2-4 a c} b^2-124 \sqrt {2} a^2 A c^{5/2} b+8 \sqrt {2} a^2 B c^{3/2} \sqrt {b^2-4 a c} b+56 \sqrt {2} a^3 B c^{5/2}+60 \sqrt {2} a^2 A c^{5/2} \sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{8 a^3 \left (b^2-4 a c\right )^{5/2} \sqrt {b+\sqrt {b^2-4 a c}}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(A + B*x)/(x^(3/2)*(a + b*x + c*x^2)^3),x]

[Out]

(-8*a^2*A*b^4 + 64*a^3*A*b^2*c - 128*a^4*A*c^2 - 25*a*A*b^5*x + 5*a^2*b^4*B*x + 194*a^2*A*b^3*c*x - 37*a^3*b^2
*B*c*x - 364*a^3*A*b*c^2*x + 44*a^4*B*c^2*x - 15*A*b^6*x^2 + 3*a*b^5*B*x^2 + 91*a*A*b^4*c*x^2 - 20*a^2*b^3*B*c
*x^2 - 25*a^2*A*b^2*c^2*x^2 - 4*a^3*b*B*c^2*x^2 - 324*a^3*A*c^3*x^2 - 30*A*b^5*c*x^3 + 6*a*b^4*B*c*x^3 + 227*a
*A*b^3*c^2*x^3 - 49*a^2*b^2*B*c^2*x^3 - 392*a^2*A*b*c^3*x^3 + 28*a^3*B*c^3*x^3 - 15*A*b^4*c^2*x^4 + 3*a*b^3*B*
c^2*x^4 + 111*a*A*b^2*c^3*x^4 - 24*a^2*b*B*c^3*x^4 - 180*a^2*A*c^4*x^4)/(4*a^3*(-b^2 + 4*a*c)^2*Sqrt[x]*(a + b
*x + c*x^2)^2) - (3*(5*Sqrt[2]*A*b^5*Sqrt[c] - Sqrt[2]*a*b^4*B*Sqrt[c] - 47*Sqrt[2]*a*A*b^3*c^(3/2) + 10*Sqrt[
2]*a^2*b^2*B*c^(3/2) + 124*Sqrt[2]*a^2*A*b*c^(5/2) - 56*Sqrt[2]*a^3*B*c^(5/2) + 5*Sqrt[2]*A*b^4*Sqrt[c]*Sqrt[b
^2 - 4*a*c] - Sqrt[2]*a*b^3*B*Sqrt[c]*Sqrt[b^2 - 4*a*c] - 37*Sqrt[2]*a*A*b^2*c^(3/2)*Sqrt[b^2 - 4*a*c] + 8*Sqr
t[2]*a^2*b*B*c^(3/2)*Sqrt[b^2 - 4*a*c] + 60*Sqrt[2]*a^2*A*c^(5/2)*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*S
qrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(8*a^3*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (3*(-5*Sqrt[2]
*A*b^5*Sqrt[c] + Sqrt[2]*a*b^4*B*Sqrt[c] + 47*Sqrt[2]*a*A*b^3*c^(3/2) - 10*Sqrt[2]*a^2*b^2*B*c^(3/2) - 124*Sqr
t[2]*a^2*A*b*c^(5/2) + 56*Sqrt[2]*a^3*B*c^(5/2) + 5*Sqrt[2]*A*b^4*Sqrt[c]*Sqrt[b^2 - 4*a*c] - Sqrt[2]*a*b^3*B*
Sqrt[c]*Sqrt[b^2 - 4*a*c] - 37*Sqrt[2]*a*A*b^2*c^(3/2)*Sqrt[b^2 - 4*a*c] + 8*Sqrt[2]*a^2*b*B*c^(3/2)*Sqrt[b^2
- 4*a*c] + 60*Sqrt[2]*a^2*A*c^(5/2)*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*
a*c]]])/(8*a^3*(b^2 - 4*a*c)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]])

________________________________________________________________________________________

fricas [B]  time = 71.08, size = 12534, normalized size = 18.88

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(3/2)/(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

