\(\int \frac {A+B x}{x^{3/2} (a+b x+c x^2)^2} \, dx\) [1021]

Optimal result

Integrand size = 23, antiderivative size = 406 \[ \int \frac {A+B x}{x^{3/2} \left (a+b x+c x^2\right )^2} \, dx=-\frac {3 A b^2-a b B-10 a A c}{a^2 \left (b^2-4 a c\right ) \sqrt {x}}+\frac {A b^2-a b B-2 a A c+(A b-2 a B) c x}{a \left (b^2-4 a c\right ) \sqrt {x} \left (a+b x+c x^2\right )}+\frac {\sqrt {c} \left (a B \left (b^2-12 a c+b \sqrt {b^2-4 a c}\right )-A \left (3 b^3-16 a b c+3 b^2 \sqrt {b^2-4 a c}-10 a c \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (a B \left (b^2-12 a c-b \sqrt {b^2-4 a c}\right )-A \left (3 b^3-16 a b c-3 b^2 \sqrt {b^2-4 a c}+10 a c \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}} \]

[Out]

(10*A*a*c-3*A*b^2+B*a*b)/a^2/(-4*a*c+b^2)/x^(1/2)+(A*b^2-a*b*B-2*A*a*c+(A*b-2*B*a)*c*x)/a/(-4*a*c+b^2)/(c*x^2+
b*x+a)/x^(1/2)+1/2*arctan(2^(1/2)*c^(1/2)*x^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2))*c^(1/2)*(a*B*(b^2-12*a*c+b*(-4
*a*c+b^2)^(1/2))-A*(3*b^3-16*a*b*c+3*b^2*(-4*a*c+b^2)^(1/2)-10*a*c*(-4*a*c+b^2)^(1/2)))/a^2/(-4*a*c+b^2)^(3/2)
*2^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)-1/2*arctan(2^(1/2)*c^(1/2)*x^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2))*c^(1/2)
*(a*B*(b^2-12*a*c-b*(-4*a*c+b^2)^(1/2))-A*(3*b^3-16*a*b*c-3*b^2*(-4*a*c+b^2)^(1/2)+10*a*c*(-4*a*c+b^2)^(1/2)))
/a^2/(-4*a*c+b^2)^(3/2)*2^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {836, 842, 840, 1180, 211} \[ \int \frac {A+B x}{x^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\frac {\sqrt {c} \left (a B \left (b \sqrt {b^2-4 a c}-12 a c+b^2\right )-A \left (3 b^2 \sqrt {b^2-4 a c}-10 a c \sqrt {b^2-4 a c}-16 a b c+3 b^3\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (a B \left (-b \sqrt {b^2-4 a c}-12 a c+b^2\right )-A \left (-3 b^2 \sqrt {b^2-4 a c}+10 a c \sqrt {b^2-4 a c}-16 a b c+3 b^3\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt {\sqrt {b^2-4 a c}+b}}-\frac {-10 a A c-a b B+3 A b^2}{a^2 \sqrt {x} \left (b^2-4 a c\right )}+\frac {c x (A b-2 a B)-2 a A c-a b B+A b^2}{a \sqrt {x} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]

[In]

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

[Out]

-((3*A*b^2 - a*b*B - 10*a*A*c)/(a^2*(b^2 - 4*a*c)*Sqrt[x])) + (A*b^2 - a*b*B - 2*a*A*c + (A*b - 2*a*B)*c*x)/(a
*(b^2 - 4*a*c)*Sqrt[x]*(a + b*x + c*x^2)) + (Sqrt[c]*(a*B*(b^2 - 12*a*c + b*Sqrt[b^2 - 4*a*c]) - A*(3*b^3 - 16
*a*b*c + 3*b^2*Sqrt[b^2 - 4*a*c] - 10*a*c*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b
^2 - 4*a*c]]])/(Sqrt[2]*a^2*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[c]*(a*B*(b^2 - 12*a*c - b
*Sqrt[b^2 - 4*a*c]) - A*(3*b^3 - 16*a*b*c - 3*b^2*Sqrt[b^2 - 4*a*c] + 10*a*c*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[
2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*a^2*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]
)