1/8*(3*sqrt(1/2)*((a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*x^5 + 2*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3
)*x^4 + (a^3*b^6 - 6*a^4*b^4*c + 32*a^6*c^3)*x^3 + 2*(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*x^2 + (a^5*b^4 - 8
*a^6*b^2*c + 16*a^7*c^2)*x)*sqrt(-(B^2*a^2*b^9 - 10*A*B*a*b^10 + 25*A^2*b^11 + 1680*(4*A*B*a^6 - 11*A^2*a^5*b)
*c^5 + 840*(2*B^2*a^6*b - 16*A*B*a^5*b^2 + 33*A^2*a^4*b^3)*c^4 - 105*(8*B^2*a^5*b^3 - 68*A*B*a^4*b^4 + 143*A^2
*a^3*b^5)*c^3 + 3*(63*B^2*a^4*b^5 - 574*A*B*a^3*b^6 + 1298*A^2*a^2*b^7)*c^2 - 3*(7*B^2*a^3*b^7 - 68*A*B*a^2*b^
8 + 165*A^2*a*b^9)*c + (a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 102
4*a^12*c^5)*sqrt((B^4*a^4*b^8 - 20*A*B^3*a^3*b^9 + 150*A^2*B^2*a^2*b^10 - 500*A^3*B*a*b^11 + 625*A^4*b^12 + 50
625*A^4*a^6*c^6 - 450*(49*A^2*B^2*a^7 - 382*A^3*B*a^6*b + 694*A^4*a^5*b^2)*c^5 + (2401*B^4*a^8 - 37436*A*B^3*a
^7*b + 218886*A^2*B^2*a^6*b^2 - 577016*A^3*B*a^5*b^3 + 591886*A^4*a^4*b^4)*c^4 - 2*(539*B^4*a^7*b^2 - 9298*A*B
^3*a^6*b^3 + 59592*A^2*B^2*a^5*b^4 - 168016*A^3*B*a^4*b^5 + 175655*A^4*a^3*b^6)*c^3 + 3*(73*B^4*a^6*b^4 - 1344
*A*B^3*a^5*b^5 + 9228*A^2*B^2*a^4*b^6 - 27980*A^3*B*a^3*b^7 + 31575*A^4*a^2*b^8)*c^2 - 2*(11*B^4*a^5*b^6 - 214
*A*B^3*a^4*b^7 + 1560*A^2*B^2*a^3*b^8 - 5050*A^3*B*a^2*b^9 + 6125*A^4*a*b^10)*c)/(a^14*b^10 - 20*a^15*b^8*c +
160*a^16*b^6*c^2 - 640*a^17*b^4*c^3 + 1280*a^18*b^2*c^4 - 1024*a^19*c^5)))/(a^7*b^10 - 20*a^8*b^8*c + 160*a^9*
b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5))*log(27/2*sqrt(1/2)*(B^3*a^3*b^14 - 15*A*B^2*a
^2*b^15 + 75*A^2*B*a*b^16 - 125*A^3*b^17 + 57600*(7*A^2*B*a^9 - 23*A^3*a^8*b)*c^8 - 64*(1372*B^3*a^10 - 15204*
A*B^2*a^9*b + 61326*A^2*B*a^8*b^2 - 88823*A^3*a^7*b^3)*c^7 + 16*(7112*B^3*a^9*b^2 - 83292*A*B^2*a^8*b^3 + 3303
00*A^2*B*a^7*b^4 - 446671*A^3*a^6*b^5)*c^6 - 4*(15920*B^3*a^8*b^4 - 197004*A*B^2*a^7*b^5 + 811446*A^2*B*a^6*b^
6 - 1115785*A^3*a^5*b^7)*c^5 + 3*(6696*B^3*a^7*b^6 - 87308*A*B^2*a^6*b^7 + 377471*A^2*B*a^5*b^8 - 541178*A^3*a
^4*b^9)*c^4 - 3*(1295*B^3*a^6*b^8 - 17704*A*B^2*a^5*b^9 + 80329*A^2*B*a^4*b^10 - 120911*A^3*a^3*b^11)*c^3 + (4
64*B^3*a^5*b^10 - 6609*A*B^2*a^4*b^11 + 31317*A^2*B*a^3*b^12 - 49360*A^3*a^2*b^13)*c^2 - (32*B^3*a^4*b^12 - 47
1*A*B^2*a^3*b^13 + 2310*A^2*B*a^2*b^14 - 3775*A^3*a*b^15)*c - (B*a^8*b^15 - 5*A*a^7*b^16 - 122880*A*a^15*c^8 -
4096*(11*B*a^15*b - 79*A*a^14*b^2)*c^7 + 1536*(44*B*a^14*b^3 - 223*A*a^13*b^4)*c^6 - 256*(169*B*a^13*b^5 - 77
0*A*a^12*b^6)*c^5 + 480*(32*B*a^12*b^7 - 143*A*a^11*b^8)*c^4 - 80*(41*B*a^11*b^9 - 187*A*a^10*b^10)*c^3 + 2*(2
12*B*a^10*b^11 - 1003*A*a^9*b^12)*c^2 - (31*B*a^9*b^13 - 152*A*a^8*b^14)*c)*sqrt((B^4*a^4*b^8 - 20*A*B^3*a^3*b
^9 + 150*A^2*B^2*a^2*b^10 - 500*A^3*B*a*b^11 + 625*A^4*b^12 + 50625*A^4*a^6*c^6 - 450*(49*A^2*B^2*a^7 - 382*A^
3*B*a^6*b + 694*A^4*a^5*b^2)*c^5 + (2401*B^4*a^8 - 37436*A*B^3*a^7*b + 218886*A^2*B^2*a^6*b^2 - 577016*A^3*B*a
^5*b^3 + 591886*A^4*a^4*b^4)*c^4 - 2*(539*B^4*a^7*b^2 - 9298*A*B^3*a^6*b^3 + 59592*A^2*B^2*a^5*b^4 - 168016*A^
3*B*a^4*b^5 + 175655*A^4*a^3*b^6)*c^3 + 3*(73*B^4*a^6*b^4 - 1344*A*B^3*a^5*b^5 + 9228*A^2*B^2*a^4*b^6 - 27980*
A^3*B*a^3*b^7 + 31575*A^4*a^2*b^8)*c^2 - 2*(11*B^4*a^5*b^6 - 214*A*B^3*a^4*b^7 + 1560*A^2*B^2*a^3*b^8 - 5050*A
^3*B*a^2*b^9 + 6125*A^4*a*b^10)*c)/(a^14*b^10 - 20*a^15*b^8*c + 160*a^16*b^6*c^2 - 640*a^17*b^4*c^3 + 1280*a^1
8*b^2*c^4 - 1024*a^19*c^5)))*sqrt(-(B^2*a^2*b^9 - 10*A*B*a*b^10 + 25*A^2*b^11 + 1680*(4*A*B*a^6 - 11*A^2*a^5*b
)*c^5 + 840*(2*B^2*a^6*b - 16*A*B*a^5*b^2 + 33*A^2*a^4*b^3)*c^4 - 105*(8*B^2*a^5*b^3 - 68*A*B*a^4*b^4 + 143*A^
2*a^3*b^5)*c^3 + 3*(63*B^2*a^4*b^5 - 574*A*B*a^3*b^6 + 1298*A^2*a^2*b^7)*c^2 - 3*(7*B^2*a^3*b^7 - 68*A*B*a^2*b
^8 + 165*A^2*a*b^9)*c + (a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 10
24*a^12*c^5)*sqrt((B^4*a^4*b^8 - 20*A*B^3*a^3*b^9 + 150*A^2*B^2*a^2*b^10 - 500*A^3*B*a*b^11 + 625*A^4*b^12 + 5
0625*A^4*a^6*c^6 - 450*(49*A^2*B^2*a^7 - 382*A^3*B*a^6*b + 694*A^4*a^5*b^2)*c^5 + (2401*B^4*a^8 - 37436*A*B^3*
a^7*b + 218886*A^2*B^2*a^6*b^2 - 577016*A^3*B*a^5*b^3 + 591886*A^4*a^4*b^4)*c^4 - 2*(539*B^4*a^7*b^2 - 9298*A*
B^3*a^6*b^3 + 59592*A^2*B^2*a^5*b^4 - 168016*A^3*B*a^4*b^5 + 175655*A^4*a^3*b^6)*c^3 + 3*(73*B^4*a^6*b^4 - 134
4*A*B^3*a^5*b^5 + 9228*A^2*B^2*a^4*b^6 - 27980*A^3*B*a^3*b^7 + 31575*A^4*a^2*b^8)*c^2 - 2*(11*B^4*a^5*b^6 - 21
4*A*B^3*a^4*b^7 + 1560*A^2*B^2*a^3*b^8 - 5050*A^3*B*a^2*b^9 + 6125*A^4*a*b^10)*c)/(a^14*b^10 - 20*a^15*b^8*c +
160*a^16*b^6*c^2 - 640*a^17*b^4*c^3 + 1280*a^18*b^2*c^4 - 1024*a^19*c^5)))/(a^7*b^10 - 20*a^8*b^8*c + 160*a^9
*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5)) + 27*(810000*A^4*a^5*c^9 + 405000*(2*A^3*B*a
^5*b - 7*A^4*a^4*b^2)*c^8 - (38416*B^4*a^7 - 422576*A*B^3*a^6*b + 1439376*A^2*B^2*a^5*b^2 - 1018856*A^3*B*a^4*
b^3 - 1957349*A^4*a^3*b^4)*c^7 + (19208*B^4*a^6*b^2 - 239896*A*B^3*a^5*b^3 + 955704*A^2*B^2*a^4*b^4 - 1067347*
A^3*B*a^3*b^5 - 571030*A^4*a^2*b^6)*c^6 - (4189*B^4*a^5*b^4 - 56807*A*B^3*a^4*b^5 + 251349*A^2*B^2*a^3*b^6 - 3
44630*A^3*B*a^2*b^7 - 77825*A^4*a*b^8)*c^5 + 3*(149*B^4*a^4*b^6 - 2161*A*B^3*a^3*b^7 + 10380*A^2*B^2*a^2*b^8 -
16225*A^3*B*a*b^9 - 1375*A^4*b^10)*c^4 - 21*(B^4*a^3*b^8 - 15*A*B^3*a^2*b^9 + 75*A^2*B^2*a*b^10 - 125*A^3*B*b
^11)*c^3)*sqrt(x)) - 3*sqrt(1/2)*((a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*x^5 + 2*(a^3*b^5*c - 8*a^4*b^3*c^
2 + 16*a^5*b*c^3)*x^4 + (a^3*b^6 - 6*a^4*b^4*c + 32*a^6*c^3)*x^3 + 2*(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*x^
2 + (a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*x)*sqrt(-(B^2*a^2*b^9 - 10*A*B*a*b^10 + 25*A^2*b^11 + 1680*(4*A*B*a^6
- 11*A^2*a^5*b)*c^5 + 840*(2*B^2*a^6*b - 16*A*B*a^5*b^2 + 33*A^2*a^4*b^3)*c^4 - 105*(8*B^2*a^5*b^3 - 68*A*B*a
^4*b^4 + 143*A^2*a^3*b^5)*c^3 + 3*(63*B^2*a^4*b^5 - 574*A*B*a^3*b^6 + 1298*A^2*a^2*b^7)*c^2 - 3*(7*B^2*a^3*b^7
- 68*A*B*a^2*b^8 + 165*A^2*a*b^9)*c + (a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^
11*b^2*c^4 - 1024*a^12*c^5)*sqrt((B^4*a^4*b^8 - 20*A*B^3*a^3*b^9 + 150*A^2*B^2*a^2*b^10 - 500*A^3*B*a*b^11 + 6
25*A^4*b^12 + 50625*A^4*a^6*c^6 - 450*(49*A^2*B^2*a^7 - 382*A^3*B*a^6*b + 694*A^4*a^5*b^2)*c^5 + (2401*B^4*a^8
- 37436*A*B^3*a^7*b + 218886*A^2*B^2*a^6*b^2 - 577016*A^3*B*a^5*b^3 + 591886*A^4*a^4*b^4)*c^4 - 2*(539*B^4*a^
7*b^2 - 9298*A*B^3*a^6*b^3 + 59592*A^2*B^2*a^5*b^4 - 168016*A^3*B*a^4*b^5 + 175655*A^4*a^3*b^6)*c^3 + 3*(73*B^
4*a^6*b^4 - 1344*A*B^3*a^5*b^5 + 9228*A^2*B^2*a^4*b^6 - 27980*A^3*B*a^3*b^7 + 31575*A^4*a^2*b^8)*c^2 - 2*(11*B
^4*a^5*b^6 - 214*A*B^3*a^4*b^7 + 1560*A^2*B^2*a^3*b^8 - 5050*A^3*B*a^2*b^9 + 6125*A^4*a*b^10)*c)/(a^14*b^10 -
20*a^15*b^8*c + 160*a^16*b^6*c^2 - 640*a^17*b^4*c^3 + 1280*a^18*b^2*c^4 - 1024*a^19*c^5)))/(a^7*b^10 - 20*a^8*
b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5))*log(-27/2*sqrt(1/2)*(B^3*a^3*
b^14 - 15*A*B^2*a^2*b^15 + 75*A^2*B*a*b^16 - 125*A^3*b^17 + 57600*(7*A^2*B*a^9 - 23*A^3*a^8*b)*c^8 - 64*(1372*
B^3*a^10 - 15204*A*B^2*a^9*b + 61326*A^2*B*a^8*b^2 - 88823*A^3*a^7*b^3)*c^7 + 16*(7112*B^3*a^9*b^2 - 83292*A*B
^2*a^8*b^3 + 330300*A^2*B*a^7*b^4 - 446671*A^3*a^6*b^5)*c^6 - 4*(15920*B^3*a^8*b^4 - 197004*A*B^2*a^7*b^5 + 81
1446*A^2*B*a^6*b^6 - 1115785*A^3*a^5*b^7)*c^5 + 3*(6696*B^3*a^7*b^6 - 87308*A*B^2*a^6*b^7 + 377471*A^2*B*a^5*b
^8 - 541178*A^3*a^4*b^9)*c^4 - 3*(1295*B^3*a^6*b^8 - 17704*A*B^2*a^5*b^9 + 80329*A^2*B*a^4*b^10 - 120911*A^3*a
^3*b^11)*c^3 + (464*B^3*a^5*b^10 - 6609*A*B^2*a^4*b^11 + 31317*A^2*B*a^3*b^12 - 49360*A^3*a^2*b^13)*c^2 - (32*
B^3*a^4*b^12 - 471*A*B^2*a^3*b^13 + 2310*A^2*B*a^2*b^14 - 3775*A^3*a*b^15)*c - (B*a^8*b^15 - 5*A*a^7*b^16 - 12
2880*A*a^15*c^8 - 4096*(11*B*a^15*b - 79*A*a^14*b^2)*c^7 + 1536*(44*B*a^14*b^3 - 223*A*a^13*b^4)*c^6 - 256*(16
9*B*a^13*b^5 - 770*A*a^12*b^6)*c^5 + 480*(32*B*a^12*b^7 - 143*A*a^11*b^8)*c^4 - 80*(41*B*a^11*b^9 - 187*A*a^10
*b^10)*c^3 + 2*(212*B*a^10*b^11 - 1003*A*a^9*b^12)*c^2 - (31*B*a^9*b^13 - 152*A*a^8*b^14)*c)*sqrt((B^4*a^4*b^8
- 20*A*B^3*a^3*b^9 + 150*A^2*B^2*a^2*b^10 - 500*A^3*B*a*b^11 + 625*A^4*b^12 + 50625*A^4*a^6*c^6 - 450*(49*A^2
*B^2*a^7 - 382*A^3*B*a^6*b + 694*A^4*a^5*b^2)*c^5 + (2401*B^4*a^8 - 37436*A*B^3*a^7*b + 218886*A^2*B^2*a^6*b^2
- 577016*A^3*B*a^5*b^3 + 591886*A^4*a^4*b^4)*c^4 - 2*(539*B^4*a^7*b^2 - 9298*A*B^3*a^6*b^3 + 59592*A^2*B^2*a^
5*b^4 - 168016*A^3*B*a^4*b^5 + 175655*A^4*a^3*b^6)*c^3 + 3*(73*B^4*a^6*b^4 - 1344*A*B^3*a^5*b^5 + 9228*A^2*B^2
*a^4*b^6 - 27980*A^3*B*a^3*b^7 + 31575*A^4*a^2*b^8)*c^2 - 2*(11*B^4*a^5*b^6 - 214*A*B^3*a^4*b^7 + 1560*A^2*B^2
*a^3*b^8 - 5050*A^3*B*a^2*b^9 + 6125*A^4*a*b^10)*c)/(a^14*b^10 - 20*a^15*b^8*c + 160*a^16*b^6*c^2 - 640*a^17*b
^4*c^3 + 1280*a^18*b^2*c^4 - 1024*a^19*c^5)))*sqrt(-(B^2*a^2*b^9 - 10*A*B*a*b^10 + 25*A^2*b^11 + 1680*(4*A*B*a
^6 - 11*A^2*a^5*b)*c^5 + 840*(2*B^2*a^6*b - 16*A*B*a^5*b^2 + 33*A^2*a^4*b^3)*c^4 - 105*(8*B^2*a^5*b^3 - 68*A*B
*a^4*b^4 + 143*A^2*a^3*b^5)*c^3 + 3*(63*B^2*a^4*b^5 - 574*A*B*a^3*b^6 + 1298*A^2*a^2*b^7)*c^2 - 3*(7*B^2*a^3*b
^7 - 68*A*B*a^2*b^8 + 165*A^2*a*b^9)*c + (a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*
a^11*b^2*c^4 - 1024*a^12*c^5)*sqrt((B^4*a^4*b^8 - 20*A*B^3*a^3*b^9 + 150*A^2*B^2*a^2*b^10 - 500*A^3*B*a*b^11 +
625*A^4*b^12 + 50625*A^4*a^6*c^6 - 450*(49*A^2*B^2*a^7 - 382*A^3*B*a^6*b + 694*A^4*a^5*b^2)*c^5 + (2401*B^4*a
^8 - 37436*A*B^3*a^7*b + 218886*A^2*B^2*a^6*b^2 - 577016*A^3*B*a^5*b^3 + 591886*A^4*a^4*b^4)*c^4 - 2*(539*B^4*
a^7*b^2 - 9298*A*B^3*a^6*b^3 + 59592*A^2*B^2*a^5*b^4 - 168016*A^3*B*a^4*b^5 + 175655*A^4*a^3*b^6)*c^3 + 3*(73*
B^4*a^6*b^4 - 1344*A*B^3*a^5*b^5 + 9228*A^2*B^2*a^4*b^6 - 27980*A^3*B*a^3*b^7 + 31575*A^4*a^2*b^8)*c^2 - 2*(11
*B^4*a^5*b^6 - 214*A*B^3*a^4*b^7 + 1560*A^2*B^2*a^3*b^8 - 5050*A^3*B*a^2*b^9 + 6125*A^4*a*b^10)*c)/(a^14*b^10
- 20*a^15*b^8*c + 160*a^16*b^6*c^2 - 640*a^17*b^4*c^3 + 1280*a^18*b^2*c^4 - 1024*a^19*c^5)))/(a^7*b^10 - 20*a^
8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5)) + 27*(810000*A^4*a^5*c^9 +
405000*(2*A^3*B*a^5*b - 7*A^4*a^4*b^2)*c^8 - (38416*B^4*a^7 - 422576*A*B^3*a^6*b + 1439376*A^2*B^2*a^5*b^2 - 1
018856*A^3*B*a^4*b^3 - 1957349*A^4*a^3*b^4)*c^7 + (19208*B^4*a^6*b^2 - 239896*A*B^3*a^5*b^3 + 955704*A^2*B^2*a
^4*b^4 - 1067347*A^3*B*a^3*b^5 - 571030*A^4*a^2*b^6)*c^6 - (4189*B^4*a^5*b^4 - 56807*A*B^3*a^4*b^5 + 251349*A^
2*B^2*a^3*b^6 - 344630*A^3*B*a^2*b^7 - 77825*A^4*a*b^8)*c^5 + 3*(149*B^4*a^4*b^6 - 2161*A*B^3*a^3*b^7 + 10380*
A^2*B^2*a^2*b^8 - 16225*A^3*B*a*b^9 - 1375*A^4*b^10)*c^4 - 21*(B^4*a^3*b^8 - 15*A*B^3*a^2*b^9 + 75*A^2*B^2*a*b
^10 - 125*A^3*B*b^11)*c^3)*sqrt(x)) + 3*sqrt(1/2)*((a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*x^5 + 2*(a^3*b^5
*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*x^4 + (a^3*b^6 - 6*a^4*b^4*c + 32*a^6*c^3)*x^3 + 2*(a^4*b^5 - 8*a^5*b^3*c +
16*a^6*b*c^2)*x^2 + (a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*x)*sqrt(-(B^2*a^2*b^9 - 10*A*B*a*b^10 + 25*A^2*b^11
+ 1680*(4*A*B*a^6 - 11*A^2*a^5*b)*c^5 + 840*(2*B^2*a^6*b - 16*A*B*a^5*b^2 + 33*A^2*a^4*b^3)*c^4 - 105*(8*B^2*a
^5*b^3 - 68*A*B*a^4*b^4 + 143*A^2*a^3*b^5)*c^3 + 3*(63*B^2*a^4*b^5 - 574*A*B*a^3*b^6 + 1298*A^2*a^2*b^7)*c^2 -
3*(7*B^2*a^3*b^7 - 68*A*B*a^2*b^8 + 165*A^2*a*b^9)*c - (a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*
b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5)*sqrt((B^4*a^4*b^8 - 20*A*B^3*a^3*b^9 + 150*A^2*B^2*a^2*b^10 - 500
*A^3*B*a*b^11 + 625*A^4*b^12 + 50625*A^4*a^6*c^6 - 450*(49*A^2*B^2*a^7 - 382*A^3*B*a^6*b + 694*A^4*a^5*b^2)*c^
5 + (2401*B^4*a^8 - 37436*A*B^3*a^7*b + 218886*A^2*B^2*a^6*b^2 - 577016*A^3*B*a^5*b^3 + 591886*A^4*a^4*b^4)*c^
4 - 2*(539*B^4*a^7*b^2 - 9298*A*B^3*a^6*b^3 + 59592*A^2*B^2*a^5*b^4 - 168016*A^3*B*a^4*b^5 + 175655*A^4*a^3*b^
6)*c^3 + 3*(73*B^4*a^6*b^4 - 1344*A*B^3*a^5*b^5 + 9228*A^2*B^2*a^4*b^6 - 27980*A^3*B*a^3*b^7 + 31575*A^4*a^2*b
^8)*c^2 - 2*(11*B^4*a^5*b^6 - 214*A*B^3*a^4*b^7 + 1560*A^2*B^2*a^3*b^8 - 5050*A^3*B*a^2*b^9 + 6125*A^4*a*b^10)
*c)/(a^14*b^10 - 20*a^15*b^8*c + 160*a^16*b^6*c^2 - 640*a^17*b^4*c^3 + 1280*a^18*b^2*c^4 - 1024*a^19*c^5)))/(a
^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5))*log(27/2*sqr
t(1/2)*(B^3*a^3*b^14 - 15*A*B^2*a^2*b^15 + 75*A^2*B*a*b^16 - 125*A^3*b^17 + 57600*(7*A^2*B*a^9 - 23*A^3*a^8*b)
*c^8 - 64*(1372*B^3*a^10 - 15204*A*B^2*a^9*b + 61326*A^2*B*a^8*b^2 - 88823*A^3*a^7*b^3)*c^7 + 16*(7112*B^3*a^9
*b^2 - 83292*A*B^2*a^8*b^3 + 330300*A^2*B*a^7*b^4 - 446671*A^3*a^6*b^5)*c^6 - 4*(15920*B^3*a^8*b^4 - 197004*A*
B^2*a^7*b^5 + 811446*A^2*B*a^6*b^6 - 1115785*A^3*a^5*b^7)*c^5 + 3*(6696*B^3*a^7*b^6 - 87308*A*B^2*a^6*b^7 + 37
7471*A^2*B*a^5*b^8 - 541178*A^3*a^4*b^9)*c^4 - 3*(1295*B^3*a^6*b^8 - 17704*A*B^2*a^5*b^9 + 80329*A^2*B*a^4*b^1
0 - 120911*A^3*a^3*b^11)*c^3 + (464*B^3*a^5*b^10 - 6609*A*B^2*a^4*b^11 + 31317*A^2*B*a^3*b^12 - 49360*A^3*a^2*
b^13)*c^2 - (32*B^3*a^4*b^12 - 471*A*B^2*a^3*b^13 + 2310*A^2*B*a^2*b^14 - 3775*A^3*a*b^15)*c + (B*a^8*b^15 - 5
*A*a^7*b^16 - 122880*A*a^15*c^8 - 4096*(11*B*a^15*b - 79*A*a^14*b^2)*c^7 + 1536*(44*B*a^14*b^3 - 223*A*a^13*b^
4)*c^6 - 256*(169*B*a^13*b^5 - 770*A*a^12*b^6)*c^5 + 480*(32*B*a^12*b^7 - 143*A*a^11*b^8)*c^4 - 80*(41*B*a^11*
b^9 - 187*A*a^10*b^10)*c^3 + 2*(212*B*a^10*b^11 - 1003*A*a^9*b^12)*c^2 - (31*B*a^9*b^13 - 152*A*a^8*b^14)*c)*s
qrt((B^4*a^4*b^8 - 20*A*B^3*a^3*b^9 + 150*A^2*B^2*a^2*b^10 - 500*A^3*B*a*b^11 + 625*A^4*b^12 + 50625*A^4*a^6*c
^6 - 450*(49*A^2*B^2*a^7 - 382*A^3*B*a^6*b + 694*A^4*a^5*b^2)*c^5 + (2401*B^4*a^8 - 37436*A*B^3*a^7*b + 218886
*A^2*B^2*a^6*b^2 - 577016*A^3*B*a^5*b^3 + 591886*A^4*a^4*b^4)*c^4 - 2*(539*B^4*a^7*b^2 - 9298*A*B^3*a^6*b^3 +
59592*A^2*B^2*a^5*b^4 - 168016*A^3*B*a^4*b^5 + 175655*A^4*a^3*b^6)*c^3 + 3*(73*B^4*a^6*b^4 - 1344*A*B^3*a^5*b^
5 + 9228*A^2*B^2*a^4*b^6 - 27980*A^3*B*a^3*b^7 + 31575*A^4*a^2*b^8)*c^2 - 2*(11*B^4*a^5*b^6 - 214*A*B^3*a^4*b^
7 + 1560*A^2*B^2*a^3*b^8 - 5050*A^3*B*a^2*b^9 + 6125*A^4*a*b^10)*c)/(a^14*b^10 - 20*a^15*b^8*c + 160*a^16*b^6*
c^2 - 640*a^17*b^4*c^3 + 1280*a^18*b^2*c^4 - 1024*a^19*c^5)))*sqrt(-(B^2*a^2*b^9 - 10*A*B*a*b^10 + 25*A^2*b^11
+ 1680*(4*A*B*a^6 - 11*A^2*a^5*b)*c^5 + 840*(2*B^2*a^6*b - 16*A*B*a^5*b^2 + 33*A^2*a^4*b^3)*c^4 - 105*(8*B^2*
a^5*b^3 - 68*A*B*a^4*b^4 + 143*A^2*a^3*b^5)*c^3 + 3*(63*B^2*a^4*b^5 - 574*A*B*a^3*b^6 + 1298*A^2*a^2*b^7)*c^2
- 3*(7*B^2*a^3*b^7 - 68*A*B*a^2*b^8 + 165*A^2*a*b^9)*c - (a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10
*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5)*sqrt((B^4*a^4*b^8 - 20*A*B^3*a^3*b^9 + 150*A^2*B^2*a^2*b^10 - 50
0*A^3*B*a*b^11 + 625*A^4*b^12 + 50625*A^4*a^6*c^6 - 450*(49*A^2*B^2*a^7 - 382*A^3*B*a^6*b + 694*A^4*a^5*b^2)*c
^5 + (2401*B^4*a^8 - 37436*A*B^3*a^7*b + 218886*A^2*B^2*a^6*b^2 - 577016*A^3*B*a^5*b^3 + 591886*A^4*a^4*b^4)*c
^4 - 2*(539*B^4*a^7*b^2 - 9298*A*B^3*a^6*b^3 + 59592*A^2*B^2*a^5*b^4 - 168016*A^3*B*a^4*b^5 + 175655*A^4*a^3*b
^6)*c^3 + 3*(73*B^4*a^6*b^4 - 1344*A*B^3*a^5*b^5 + 9228*A^2*B^2*a^4*b^6 - 27980*A^3*B*a^3*b^7 + 31575*A^4*a^2*
b^8)*c^2 - 2*(11*B^4*a^5*b^6 - 214*A*B^3*a^4*b^7 + 1560*A^2*B^2*a^3*b^8 - 5050*A^3*B*a^2*b^9 + 6125*A^4*a*b^10
)*c)/(a^14*b^10 - 20*a^15*b^8*c + 160*a^16*b^6*c^2 - 640*a^17*b^4*c^3 + 1280*a^18*b^2*c^4 - 1024*a^19*c^5)))/(
a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5)) + 27*(81000
0*A^4*a^5*c^9 + 405000*(2*A^3*B*a^5*b - 7*A^4*a^4*b^2)*c^8 - (38416*B^4*a^7 - 422576*A*B^3*a^6*b + 1439376*A^2
*B^2*a^5*b^2 - 1018856*A^3*B*a^4*b^3 - 1957349*A^4*a^3*b^4)*c^7 + (19208*B^4*a^6*b^2 - 239896*A*B^3*a^5*b^3 +
955704*A^2*B^2*a^4*b^4 - 1067347*A^3*B*a^3*b^5 - 571030*A^4*a^2*b^6)*c^6 - (4189*B^4*a^5*b^4 - 56807*A*B^3*a^4
*b^5 + 251349*A^2*B^2*a^3*b^6 - 344630*A^3*B*a^2*b^7 - 77825*A^4*a*b^8)*c^5 + 3*(149*B^4*a^4*b^6 - 2161*A*B^3*
a^3*b^7 + 10380*A^2*B^2*a^2*b^8 - 16225*A^3*B*a*b^9 - 1375*A^4*b^10)*c^4 - 21*(B^4*a^3*b^8 - 15*A*B^3*a^2*b^9
+ 75*A^2*B^2*a*b^10 - 125*A^3*B*b^11)*c^3)*sqrt(x)) - 3*sqrt(1/2)*((a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*
x^5 + 2*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*x^4 + (a^3*b^6 - 6*a^4*b^4*c + 32*a^6*c^3)*x^3 + 2*(a^4*b^5
- 8*a^5*b^3*c + 16*a^6*b*c^2)*x^2 + (a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*x)*sqrt(-(B^2*a^2*b^9 - 10*A*B*a*b^1
0 + 25*A^2*b^11 + 1680*(4*A*B*a^6 - 11*A^2*a^5*b)*c^5 + 840*(2*B^2*a^6*b - 16*A*B*a^5*b^2 + 33*A^2*a^4*b^3)*c^
4 - 105*(8*B^2*a^5*b^3 - 68*A*B*a^4*b^4 + 143*A^2*a^3*b^5)*c^3 + 3*(63*B^2*a^4*b^5 - 574*A*B*a^3*b^6 + 1298*A^
2*a^2*b^7)*c^2 - 3*(7*B^2*a^3*b^7 - 68*A*B*a^2*b^8 + 165*A^2*a*b^9)*c - (a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6
*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5)*sqrt((B^4*a^4*b^8 - 20*A*B^3*a^3*b^9 + 150*A^2*B^
2*a^2*b^10 - 500*A^3*B*a*b^11 + 625*A^4*b^12 + 50625*A^4*a^6*c^6 - 450*(49*A^2*B^2*a^7 - 382*A^3*B*a^6*b + 694
*A^4*a^5*b^2)*c^5 + (2401*B^4*a^8 - 37436*A*B^3*a^7*b + 218886*A^2*B^2*a^6*b^2 - 577016*A^3*B*a^5*b^3 + 591886
*A^4*a^4*b^4)*c^4 - 2*(539*B^4*a^7*b^2 - 9298*A*B^3*a^6*b^3 + 59592*A^2*B^2*a^5*b^4 - 168016*A^3*B*a^4*b^5 + 1
75655*A^4*a^3*b^6)*c^3 + 3*(73*B^4*a^6*b^4 - 1344*A*B^3*a^5*b^5 + 9228*A^2*B^2*a^4*b^6 - 27980*A^3*B*a^3*b^7 +
31575*A^4*a^2*b^8)*c^2 - 2*(11*B^4*a^5*b^6 - 214*A*B^3*a^4*b^7 + 1560*A^2*B^2*a^3*b^8 - 5050*A^3*B*a^2*b^9 +
6125*A^4*a*b^10)*c)/(a^14*b^10 - 20*a^15*b^8*c + 160*a^16*b^6*c^2 - 640*a^17*b^4*c^3 + 1280*a^18*b^2*c^4 - 102
4*a^19*c^5)))/(a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^
5))*log(-27/2*sqrt(1/2)*(B^3*a^3*b^14 - 15*A*B^2*a^2*b^15 + 75*A^2*B*a*b^16 - 125*A^3*b^17 + 57600*(7*A^2*B*a^
9 - 23*A^3*a^8*b)*c^8 - 64*(1372*B^3*a^10 - 15204*A*B^2*a^9*b + 61326*A^2*B*a^8*b^2 - 88823*A^3*a^7*b^3)*c^7 +
16*(7112*B^3*a^9*b^2 - 83292*A*B^2*a^8*b^3 + 330300*A^2*B*a^7*b^4 - 446671*A^3*a^6*b^5)*c^6 - 4*(15920*B^3*a^
8*b^4 - 197004*A*B^2*a^7*b^5 + 811446*A^2*B*a^6*b^6 - 1115785*A^3*a^5*b^7)*c^5 + 3*(6696*B^3*a^7*b^6 - 87308*A
*B^2*a^6*b^7 + 377471*A^2*B*a^5*b^8 - 541178*A^3*a^4*b^9)*c^4 - 3*(1295*B^3*a^6*b^8 - 17704*A*B^2*a^5*b^9 + 80
329*A^2*B*a^4*b^10 - 120911*A^3*a^3*b^11)*c^3 + (464*B^3*a^5*b^10 - 6609*A*B^2*a^4*b^11 + 31317*A^2*B*a^3*b^12
- 49360*A^3*a^2*b^13)*c^2 - (32*B^3*a^4*b^12 - 471*A*B^2*a^3*b^13 + 2310*A^2*B*a^2*b^14 - 3775*A^3*a*b^15)*c
+ (B*a^8*b^15 - 5*A*a^7*b^16 - 122880*A*a^15*c^8 - 4096*(11*B*a^15*b - 79*A*a^14*b^2)*c^7 + 1536*(44*B*a^14*b^
3 - 223*A*a^13*b^4)*c^6 - 256*(169*B*a^13*b^5 - 770*A*a^12*b^6)*c^5 + 480*(32*B*a^12*b^7 - 143*A*a^11*b^8)*c^4
- 80*(41*B*a^11*b^9 - 187*A*a^10*b^10)*c^3 + 2*(212*B*a^10*b^11 - 1003*A*a^9*b^12)*c^2 - (31*B*a^9*b^13 - 152
*A*a^8*b^14)*c)*sqrt((B^4*a^4*b^8 - 20*A*B^3*a^3*b^9 + 150*A^2*B^2*a^2*b^10 - 500*A^3*B*a*b^11 + 625*A^4*b^12
+ 50625*A^4*a^6*c^6 - 450*(49*A^2*B^2*a^7 - 382*A^3*B*a^6*b + 694*A^4*a^5*b^2)*c^5 + (2401*B^4*a^8 - 37436*A*B
^3*a^7*b + 218886*A^2*B^2*a^6*b^2 - 577016*A^3*B*a^5*b^3 + 591886*A^4*a^4*b^4)*c^4 - 2*(539*B^4*a^7*b^2 - 9298
*A*B^3*a^6*b^3 + 59592*A^2*B^2*a^5*b^4 - 168016*A^3*B*a^4*b^5 + 175655*A^4*a^3*b^6)*c^3 + 3*(73*B^4*a^6*b^4 -
1344*A*B^3*a^5*b^5 + 9228*A^2*B^2*a^4*b^6 - 27980*A^3*B*a^3*b^7 + 31575*A^4*a^2*b^8)*c^2 - 2*(11*B^4*a^5*b^6 -
214*A*B^3*a^4*b^7 + 1560*A^2*B^2*a^3*b^8 - 5050*A^3*B*a^2*b^9 + 6125*A^4*a*b^10)*c)/(a^14*b^10 - 20*a^15*b^8*
c + 160*a^16*b^6*c^2 - 640*a^17*b^4*c^3 + 1280*a^18*b^2*c^4 - 1024*a^19*c^5)))*sqrt(-(B^2*a^2*b^9 - 10*A*B*a*b
^10 + 25*A^2*b^11 + 1680*(4*A*B*a^6 - 11*A^2*a^5*b)*c^5 + 840*(2*B^2*a^6*b - 16*A*B*a^5*b^2 + 33*A^2*a^4*b^3)*
c^4 - 105*(8*B^2*a^5*b^3 - 68*A*B*a^4*b^4 + 143*A^2*a^3*b^5)*c^3 + 3*(63*B^2*a^4*b^5 - 574*A*B*a^3*b^6 + 1298*
A^2*a^2*b^7)*c^2 - 3*(7*B^2*a^3*b^7 - 68*A*B*a^2*b^8 + 165*A^2*a*b^9)*c - (a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b
^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*c^5)*sqrt((B^4*a^4*b^8 - 20*A*B^3*a^3*b^9 + 150*A^2*
B^2*a^2*b^10 - 500*A^3*B*a*b^11 + 625*A^4*b^12 + 50625*A^4*a^6*c^6 - 450*(49*A^2*B^2*a^7 - 382*A^3*B*a^6*b + 6
94*A^4*a^5*b^2)*c^5 + (2401*B^4*a^8 - 37436*A*B^3*a^7*b + 218886*A^2*B^2*a^6*b^2 - 577016*A^3*B*a^5*b^3 + 5918
86*A^4*a^4*b^4)*c^4 - 2*(539*B^4*a^7*b^2 - 9298*A*B^3*a^6*b^3 + 59592*A^2*B^2*a^5*b^4 - 168016*A^3*B*a^4*b^5 +
175655*A^4*a^3*b^6)*c^3 + 3*(73*B^4*a^6*b^4 - 1344*A*B^3*a^5*b^5 + 9228*A^2*B^2*a^4*b^6 - 27980*A^3*B*a^3*b^7
+ 31575*A^4*a^2*b^8)*c^2 - 2*(11*B^4*a^5*b^6 - 214*A*B^3*a^4*b^7 + 1560*A^2*B^2*a^3*b^8 - 5050*A^3*B*a^2*b^9
+ 6125*A^4*a*b^10)*c)/(a^14*b^10 - 20*a^15*b^8*c + 160*a^16*b^6*c^2 - 640*a^17*b^4*c^3 + 1280*a^18*b^2*c^4 - 1
024*a^19*c^5)))/(a^7*b^10 - 20*a^8*b^8*c + 160*a^9*b^6*c^2 - 640*a^10*b^4*c^3 + 1280*a^11*b^2*c^4 - 1024*a^12*
c^5)) + 27*(810000*A^4*a^5*c^9 + 405000*(2*A^3*B*a^5*b - 7*A^4*a^4*b^2)*c^8 - (38416*B^4*a^7 - 422576*A*B^3*a^
6*b + 1439376*A^2*B^2*a^5*b^2 - 1018856*A^3*B*a^4*b^3 - 1957349*A^4*a^3*b^4)*c^7 + (19208*B^4*a^6*b^2 - 239896
*A*B^3*a^5*b^3 + 955704*A^2*B^2*a^4*b^4 - 1067347*A^3*B*a^3*b^5 - 571030*A^4*a^2*b^6)*c^6 - (4189*B^4*a^5*b^4
- 56807*A*B^3*a^4*b^5 + 251349*A^2*B^2*a^3*b^6 - 344630*A^3*B*a^2*b^7 - 77825*A^4*a*b^8)*c^5 + 3*(149*B^4*a^4*
b^6 - 2161*A*B^3*a^3*b^7 + 10380*A^2*B^2*a^2*b^8 - 16225*A^3*B*a*b^9 - 1375*A^4*b^10)*c^4 - 21*(B^4*a^3*b^8 -
15*A*B^3*a^2*b^9 + 75*A^2*B^2*a*b^10 - 125*A^3*B*b^11)*c^3)*sqrt(x)) - 2*(8*A*a^2*b^4 - 64*A*a^3*b^2*c + 128*A
*a^4*c^2 + 3*(60*A*a^2*c^4 + (8*B*a^2*b - 37*A*a*b^2)*c^3 - (B*a*b^3 - 5*A*b^4)*c^2)*x^4 - (28*(B*a^3 - 14*A*a
^2*b)*c^3 - (49*B*a^2*b^2 - 227*A*a*b^3)*c^2 + 6*(B*a*b^4 - 5*A*b^5)*c)*x^3 - (3*B*a*b^5 - 15*A*b^6 - 324*A*a^
3*c^3 - (4*B*a^3*b + 25*A*a^2*b^2)*c^2 - (20*B*a^2*b^3 - 91*A*a*b^4)*c)*x^2 - (5*B*a^2*b^4 - 25*A*a*b^5 + 4*(1
1*B*a^4 - 91*A*a^3*b)*c^2 - (37*B*a^3*b^2 - 194*A*a^2*b^3)*c)*x)*sqrt(x))/((a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a
^5*c^4)*x^5 + 2*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*x^4 + (a^3*b^6 - 6*a^4*b^4*c + 32*a^6*c^3)*x^3 + 2*
(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*x^2 + (a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*x)

________________________________________________________________________________________

giac [B]  time = 3.36, size = 9534, normalized size = 14.36

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(3/2)/(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

-3/32*((10*b^6*c^2 - 114*a*b^4*c^3 + 416*a^2*b^2*c^4 - 480*a^3*c^5 - 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sq
rt(b^2 - 4*a*c)*c)*b^6 + 57*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c + 10*sqrt(2)*sqr
t(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5*c - 208*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a
*c)*c)*a^2*b^2*c^2 - 74*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^2 - 5*sqrt(2)*sqrt(b
^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c^2 + 240*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*
c)*c)*a^3*c^3 + 120*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^3 + 37*sqrt(2)*sqrt(b^2
- 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^3 - 60*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)
*c)*a^2*c^4 - 10*(b^2 - 4*a*c)*b^4*c^2 + 74*(b^2 - 4*a*c)*a*b^2*c^3 - 120*(b^2 - 4*a*c)*a^2*c^4)*(a^3*b^4 - 8*
a^4*b^2*c + 16*a^5*c^2)^2*A - (2*a*b^5*c^2 - 24*a^2*b^3*c^3 + 64*a^3*b*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*
c + sqrt(b^2 - 4*a*c)*c)*a*b^5 + 12*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c + 2*sq
rt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c - 32*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt
(b^2 - 4*a*c)*c)*a^3*b*c^2 - 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^2 - sqrt(2
)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^2 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^
2 - 4*a*c)*c)*a^2*b*c^3 - 2*(b^2 - 4*a*c)*a*b^3*c^2 + 16*(b^2 - 4*a*c)*a^2*b*c^3)*(a^3*b^4 - 8*a^4*b^2*c + 16*
a^5*c^2)^2*B + 2*(5*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^11 - 102*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c
)*c)*a^4*b^9*c - 10*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^10*c - 10*a^3*b^11*c + 836*sqrt(2)*sqrt(b*c
+ sqrt(b^2 - 4*a*c)*c)*a^5*b^7*c^2 + 164*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^8*c^2 + 5*sqrt(2)*sqrt(
b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^9*c^2 + 204*a^4*b^9*c^2 - 3440*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^
5*c^3 - 1016*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^6*c^3 - 82*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*
a^4*b^7*c^3 - 1672*a^5*b^7*c^3 + 7104*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^3*c^4 + 2816*sqrt(2)*sqrt(
b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^4*c^4 + 508*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^5*c^4 + 6880*a^6*b^
5*c^4 - 5888*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^8*b*c^5 - 2944*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*
a^7*b^2*c^5 - 1408*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^3*c^5 - 14208*a^7*b^3*c^5 + 1472*sqrt(2)*sqrt
(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b*c^6 + 11776*a^8*b*c^6 + 10*(b^2 - 4*a*c)*a^3*b^9*c - 164*(b^2 - 4*a*c)*a^4*b
^7*c^2 + 1016*(b^2 - 4*a*c)*a^5*b^5*c^3 - 2816*(b^2 - 4*a*c)*a^6*b^3*c^4 + 2944*(b^2 - 4*a*c)*a^7*b*c^5)*A*abs
(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2) - 2*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^10 - 21*sqrt(2)*sqrt(b
*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^8*c - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^9*c - 2*a^4*b^10*c + 184
*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^6*c^2 + 34*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^7*c^2
+ sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^8*c^2 + 42*a^5*b^8*c^2 - 832*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a
*c)*c)*a^7*b^4*c^3 - 232*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^5*c^3 - 17*sqrt(2)*sqrt(b*c + sqrt(b^2
- 4*a*c)*c)*a^5*b^6*c^3 - 368*a^6*b^6*c^3 + 1920*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^8*b^2*c^4 + 736*sqr
t(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^3*c^4 + 116*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^4*c^4 + 1
664*a^7*b^4*c^4 - 1792*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^9*c^5 - 896*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a
*c)*c)*a^8*b*c^5 - 368*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^2*c^5 - 3840*a^8*b^2*c^5 + 448*sqrt(2)*sq
rt(b*c + sqrt(b^2 - 4*a*c)*c)*a^8*c^6 + 3584*a^9*c^6 + 2*(b^2 - 4*a*c)*a^4*b^8*c - 34*(b^2 - 4*a*c)*a^5*b^6*c^
2 + 232*(b^2 - 4*a*c)*a^6*b^4*c^3 - 736*(b^2 - 4*a*c)*a^7*b^2*c^4 + 896*(b^2 - 4*a*c)*a^8*c^5)*B*abs(a^3*b^4 -
8*a^4*b^2*c + 16*a^5*c^2) + (10*a^6*b^14*c^2 - 254*a^7*b^12*c^3 + 2712*a^8*b^10*c^4 - 15552*a^9*b^8*c^5 + 504
32*a^10*b^6*c^6 - 87552*a^11*b^4*c^7 + 63488*a^12*b^2*c^8 - 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 -
4*a*c)*c)*a^6*b^14 + 127*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^12*c + 10*sqrt(2)*sqr
t(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^13*c - 1356*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2
- 4*a*c)*c)*a^8*b^10*c^2 - 214*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^11*c^2 - 5*sqr
t(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^12*c^2 + 7776*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c
+ sqrt(b^2 - 4*a*c)*c)*a^9*b^8*c^3 + 1856*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^8*b^9*c^
3 + 107*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^10*c^3 - 25216*sqrt(2)*sqrt(b^2 - 4*a*
c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^10*b^6*c^4 - 8128*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*
c)*a^9*b^7*c^4 - 928*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^8*b^8*c^4 + 43776*sqrt(2)*sqr
t(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^11*b^4*c^5 + 17920*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(
b^2 - 4*a*c)*c)*a^10*b^5*c^5 + 4064*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^9*b^6*c^5 - 31
744*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^12*b^2*c^6 - 15872*sqrt(2)*sqrt(b^2 - 4*a*c)*s
qrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^11*b^3*c^6 - 8960*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a
^10*b^4*c^6 + 7936*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^11*b^2*c^7 - 10*(b^2 - 4*a*c)*a
^6*b^12*c^2 + 214*(b^2 - 4*a*c)*a^7*b^10*c^3 - 1856*(b^2 - 4*a*c)*a^8*b^8*c^4 + 8128*(b^2 - 4*a*c)*a^9*b^6*c^5
- 17920*(b^2 - 4*a*c)*a^10*b^4*c^6 + 15872*(b^2 - 4*a*c)*a^11*b^2*c^7)*A - (2*a^7*b^13*c^2 - 52*a^8*b^11*c^3
+ 624*a^9*b^9*c^4 - 4224*a^10*b^7*c^5 + 16384*a^11*b^5*c^6 - 33792*a^12*b^3*c^7 + 28672*a^13*b*c^8 - sqrt(2)*s
qrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^13 + 26*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 -
4*a*c)*c)*a^8*b^11*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^12*c - 312*sqrt(2)*s
qrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^9*b^9*c^2 - 44*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^
2 - 4*a*c)*c)*a^8*b^10*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^11*c^2 + 2112*sqr
t(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^10*b^7*c^3 + 448*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c +
sqrt(b^2 - 4*a*c)*c)*a^9*b^8*c^3 + 22*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^8*b^9*c^3 -
8192*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^11*b^5*c^4 - 2432*sqrt(2)*sqrt(b^2 - 4*a*c)*
sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^10*b^6*c^4 - 224*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a
^9*b^7*c^4 + 16896*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^12*b^3*c^5 + 6656*sqrt(2)*sqrt(
b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^11*b^4*c^5 + 1216*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2
- 4*a*c)*c)*a^10*b^5*c^5 - 14336*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^13*b*c^6 - 7168*
sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^12*b^2*c^6 - 3328*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b
*c + sqrt(b^2 - 4*a*c)*c)*a^11*b^3*c^6 + 3584*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^12*b
*c^7 - 2*(b^2 - 4*a*c)*a^7*b^11*c^2 + 44*(b^2 - 4*a*c)*a^8*b^9*c^3 - 448*(b^2 - 4*a*c)*a^9*b^7*c^4 + 2432*(b^2
- 4*a*c)*a^10*b^5*c^5 - 6656*(b^2 - 4*a*c)*a^11*b^3*c^6 + 7168*(b^2 - 4*a*c)*a^12*b*c^7)*B)*arctan(2*sqrt(1/2
)*sqrt(x)/sqrt((a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2 + sqrt((a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)^2 - 4*(a^4*
b^4 - 8*a^5*b^2*c + 16*a^6*c^2)*(a^3*b^4*c - 8*a^4*b^2*c^2 + 16*a^5*c^3)))/(a^3*b^4*c - 8*a^4*b^2*c^2 + 16*a^5
*c^3)))/((a^7*b^10 - 20*a^8*b^8*c - 2*a^7*b^9*c + 160*a^9*b^6*c^2 + 32*a^8*b^7*c^2 + a^7*b^8*c^2 - 640*a^10*b^
4*c^3 - 192*a^9*b^5*c^3 - 16*a^8*b^6*c^3 + 1280*a^11*b^2*c^4 + 512*a^10*b^3*c^4 + 96*a^9*b^4*c^4 - 1024*a^12*c
^5 - 512*a^11*b*c^5 - 256*a^10*b^2*c^5 + 256*a^11*c^6)*abs(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*abs(c)) + 3/32*
((10*b^6*c^2 - 114*a*b^4*c^3 + 416*a^2*b^2*c^4 - 480*a^3*c^5 - 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2
- 4*a*c)*c)*b^6 + 57*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^4*c + 10*sqrt(2)*sqrt(b^2
- 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^5*c - 208*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)
*a^2*b^2*c^2 - 74*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^3*c^2 - 5*sqrt(2)*sqrt(b^2 - 4
*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^4*c^2 + 240*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*
a^3*c^3 + 120*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b*c^3 + 37*sqrt(2)*sqrt(b^2 - 4*a*
c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^2*c^3 - 60*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^
2*c^4 - 10*(b^2 - 4*a*c)*b^4*c^2 + 74*(b^2 - 4*a*c)*a*b^2*c^3 - 120*(b^2 - 4*a*c)*a^2*c^4)*(a^3*b^4 - 8*a^4*b^
2*c + 16*a^5*c^2)^2*A - (2*a*b^5*c^2 - 24*a^2*b^3*c^3 + 64*a^3*b*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sq
rt(b^2 - 4*a*c)*c)*a*b^5 + 12*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c + 2*sqrt(2)*
sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^4*c - 32*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 -
4*a*c)*c)*a^3*b*c^2 - 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^2 - sqrt(2)*sqrt
(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^3*c^2 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*
a*c)*c)*a^2*b*c^3 - 2*(b^2 - 4*a*c)*a*b^3*c^2 + 16*(b^2 - 4*a*c)*a^2*b*c^3)*(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^
2)^2*B - 2*(5*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^11 - 102*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a
^4*b^9*c - 10*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^10*c + 10*a^3*b^11*c + 836*sqrt(2)*sqrt(b*c - sqrt
(b^2 - 4*a*c)*c)*a^5*b^7*c^2 + 164*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b^8*c^2 + 5*sqrt(2)*sqrt(b*c -
sqrt(b^2 - 4*a*c)*c)*a^3*b^9*c^2 - 204*a^4*b^9*c^2 - 3440*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^6*b^5*c^3
- 1016*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^5*b^6*c^3 - 82*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b^
7*c^3 + 1672*a^5*b^7*c^3 + 7104*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^7*b^3*c^4 + 2816*sqrt(2)*sqrt(b*c -
sqrt(b^2 - 4*a*c)*c)*a^6*b^4*c^4 + 508*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^5*b^5*c^4 - 6880*a^6*b^5*c^4
- 5888*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^8*b*c^5 - 2944*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^7*b^
2*c^5 - 1408*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^6*b^3*c^5 + 14208*a^7*b^3*c^5 + 1472*sqrt(2)*sqrt(b*c -
sqrt(b^2 - 4*a*c)*c)*a^7*b*c^6 - 11776*a^8*b*c^6 - 10*(b^2 - 4*a*c)*a^3*b^9*c + 164*(b^2 - 4*a*c)*a^4*b^7*c^2
- 1016*(b^2 - 4*a*c)*a^5*b^5*c^3 + 2816*(b^2 - 4*a*c)*a^6*b^3*c^4 - 2944*(b^2 - 4*a*c)*a^7*b*c^5)*A*abs(a^3*b
^4 - 8*a^4*b^2*c + 16*a^5*c^2) + 2*(sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b^10 - 21*sqrt(2)*sqrt(b*c - s
qrt(b^2 - 4*a*c)*c)*a^5*b^8*c - 2*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b^9*c + 2*a^4*b^10*c + 184*sqrt(
2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^6*b^6*c^2 + 34*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^5*b^7*c^2 + sqrt
(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b^8*c^2 - 42*a^5*b^8*c^2 - 832*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)
*a^7*b^4*c^3 - 232*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^6*b^5*c^3 - 17*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*
c)*c)*a^5*b^6*c^3 + 368*a^6*b^6*c^3 + 1920*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^8*b^2*c^4 + 736*sqrt(2)*s
qrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^7*b^3*c^4 + 116*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^6*b^4*c^4 - 1664*a^
7*b^4*c^4 - 1792*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^9*c^5 - 896*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)
*a^8*b*c^5 - 368*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^7*b^2*c^5 + 3840*a^8*b^2*c^5 + 448*sqrt(2)*sqrt(b*c
- sqrt(b^2 - 4*a*c)*c)*a^8*c^6 - 3584*a^9*c^6 - 2*(b^2 - 4*a*c)*a^4*b^8*c + 34*(b^2 - 4*a*c)*a^5*b^6*c^2 - 23
2*(b^2 - 4*a*c)*a^6*b^4*c^3 + 736*(b^2 - 4*a*c)*a^7*b^2*c^4 - 896*(b^2 - 4*a*c)*a^8*c^5)*B*abs(a^3*b^4 - 8*a^4
*b^2*c + 16*a^5*c^2) + (10*a^6*b^14*c^2 - 254*a^7*b^12*c^3 + 2712*a^8*b^10*c^4 - 15552*a^9*b^8*c^5 + 50432*a^1
0*b^6*c^6 - 87552*a^11*b^4*c^7 + 63488*a^12*b^2*c^8 - 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)
*c)*a^6*b^14 + 127*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^7*b^12*c + 10*sqrt(2)*sqrt(b^2
- 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^6*b^13*c - 1356*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a
*c)*c)*a^8*b^10*c^2 - 214*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^7*b^11*c^2 - 5*sqrt(2)*s
qrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^6*b^12*c^2 + 7776*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt
(b^2 - 4*a*c)*c)*a^9*b^8*c^3 + 1856*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^8*b^9*c^3 + 10
7*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^7*b^10*c^3 - 25216*sqrt(2)*sqrt(b^2 - 4*a*c)*sqr
t(b*c - sqrt(b^2 - 4*a*c)*c)*a^10*b^6*c^4 - 8128*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^9
*b^7*c^4 - 928*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^8*b^8*c^4 + 43776*sqrt(2)*sqrt(b^2
- 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^11*b^4*c^5 + 17920*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 -
4*a*c)*c)*a^10*b^5*c^5 + 4064*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^9*b^6*c^5 - 31744*sq
rt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^12*b^2*c^6 - 15872*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*
c - sqrt(b^2 - 4*a*c)*c)*a^11*b^3*c^6 - 8960*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^10*b^
4*c^6 + 7936*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^11*b^2*c^7 - 10*(b^2 - 4*a*c)*a^6*b^1
2*c^2 + 214*(b^2 - 4*a*c)*a^7*b^10*c^3 - 1856*(b^2 - 4*a*c)*a^8*b^8*c^4 + 8128*(b^2 - 4*a*c)*a^9*b^6*c^5 - 179
20*(b^2 - 4*a*c)*a^10*b^4*c^6 + 15872*(b^2 - 4*a*c)*a^11*b^2*c^7)*A - (2*a^7*b^13*c^2 - 52*a^8*b^11*c^3 + 624*
a^9*b^9*c^4 - 4224*a^10*b^7*c^5 + 16384*a^11*b^5*c^6 - 33792*a^12*b^3*c^7 + 28672*a^13*b*c^8 - sqrt(2)*sqrt(b^
2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^7*b^13 + 26*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c
)*c)*a^8*b^11*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^7*b^12*c - 312*sqrt(2)*sqrt(b^
2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^9*b^9*c^2 - 44*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*
a*c)*c)*a^8*b^10*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^7*b^11*c^2 + 2112*sqrt(2)*s
qrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^10*b^7*c^3 + 448*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(
b^2 - 4*a*c)*c)*a^9*b^8*c^3 + 22*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^8*b^9*c^3 - 8192*
sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^11*b^5*c^4 - 2432*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b
*c - sqrt(b^2 - 4*a*c)*c)*a^10*b^6*c^4 - 224*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^9*b^7
*c^4 + 16896*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^12*b^3*c^5 + 6656*sqrt(2)*sqrt(b^2 -
4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^11*b^4*c^5 + 1216*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a
*c)*c)*a^10*b^5*c^5 - 14336*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^13*b*c^6 - 7168*sqrt(2
)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^12*b^2*c^6 - 3328*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - s
qrt(b^2 - 4*a*c)*c)*a^11*b^3*c^6 + 3584*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^12*b*c^7 -
2*(b^2 - 4*a*c)*a^7*b^11*c^2 + 44*(b^2 - 4*a*c)*a^8*b^9*c^3 - 448*(b^2 - 4*a*c)*a^9*b^7*c^4 + 2432*(b^2 - 4*a
*c)*a^10*b^5*c^5 - 6656*(b^2 - 4*a*c)*a^11*b^3*c^6 + 7168*(b^2 - 4*a*c)*a^12*b*c^7)*B)*arctan(2*sqrt(1/2)*sqrt
(x)/sqrt((a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2 - sqrt((a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)^2 - 4*(a^4*b^4 -
8*a^5*b^2*c + 16*a^6*c^2)*(a^3*b^4*c - 8*a^4*b^2*c^2 + 16*a^5*c^3)))/(a^3*b^4*c - 8*a^4*b^2*c^2 + 16*a^5*c^3))
)/((a^7*b^10 - 20*a^8*b^8*c - 2*a^7*b^9*c + 160*a^9*b^6*c^2 + 32*a^8*b^7*c^2 + a^7*b^8*c^2 - 640*a^10*b^4*c^3
- 192*a^9*b^5*c^3 - 16*a^8*b^6*c^3 + 1280*a^11*b^2*c^4 + 512*a^10*b^3*c^4 + 96*a^9*b^4*c^4 - 1024*a^12*c^5 - 5
12*a^11*b*c^5 - 256*a^10*b^2*c^5 + 256*a^11*c^6)*abs(a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*abs(c)) + 1/4*(3*B*a*
b^3*c^2*x^(7/2) - 7*A*b^4*c^2*x^(7/2) - 24*B*a^2*b*c^3*x^(7/2) + 47*A*a*b^2*c^3*x^(7/2) - 52*A*a^2*c^4*x^(7/2)
+ 6*B*a*b^4*c*x^(5/2) - 14*A*b^5*c*x^(5/2) - 49*B*a^2*b^2*c^2*x^(5/2) + 99*A*a*b^3*c^2*x^(5/2) + 28*B*a^3*c^3
*x^(5/2) - 136*A*a^2*b*c^3*x^(5/2) + 3*B*a*b^5*x^(3/2) - 7*A*b^6*x^(3/2) - 20*B*a^2*b^3*c*x^(3/2) + 43*A*a*b^4
*c*x^(3/2) - 4*B*a^3*b*c^2*x^(3/2) - 25*A*a^2*b^2*c^2*x^(3/2) - 68*A*a^3*c^3*x^(3/2) + 5*B*a^2*b^4*sqrt(x) - 9
*A*a*b^5*sqrt(x) - 37*B*a^3*b^2*c*sqrt(x) + 66*A*a^2*b^3*c*sqrt(x) + 44*B*a^4*c^2*sqrt(x) - 108*A*a^3*b*c^2*sq
rt(x))/((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*(c*x^2 + b*x + a)^2) - 2*A/(a^3*sqrt(x))

________________________________________________________________________________________

maple [B]  time = 0.12, size = 2918, normalized size = 4.39 \begin {gather*} \text {output too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/x^(3/2)/(c*x^2+b*x+a)^3,x)

[Out]

-3/8/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(2^(1/
2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))*B*b^4+93/2/a/(16*a^2*c^2-8*a*b^2*c+b^4)*c^3/(-4*a*c+b^2)^(1/2)*
2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))*A*b-141
/8/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(2^(
1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))*A*b^3+15/4/a/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2/(-4*a*c+b^2)^(1/
2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))*B*b^
2-3/8/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(2^
(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))*B*b^4+93/2/a/(16*a^2*c^2-8*a*b^2*c+b^4)*c^3/(-4*a*c+b^2)^(1
/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))*A*b-14
1/8/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(2^(1
/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))*A*b^3+15/4/a/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2/(-4*a*c+b^2)^(1/2)
*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))*B*b^2+15/
8/a^3/(16*a^2*c^2-8*a*b^2*c+b^4)*c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(2^(1/2)/
((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))*A*b^5+15/8/a^3/(16*a^2*c^2-8*a*b^2*c+b^4)*c/(-4*a*c+b^2)^(1/2)*2^(
1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))*A*b^5+5/4/
a/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(1/2)*B*b^4-13/a/(c*x^2+b*x+a)^2*c^4/(16*a^2*c^2-8*a*b^2*c+b^4)
*x^(7/2)*A-7/4/a^3/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(3/2)*A*b^6+3/8/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)
*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))*B*b^3+1
11/8/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(2^(1/2)/((b+(-4*a*c+b^
2)^(1/2))*c)^(1/2)*c*x^(1/2))*A*b^2+3/a/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/
2)*arctanh(2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))*b*B-3/8/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c*2^(1/
2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))*B*b^3-15/8/a
^3/(16*a^2*c^2-8*a*b^2*c+b^4)*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(2^(1/2)/((b+(-4*a*c+b^2)^(1/2)
)*c)^(1/2)*c*x^(1/2))*A*b^4-3/a/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan
(2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))*b*B-111/8/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2*2^(1/2)/((-b
+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))*A*b^2+3/4/a^2/(c*x^
2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(3/2)*B*b^5-9/4/a^2/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(1/2)
*A*b^5+11*a/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(1/2)*B*c^2-1/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b
^4)*x^(3/2)*B*b*c^2-37/4/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(1/2)*B*b^2*c-27/(c*x^2+b*x+a)^2/(16*a^2
*c^2-8*a*b^2*c+b^4)*x^(1/2)*A*b*c^2+15/8/a^3/(16*a^2*c^2-8*a*b^2*c+b^4)*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^
(1/2)*arctanh(2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))*A*b^4-2*A/a^3/x^(1/2)+7/(c*x^2+b*x+a)^2*c^3
/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(5/2)*B-17/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(3/2)*A*c^3+99/4/a^2/(c*
x^2+b*x+a)^2*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(5/2)*A*b^3-49/4/a/(c*x^2+b*x+a)^2*c^2/(16*a^2*c^2-8*a*b^2*c+b^4
)*x^(5/2)*B*b^2+3/2/a^2/(c*x^2+b*x+a)^2*c/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(5/2)*B*b^4-21/(16*a^2*c^2-8*a*b^2*c+b^
4)*c^3/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(
1/2)*c*x^(1/2))*B+3/4/a^2/(c*x^2+b*x+a)^2*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(7/2)*B*b^3-34/a/(c*x^2+b*x+a)^2*c^
3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(5/2)*A*b-21/(16*a^2*c^2-8*a*b^2*c+b^4)*c^3/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4
*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))*B-25/4/a/(c*x^2+b*x+a)^
2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(3/2)*A*b^2*c^2+43/4/a^2/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(3/2)*A*b
^4*c-5/a/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(3/2)*B*b^3*c+45/2/a/(16*a^2*c^2-8*a*b^2*c+b^4)*c^3*2^(1
/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))*A-45/2/a/(1
6*a^2*c^2-8*a*b^2*c+b^4)*c^3*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c
)^(1/2)*c*x^(1/2))*A-7/2/a^3/(c*x^2+b*x+a)^2*c/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(5/2)*A*b^5-7/4/a^3/(c*x^2+b*x+a)^
2*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(7/2)*A*b^4+33/2/a/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(1/2)*A*b^3
*c+47/4/a^2/(c*x^2+b*x+a)^2*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^(7/2)*A*b^2-6/a/(c*x^2+b*x+a)^2*c^3/(16*a^2*c^2-8
*a*b^2*c+b^4)*x^(7/2)*b*B

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(3/2)/(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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mupad [B]  time = 8.04, size = 29137, normalized size = 43.88

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/(x^(3/2)*(a + b*x + c*x^2)^3),x)

[Out]

- atan(((x^(1/2)*(33973862400*A^2*a^20*c^14 - 7398752256*B^2*a^21*c^13 - 28800*A^2*a^9*b^22*c^3 + 1232640*A^2*
a^10*b^20*c^4 - 23879808*A^2*a^11*b^18*c^5 + 275975424*A^2*a^12*b^16*c^6 - 2109763584*A^2*a^13*b^14*c^7 + 1117
1856384*A^2*a^14*b^12*c^8 - 41653370880*A^2*a^15*b^10*c^9 + 108726976512*A^2*a^16*b^8*c^10 - 192980975616*A^2*
a^17*b^6*c^11 + 218414186496*A^2*a^18*b^4*c^12 - 137631891456*A^2*a^19*b^2*c^13 - 1152*B^2*a^11*b^20*c^3 + 506
88*B^2*a^12*b^18*c^4 - 1025280*B^2*a^13*b^16*c^5 + 12496896*B^2*a^14*b^14*c^6 - 101744640*B^2*a^15*b^12*c^7 +
579796992*B^2*a^16*b^10*c^8 - 2346319872*B^2*a^17*b^8*c^9 + 6653214720*B^2*a^18*b^6*c^10 - 12608077824*B^2*a^1
9*b^4*c^11 + 14344519680*B^2*a^20*b^2*c^12 + 11520*A*B*a^10*b^21*c^3 - 499968*A*B*a^11*b^19*c^4 + 9900288*A*B*
a^12*b^17*c^5 - 117559296*A*B*a^13*b^15*c^6 + 925433856*A*B*a^14*b^13*c^7 - 5038866432*A*B*a^15*b^11*c^8 + 191
91693312*A*B*a^16*b^9*c^9 - 50422874112*A*B*a^17*b^7*c^10 + 87350575104*A*B*a^18*b^5*c^11 - 89992986624*A*B*a^
19*b^3*c^12 + 41825599488*A*B*a^20*b*c^13) + (-(9*(25*A^2*b^21 + B^2*a^2*b^19 + 25*A^2*b^6*(-(4*a*c - b^2)^15)
^(1/2) - 10*A*B*a*b^20 + 17794*A^2*a^2*b^17*c^2 - 188095*A^2*a^3*b^15*c^3 + 1299860*A^2*a^4*b^13*c^4 - 6126640
*A^2*a^5*b^11*c^5 + 19905600*A^2*a^6*b^9*c^6 - 43904256*A^2*a^7*b^7*c^7 + 62684160*A^2*a^8*b^5*c^8 - 52039680*
A^2*a^9*b^3*c^9 - 225*A^2*a^3*c^3*(-(4*a*c - b^2)^15)^(1/2) + B^2*a^2*b^4*(-(4*a*c - b^2)^15)^(1/2) + 769*B^2*
a^4*b^15*c^2 - 8620*B^2*a^5*b^13*c^3 + 63440*B^2*a^6*b^11*c^4 - 316864*B^2*a^7*b^9*c^5 + 1069824*B^2*a^8*b^7*c
^6 - 2343936*B^2*a^9*b^5*c^7 + 3010560*B^2*a^10*b^3*c^8 + 49*B^2*a^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 6881280*A
*B*a^11*c^10 - 995*A^2*a*b^19*c + 18923520*A^2*a^10*b*c^10 - 41*B^2*a^3*b^17*c - 1720320*B^2*a^11*b*c^9 + 694*
A^2*a^2*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) - 7402*A*B*a^3*b^16*c^2 + 80620*A*B*a^4*b^14*c^3 - 575120*A*B*a^5*b^
12*c^4 + 2791360*A*B*a^6*b^10*c^5 - 9267456*A*B*a^7*b^8*c^6 + 20579328*A*B*a^8*b^6*c^7 - 28815360*A*B*a^9*b^4*
c^8 + 22364160*A*B*a^10*b^2*c^9 - 245*A^2*a*b^4*c*(-(4*a*c - b^2)^15)^(1/2) - 11*B^2*a^3*b^2*c*(-(4*a*c - b^2)
^15)^(1/2) - 10*A*B*a*b^5*(-(4*a*c - b^2)^15)^(1/2) + 404*A*B*a^2*b^18*c + 104*A*B*a^2*b^3*c*(-(4*a*c - b^2)^1
5)^(1/2) - 382*A*B*a^3*b*c^2*(-(4*a*c - b^2)^15)^(1/2)))/(128*(a^7*b^20 + 1048576*a^17*c^10 - 40*a^8*b^18*c +
720*a^9*b^16*c^2 - 7680*a^10*b^14*c^3 + 53760*a^11*b^12*c^4 - 258048*a^12*b^10*c^5 + 860160*a^13*b^8*c^6 - 196
6080*a^14*b^6*c^7 + 2949120*a^15*b^4*c^8 - 2621440*a^16*b^2*c^9)))^(1/2)*(x^(1/2)*(-(9*(25*A^2*b^21 + B^2*a^2*
b^19 + 25*A^2*b^6*(-(4*a*c - b^2)^15)^(1/2) - 10*A*B*a*b^20 + 17794*A^2*a^2*b^17*c^2 - 188095*A^2*a^3*b^15*c^3
+ 1299860*A^2*a^4*b^13*c^4 - 6126640*A^2*a^5*b^11*c^5 + 19905600*A^2*a^6*b^9*c^6 - 43904256*A^2*a^7*b^7*c^7 +
62684160*A^2*a^8*b^5*c^8 - 52039680*A^2*a^9*b^3*c^9 - 225*A^2*a^3*c^3*(-(4*a*c - b^2)^15)^(1/2) + B^2*a^2*b^4
*(-(4*a*c - b^2)^15)^(1/2) + 769*B^2*a^4*b^15*c^2 - 8620*B^2*a^5*b^13*c^3 + 63440*B^2*a^6*b^11*c^4 - 316864*B^
2*a^7*b^9*c^5 + 1069824*B^2*a^8*b^7*c^6 - 2343936*B^2*a^9*b^5*c^7 + 3010560*B^2*a^10*b^3*c^8 + 49*B^2*a^4*c^2*
(-(4*a*c - b^2)^15)^(1/2) - 6881280*A*B*a^11*c^10 - 995*A^2*a*b^19*c + 18923520*A^2*a^10*b*c^10 - 41*B^2*a^3*b
^17*c - 1720320*B^2*a^11*b*c^9 + 694*A^2*a^2*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) - 7402*A*B*a^3*b^16*c^2 + 80620
*A*B*a^4*b^14*c^3 - 575120*A*B*a^5*b^12*c^4 + 2791360*A*B*a^6*b^10*c^5 - 9267456*A*B*a^7*b^8*c^6 + 20579328*A*
B*a^8*b^6*c^7 - 28815360*A*B*a^9*b^4*c^8 + 22364160*A*B*a^10*b^2*c^9 - 245*A^2*a*b^4*c*(-(4*a*c - b^2)^15)^(1/
2) - 11*B^2*a^3*b^2*c*(-(4*a*c - b^2)^15)^(1/2) - 10*A*B*a*b^5*(-(4*a*c - b^2)^15)^(1/2) + 404*A*B*a^2*b^18*c
+ 104*A*B*a^2*b^3*c*(-(4*a*c - b^2)^15)^(1/2) - 382*A*B*a^3*b*c^2*(-(4*a*c - b^2)^15)^(1/2)))/(128*(a^7*b^20 +
1048576*a^17*c^10 - 40*a^8*b^18*c + 720*a^9*b^16*c^2 - 7680*a^10*b^14*c^3 + 53760*a^11*b^12*c^4 - 258048*a^12
*b^10*c^5 + 860160*a^13*b^8*c^6 - 1966080*a^14*b^6*c^7 + 2949120*a^15*b^4*c^8 - 2621440*a^16*b^2*c^9)))^(1/2)*
(34359738368*a^26*b*c^13 - 8192*a^15*b^23*c^2 + 360448*a^16*b^21*c^3 - 7208960*a^17*b^19*c^4 + 86507520*a^18*b
^17*c^5 - 692060160*a^19*b^15*c^6 + 3875536896*a^20*b^13*c^7 - 15502147584*a^21*b^11*c^8 + 44291850240*a^22*b^
9*c^9 - 88583700480*a^23*b^7*c^10 + 118111600640*a^24*b^5*c^11 - 94489280512*a^25*b^3*c^12) - 22548578304*B*a^
24*c^13 + 74088185856*A*a^23*b*c^13 - 15360*A*a^12*b^23*c^2 + 681984*A*a^13*b^21*c^3 - 13774848*A*a^14*b^19*c^
4 + 167067648*A*a^15*b^17*c^5 - 1351876608*A*a^16*b^15*c^6 + 7662993408*A*a^17*b^13*c^7 - 31048335360*A*a^18*b
^11*c^8 + 89917489152*A*a^19*b^9*c^9 - 182401892352*A*a^20*b^7*c^10 + 246826401792*A*a^21*b^5*c^11 - 200521285
632*A*a^22*b^3*c^12 + 3072*B*a^13*b^22*c^2 - 138240*B*a^14*b^20*c^3 + 2850816*B*a^15*b^18*c^4 - 35536896*B*a^1
6*b^16*c^5 + 297271296*B*a^17*b^14*c^6 - 1750597632*B*a^18*b^12*c^7 + 7398752256*B*a^19*b^10*c^8 - 22422749184
*B*a^20*b^8*c^9 + 47714402304*B*a^21*b^6*c^10 - 67847061504*B*a^22*b^4*c^11 + 57982058496*B*a^23*b^2*c^12))*(-
(9*(25*A^2*b^21 + B^2*a^2*b^19 + 25*A^2*b^6*(-(4*a*c - b^2)^15)^(1/2) - 10*A*B*a*b^20 + 17794*A^2*a^2*b^17*c^2
- 188095*A^2*a^3*b^15*c^3 + 1299860*A^2*a^4*b^13*c^4 - 6126640*A^2*a^5*b^11*c^5 + 19905600*A^2*a^6*b^9*c^6 -
43904256*A^2*a^7*b^7*c^7 + 62684160*A^2*a^8*b^5*c^8 - 52039680*A^2*a^9*b^3*c^9 - 225*A^2*a^3*c^3*(-(4*a*c - b^
2)^15)^(1/2) + B^2*a^2*b^4*(-(4*a*c - b^2)^15)^(1/2) + 769*B^2*a^4*b^15*c^2 - 8620*B^2*a^5*b^13*c^3 + 63440*B^
2*a^6*b^11*c^4 - 316864*B^2*a^7*b^9*c^5 + 1069824*B^2*a^8*b^7*c^6 - 2343936*B^2*a^9*b^5*c^7 + 3010560*B^2*a^10
*b^3*c^8 + 49*B^2*a^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 6881280*A*B*a^11*c^10 - 995*A^2*a*b^19*c + 18923520*A^2*
a^10*b*c^10 - 41*B^2*a^3*b^17*c - 1720320*B^2*a^11*b*c^9 + 694*A^2*a^2*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) - 740
2*A*B*a^3*b^16*c^2 + 80620*A*B*a^4*b^14*c^3 - 575120*A*B*a^5*b^12*c^4 + 2791360*A*B*a^6*b^10*c^5 - 9267456*A*B
*a^7*b^8*c^6 + 20579328*A*B*a^8*b^6*c^7 - 28815360*A*B*a^9*b^4*c^8 + 22364160*A*B*a^10*b^2*c^9 - 245*A^2*a*b^4
*c*(-(4*a*c - b^2)^15)^(1/2) - 11*B^2*a^3*b^2*c*(-(4*a*c - b^2)^15)^(1/2) - 10*A*B*a*b^5*(-(4*a*c - b^2)^15)^(
1/2) + 404*A*B*a^2*b^18*c + 104*A*B*a^2*b^3*c*(-(4*a*c - b^2)^15)^(1/2) - 382*A*B*a^3*b*c^2*(-(4*a*c - b^2)^15
)^(1/2)))/(128*(a^7*b^20 + 1048576*a^17*c^10 - 40*a^8*b^18*c + 720*a^9*b^16*c^2 - 7680*a^10*b^14*c^3 + 53760*a
^11*b^12*c^4 - 258048*a^12*b^10*c^5 + 860160*a^13*b^8*c^6 - 1966080*a^14*b^6*c^7 + 2949120*a^15*b^4*c^8 - 2621
440*a^16*b^2*c^9)))^(1/2)*1i + (x^(1/2)*(33973862400*A^2*a^20*c^14 - 7398752256*B^2*a^21*c^13 - 28800*A^2*a^9*
b^22*c^3 + 1232640*A^2*a^10*b^20*c^4 - 23879808*A^2*a^11*b^18*c^5 + 275975424*A^2*a^12*b^16*c^6 - 2109763584*A
^2*a^13*b^14*c^7 + 11171856384*A^2*a^14*b^12*c^8 - 41653370880*A^2*a^15*b^10*c^9 + 108726976512*A^2*a^16*b^8*c
^10 - 192980975616*A^2*a^17*b^6*c^11 + 218414186496*A^2*a^18*b^4*c^12 - 137631891456*A^2*a^19*b^2*c^13 - 1152*
B^2*a^11*b^20*c^3 + 50688*B^2*a^12*b^18*c^4 - 1025280*B^2*a^13*b^16*c^5 + 12496896*B^2*a^14*b^14*c^6 - 1017446
40*B^2*a^15*b^12*c^7 + 579796992*B^2*a^16*b^10*c^8 - 2346319872*B^2*a^17*b^8*c^9 + 6653214720*B^2*a^18*b^6*c^1
0 - 12608077824*B^2*a^19*b^4*c^11 + 14344519680*B^2*a^20*b^2*c^12 + 11520*A*B*a^10*b^21*c^3 - 499968*A*B*a^11*
b^19*c^4 + 9900288*A*B*a^12*b^17*c^5 - 117559296*A*B*a^13*b^15*c^6 + 925433856*A*B*a^14*b^13*c^7 - 5038866432*
A*B*a^15*b^11*c^8 + 19191693312*A*B*a^16*b^9*c^9 - 50422874112*A*B*a^17*b^7*c^10 + 87350575104*A*B*a^18*b^5*c^
11 - 89992986624*A*B*a^19*b^3*c^12 + 41825599488*A*B*a^20*b*c^13) + (-(9*(25*A^2*b^21 + B^2*a^2*b^19 + 25*A^2*
b^6*(-(4*a*c - b^2)^15)^(1/2) - 10*A*B*a*b^20 + 17794*A^2*a^2*b^17*c^2 - 188095*A^2*a^3*b^15*c^3 + 1299860*A^2
*a^4*b^13*c^4 - 6126640*A^2*a^5*b^11*c^5 + 19905600*A^2*a^6*b^9*c^6 - 43904256*A^2*a^7*b^7*c^7 + 62684160*A^2*
a^8*b^5*c^8 - 52039680*A^2*a^9*b^3*c^9 - 225*A^2*a^3*c^3*(-(4*a*c - b^2)^15)^(1/2) + B^2*a^2*b^4*(-(4*a*c - b^
2)^15)^(1/2) + 769*B^2*a^4*b^15*c^2 - 8620*B^2*a^5*b^13*c^3 + 63440*B^2*a^6*b^11*c^4 - 316864*B^2*a^7*b^9*c^5
+ 1069824*B^2*a^8*b^7*c^6 - 2343936*B^2*a^9*b^5*c^7 + 3010560*B^2*a^10*b^3*c^8 + 49*B^2*a^4*c^2*(-(4*a*c - b^2
)^15)^(1/2) - 6881280*A*B*a^11*c^10 - 995*A^2*a*b^19*c + 18923520*A^2*a^10*b*c^10 - 41*B^2*a^3*b^17*c - 172032
0*B^2*a^11*b*c^9 + 694*A^2*a^2*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) - 7402*A*B*a^3*b^16*c^2 + 80620*A*B*a^4*b^14*
c^3 - 575120*A*B*a^5*b^12*c^4 + 2791360*A*B*a^6*b^10*c^5 - 9267456*A*B*a^7*b^8*c^6 + 20579328*A*B*a^8*b^6*c^7
- 28815360*A*B*a^9*b^4*c^8 + 22364160*A*B*a^10*b^2*c^9 - 245*A^2*a*b^4*c*(-(4*a*c - b^2)^15)^(1/2) - 11*B^2*a^
3*b^2*c*(-(4*a*c - b^2)^15)^(1/2) - 10*A*B*a*b^5*(-(4*a*c - b^2)^15)^(1/2) + 404*A*B*a^2*b^18*c + 104*A*B*a^2*
b^3*c*(-(4*a*c - b^2)^15)^(1/2) - 382*A*B*a^3*b*c^2*(-(4*a*c - b^2)^15)^(1/2)))/(128*(a^7*b^20 + 1048576*a^17*
c^10 - 40*a^8*b^18*c + 720*a^9*b^16*c^2 - 7680*a^10*b^14*c^3 + 53760*a^11*b^12*c^4 - 258048*a^12*b^10*c^5 + 86
0160*a^13*b^8*c^6 - 1966080*a^14*b^6*c^7 + 2949120*a^15*b^4*c^8 - 2621440*a^16*b^2*c^9)))^(1/2)*(x^(1/2)*(-(9*
(25*A^2*b^21 + B^2*a^2*b^19 + 25*A^2*b^6*(-(4*a*c - b^2)^15)^(1/2) - 10*A*B*a*b^20 + 17794*A^2*a^2*b^17*c^2 -
188095*A^2*a^3*b^15*c^3 + 1299860*A^2*a^4*b^13*c^4 - 6126640*A^2*a^5*b^11*c^5 + 19905600*A^2*a^6*b^9*c^6 - 439
04256*A^2*a^7*b^7*c^7 + 62684160*A^2*a^8*b^5*c^8 - 52039680*A^2*a^9*b^3*c^9 - 225*A^2*a^3*c^3*(-(4*a*c - b^2)^
15)^(1/2) + B^2*a^2*b^4*(-(4*a*c - b^2)^15)^(1/2) + 769*B^2*a^4*b^15*c^2 - 8620*B^2*a^5*b^13*c^3 + 63440*B^2*a
^6*b^11*c^4 - 316864*B^2*a^7*b^9*c^5 + 1069824*B^2*a^8*b^7*c^6 - 2343936*B^2*a^9*b^5*c^7 + 3010560*B^2*a^10*b^
3*c^8 + 49*B^2*a^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 6881280*A*B*a^11*c^10 - 995*A^2*a*b^19*c + 18923520*A^2*a^1
0*b*c^10 - 41*B^2*a^3*b^17*c - 1720320*B^2*a^11*b*c^9 + 694*A^2*a^2*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) - 7402*A
*B*a^3*b^16*c^2 + 80620*A*B*a^4*b^14*c^3 - 575120*A*B*a^5*b^12*c^4 + 2791360*A*B*a^6*b^10*c^5 - 9267456*A*B*a^
7*b^8*c^6 + 20579328*A*B*a^8*b^6*c^7 - 28815360*A*B*a^9*b^4*c^8 + 22364160*A*B*a^10*b^2*c^9 - 245*A^2*a*b^4*c*
(-(4*a*c - b^2)^15)^(1/2) - 11*B^2*a^3*b^2*c*(-(4*a*c - b^2)^15)^(1/2) - 10*A*B*a*b^5*(-(4*a*c - b^2)^15)^(1/2
) + 404*A*B*a^2*b^18*c + 104*A*B*a^2*b^3*c*(-(4*a*c - b^2)^15)^(1/2) - 382*A*B*a^3*b*c^2*(-(4*a*c - b^2)^15)^(
1/2)))/(128*(a^7*b^20 + 1048576*a^17*c^10 - 40*a^8*b^18*c + 720*a^9*b^16*c^2 - 7680*a^10*b^14*c^3 + 53760*a^11
*b^12*c^4 - 258048*a^12*b^10*c^5 + 860160*a^13*b^8*c^6 - 1966080*a^14*b^6*c^7 + 2949120*a^15*b^4*c^8 - 2621440
*a^16*b^2*c^9)))^(1/2)*(34359738368*a^26*b*c^13 - 8192*a^15*b^23*c^2 + 360448*a^16*b^21*c^3 - 7208960*a^17*b^1
9*c^4 + 86507520*a^18*b^17*c^5 - 692060160*a^19*b^15*c^6 + 3875536896*a^20*b^13*c^7 - 15502147584*a^21*b^11*c^
8 + 44291850240*a^22*b^9*c^9 - 88583700480*a^23*b^7*c^10 + 118111600640*a^24*b^5*c^11 - 94489280512*a^25*b^3*c
^12) + 22548578304*B*a^24*c^13 - 74088185856*A*a^23*b*c^13 + 15360*A*a^12*b^23*c^2 - 681984*A*a^13*b^21*c^3 +
13774848*A*a^14*b^19*c^4 - 167067648*A*a^15*b^17*c^5 + 1351876608*A*a^16*b^15*c^6 - 7662993408*A*a^17*b^13*c^7
+ 31048335360*A*a^18*b^11*c^8 - 89917489152*A*a^19*b^9*c^9 + 182401892352*A*a^20*b^7*c^10 - 246826401792*A*a^
21*b^5*c^11 + 200521285632*A*a^22*b^3*c^12 - 3072*B*a^13*b^22*c^2 + 138240*B*a^14*b^20*c^3 - 2850816*B*a^15*b^
18*c^4 + 35536896*B*a^16*b^16*c^5 - 297271296*B*a^17*b^14*c^6 + 1750597632*B*a^18*b^12*c^7 - 7398752256*B*a^19
*b^10*c^8 + 22422749184*B*a^20*b^8*c^9 - 47714402304*B*a^21*b^6*c^10 + 67847061504*B*a^22*b^4*c^11 - 579820584
96*B*a^23*b^2*c^12))*(-(9*(25*A^2*b^21 + B^2*a^2*b^19 + 25*A^2*b^6*(-(4*a*c - b^2)^15)^(1/2) - 10*A*B*a*b^20 +
17794*A^2*a^2*b^17*c^2 - 188095*A^2*a^3*b^15*c^3 + 1299860*A^2*a^4*b^13*c^4 - 6126640*A^2*a^5*b^11*c^5 + 1990
5600*A^2*a^6*b^9*c^6 - 43904256*A^2*a^7*b^7*c^7 + 62684160*A^2*a^8*b^5*c^8 - 52039680*A^2*a^9*b^3*c^9 - 225*A^
2*a^3*c^3*(-(4*a*c - b^2)^15)^(1/2) + B^2*a^2*b^4*(-(4*a*c - b^2)^15)^(1/2) + 769*B^2*a^4*b^15*c^2 - 8620*B^2*
a^5*b^13*c^3 + 63440*B^2*a^6*b^11*c^4 - 316864*B^2*a^7*b^9*c^5 + 1069824*B^2*a^8*b^7*c^6 - 2343936*B^2*a^9*b^5
*c^7 + 3010560*B^2*a^10*b^3*c^8 + 49*B^2*a^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 6881280*A*B*a^11*c^10 - 995*A^2*a
*b^19*c + 18923520*A^2*a^10*b*c^10 - 41*B^2*a^3*b^17*c - 1720320*B^2*a^11*b*c^9 + 694*A^2*a^2*b^2*c^2*(-(4*a*c
- b^2)^15)^(1/2) - 7402*A*B*a^3*b^16*c^2 + 80620*A*B*a^4*b^14*c^3 - 575120*A*B*a^5*b^12*c^4 + 2791360*A*B*a^6
*b^10*c^5 - 9267456*A*B*a^7*b^8*c^6 + 20579328*A*B*a^8*b^6*c^7 - 28815360*A*B*a^9*b^4*c^8 + 22364160*A*B*a^10*
b^2*c^9 - 245*A^2*a*b^4*c*(-(4*a*c - b^2)^15)^(1/2) - 11*B^2*a^3*b^2*c*(-(4*a*c - b^2)^15)^(1/2) - 10*A*B*a*b^
5*(-(4*a*c - b^2)^15)^(1/2) + 404*A*B*a^2*b^18*c + 104*A*B*a^2*b^3*c*(-(4*a*c - b^2)^15)^(1/2) - 382*A*B*a^3*b
*c^2*(-(4*a*c - b^2)^15)^(1/2)))/(128*(a^7*b^20 + 1048576*a^17*c^10 - 40*a^8*b^18*c + 720*a^9*b^16*c^2 - 7680*
a^10*b^14*c^3 + 53760*a^11*b^12*c^4 - 258048*a^12*b^10*c^5 + 860160*a^13*b^8*c^6 - 1966080*a^14*b^6*c^7 + 2949
120*a^15*b^4*c^8 - 2621440*a^16*b^2*c^9)))^(1/2)*1i)/((x^(1/2)*(33973862400*A^2*a^20*c^14 - 7398752256*B^2*a^2
1*c^13 - 28800*A^2*a^9*b^22*c^3 + 1232640*A^2*a^10*b^20*c^4 - 23879808*A^2*a^11*b^18*c^5 + 275975424*A^2*a^12*
b^16*c^6 - 2109763584*A^2*a^13*b^14*c^7 + 11171856384*A^2*a^14*b^12*c^8 - 41653370880*A^2*a^15*b^10*c^9 + 1087
26976512*A^2*a^16*b^8*c^10 - 192980975616*A^2*a^17*b^6*c^11 + 218414186496*A^2*a^18*b^4*c^12 - 137631891456*A^
2*a^19*b^2*c^13 - 1152*B^2*a^11*b^20*c^3 + 50688*B^2*a^12*b^18*c^4 - 1025280*B^2*a^13*b^16*c^5 + 12496896*B^2*
a^14*b^14*c^6 - 101744640*B^2*a^15*b^12*c^7 + 579796992*B^2*a^16*b^10*c^8 - 2346319872*B^2*a^17*b^8*c^9 + 6653
214720*B^2*a^18*b^6*c^10 - 12608077824*B^2*a^19*b^4*c^11 + 14344519680*B^2*a^20*b^2*c^12 + 11520*A*B*a^10*b^21
*c^3 - 499968*A*B*a^11*b^19*c^4 + 9900288*A*B*a^12*b^17*c^5 - 117559296*A*B*a^13*b^15*c^6 + 925433856*A*B*a^14
*b^13*c^7 - 5038866432*A*B*a^15*b^11*c^8 + 19191693312*A*B*a^16*b^9*c^9 - 50422874112*A*B*a^17*b^7*c^10 + 8735
0575104*A*B*a^18*b^5*c^11 - 89992986624*A*B*a^19*b^3*c^12 + 41825599488*A*B*a^20*b*c^13) + (-(9*(25*A^2*b^21 +
B^2*a^2*b^19 + 25*A^2*b^6*(-(4*a*c - b^2)^15)^(1/2) - 10*A*B*a*b^20 + 17794*A^2*a^2*b^17*c^2 - 188095*A^2*a^3
*b^15*c^3 + 1299860*A^2*a^4*b^13*c^4 - 6126640*A^2*a^5*b^11*c^5 + 19905600*A^2*a^6*b^9*c^6 - 43904256*A^2*a^7*
b^7*c^7 + 62684160*A^2*a^8*b^5*c^8 - 52039680*A^2*a^9*b^3*c^9 - 225*A^2*a^3*c^3*(-(4*a*c - b^2)^15)^(1/2) + B^
2*a^2*b^4*(-(4*a*c - b^2)^15)^(1/2) + 769*B^2*a^4*b^15*c^2 - 8620*B^2*a^5*b^13*c^3 + 63440*B^2*a^6*b^11*c^4 -
316864*B^2*a^7*b^9*c^5 + 1069824*B^2*a^8*b^7*c^6 - 2343936*B^2*a^9*b^5*c^7 + 3010560*B^2*a^10*b^3*c^8 + 49*B^2
*a^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 6881280*A*B*a^11*c^10 - 995*A^2*a*b^19*c + 18923520*A^2*a^10*b*c^10 - 41*
B^2*a^3*b^17*c - 1720320*B^2*a^11*b*c^9 + 694*A^2*a^2*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) - 7402*A*B*a^3*b^16*c^
2 + 80620*A*B*a^4*b^14*c^3 - 575120*A*B*a^5*b^12*c^4 + 2791360*A*B*a^6*b^10*c^5 - 9267456*A*B*a^7*b^8*c^6 + 20
579328*A*B*a^8*b^6*c^7 - 28815360*A*B*a^9*b^4*c^8 + 22364160*A*B*a^10*b^2*c^9 - 245*A^2*a*b^4*c*(-(4*a*c - b^2
)^15)^(1/2) - 11*B^2*a^3*b^2*c*(-(4*a*c - b^2)^15)^(1/2) - 10*A*B*a*b^5*(-(4*a*c - b^2)^15)^(1/2) + 404*A*B*a^
2*b^18*c + 104*A*B*a^2*b^3*c*(-(4*a*c - b^2)^15)^(1/2) - 382*A*B*a^3*b*c^2*(-(4*a*c - b^2)^15)^(1/2)))/(128*(a
^7*b^20 + 1048576*a^17*c^10 - 40*a^8*b^18*c + 720*a^9*b^16*c^2 - 7680*a^10*b^14*c^3 + 53760*a^11*b^12*c^4 - 25
8048*a^12*b^10*c^5 + 860160*a^13*b^8*c^6 - 1966080*a^14*b^6*c^7 + 2949120*a^15*b^4*c^8 - 2621440*a^16*b^2*c^9)
))^(1/2)*(x^(1/2)*(-(9*(25*A^2*b^21 + B^2*a^2*b^19 + 25*A^2*b^6*(-(4*a*c - b^2)^15)^(1/2) - 10*A*B*a*b^20 + 17
794*A^2*a^2*b^17*c^2 - 188095*A^2*a^3*b^15*c^3 + 1299860*A^2*a^4*b^13*c^4 - 6126640*A^2*a^5*b^11*c^5 + 1990560
0*A^2*a^6*b^9*c^6 - 43904256*A^2*a^7*b^7*c^7 + 62684160*A^2*a^8*b^5*c^8 - 52039680*A^2*a^9*b^3*c^9 - 225*A^2*a
^3*c^3*(-(4*a*c - b^2)^15)^(1/2) + B^2*a^2*b^4*(-(4*a*c - b^2)^15)^(1/2) + 769*B^2*a^4*b^15*c^2 - 8620*B^2*a^5
*b^13*c^3 + 63440*B^2*a^6*b^11*c^4 - 316864*B^2*a^7*b^9*c^5 + 1069824*B^2*a^8*b^7*c^6 - 2343936*B^2*a^9*b^5*c^
7 + 3010560*B^2*a^10*b^3*c^8 + 49*B^2*a^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 6881280*A*B*a^11*c^10 - 995*A^2*a*b^
19*c + 18923520*A^2*a^10*b*c^10 - 41*B^2*a^3*b^17*c - 1720320*B^2*a^11*b*c^9 + 694*A^2*a^2*b^2*c^2*(-(4*a*c -
b^2)^15)^(1/2) - 7402*A*B*a^3*b^16*c^2 + 80620*A*B*a^4*b^14*c^3 - 575120*A*B*a^5*b^12*c^4 + 2791360*A*B*a^6*b^
10*c^5 - 9267456*A*B*a^7*b^8*c^6 + 20579328*A*B*a^8*b^6*c^7 - 28815360*A*B*a^9*b^4*c^8 + 22364160*A*B*a^10*b^2
*c^9 - 245*A^2*a*b^4*c*(-(4*a*c - b^2)^15)^(1/2) - 11*B^2*a^3*b^2*c*(-(4*a*c - b^2)^15)^(1/2) - 10*A*B*a*b^5*(
-(4*a*c - b^2)^15)^(1/2) + 404*A*B*a^2*b^18*c + 104*A*B*a^2*b^3*c*(-(4*a*c - b^2)^15)^(1/2) - 382*A*B*a^3*b*c^
2*(-(4*a*c - b^2)^15)^(1/2)))/(128*(a^7*b^20 + 1048576*a^17*c^10 - 40*a^8*b^18*c + 720*a^9*b^16*c^2 - 7680*a^1
0*b^14*c^3 + 53760*a^11*b^12*c^4 - 258048*a^12*b^10*c^5 + 860160*a^13*b^8*c^6 - 1966080*a^14*b^6*c^7 + 2949120
*a^15*b^4*c^8 - 2621440*a^16*b^2*c^9)))^(1/2)*(34359738368*a^26*b*c^13 - 8192*a^15*b^23*c^2 + 360448*a^16*b^21
*c^3 - 7208960*a^17*b^19*c^4 + 86507520*a^18*b^17*c^5 - 692060160*a^19*b^15*c^6 + 3875536896*a^20*b^13*c^7 - 1
5502147584*a^21*b^11*c^8 + 44291850240*a^22*b^9*c^9 - 88583700480*a^23*b^7*c^10 + 118111600640*a^24*b^5*c^11 -
94489280512*a^25*b^3*c^12) - 22548578304*B*a^24*c^13 + 74088185856*A*a^23*b*c^13 - 15360*A*a^12*b^23*c^2 + 68
1984*A*a^13*b^21*c^3 - 13774848*A*a^14*b^19*c^4 + 167067648*A*a^15*b^17*c^5 - 1351876608*A*a^16*b^15*c^6 + 766
2993408*A*a^17*b^13*c^7 - 31048335360*A*a^18*b^11*c^8 + 89917489152*A*a^19*b^9*c^9 - 182401892352*A*a^20*b^7*c
^10 + 246826401792*A*a^21*b^5*c^11 - 200521285632*A*a^22*b^3*c^12 + 3072*B*a^13*b^22*c^2 - 138240*B*a^14*b^20*
c^3 + 2850816*B*a^15*b^18*c^4 - 35536896*B*a^16*b^16*c^5 + 297271296*B*a^17*b^14*c^6 - 1750597632*B*a^18*b^12*
c^7 + 7398752256*B*a^19*b^10*c^8 - 22422749184*B*a^20*b^8*c^9 + 47714402304*B*a^21*b^6*c^10 - 67847061504*B*a^
22*b^4*c^11 + 57982058496*B*a^23*b^2*c^12))*(-(9*(25*A^2*b^21 + B^2*a^2*b^19 + 25*A^2*b^6*(-(4*a*c - b^2)^15)^
(1/2) - 10*A*B*a*b^20 + 17794*A^2*a^2*b^17*c^2 - 188095*A^2*a^3*b^15*c^3 + 1299860*A^2*a^4*b^13*c^4 - 6126640*
A^2*a^5*b^11*c^5 + 19905600*A^2*a^6*b^9*c^6 - 43904256*A^2*a^7*b^7*c^7 + 62684160*A^2*a^8*b^5*c^8 - 52039680*A
^2*a^9*b^3*c^9 - 225*A^2*a^3*c^3*(-(4*a*c - b^2)^15)^(1/2) + B^2*a^2*b^4*(-(4*a*c - b^2)^15)^(1/2) + 769*B^2*a
^4*b^15*c^2 - 8620*B^2*a^5*b^13*c^3 + 63440*B^2*a^6*b^11*c^4 - 316864*B^2*a^7*b^9*c^5 + 1069824*B^2*a^8*b^7*c^
6 - 2343936*B^2*a^9*b^5*c^7 + 3010560*B^2*a^10*b^3*c^8 + 49*B^2*a^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 6881280*A*
B*a^11*c^10 - 995*A^2*a*b^19*c + 18923520*A^2*a^10*b*c^10 - 41*B^2*a^3*b^17*c - 1720320*B^2*a^11*b*c^9 + 694*A
^2*a^2*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) - 7402*A*B*a^3*b^16*c^2 + 80620*A*B*a^4*b^14*c^3 - 575120*A*B*a^5*b^1
2*c^4 + 2791360*A*B*a^6*b^10*c^5 - 9267456*A*B*a^7*b^8*c^6 + 20579328*A*B*a^8*b^6*c^7 - 28815360*A*B*a^9*b^4*c
^8 + 22364160*A*B*a^10*b^2*c^9 - 245*A^2*a*b^4*c*(-(4*a*c - b^2)^15)^(1/2) - 11*B^2*a^3*b^2*c*(-(4*a*c - b^2)^
15)^(1/2) - 10*A*B*a*b^5*(-(4*a*c - b^2)^15)^(1/2) + 404*A*B*a^2*b^18*c + 104*A*B*a^2*b^3*c*(-(4*a*c - b^2)^15
)^(1/2) - 382*A*B*a^3*b*c^2*(-(4*a*c - b^2)^15)^(1/2)))/(128*(a^7*b^20 + 1048576*a^17*c^10 - 40*a^8*b^18*c + 7
20*a^9*b^16*c^2 - 7680*a^10*b^14*c^3 + 53760*a^11*b^12*c^4 - 258048*a^12*b^10*c^5 + 860160*a^13*b^8*c^6 - 1966
080*a^14*b^6*c^7 + 2949120*a^15*b^4*c^8 - 2621440*a^16*b^2*c^9)))^(1/2) - (x^(1/2)*(33973862400*A^2*a^20*c^14
- 7398752256*B^2*a^21*c^13 - 28800*A^2*a^9*b^22*c^3 + 1232640*A^2*a^10*b^20*c^4 - 23879808*A^2*a^11*b^18*c^5 +
275975424*A^2*a^12*b^16*c^6 - 2109763584*A^2*a^13*b^14*c^7 + 11171856384*A^2*a^14*b^12*c^8 - 41653370880*A^2*
a^15*b^10*c^9 + 108726976512*A^2*a^16*b^8*c^10 - 192980975616*A^2*a^17*b^6*c^11 + 218414186496*A^2*a^18*b^4*c^
12 - 137631891456*A^2*a^19*b^2*c^13 - 1152*B^2*a^11*b^20*c^3 + 50688*B^2*a^12*b^18*c^4 - 1025280*B^2*a^13*b^16
*c^5 + 12496896*B^2*a^14*b^14*c^6 - 101744640*B^2*a^15*b^12*c^7 + 579796992*B^2*a^16*b^10*c^8 - 2346319872*B^2
*a^17*b^8*c^9 + 6653214720*B^2*a^18*b^6*c^10 - 12608077824*B^2*a^19*b^4*c^11 + 14344519680*B^2*a^20*b^2*c^12 +
11520*A*B*a^10*b^21*c^3 - 499968*A*B*a^11*b^19*c^4 + 9900288*A*B*a^12*b^17*c^5 - 117559296*A*B*a^13*b^15*c^6
+ 925433856*A*B*a^14*b^13*c^7 - 5038866432*A*B*a^15*b^11*c^8 + 19191693312*A*B*a^16*b^9*c^9 - 50422874112*A*B*
a^17*b^7*c^10 + 87350575104*A*B*a^18*b^5*c^11 - 89992986624*A*B*a^19*b^3*c^12 + 41825599488*A*B*a^20*b*c^13) +
(-(9*(25*A^2*b^21 + B^2*a^2*b^19 + 25*A^2*b^6*(-(4*a*c - b^2)^15)^(1/2) - 10*A*B*a*b^20 + 17794*A^2*a^2*b^17*
c^2 - 188095*A^2*a^3*b^15*c^3 + 1299860*A^2*a^4*b^13*c^4 - 6126640*A^2*a^5*b^11*c^5 + 19905600*A^2*a^6*b^9*c^6
- 43904256*A^2*a^7*b^7*c^7 + 62684160*A^2*a^8*b^5*c^8 - 52039680*A^2*a^9*b^3*c^9 - 225*A^2*a^3*c^3*(-(4*a*c -
b^2)^15)^(1/2) + B^2*a^2*b^4*(-(4*a*c - b^2)^15)^(1/2) + 769*B^2*a^4*b^15*c^2 - 8620*B^2*a^5*b^13*c^3 + 63440
*B^2*a^6*b^11*c^4 - 316864*B^2*a^7*b^9*c^5 + 1069824*B^2*a^8*b^7*c^6 - 2343936*B^2*a^9*b^5*c^7 + 3010560*B^2*a
^10*b^3*c^8 + 49*B^2*a^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 6881280*A*B*a^11*c^10 - 995*A^2*a*b^19*c + 18923520*A
^2*a^10*b*c^10 - 41*B^2*a^3*b^17*c - 1720320*B^2*a^11*b*c^9 + 694*A^2*a^2*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) -
7402*A*B*a^3*b^16*c^2 + 80620*A*B*a^4*b^14*c^3 - 575120*A*B*a^5*b^12*c^4 + 2791360*A*B*a^6*b^10*c^5 - 9267456*
A*B*a^7*b^8*c^6 + 20579328*A*B*a^8*b^6*c^7 - 28815360*A*B*a^9*b^4*c^8 + 22364160*A*B*a^10*b^2*c^9 - 245*A^2*a*
b^4*c*(-(4*a*c - b^2)^15)^(1/2) - 11*B^2*a^3*b^2*c*(-(4*a*c - b^2)^15)^(1/2) - 10*A*B*a*b^5*(-(4*a*c - b^2)^15
)^(1/2) + 404*A*B*a^2*b^18*c + 104*A*B*a^2*b^3*c*(-(4*a*c - b^2)^15)^(1/2) - 382*A*B*a^3*b*c^2*(-(4*a*c - b^2)
^15)^(1/2)))/(128*(a^7*b^20 + 1048576*a^17*c^10 - 40*a^8*b^18*c + 720*a^9*b^16*c^2 - 7680*a^10*b^14*c^3 + 5376
0*a^11*b^12*c^4 - 258048*a^12*b^10*c^5 + 860160*a^13*b^8*c^6 - 1966080*a^14*b^6*c^7 + 2949120*a^15*b^4*c^8 - 2
621440*a^16*b^2*c^9)))^(1/2)*(x^(1/2)*(-(9*(25*A^2*b^21 + B^2*a^2*b^19 + 25*A^2*b^6*(-(4*a*c - b^2)^15)^(1/2)
- 10*A*B*a*b^20 + 17794*A^2*a^2*b^17*c^2 - 188095*A^2*a^3*b^15*c^3 + 1299860*A^2*a^4*b^13*c^4 - 6126640*A^2*a^
5*b^11*c^5 + 19905600*A^2*a^6*b^9*c^6 - 43904256*A^2*a^7*b^7*c^7 + 62684160*A^2*a^8*b^5*c^8 - 52039680*A^2*a^9
*b^3*c^9 - 225*A^2*a^3*c^3*(-(4*a*c - b^2)^15)^(1/2) + B^2*a^2*b^4*(-(4*a*c - b^2)^15)^(1/2) + 769*B^2*a^4*b^1
5*c^2 - 8620*B^2*a^5*b^13*c^3 + 63440*B^2*a^6*b^11*c^4 - 316864*B^2*a^7*b^9*c^5 + 1069824*B^2*a^8*b^7*c^6 - 23
43936*B^2*a^9*b^5*c^7 + 3010560*B^2*a^10*b^3*c^8 + 49*B^2*a^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 6881280*A*B*a^11
*c^10 - 995*A^2*a*b^19*c + 18923520*A^2*a^10*b*c^10 - 41*B^2*a^3*b^17*c - 1720320*B^2*a^11*b*c^9 + 694*A^2*a^2
*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) - 7402*A*B*a^3*b^16*c^2 + 80620*A*B*a^4*b^14*c^3 - 575120*A*B*a^5*b^12*c^4
+ 2791360*A*B*a^6*b^10*c^5 - 9267456*A*B*a^7*b^8*c^6 + 20579328*A*B*a^8*b^6*c^7 - 28815360*A*B*a^9*b^4*c^8 + 2
2364160*A*B*a^10*b^2*c^9 - 245*A^2*a*b^4*c*(-(4*a*c - b^2)^15)^(1/2) - 11*B^2*a^3*b^2*c*(-(4*a*c - b^2)^15)^(1
/2) - 10*A*B*a*b^5*(-(4*a*c - b^2)^15)^(1/2) + 404*A*B*a^2*b^18*c + 104*A*B*a^2*b^3*c*(-(4*a*c - b^2)^15)^(1/2
) - 382*A*B*a^3*b*c^2*(-(4*a*c - b^2)^15)^(1/2)))/(128*(a^7*b^20 + 1048576*a^17*c^10 - 40*a^8*b^18*c + 720*a^9
*b^16*c^2 - 7680*a^10*b^14*c^3 + 53760*a^11*b^12*c^4 - 258048*a^12*b^10*c^5 + 860160*a^13*b^8*c^6 - 1966080*a^
14*b^6*c^7 + 2949120*a^15*b^4*c^8 - 2621440*a^16*b^2*c^9)))^(1/2)*(34359738368*a^26*b*c^13 - 8192*a^15*b^23*c^
2 + 360448*a^16*b^21*c^3 - 7208960*a^17*b^19*c^4 + 86507520*a^18*b^17*c^5 - 692060160*a^19*b^15*c^6 + 38755368
96*a^20*b^13*c^7 - 15502147584*a^21*b^11*c^8 + 44291850240*a^22*b^9*c^9 - 88583700480*a^23*b^7*c^10 + 11811160
0640*a^24*b^5*c^11 - 94489280512*a^25*b^3*c^12) + 22548578304*B*a^24*c^13 - 74088185856*A*a^23*b*c^13 + 15360*
A*a^12*b^23*c^2 - 681984*A*a^13*b^21*c^3 + 13774848*A*a^14*b^19*c^4 - 167067648*A*a^15*b^17*c^5 + 1351876608*A
*a^16*b^15*c^6 - 7662993408*A*a^17*b^13*c^7 + 31048335360*A*a^18*b^11*c^8 - 89917489152*A*a^19*b^9*c^9 + 18240
1892352*A*a^20*b^7*c^10 - 246826401792*A*a^21*b^5*c^11 + 200521285632*A*a^22*b^3*c^12 - 3072*B*a^13*b^22*c^2 +
138240*B*a^14*b^20*c^3 - 2850816*B*a^15*b^18*c^4 + 35536896*B*a^16*b^16*c^5 - 297271296*B*a^17*b^14*c^6 + 175
0597632*B*a^18*b^12*c^7 - 7398752256*B*a^19*b^10*c^8 + 22422749184*B*a^20*b^8*c^9 - 47714402304*B*a^21*b^6*c^1
0 + 67847061504*B*a^22*b^4*c^11 - 57982058496*B*a^23*b^2*c^12))*(-(9*(25*A^2*b^21 + B^2*a^2*b^19 + 25*A^2*b^6*
(-(4*a*c - b^2)^15)^(1/2) - 10*A*B*a*b^20 + 17794*A^2*a^2*b^17*c^2 - 188095*A^2*a^3*b^15*c^3 + 1299860*A^2*a^4
*b^13*c^4 - 6126640*A^2*a^5*b^11*c^5 + 19905600*A^2*a^6*b^9*c^6 - 43904256*A^2*a^7*b^7*c^7 + 62684160*A^2*a^8*
b^5*c^8 - 52039680*A^2*a^9*b^3*c^9 - 225*A^2*a^3*c^3*(-(4*a*c - b^2)^15)^(1/2) + B^2*a^2*b^4*(-(4*a*c - b^2)^1
5)^(1/2) + 769*B^2*a^4*b^15*c^2 - 8620*B^2*a^5*b^13*c^3 + 63440*B^2*a^6*b^11*c^4 - 316864*B^2*a^7*b^9*c^5 + 10
69824*B^2*a^8*b^7*c^6 - 2343936*B^2*a^9*b^5*c^7 + 3010560*B^2*a^10*b^3*c^8 + 49*B^2*a^4*c^2*(-(4*a*c - b^2)^15
)^(1/2) - 6881280*A*B*a^11*c^10 - 995*A^2*a*b^19*c + 18923520*A^2*a^10*b*c^10 - 41*B^2*a^3*b^17*c - 1720320*B^
2*a^11*b*c^9 + 694*A^2*a^2*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) - 7402*A*B*a^3*b^16*c^2 + 80620*A*B*a^4*b^14*c^3
- 575120*A*B*a^5*b^12*c^4 + 2791360*A*B*a^6*b^10*c^5 - 9267456*A*B*a^7*b^8*c^6 + 20579328*A*B*a^8*b^6*c^7 - 28
815360*A*B*a^9*b^4*c^8 + 22364160*A*B*a^10*b^2*c^9 - 245*A^2*a*b^4*c*(-(4*a*c - b^2)^15)^(1/2) - 11*B^2*a^3*b^
2*c*(-(4*a*c - b^2)^15)^(1/2) - 10*A*B*a*b^5*(-(4*a*c - b^2)^15)^(1/2) + 404*A*B*a^2*b^18*c + 104*A*B*a^2*b^3*
c*(-(4*a*c - b^2)^15)^(1/2) - 382*A*B*a^3*b*c^2*(-(4*a*c - b^2)^15)^(1/2)))/(128*(a^7*b^20 + 1048576*a^17*c^10
- 40*a^8*b^18*c + 720*a^9*b^16*c^2 - 7680*a^10*b^14*c^3 + 53760*a^11*b^12*c^4 - 258048*a^12*b^10*c^5 + 860160
*a^13*b^8*c^6 - 1966080*a^14*b^6*c^7 + 2949120*a^15*b^4*c^8 - 2621440*a^16*b^2*c^9)))^(1/2) + 47775744000*A^3*
a^17*c^14 + 712800*A^3*a^9*b^16*c^6 - 23142240*A^3*a^10*b^14*c^7 + 328157568*A^3*a^11*b^12*c^8 - 2652784128*A^
3*a^12*b^10*c^9 + 13361338368*A^3*a^13*b^8*c^10 - 42897973248*A^3*a^14*b^6*c^11 + 85645099008*A^3*a^15*b^4*c^1
2 - 97090928640*A^3*a^16*b^2*c^13 - 18144*B^3*a^11*b^15*c^5 + 622080*B^3*a^12*b^13*c^6 - 9220608*B^3*a^13*b^11
*c^7 + 76640256*B^3*a^14*b^9*c^8 - 384638976*B^3*a^15*b^7*c^9 + 1160773632*B^3*a^16*b^5*c^10 - 1942880256*B^3*
a^17*b^3*c^11 + 10404495360*A*B^2*a^18*c^13 + 1387266048*B^3*a^18*b*c^12 - 26966753280*A^2*B*a^17*b*c^13 + 181
440*A*B^2*a^10*b^16*c^5 - 6083424*A*B^2*a^11*b^14*c^6 + 88656768*A*B^2*a^12*b^12*c^7 - 731026944*A*B^2*a^13*b^
10*c^8 + 3713071104*A*B^2*a^14*b^8*c^9 - 11822505984*A*B^2*a^15*b^6*c^10 + 22839459840*A*B^2*a^16*b^4*c^11 - 2
4132059136*A*B^2*a^17*b^2*c^12 - 453600*A^2*B*a^9*b^17*c^5 + 14722560*A^2*B*a^10*b^15*c^6 - 208303488*A^2*B*a^
11*b^13*c^7 + 1675717632*A^2*B*a^12*b^11*c^8 - 8368883712*A^2*B*a^13*b^9*c^9 + 26512883712*A^2*B*a^14*b^7*c^10
- 51887112192*A^2*B*a^15*b^5*c^11 + 57139789824*A^2*B*a^16*b^3*c^12))*(-(9*(25*A^2*b^21 + B^2*a^2*b^19 + 25*A
^2*b^6*(-(4*a*c - b^2)^15)^(1/2) - 10*A*B*a*b^20 + 17794*A^2*a^2*b^17*c^2 - 188095*A^2*a^3*b^15*c^3 + 1299860*
A^2*a^4*b^13*c^4 - 6126640*A^2*a^5*b^11*c^5 + 19905600*A^2*a^6*b^9*c^6 - 43904256*A^2*a^7*b^7*c^7 + 62684160*A
^2*a^8*b^5*c^8 - 52039680*A^2*a^9*b^3*c^9 - 225*A^2*a^3*c^3*(-(4*a*c - b^2)^15)^(1/2) + B^2*a^2*b^4*(-(4*a*c -
b^2)^15)^(1/2) + 769*B^2*a^4*b^15*c^2 - 8620*B^2*a^5*b^13*c^3 + 63440*B^2*a^6*b^11*c^4 - 316864*B^2*a^7*b^9*c
^5 + 1069824*B^2*a^8*b^7*c^6 - 2343936*B^2*a^9*b^5*c^7 + 3010560*B^2*a^10*b^3*c^8 + 49*B^2*a^4*c^2*(-(4*a*c -
b^2)^15)^(1/2) - 6881280*A*B*a^11*c^10 - 995*A^2*a*b^19*c + 18923520*A^2*a^10*b*c^10 - 41*B^2*a^3*b^17*c - 172
0320*B^2*a^11*b*c^9 + 694*A^2*a^2*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) - 7402*A*B*a^3*b^16*c^2 + 80620*A*B*a^4*b^
14*c^3 - 575120*A*B*a^5*b^12*c^4 + 2791360*A*B*a^6*b^10*c^5 - 9267456*A*B*a^7*b^8*c^6 + 20579328*A*B*a^8*b^6*c
^7 - 28815360*A*B*a^9*b^4*c^8 + 22364160*A*B*a^10*b^2*c^9 - 245*A^2*a*b^4*c*(-(4*a*c - b^2)^15)^(1/2) - 11*B^2
*a^3*b^2*c*(-(4*a*c - b^2)^15)^(1/2) - 10*A*B*a*b^5*(-(4*a*c - b^2)^15)^(1/2) + 404*A*B*a^2*b^18*c + 104*A*B*a
^2*b^3*c*(-(4*a*c - b^2)^15)^(1/2) - 382*A*B*a^3*b*c^2*(-(4*a*c - b^2)^15)^(1/2)))/(128*(a^7*b^20 + 1048576*a^
17*c^10 - 40*a^8*b^18*c + 720*a^9*b^16*c^2 - 7680*a^10*b^14*c^3 + 53760*a^11*b^12*c^4 - 258048*a^12*b^10*c^5 +
860160*a^13*b^8*c^6 - 1966080*a^14*b^6*c^7 + 2949120*a^15*b^4*c^8 - 2621440*a^16*b^2*c^9)))^(1/2)*2i - atan((
(x^(1/2)*(33973862400*A^2*a^20*c^14 - 7398752256*B^2*a^21*c^13 - 28800*A^2*a^9*b^22*c^3 + 1232640*A^2*a^10*b^2
0*c^4 - 23879808*A^2*a^11*b^18*c^5 + 275975424*A^2*a^12*b^16*c^6 - 2109763584*A^2*a^13*b^14*c^7 + 11171856384*
A^2*a^14*b^12*c^8 - 41653370880*A^2*a^15*b^10*c^9 + 108726976512*A^2*a^16*b^8*c^10 - 192980975616*A^2*a^17*b^6
*c^11 + 218414186496*A^2*a^18*b^4*c^12 - 137631891456*A^2*a^19*b^2*c^13 - 1152*B^2*a^11*b^20*c^3 + 50688*B^2*a
^12*b^18*c^4 - 1025280*B^2*a^13*b^16*c^5 + 12496896*B^2*a^14*b^14*c^6 - 101744640*B^2*a^15*b^12*c^7 + 57979699
2*B^2*a^16*b^10*c^8 - 2346319872*B^2*a^17*b^8*c^9 + 6653214720*B^2*a^18*b^6*c^10 - 12608077824*B^2*a^19*b^4*c^
11 + 14344519680*B^2*a^20*b^2*c^12 + 11520*A*B*a^10*b^21*c^3 - 499968*A*B*a^11*b^19*c^4 + 9900288*A*B*a^12*b^1
7*c^5 - 117559296*A*B*a^13*b^15*c^6 + 925433856*A*B*a^14*b^13*c^7 - 5038866432*A*B*a^15*b^11*c^8 + 19191693312
*A*B*a^16*b^9*c^9 - 50422874112*A*B*a^17*b^7*c^10 + 87350575104*A*B*a^18*b^5*c^11 - 89992986624*A*B*a^19*b^3*c
^12 + 41825599488*A*B*a^20*b*c^13) + (-(9*(25*A^2*b^21 + B^2*a^2*b^19 - 25*A^2*b^6*(-(4*a*c - b^2)^15)^(1/2) -
10*A*B*a*b^20 + 17794*A^2*a^2*b^17*c^2 - 188095*A^2*a^3*b^15*c^3 + 1299860*A^2*a^4*b^13*c^4 - 6126640*A^2*a^5
*b^11*c^5 + 19905600*A^2*a^6*b^9*c^6 - 43904256*A^2*a^7*b^7*c^7 + 62684160*A^2*a^8*b^5*c^8 - 52039680*A^2*a^9*
b^3*c^9 + 225*A^2*a^3*c^3*(-(4*a*c - b^2)^15)^(1/2) - B^2*a^2*b^4*(-(4*a*c - b^2)^15)^(1/2) + 769*B^2*a^4*b^15
*c^2 - 8620*B^2*a^5*b^13*c^3 + 63440*B^2*a^6*b^11*c^4 - 316864*B^2*a^7*b^9*c^5 + 1069824*B^2*a^8*b^7*c^6 - 234
3936*B^2*a^9*b^5*c^7 + 3010560*B^2*a^10*b^3*c^8 - 49*B^2*a^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 6881280*A*B*a^11*
c^10 - 995*A^2*a*b^19*c + 18923520*A^2*a^10*b*c^10 - 41*B^2*a^3*b^17*c - 1720320*B^2*a^11*b*c^9 - 694*A^2*a^2*
b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) - 7402*A*B*a^3*b^16*c^2 + 80620*A*B*a^4*b^14*c^3 - 575120*A*B*a^5*b^12*c^4 +
2791360*A*B*a^6*b^10*c^5 - 9267456*A*B*a^7*b^8*c^6 + 20579328*A*B*a^8*b^6*c^7 - 28815360*A*B*a^9*b^4*c^8 + 22
364160*A*B*a^10*b^2*c^9 + 245*A^2*a*b^4*c*(-(4*a*c - b^2)^15)^(1/2) + 11*B^2*a^3*b^2*c*(-(4*a*c - b^2)^15)^(1/
2) + 10*A*B*a*b^5*(-(4*a*c - b^2)^15)^(1/2) + 404*A*B*a^2*b^18*c - 104*A*B*a^2*b^3*c*(-(4*a*c - b^2)^15)^(1/2)
+ 382*A*B*a^3*b*c^2*(-(4*a*c - b^2)^15)^(1/2)))/(128*(a^7*b^20 + 1048576*a^17*c^10 - 40*a^8*b^18*c + 720*a^9*
b^16*c^2 - 7680*a^10*b^14*c^3 + 53760*a^11*b^12*c^4 - 258048*a^12*b^10*c^5 + 860160*a^13*b^8*c^6 - 1966080*a^1
4*b^6*c^7 + 2949120*a^15*b^4*c^8 - 2621440*a^16*b^2*c^9)))^(1/2)*(x^(1/2)*(-(9*(25*A^2*b^21 + B^2*a^2*b^19 - 2
5*A^2*b^6*(-(4*a*c - b^2)^15)^(1/2) - 10*A*B*a*b^20 + 17794*A^2*a^2*b^17*c^2 - 188095*A^2*a^3*b^15*c^3 + 12998
60*A^2*a^4*b^13*c^4 - 6126640*A^2*a^5*b^11*c^5 + 19905600*A^2*a^6*b^9*c^6 - 43904256*A^2*a^7*b^7*c^7 + 6268416
0*A^2*a^8*b^5*c^8 - 52039680*A^2*a^9*b^3*c^9 + 225*A^2*a^3*c^3*(-(4*a*c - b^2)^15)^(1/2) - B^2*a^2*b^4*(-(4*a*
c - b^2)^15)^(1/2) + 769*B^2*a^4*b^15*c^2 - 8620*B^2*a^5*b^13*c^3 + 63440*B^2*a^6*b^11*c^4 - 316864*B^2*a^7*b^
9*c^5 + 1069824*B^2*a^8*b^7*c^6 - 2343936*B^2*a^9*b^5*c^7 + 3010560*B^2*a^10*b^3*c^8 - 49*B^2*a^4*c^2*(-(4*a*c
- b^2)^15)^(1/2) - 6881280*A*B*a^11*c^10 - 995*A^2*a*b^19*c + 18923520*A^2*a^10*b*c^10 - 41*B^2*a^3*b^17*c -
1720320*B^2*a^11*b*c^9 - 694*A^2*a^2*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) - 7402*A*B*a^3*b^16*c^2 + 80620*A*B*a^4
*b^14*c^3 - 575120*A*B*a^5*b^12*c^4 + 2791360*A*B*a^6*b^10*c^5 - 9267456*A*B*a^7*b^8*c^6 + 20579328*A*B*a^8*b^
6*c^7 - 28815360*A*B*a^9*b^4*c^8 + 22364160*A*B*a^10*b^2*c^9 + 245*A^2*a*b^4*c*(-(4*a*c - b^2)^15)^(1/2) + 11*
B^2*a^3*b^2*c*(-(4*a*c - b^2)^15)^(1/2) + 10*A*B*a*b^5*(-(4*a*c - b^2)^15)^(1/2) + 404*A*B*a^2*b^18*c - 104*A*
B*a^2*b^3*c*(-(4*a*c - b^2)^15)^(1/2) + 382*A*B*a^3*b*c^2*(-(4*a*c - b^2)^15)^(1/2)))/(128*(a^7*b^20 + 1048576
*a^17*c^10 - 40*a^8*b^18*c + 720*a^9*b^16*c^2 - 7680*a^10*b^14*c^3 + 53760*a^11*b^12*c^4 - 258048*a^12*b^10*c^
5 + 860160*a^13*b^8*c^6 - 1966080*a^14*b^6*c^7 + 2949120*a^15*b^4*c^8 - 2621440*a^16*b^2*c^9)))^(1/2)*(3435973
8368*a^26*b*c^13 - 8192*a^15*b^23*c^2 + 360448*a^16*b^21*c^3 - 7208960*a^17*b^19*c^4 + 86507520*a^18*b^17*c^5
- 692060160*a^19*b^15*c^6 + 3875536896*a^20*b^13*c^7 - 15502147584*a^21*b^11*c^8 + 44291850240*a^22*b^9*c^9 -
88583700480*a^23*b^7*c^10 + 118111600640*a^24*b^5*c^11 - 94489280512*a^25*b^3*c^12) - 22548578304*B*a^24*c^13
+ 74088185856*A*a^23*b*c^13 - 15360*A*a^12*b^23*c^2 + 681984*A*a^13*b^21*c^3 - 13774848*A*a^14*b^19*c^4 + 1670
67648*A*a^15*b^17*c^5 - 1351876608*A*a^16*b^15*c^6 + 7662993408*A*a^17*b^13*c^7 - 31048335360*A*a^18*b^11*c^8
+ 89917489152*A*a^19*b^9*c^9 - 182401892352*A*a^20*b^7*c^10 + 246826401792*A*a^21*b^5*c^11 - 200521285632*A*a^
22*b^3*c^12 + 3072*B*a^13*b^22*c^2 - 138240*B*a^14*b^20*c^3 + 2850816*B*a^15*b^18*c^4 - 35536896*B*a^16*b^16*c
^5 + 297271296*B*a^17*b^14*c^6 - 1750597632*B*a^18*b^12*c^7 + 7398752256*B*a^19*b^10*c^8 - 22422749184*B*a^20*
b^8*c^9 + 47714402304*B*a^21*b^6*c^10 - 67847061504*B*a^22*b^4*c^11 + 57982058496*B*a^23*b^2*c^12))*(-(9*(25*A
^2*b^21 + B^2*a^2*b^19 - 25*A^2*b^6*(-(4*a*c - b^2)^15)^(1/2) - 10*A*B*a*b^20 + 17794*A^2*a^2*b^17*c^2 - 18809
5*A^2*a^3*b^15*c^3 + 1299860*A^2*a^4*b^13*c^4 - 6126640*A^2*a^5*b^11*c^5 + 19905600*A^2*a^6*b^9*c^6 - 43904256
*A^2*a^7*b^7*c^7 + 62684160*A^2*a^8*b^5*c^8 - 52039680*A^2*a^9*b^3*c^9 + 225*A^2*a^3*c^3*(-(4*a*c - b^2)^15)^(
1/2) - B^2*a^2*b^4*(-(4*a*c - b^2)^15)^(1/2) + 769*B^2*a^4*b^15*c^2 - 8620*B^2*a^5*b^13*c^3 + 63440*B^2*a^6*b^
11*c^4 - 316864*B^2*a^7*b^9*c^5 + 1069824*B^2*a^8*b^7*c^6 - 2343936*B^2*a^9*b^5*c^7 + 3010560*B^2*a^10*b^3*c^8
- 49*B^2*a^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 6881280*A*B*a^11*c^10 - 995*A^2*a*b^19*c + 18923520*A^2*a^10*b*c
^10 - 41*B^2*a^3*b^17*c - 1720320*B^2*a^11*b*c^9 - 694*A^2*a^2*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) - 7402*A*B*a^
3*b^16*c^2 + 80620*A*B*a^4*b^14*c^3 - 575120*A*B*a^5*b^12*c^4 + 2791360*A*B*a^6*b^10*c^5 - 9267456*A*B*a^7*b^8
*c^6 + 20579328*A*B*a^8*b^6*c^7 - 28815360*A*B*a^9*b^4*c^8 + 22364160*A*B*a^10*b^2*c^9 + 245*A^2*a*b^4*c*(-(4*
a*c - b^2)^15)^(1/2) + 11*B^2*a^3*b^2*c*(-(4*a*c - b^2)^15)^(1/2) + 10*A*B*a*b^5*(-(4*a*c - b^2)^15)^(1/2) + 4
04*A*B*a^2*b^18*c - 104*A*B*a^2*b^3*c*(-(4*a*c - b^2)^15)^(1/2) + 382*A*B*a^3*b*c^2*(-(4*a*c - b^2)^15)^(1/2))
)/(128*(a^7*b^20 + 1048576*a^17*c^10 - 40*a^8*b^18*c + 720*a^9*b^16*c^2 - 7680*a^10*b^14*c^3 + 53760*a^11*b^12
*c^4 - 258048*a^12*b^10*c^5 + 860160*a^13*b^8*c^6 - 1966080*a^14*b^6*c^7 + 2949120*a^15*b^4*c^8 - 2621440*a^16
*b^2*c^9)))^(1/2)*1i + (x^(1/2)*(33973862400*A^2*a^20*c^14 - 7398752256*B^2*a^21*c^13 - 28800*A^2*a^9*b^22*c^3
+ 1232640*A^2*a^10*b^20*c^4 - 23879808*A^2*a^11*b^18*c^5 + 275975424*A^2*a^12*b^16*c^6 - 2109763584*A^2*a^13*
b^14*c^7 + 11171856384*A^2*a^14*b^12*c^8 - 41653370880*A^2*a^15*b^10*c^9 + 108726976512*A^2*a^16*b^8*c^10 - 19
2980975616*A^2*a^17*b^6*c^11 + 218414186496*A^2*a^18*b^4*c^12 - 137631891456*A^2*a^19*b^2*c^13 - 1152*B^2*a^11
*b^20*c^3 + 50688*B^2*a^12*b^18*c^4 - 1025280*B^2*a^13*b^16*c^5 + 12496896*B^2*a^14*b^14*c^6 - 101744640*B^2*a
^15*b^12*c^7 + 579796992*B^2*a^16*b^10*c^8 - 2346319872*B^2*a^17*b^8*c^9 + 6653214720*B^2*a^18*b^6*c^10 - 1260
8077824*B^2*a^19*b^4*c^11 + 14344519680*B^2*a^20*b^2*c^12 + 11520*A*B*a^10*b^21*c^3 - 499968*A*B*a^11*b^19*c^4
+ 9900288*A*B*a^12*b^17*c^5 - 117559296*A*B*a^13*b^15*c^6 + 925433856*A*B*a^14*b^13*c^7 - 5038866432*A*B*a^15
*b^11*c^8 + 19191693312*A*B*a^16*b^9*c^9 - 50422874112*A*B*a^17*b^7*c^10 + 87350575104*A*B*a^18*b^5*c^11 - 899
92986624*A*B*a^19*b^3*c^12 + 41825599488*A*B*a^20*b*c^13) + (-(9*(25*A^2*b^21 + B^2*a^2*b^19 - 25*A^2*b^6*(-(4
*a*c - b^2)^15)^(1/2) - 10*A*B*a*b^20 + 17794*A^2*a^2*b^17*c^2 - 188095*A^2*a^3*b^15*c^3 + 1299860*A^2*a^4*b^1
3*c^4 - 6126640*A^2*a^5*b^11*c^5 + 19905600*A^2*a^6*b^9*c^6 - 43904256*A^2*a^7*b^7*c^7 + 62684160*A^2*a^8*b^5*
c^8 - 52039680*A^2*a^9*b^3*c^9 + 225*A^2*a^3*c^3*(-(4*a*c - b^2)^15)^(1/2) - B^2*a^2*b^4*(-(4*a*c - b^2)^15)^(
1/2) + 769*B^2*a^4*b^15*c^2 - 8620*B^2*a^5*b^13*c^3 + 63440*B^2*a^6*b^11*c^4 - 316864*B^2*a^7*b^9*c^5 + 106982
4*B^2*a^8*b^7*c^6 - 2343936*B^2*a^9*b^5*c^7 + 3010560*B^2*a^10*b^3*c^8 - 49*B^2*a^4*c^2*(-(4*a*c - b^2)^15)^(1
/2) - 6881280*A*B*a^11*c^10 - 995*A^2*a*b^19*c + 18923520*A^2*a^10*b*c^10 - 41*B^2*a^3*b^17*c - 1720320*B^2*a^
11*b*c^9 - 694*A^2*a^2*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) - 7402*A*B*a^3*b^16*c^2 + 80620*A*B*a^4*b^14*c^3 - 57
5120*A*B*a^5*b^12*c^4 + 2791360*A*B*a^6*b^10*c^5 - 9267456*A*B*a^7*b^8*c^6 + 20579328*A*B*a^8*b^6*c^7 - 288153
60*A*B*a^9*b^4*c^8 + 22364160*A*B*a^10*b^2*c^9 + 245*A^2*a*b^4*c*(-(4*a*c - b^2)^15)^(1/2) + 11*B^2*a^3*b^2*c*
(-(4*a*c - b^2)^15)^(1/2) + 10*A*B*a*b^5*(-(4*a*c - b^2)^15)^(1/2) + 404*A*B*a^2*b^18*c - 104*A*B*a^2*b^3*c*(-
(4*a*c - b^2)^15)^(1/2) + 382*A*B*a^3*b*c^2*(-(4*a*c - b^2)^15)^(1/2)))/(128*(a^7*b^20 + 1048576*a^17*c^10 - 4
0*a^8*b^18*c + 720*a^9*b^16*c^2 - 7680*a^10*b^14*c^3 + 53760*a^11*b^12*c^4 - 258048*a^12*b^10*c^5 + 860160*a^1
3*b^8*c^6 - 1966080*a^14*b^6*c^7 + 2949120*a^15*b^4*c^8 - 2621440*a^16*b^2*c^9)))^(1/2)*(x^(1/2)*(-(9*(25*A^2*
b^21 + B^2*a^2*b^19 - 25*A^2*b^6*(-(4*a*c - b^2)^15)^(1/2) - 10*A*B*a*b^20 + 17794*A^2*a^2*b^17*c^2 - 188095*A
^2*a^3*b^15*c^3 + 1299860*A^2*a^4*b^13*c^4 - 6126640*A^2*a^5*b^11*c^5 + 19905600*A^2*a^6*b^9*c^6 - 43904256*A^
2*a^7*b^7*c^7 + 62684160*A^2*a^8*b^5*c^8 - 52039680*A^2*a^9*b^3*c^9 + 225*A^2*a^3*c^3*(-(4*a*c - b^2)^15)^(1/2
) - B^2*a^2*b^4*(-(4*a*c - b^2)^15)^(1/2) + 769*B^2*a^4*b^15*c^2 - 8620*B^2*a^5*b^13*c^3 + 63440*B^2*a^6*b^11*
c^4 - 316864*B^2*a^7*b^9*c^5 + 1069824*B^2*a^8*b^7*c^6 - 2343936*B^2*a^9*b^5*c^7 + 3010560*B^2*a^10*b^3*c^8 -
49*B^2*a^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 6881280*A*B*a^11*c^10 - 995*A^2*a*b^19*c + 18923520*A^2*a^10*b*c^10
- 41*B^2*a^3*b^17*c - 1720320*B^2*a^11*b*c^9 - 694*A^2*a^2*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) - 7402*A*B*a^3*b
^16*c^2 + 80620*A*B*a^4*b^14*c^3 - 575120*A*B*a^5*b^12*c^4 + 2791360*A*B*a^6*b^10*c^5 - 9267456*A*B*a^7*b^8*c^
6 + 20579328*A*B*a^8*b^6*c^7 - 28815360*A*B*a^9*b^4*c^8 + 22364160*A*B*a^10*b^2*c^9 + 245*A^2*a*b^4*c*(-(4*a*c
- b^2)^15)^(1/2) + 11*B^2*a^3*b^2*c*(-(4*a*c - b^2)^15)^(1/2) + 10*A*B*a*b^5*(-(4*a*c - b^2)^15)^(1/2) + 404*
A*B*a^2*b^18*c - 104*A*B*a^2*b^3*c*(-(4*a*c - b^2)^15)^(1/2) + 382*A*B*a^3*b*c^2*(-(4*a*c - b^2)^15)^(1/2)))/(
128*(a^7*b^20 + 1048576*a^17*c^10 - 40*a^8*b^18*c + 720*a^9*b^16*c^2 - 7680*a^10*b^14*c^3 + 53760*a^11*b^12*c^
4 - 258048*a^12*b^10*c^5 + 860160*a^13*b^8*c^6 - 1966080*a^14*b^6*c^7 + 2949120*a^15*b^4*c^8 - 2621440*a^16*b^
2*c^9)))^(1/2)*(34359738368*a^26*b*c^13 - 8192*a^15*b^23*c^2 + 360448*a^16*b^21*c^3 - 7208960*a^17*b^19*c^4 +
86507520*a^18*b^17*c^5 - 692060160*a^19*b^15*c^6 + 3875536896*a^20*b^13*c^7 - 15502147584*a^21*b^11*c^8 + 4429
1850240*a^22*b^9*c^9 - 88583700480*a^23*b^7*c^10 + 118111600640*a^24*b^5*c^11 - 94489280512*a^25*b^3*c^12) + 2
2548578304*B*a^24*c^13 - 74088185856*A*a^23*b*c^13 + 15360*A*a^12*b^23*c^2 - 681984*A*a^13*b^21*c^3 + 13774848
*A*a^14*b^19*c^4 - 167067648*A*a^15*b^17*c^5 + 1351876608*A*a^16*b^15*c^6 - 7662993408*A*a^17*b^13*c^7 + 31048
335360*A*a^18*b^11*c^8 - 89917489152*A*a^19*b^9*c^9 + 182401892352*A*a^20*b^7*c^10 - 246826401792*A*a^21*b^5*c
^11 + 200521285632*A*a^22*b^3*c^12 - 3072*B*a^13*b^22*c^2 + 138240*B*a^14*b^20*c^3 - 2850816*B*a^15*b^18*c^4 +
35536896*B*a^16*b^16*c^5 - 297271296*B*a^17*b^14*c^6 + 1750597632*B*a^18*b^12*c^7 - 7398752256*B*a^19*b^10*c^
8 + 22422749184*B*a^20*b^8*c^9 - 47714402304*B*a^21*b^6*c^10 + 67847061504*B*a^22*b^4*c^11 - 57982058496*B*a^2
3*b^2*c^12))*(-(9*(25*A^2*b^21 + B^2*a^2*b^19 - 25*A^2*b^6*(-(4*a*c - b^2)^15)^(1/2) - 10*A*B*a*b^20 + 17794*A
^2*a^2*b^17*c^2 - 188095*A^2*a^3*b^15*c^3 + 1299860*A^2*a^4*b^13*c^4 - 6126640*A^2*a^5*b^11*c^5 + 19905600*A^2
*a^6*b^9*c^6 - 43904256*A^2*a^7*b^7*c^7 + 62684160*A^2*a^8*b^5*c^8 - 52039680*A^2*a^9*b^3*c^9 + 225*A^2*a^3*c^
3*(-(4*a*c - b^2)^15)^(1/2) - B^2*a^2*b^4*(-(4*a*c - b^2)^15)^(1/2) + 769*B^2*a^4*b^15*c^2 - 8620*B^2*a^5*b^13
*c^3 + 63440*B^2*a^6*b^11*c^4 - 316864*B^2*a^7*b^9*c^5 + 1069824*B^2*a^8*b^7*c^6 - 2343936*B^2*a^9*b^5*c^7 + 3
010560*B^2*a^10*b^3*c^8 - 49*B^2*a^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 6881280*A*B*a^11*c^10 - 995*A^2*a*b^19*c
+ 18923520*A^2*a^10*b*c^10 - 41*B^2*a^3*b^17*c - 1720320*B^2*a^11*b*c^9 - 694*A^2*a^2*b^2*c^2*(-(4*a*c - b^2)^
15)^(1/2) - 7402*A*B*a^3*b^16*c^2 + 80620*A*B*a^4*b^14*c^3 - 575120*A*B*a^5*b^12*c^4 + 2791360*A*B*a^6*b^10*c^
5 - 9267456*A*B*a^7*b^8*c^6 + 20579328*A*B*a^8*b^6*c^7 - 28815360*A*B*a^9*b^4*c^8 + 22364160*A*B*a^10*b^2*c^9
+ 245*A^2*a*b^4*c*(-(4*a*c - b^2)^15)^(1/2) + 11*B^2*a^3*b^2*c*(-(4*a*c - b^2)^15)^(1/2) + 10*A*B*a*b^5*(-(4*a
*c - b^2)^15)^(1/2) + 404*A*B*a^2*b^18*c - 104*A*B*a^2*b^3*c*(-(4*a*c - b^2)^15)^(1/2) + 382*A*B*a^3*b*c^2*(-(
4*a*c - b^2)^15)^(1/2)))/(128*(a^7*b^20 + 1048576*a^17*c^10 - 40*a^8*b^18*c + 720*a^9*b^16*c^2 - 7680*a^10*b^1
4*c^3 + 53760*a^11*b^12*c^4 - 258048*a^12*b^10*c^5 + 860160*a^13*b^8*c^6 - 1966080*a^14*b^6*c^7 + 2949120*a^15
*b^4*c^8 - 2621440*a^16*b^2*c^9)))^(1/2)*1i)/((x^(1/2)*(33973862400*A^2*a^20*c^14 - 7398752256*B^2*a^21*c^13 -
28800*A^2*a^9*b^22*c^3 + 1232640*A^2*a^10*b^20*c^4 - 23879808*A^2*a^11*b^18*c^5 + 275975424*A^2*a^12*b^16*c^6
- 2109763584*A^2*a^13*b^14*c^7 + 11171856384*A^2*a^14*b^12*c^8 - 41653370880*A^2*a^15*b^10*c^9 + 108726976512
*A^2*a^16*b^8*c^10 - 192980975616*A^2*a^17*b^6*c^11 + 218414186496*A^2*a^18*b^4*c^12 - 137631891456*A^2*a^19*b
^2*c^13 - 1152*B^2*a^11*b^20*c^3 + 50688*B^2*a^12*b^18*c^4 - 1025280*B^2*a^13*b^16*c^5 + 12496896*B^2*a^14*b^1
4*c^6 - 101744640*B^2*a^15*b^12*c^7 + 579796992*B^2*a^16*b^10*c^8 - 2346319872*B^2*a^17*b^8*c^9 + 6653214720*B
^2*a^18*b^6*c^10 - 12608077824*B^2*a^19*b^4*c^11 + 14344519680*B^2*a^20*b^2*c^12 + 11520*A*B*a^10*b^21*c^3 - 4
99968*A*B*a^11*b^19*c^4 + 9900288*A*B*a^12*b^17*c^5 - 117559296*A*B*a^13*b^15*c^6 + 925433856*A*B*a^14*b^13*c^
7 - 5038866432*A*B*a^15*b^11*c^8 + 19191693312*A*B*a^16*b^9*c^9 - 50422874112*A*B*a^17*b^7*c^10 + 87350575104*
A*B*a^18*b^5*c^11 - 89992986624*A*B*a^19*b^3*c^12 + 41825599488*A*B*a^20*b*c^13) + (-(9*(25*A^2*b^21 + B^2*a^2
*b^19 - 25*A^2*b^6*(-(4*a*c - b^2)^15)^(1/2) - 10*A*B*a*b^20 + 17794*A^2*a^2*b^17*c^2 - 188095*A^2*a^3*b^15*c^
3 + 1299860*A^2*a^4*b^13*c^4 - 6126640*A^2*a^5*b^11*c^5 + 19905600*A^2*a^6*b^9*c^6 - 43904256*A^2*a^7*b^7*c^7
+ 62684160*A^2*a^8*b^5*c^8 - 52039680*A^2*a^9*b^3*c^9 + 225*A^2*a^3*c^3*(-(4*a*c - b^2)^15)^(1/2) - B^2*a^2*b^
4*(-(4*a*c - b^2)^15)^(1/2) + 769*B^2*a^4*b^15*c^2 - 8620*B^2*a^5*b^13*c^3 + 63440*B^2*a^6*b^11*c^4 - 316864*B
^2*a^7*b^9*c^5 + 1069824*B^2*a^8*b^7*c^6 - 2343936*B^2*a^9*b^5*c^7 + 3010560*B^2*a^10*b^3*c^8 - 49*B^2*a^4*c^2
*(-(4*a*c - b^2)^15)^(1/2) - 6881280*A*B*a^11*c^10 - 995*A^2*a*b^19*c + 18923520*A^2*a^10*b*c^10 - 41*B^2*a^3*
b^17*c - 1720320*B^2*a^11*b*c^9 - 694*A^2*a^2*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) - 7402*A*B*a^3*b^16*c^2 + 8062
0*A*B*a^4*b^14*c^3 - 575120*A*B*a^5*b^12*c^4 + 2791360*A*B*a^6*b^10*c^5 - 9267456*A*B*a^7*b^8*c^6 + 20579328*A
*B*a^8*b^6*c^7 - 28815360*A*B*a^9*b^4*c^8 + 22364160*A*B*a^10*b^2*c^9 + 245*A^2*a*b^4*c*(-(4*a*c - b^2)^15)^(1
/2) + 11*B^2*a^3*b^2*c*(-(4*a*c - b^2)^15)^(1/2) + 10*A*B*a*b^5*(-(4*a*c - b^2)^15)^(1/2) + 404*A*B*a^2*b^18*c
- 104*A*B*a^2*b^3*c*(-(4*a*c - b^2)^15)^(1/2) + 382*A*B*a^3*b*c^2*(-(4*a*c - b^2)^15)^(1/2)))/(128*(a^7*b^20
+ 1048576*a^17*c^10 - 40*a^8*b^18*c + 720*a^9*b^16*c^2 - 7680*a^10*b^14*c^3 + 53760*a^11*b^12*c^4 - 258048*a^1
2*b^10*c^5 + 860160*a^13*b^8*c^6 - 1966080*a^14*b^6*c^7 + 2949120*a^15*b^4*c^8 - 2621440*a^16*b^2*c^9)))^(1/2)
*(x^(1/2)*(-(9*(25*A^2*b^21 + B^2*a^2*b^19 - 25*A^2*b^6*(-(4*a*c - b^2)^15)^(1/2) - 10*A*B*a*b^20 + 17794*A^2*
a^2*b^17*c^2 - 188095*A^2*a^3*b^15*c^3 + 1299860*A^2*a^4*b^13*c^4 - 6126640*A^2*a^5*b^11*c^5 + 19905600*A^2*a^
6*b^9*c^6 - 43904256*A^2*a^7*b^7*c^7 + 62684160*A^2*a^8*b^5*c^8 - 52039680*A^2*a^9*b^3*c^9 + 225*A^2*a^3*c^3*(
-(4*a*c - b^2)^15)^(1/2) - B^2*a^2*b^4*(-(4*a*c - b^2)^15)^(1/2) + 769*B^2*a^4*b^15*c^2 - 8620*B^2*a^5*b^13*c^
3 + 63440*B^2*a^6*b^11*c^4 - 316864*B^2*a^7*b^9*c^5 + 1069824*B^2*a^8*b^7*c^6 - 2343936*B^2*a^9*b^5*c^7 + 3010
560*B^2*a^10*b^3*c^8 - 49*B^2*a^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 6881280*A*B*a^11*c^10 - 995*A^2*a*b^19*c + 1
8923520*A^2*a^10*b*c^10 - 41*B^2*a^3*b^17*c - 1720320*B^2*a^11*b*c^9 - 694*A^2*a^2*b^2*c^2*(-(4*a*c - b^2)^15)
^(1/2) - 7402*A*B*a^3*b^16*c^2 + 80620*A*B*a^4*b^14*c^3 - 575120*A*B*a^5*b^12*c^4 + 2791360*A*B*a^6*b^10*c^5 -
9267456*A*B*a^7*b^8*c^6 + 20579328*A*B*a^8*b^6*c^7 - 28815360*A*B*a^9*b^4*c^8 + 22364160*A*B*a^10*b^2*c^9 + 2
45*A^2*a*b^4*c*(-(4*a*c - b^2)^15)^(1/2) + 11*B^2*a^3*b^2*c*(-(4*a*c - b^2)^15)^(1/2) + 10*A*B*a*b^5*(-(4*a*c
- b^2)^15)^(1/2) + 404*A*B*a^2*b^18*c - 104*A*B*a^2*b^3*c*(-(4*a*c - b^2)^15)^(1/2) + 382*A*B*a^3*b*c^2*(-(4*a
*c - b^2)^15)^(1/2)))/(128*(a^7*b^20 + 1048576*a^17*c^10 - 40*a^8*b^18*c + 720*a^9*b^16*c^2 - 7680*a^10*b^14*c
^3 + 53760*a^11*b^12*c^4 - 258048*a^12*b^10*c^5 + 860160*a^13*b^8*c^6 - 1966080*a^14*b^6*c^7 + 2949120*a^15*b^
4*c^8 - 2621440*a^16*b^2*c^9)))^(1/2)*(34359738368*a^26*b*c^13 - 8192*a^15*b^23*c^2 + 360448*a^16*b^21*c^3 - 7
208960*a^17*b^19*c^4 + 86507520*a^18*b^17*c^5 - 692060160*a^19*b^15*c^6 + 3875536896*a^20*b^13*c^7 - 155021475
84*a^21*b^11*c^8 + 44291850240*a^22*b^9*c^9 - 88583700480*a^23*b^7*c^10 + 118111600640*a^24*b^5*c^11 - 9448928
0512*a^25*b^3*c^12) - 22548578304*B*a^24*c^13 + 74088185856*A*a^23*b*c^13 - 15360*A*a^12*b^23*c^2 + 681984*A*a
^13*b^21*c^3 - 13774848*A*a^14*b^19*c^4 + 167067648*A*a^15*b^17*c^5 - 1351876608*A*a^16*b^15*c^6 + 7662993408*
A*a^17*b^13*c^7 - 31048335360*A*a^18*b^11*c^8 + 89917489152*A*a^19*b^9*c^9 - 182401892352*A*a^20*b^7*c^10 + 24
6826401792*A*a^21*b^5*c^11 - 200521285632*A*a^22*b^3*c^12 + 3072*B*a^13*b^22*c^2 - 138240*B*a^14*b^20*c^3 + 28
50816*B*a^15*b^18*c^4 - 35536896*B*a^16*b^16*c^5 + 297271296*B*a^17*b^14*c^6 - 1750597632*B*a^18*b^12*c^7 + 73
98752256*B*a^19*b^10*c^8 - 22422749184*B*a^20*b^8*c^9 + 47714402304*B*a^21*b^6*c^10 - 67847061504*B*a^22*b^4*c
^11 + 57982058496*B*a^23*b^2*c^12))*(-(9*(25*A^2*b^21 + B^2*a^2*b^19 - 25*A^2*b^6*(-(4*a*c - b^2)^15)^(1/2) -
10*A*B*a*b^20 + 17794*A^2*a^2*b^17*c^2 - 188095*A^2*a^3*b^15*c^3 + 1299860*A^2*a^4*b^13*c^4 - 6126640*A^2*a^5*
b^11*c^5 + 19905600*A^2*a^6*b^9*c^6 - 43904256*A^2*a^7*b^7*c^7 + 62684160*A^2*a^8*b^5*c^8 - 52039680*A^2*a^9*b
^3*c^9 + 225*A^2*a^3*c^3*(-(4*a*c - b^2)^15)^(1/2) - B^2*a^2*b^4*(-(4*a*c - b^2)^15)^(1/2) + 769*B^2*a^4*b^15*
c^2 - 8620*B^2*a^5*b^13*c^3 + 63440*B^2*a^6*b^11*c^4 - 316864*B^2*a^7*b^9*c^5 + 1069824*B^2*a^8*b^7*c^6 - 2343
936*B^2*a^9*b^5*c^7 + 3010560*B^2*a^10*b^3*c^8 - 49*B^2*a^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 6881280*A*B*a^11*c
^10 - 995*A^2*a*b^19*c + 18923520*A^2*a^10*b*c^10 - 41*B^2*a^3*b^17*c - 1720320*B^2*a^11*b*c^9 - 694*A^2*a^2*b
^2*c^2*(-(4*a*c - b^2)^15)^(1/2) - 7402*A*B*a^3*b^16*c^2 + 80620*A*B*a^4*b^14*c^3 - 575120*A*B*a^5*b^12*c^4 +
2791360*A*B*a^6*b^10*c^5 - 9267456*A*B*a^7*b^8*c^6 + 20579328*A*B*a^8*b^6*c^7 - 28815360*A*B*a^9*b^4*c^8 + 223
64160*A*B*a^10*b^2*c^9 + 245*A^2*a*b^4*c*(-(4*a*c - b^2)^15)^(1/2) + 11*B^2*a^3*b^2*c*(-(4*a*c - b^2)^15)^(1/2
) + 10*A*B*a*b^5*(-(4*a*c - b^2)^15)^(1/2) + 404*A*B*a^2*b^18*c - 104*A*B*a^2*b^3*c*(-(4*a*c - b^2)^15)^(1/2)
+ 382*A*B*a^3*b*c^2*(-(4*a*c - b^2)^15)^(1/2)))/(128*(a^7*b^20 + 1048576*a^17*c^10 - 40*a^8*b^18*c + 720*a^9*b
^16*c^2 - 7680*a^10*b^14*c^3 + 53760*a^11*b^12*c^4 - 258048*a^12*b^10*c^5 + 860160*a^13*b^8*c^6 - 1966080*a^14
*b^6*c^7 + 2949120*a^15*b^4*c^8 - 2621440*a^16*b^2*c^9)))^(1/2) - (x^(1/2)*(33973862400*A^2*a^20*c^14 - 739875
2256*B^2*a^21*c^13 - 28800*A^2*a^9*b^22*c^3 + 1232640*A^2*a^10*b^20*c^4 - 23879808*A^2*a^11*b^18*c^5 + 2759754
24*A^2*a^12*b^16*c^6 - 2109763584*A^2*a^13*b^14*c^7 + 11171856384*A^2*a^14*b^12*c^8 - 41653370880*A^2*a^15*b^1
0*c^9 + 108726976512*A^2*a^16*b^8*c^10 - 192980975616*A^2*a^17*b^6*c^11 + 218414186496*A^2*a^18*b^4*c^12 - 137
631891456*A^2*a^19*b^2*c^13 - 1152*B^2*a^11*b^20*c^3 + 50688*B^2*a^12*b^18*c^4 - 1025280*B^2*a^13*b^16*c^5 + 1
2496896*B^2*a^14*b^14*c^6 - 101744640*B^2*a^15*b^12*c^7 + 579796992*B^2*a^16*b^10*c^8 - 2346319872*B^2*a^17*b^
8*c^9 + 6653214720*B^2*a^18*b^6*c^10 - 12608077824*B^2*a^19*b^4*c^11 + 14344519680*B^2*a^20*b^2*c^12 + 11520*A
*B*a^10*b^21*c^3 - 499968*A*B*a^11*b^19*c^4 + 9900288*A*B*a^12*b^17*c^5 - 117559296*A*B*a^13*b^15*c^6 + 925433
856*A*B*a^14*b^13*c^7 - 5038866432*A*B*a^15*b^11*c^8 + 19191693312*A*B*a^16*b^9*c^9 - 50422874112*A*B*a^17*b^7
*c^10 + 87350575104*A*B*a^18*b^5*c^11 - 89992986624*A*B*a^19*b^3*c^12 + 41825599488*A*B*a^20*b*c^13) + (-(9*(2
5*A^2*b^21 + B^2*a^2*b^19 - 25*A^2*b^6*(-(4*a*c - b^2)^15)^(1/2) - 10*A*B*a*b^20 + 17794*A^2*a^2*b^17*c^2 - 18
8095*A^2*a^3*b^15*c^3 + 1299860*A^2*a^4*b^13*c^4 - 6126640*A^2*a^5*b^11*c^5 + 19905600*A^2*a^6*b^9*c^6 - 43904
256*A^2*a^7*b^7*c^7 + 62684160*A^2*a^8*b^5*c^8 - 52039680*A^2*a^9*b^3*c^9 + 225*A^2*a^3*c^3*(-(4*a*c - b^2)^15
)^(1/2) - B^2*a^2*b^4*(-(4*a*c - b^2)^15)^(1/2) + 769*B^2*a^4*b^15*c^2 - 8620*B^2*a^5*b^13*c^3 + 63440*B^2*a^6
*b^11*c^4 - 316864*B^2*a^7*b^9*c^5 + 1069824*B^2*a^8*b^7*c^6 - 2343936*B^2*a^9*b^5*c^7 + 3010560*B^2*a^10*b^3*
c^8 - 49*B^2*a^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 6881280*A*B*a^11*c^10 - 995*A^2*a*b^19*c + 18923520*A^2*a^10*
b*c^10 - 41*B^2*a^3*b^17*c - 1720320*B^2*a^11*b*c^9 - 694*A^2*a^2*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) - 7402*A*B
*a^3*b^16*c^2 + 80620*A*B*a^4*b^14*c^3 - 575120*A*B*a^5*b^12*c^4 + 2791360*A*B*a^6*b^10*c^5 - 9267456*A*B*a^7*
b^8*c^6 + 20579328*A*B*a^8*b^6*c^7 - 28815360*A*B*a^9*b^4*c^8 + 22364160*A*B*a^10*b^2*c^9 + 245*A^2*a*b^4*c*(-
(4*a*c - b^2)^15)^(1/2) + 11*B^2*a^3*b^2*c*(-(4*a*c - b^2)^15)^(1/2) + 10*A*B*a*b^5*(-(4*a*c - b^2)^15)^(1/2)
+ 404*A*B*a^2*b^18*c - 104*A*B*a^2*b^3*c*(-(4*a*c - b^2)^15)^(1/2) + 382*A*B*a^3*b*c^2*(-(4*a*c - b^2)^15)^(1/
2)))/(128*(a^7*b^20 + 1048576*a^17*c^10 - 40*a^8*b^18*c + 720*a^9*b^16*c^2 - 7680*a^10*b^14*c^3 + 53760*a^11*b
^12*c^4 - 258048*a^12*b^10*c^5 + 860160*a^13*b^8*c^6 - 1966080*a^14*b^6*c^7 + 2949120*a^15*b^4*c^8 - 2621440*a
^16*b^2*c^9)))^(1/2)*(x^(1/2)*(-(9*(25*A^2*b^21 + B^2*a^2*b^19 - 25*A^2*b^6*(-(4*a*c - b^2)^15)^(1/2) - 10*A*B
*a*b^20 + 17794*A^2*a^2*b^17*c^2 - 188095*A^2*a^3*b^15*c^3 + 1299860*A^2*a^4*b^13*c^4 - 6126640*A^2*a^5*b^11*c
^5 + 19905600*A^2*a^6*b^9*c^6 - 43904256*A^2*a^7*b^7*c^7 + 62684160*A^2*a^8*b^5*c^8 - 52039680*A^2*a^9*b^3*c^9
+ 225*A^2*a^3*c^3*(-(4*a*c - b^2)^15)^(1/2) - B^2*a^2*b^4*(-(4*a*c - b^2)^15)^(1/2) + 769*B^2*a^4*b^15*c^2 -
8620*B^2*a^5*b^13*c^3 + 63440*B^2*a^6*b^11*c^4 - 316864*B^2*a^7*b^9*c^5 + 1069824*B^2*a^8*b^7*c^6 - 2343936*B^
2*a^9*b^5*c^7 + 3010560*B^2*a^10*b^3*c^8 - 49*B^2*a^4*c^2*(-(4*a*c - b^2)^15)^(1/2) - 6881280*A*B*a^11*c^10 -
995*A^2*a*b^19*c + 18923520*A^2*a^10*b*c^10 - 41*B^2*a^3*b^17*c - 1720320*B^2*a^11*b*c^9 - 694*A^2*a^2*b^2*c^2
*(-(4*a*c - b^2)^15)^(1/2) - 7402*A*B*a^3*b^16*c^2 + 80620*A*B*a^4*b^14*c^3 - 575120*A*B*a^5*b^12*c^4 + 279136
0*A*B*a^6*b^10*c^5 - 9267456*A*B*a^7*b^8*c^6 + 20579328*A*B*a^8*b^6*c^7 - 28815360*A*B*a^9*b^4*c^8 + 22364160*
A*B*a^10*b^2*c^9 + 245*A^2*a*b^4*c*(-(4*a*c - b^2)^15)^(1/2) + 11*B^2*a^3*b^2*c*(-(4*a*c - b^2)^15)^(1/2) + 10
*A*B*a*b^5*(-(4*a*c - b^2)^15)^(1/2) + 404*A*B*a^2*b^18*c - 104*A*B*a^2*b^3*c*(-(4*a*c - b^2)^15)^(1/2) + 382*
A*B*a^3*b*c^2*(-(4*a*c - b^2)^15)^(1/2)))/(128*(a^7*b^20 + 1048576*a^17*c^10 - 40*a^8*b^18*c + 720*a^9*b^16*c^
2 - 7680*a^10*b^14*c^3 + 53760*a^11*b^12*c^4 - 258048*a^12*b^10*c^5 + 860160*a^13*b^8*c^6 - 1966080*a^14*b^6*c
^7 + 2949120*a^15*b^4*c^8 - 2621440*a^16*b^2*c^9)))^(1/2)*(34359738368*a^26*b*c^13 - 8192*a^15*b^23*c^2 + 3604
48*a^16*b^21*c^3 - 7208960*a^17*b^19*c^4 + 86507520*a^18*b^17*c^5 - 692060160*a^19*b^15*c^6 + 3875536896*a^20*
b^13*c^7 - 15502147584*a^21*b^11*c^8 + 44291850240*a^22*b^9*c^9 - 88583700480*a^23*b^7*c^10 + 118111600640*a^2
4*b^5*c^11 - 94489280512*a^25*b^3*c^12) + 22548578304*B*a^24*c^13 - 74088185856*A*a^23*b*c^13 + 15360*A*a^12*b
^23*c^2 - 681984*A*a^13*b^21*c^3 + 13774848*A*a^14*b^19*c^4 - 167067648*A*a^15*b^17*c^5 + 1351876608*A*a^16*b^
15*c^6 - 7662993408*A*a^17*b^13*c^7 + 31048335360*A*a^18*b^11*c^8 - 89917489152*A*a^19*b^9*c^9 + 182401892352*
A*a^20*b^7*c^10 - 246826401792*A*a^21*b^5*c^11 + 200521285632*A*a^22*b^3*c^12 - 3072*B*a^13*b^22*c^2 + 138240*
B*a^14*b^20*c^3 - 2850816*B*a^15*b^18*c^4 + 35536896*B*a^16*b^16*c^5 - 297271296*B*a^17*b^14*c^6 + 1750597632*
B*a^18*b^12*c^7 - 7398752256*B*a^19*b^10*c^8 + 22422749184*B*a^20*b^8*c^9 - 47714402304*B*a^21*b^6*c^10 + 6784
7061504*B*a^22*b^4*c^11 - 57982058496*B*a^23*b^2*c^12))*(-(9*(25*A^2*b^21 + B^2*a^2*b^19 - 25*A^2*b^6*(-(4*a*c
- b^2)^15)^(1/2) - 10*A*B*a*b^20 + 17794*A^2*a^2*b^17*c^2 - 188095*A^2*a^3*b^15*c^3 + 1299860*A^2*a^4*b^13*c^
4 - 6126640*A^2*a^5*b^11*c^5 + 19905600*A^2*a^6*b^9*c^6 - 43904256*A^2*a^7*b^7*c^7 + 62684160*A^2*a^8*b^5*c^8
- 52039680*A^2*a^9*b^3*c^9 + 225*A^2*a^3*c^3*(-(4*a*c - b^2)^15)^(1/2) - B^2*a^2*b^4*(-(4*a*c - b^2)^15)^(1/2)
+ 769*B^2*a^4*b^15*c^2 - 8620*B^2*a^5*b^13*c^3 + 63440*B^2*a^6*b^11*c^4 - 316864*B^2*a^7*b^9*c^5 + 1069824*B^
2*a^8*b^7*c^6 - 2343936*B^2*a^9*b^5*c^7 + 3010560*B^2*a^10*b^3*c^8 - 49*B^2*a^4*c^2*(-(4*a*c - b^2)^15)^(1/2)
- 6881280*A*B*a^11*c^10 - 995*A^2*a*b^19*c + 18923520*A^2*a^10*b*c^10 - 41*B^2*a^3*b^17*c - 1720320*B^2*a^11*b
*c^9 - 694*A^2*a^2*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) - 7402*A*B*a^3*b^16*c^2 + 80620*A*B*a^4*b^14*c^3 - 575120
*A*B*a^5*b^12*c^4 + 2791360*A*B*a^6*b^10*c^5 - 9267456*A*B*a^7*b^8*c^6 + 20579328*A*B*a^8*b^6*c^7 - 28815360*A
*B*a^9*b^4*c^8 + 22364160*A*B*a^10*b^2*c^9 + 245*A^2*a*b^4*c*(-(4*a*c - b^2)^15)^(1/2) + 11*B^2*a^3*b^2*c*(-(4
*a*c - b^2)^15)^(1/2) + 10*A*B*a*b^5*(-(4*a*c - b^2)^15)^(1/2) + 404*A*B*a^2*b^18*c - 104*A*B*a^2*b^3*c*(-(4*a
*c - b^2)^15)^(1/2) + 382*A*B*a^3*b*c^2*(-(4*a*c - b^2)^15)^(1/2)))/(128*(a^7*b^20 + 1048576*a^17*c^10 - 40*a^
8*b^18*c + 720*a^9*b^16*c^2 - 7680*a^10*b^14*c^3 + 53760*a^11*b^12*c^4 - 258048*a^12*b^10*c^5 + 860160*a^13*b^
8*c^6 - 1966080*a^14*b^6*c^7 + 2949120*a^15*b^4*c^8 - 2621440*a^16*b^2*c^9)))^(1/2) + 47775744000*A^3*a^17*c^1
4 + 712800*A^3*a^9*b^16*c^6 - 23142240*A^3*a^10*b^14*c^7 + 328157568*A^3*a^11*b^12*c^8 - 2652784128*A^3*a^12*b
^10*c^9 + 13361338368*A^3*a^13*b^8*c^10 - 42897973248*A^3*a^14*b^6*c^11 + 85645099008*A^3*a^15*b^4*c^12 - 9709
0928640*A^3*a^16*b^2*c^13 - 18144*B^3*a^11*b^15*c^5 + 622080*B^3*a^12*b^13*c^6 - 9220608*B^3*a^13*b^11*c^7 + 7
6640256*B^3*a^14*b^9*c^8 - 384638976*B^3*a^15*b^7*c^9 + 1160773632*B^3*a^16*b^5*c^10 - 1942880256*B^3*a^17*b^3
*c^11 + 10404495360*A*B^2*a^18*c^13 + 1387266048*B^3*a^18*b*c^12 - 26966753280*A^2*B*a^17*b*c^13 + 181440*A*B^
2*a^10*b^16*c^5 - 6083424*A*B^2*a^11*b^14*c^6 + 88656768*A*B^2*a^12*b^12*c^7 - 731026944*A*B^2*a^13*b^10*c^8 +
3713071104*A*B^2*a^14*b^8*c^9 - 11822505984*A*B^2*a^15*b^6*c^10 + 22839459840*A*B^2*a^16*b^4*c^11 - 241320591
36*A*B^2*a^17*b^2*c^12 - 453600*A^2*B*a^9*b^17*c^5 + 14722560*A^2*B*a^10*b^15*c^6 - 208303488*A^2*B*a^11*b^13*
c^7 + 1675717632*A^2*B*a^12*b^11*c^8 - 8368883712*A^2*B*a^13*b^9*c^9 + 26512883712*A^2*B*a^14*b^7*c^10 - 51887
112192*A^2*B*a^15*b^5*c^11 + 57139789824*A^2*B*a^16*b^3*c^12))*(-(9*(25*A^2*b^21 + B^2*a^2*b^19 - 25*A^2*b^6*(
-(4*a*c - b^2)^15)^(1/2) - 10*A*B*a*b^20 + 17794*A^2*a^2*b^17*c^2 - 188095*A^2*a^3*b^15*c^3 + 1299860*A^2*a^4*
b^13*c^4 - 6126640*A^2*a^5*b^11*c^5 + 19905600*A^2*a^6*b^9*c^6 - 43904256*A^2*a^7*b^7*c^7 + 62684160*A^2*a^8*b
^5*c^8 - 52039680*A^2*a^9*b^3*c^9 + 225*A^2*a^3*c^3*(-(4*a*c - b^2)^15)^(1/2) - B^2*a^2*b^4*(-(4*a*c - b^2)^15
)^(1/2) + 769*B^2*a^4*b^15*c^2 - 8620*B^2*a^5*b^13*c^3 + 63440*B^2*a^6*b^11*c^4 - 316864*B^2*a^7*b^9*c^5 + 106
9824*B^2*a^8*b^7*c^6 - 2343936*B^2*a^9*b^5*c^7 + 3010560*B^2*a^10*b^3*c^8 - 49*B^2*a^4*c^2*(-(4*a*c - b^2)^15)
^(1/2) - 6881280*A*B*a^11*c^10 - 995*A^2*a*b^19*c + 18923520*A^2*a^10*b*c^10 - 41*B^2*a^3*b^17*c - 1720320*B^2
*a^11*b*c^9 - 694*A^2*a^2*b^2*c^2*(-(4*a*c - b^2)^15)^(1/2) - 7402*A*B*a^3*b^16*c^2 + 80620*A*B*a^4*b^14*c^3 -
575120*A*B*a^5*b^12*c^4 + 2791360*A*B*a^6*b^10*c^5 - 9267456*A*B*a^7*b^8*c^6 + 20579328*A*B*a^8*b^6*c^7 - 288
15360*A*B*a^9*b^4*c^8 + 22364160*A*B*a^10*b^2*c^9 + 245*A^2*a*b^4*c*(-(4*a*c - b^2)^15)^(1/2) + 11*B^2*a^3*b^2
*c*(-(4*a*c - b^2)^15)^(1/2) + 10*A*B*a*b^5*(-(4*a*c - b^2)^15)^(1/2) + 404*A*B*a^2*b^18*c - 104*A*B*a^2*b^3*c
*(-(4*a*c - b^2)^15)^(1/2) + 382*A*B*a^3*b*c^2*(-(4*a*c - b^2)^15)^(1/2)))/(128*(a^7*b^20 + 1048576*a^17*c^10
- 40*a^8*b^18*c + 720*a^9*b^16*c^2 - 7680*a^10*b^14*c^3 + 53760*a^11*b^12*c^4 - 258048*a^12*b^10*c^5 + 860160*
a^13*b^8*c^6 - 1966080*a^14*b^6*c^7 + 2949120*a^15*b^4*c^8 - 2621440*a^16*b^2*c^9)))^(1/2)*2i - ((2*A)/a - (x^
3*(28*B*a^3*c^3 - 30*A*b^5*c + 6*B*a*b^4*c + 227*A*a*b^3*c^2 - 392*A*a^2*b*c^3 - 49*B*a^2*b^2*c^2))/(4*a^3*(b^
4 + 16*a^2*c^2 - 8*a*b^2*c)) + (x*(25*A*b^5 - 44*B*a^3*c^2 - 5*B*a*b^4 - 194*A*a*b^3*c + 364*A*a^2*b*c^2 + 37*
B*a^2*b^2*c))/(4*a^2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (x^2*(15*A*b^6 + 324*A*a^3*c^3 - 3*B*a*b^5 - 91*A*a*b^4
*c + 20*B*a^2*b^3*c + 4*B*a^3*b*c^2 + 25*A*a^2*b^2*c^2))/(4*a^3*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (3*c*x^4*(60
*A*a^2*c^3 + 5*A*b^4*c - B*a*b^3*c - 37*A*a*b^2*c^2 + 8*B*a^2*b*c^2))/(4*a^3*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))/
(x^(5/2)*(2*a*c + b^2) + a^2*x^(1/2) + c^2*x^(9/2) + 2*a*b*x^(3/2) + 2*b*c*x^(7/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x**(3/2)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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