Rule 211

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

Rule 836

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 840

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 842

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 1180

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*} \text {integral}& = \frac {A b^2-a b B-2 a A c+(A b-2 a B) c x}{a \left (b^2-4 a c\right ) \sqrt {x} \left (a+b x+c x^2\right )}-\frac {\int \frac {\frac {1}{2} \left (-3 A b^2+a b B+10 a A c\right )-\frac {3}{2} (A b-2 a B) c x}{x^{3/2} \left (a+b x+c x^2\right )} \, dx}{a \left (b^2-4 a c\right )} \\ & = -\frac {3 A b^2-a b B-10 a A c}{a^2 \left (b^2-4 a c\right ) \sqrt {x}}+\frac {A b^2-a b B-2 a A c+(A b-2 a B) c x}{a \left (b^2-4 a c\right ) \sqrt {x} \left (a+b x+c x^2\right )}-\frac {\int \frac {\frac {1}{2} \left (-a B \left (b^2-6 a c\right )+A \left (3 b^3-13 a b c\right )\right )+\frac {1}{2} c \left (3 A b^2-a b B-10 a A c\right ) x}{\sqrt {x} \left (a+b x+c x^2\right )} \, dx}{a^2 \left (b^2-4 a c\right )} \\ & = -\frac {3 A b^2-a b B-10 a A c}{a^2 \left (b^2-4 a c\right ) \sqrt {x}}+\frac {A b^2-a b B-2 a A c+(A b-2 a B) c x}{a \left (b^2-4 a c\right ) \sqrt {x} \left (a+b x+c x^2\right )}-\frac {2 \text {Subst}\left (\int \frac {\frac {1}{2} \left (-a B \left (b^2-6 a c\right )+A \left (3 b^3-13 a b c\right )\right )+\frac {1}{2} c \left (3 A b^2-a b B-10 a A c\right ) x^2}{a+b x^2+c x^4} \, dx,x,\sqrt {x}\right )}{a^2 \left (b^2-4 a c\right )} \\ & = -\frac {3 A b^2-a b B-10 a A c}{a^2 \left (b^2-4 a c\right ) \sqrt {x}}+\frac {A b^2-a b B-2 a A c+(A b-2 a B) c x}{a \left (b^2-4 a c\right ) \sqrt {x} \left (a+b x+c x^2\right )}+\frac {\left (c \left (a B \left (b^2-12 a c+b \sqrt {b^2-4 a c}\right )-A \left (3 b^3-16 a b c+3 b^2 \sqrt {b^2-4 a c}-10 a c \sqrt {b^2-4 a c}\right )\right )\right ) \text {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 )}{2 a^2 \left (b^2-4 a c\right )^{3/2}}-\frac {\left (c \left (a B \left (b^2-12 a c-b \sqrt {b^2-4 a c}\right )-A \left (3 b^3-16 a b c-3 b^2 \sqrt {b^2-4 a c}+10 a c \sqrt {b^2-4 a c}\right )\right )\right ) \text {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 )}{2 a^2 \left (b^2-4 a c\right )^{3/2}} \\ & = -\frac {3 A b^2-a b B-10 a A c}{a^2 \left (b^2-4 a c\right ) \sqrt {x}}+\frac {A b^2-a b B-2 a A c+(A b-2 a B) c x}{a \left (b^2-4 a c\right ) \sqrt {x} \left (a+b x+c x^2\right )}+\frac {\sqrt {c} \left (a B \left (b^2-12 a c+b \sqrt {b^2-4 a c}\right )-A \left (3 b^3-16 a b c+3 b^2 \sqrt {b^2-4 a c}-10 a c \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 )}{\sqrt {2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (a B \left (b^2-12 a c-b \sqrt {b^2-4 a c}\right )-A \left (3 b^3-16 a b c-3 b^2 \sqrt {b^2-4 a c}+10 a c \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 )}{\sqrt {2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.35 (sec) , antiderivative size = 399, normalized size of antiderivative = 0.98 \[ \int \frac {A+B x}{x^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\frac {-\frac {2 \left (-3 A b^2 x (b+c x)+a b B x (b+c x)+a^2 (8 A c-2 B c x)+a A \left (-2 b^2+11 b c x+10 c^2 x^2\right )\right )}{\sqrt {x} (a+x (b+c x))}-\frac {\sqrt {2} \sqrt {c} \left (a B \left (b^2-12 a c+b \sqrt {b^2-4 a c}\right )+A \left (-3 b^3+16 a b c-3 b^2 \sqrt {b^2-4 a c}+10 a c \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \sqrt {c} \left (a B \left (b^2-12 a c-b \sqrt {b^2-4 a c}\right )+A \left (-3 b^3+16 a b c+3 b^2 \sqrt {b^2-4 a c}-10 a c \sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}}{2 a^2 \left (-b^2+4 a c\right )} \]

[In]

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

[Out]

((-2*(-3*A*b^2*x*(b + c*x) + a*b*B*x*(b + c*x) + a^2*(8*A*c - 2*B*c*x) + a*A*(-2*b^2 + 11*b*c*x + 10*c^2*x^2))
)/(Sqrt[x]*(a + x*(b + c*x))) - (Sqrt[2]*Sqrt[c]*(a*B*(b^2 - 12*a*c + b*Sqrt[b^2 - 4*a*c]) + A*(-3*b^3 + 16*a*
b*c - 3*b^2*Sqrt[b^2 - 4*a*c] + 10*a*c*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2
- 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(a*B*(b^2 - 12*a*c - b*Sqrt[b^2
- 4*a*c]) + A*(-3*b^3 + 16*a*b*c + 3*b^2*Sqrt[b^2 - 4*a*c] - 10*a*c*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[
c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(2*a^2*(-b^2 + 4*a*
c))

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 383, normalized size of antiderivative = 0.94

method	result	size
derivativedivides	\(-\frac {2 \left (\frac {\frac {c \left (2 A a c -A \,b^{2}+a b B \right ) x^{\frac {3}{2}}}{8 a c -2 b^{2}}+\frac {\left (3 A a b c -A \,b^{3}-2 B \,a^{2} c +B a \,b^{2}\right ) \sqrt {x}}{8 a c -2 b^{2}}}{c \,x^{2}+b x +a}+\frac {2 c \left (-\frac {\left (10 A \sqrt {-4 a c +b^{2}}\, a c -3 A \sqrt {-4 a c +b^{2}}\, b^{2}+16 A a b c -3 A \,b^{3}+a b B \sqrt {-4 a c +b^{2}}-12 B \,a^{2} c +B a \,b^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (10 A \sqrt {-4 a c +b^{2}}\, a c -3 A \sqrt {-4 a c +b^{2}}\, b^{2}-16 A a b c +3 A \,b^{3}+a b B \sqrt {-4 a c +b^{2}}+12 B \,a^{2} c -B a \,b^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 a c -b^{2}}\right )}{a^{2}}-\frac {2 A}{a^{2} \sqrt {x}}\)	\(383\)
default	\(-\frac {2 \left (\frac {\frac {c \left (2 A a c -A \,b^{2}+a b B \right ) x^{\frac {3}{2}}}{8 a c -2 b^{2}}+\frac {\left (3 A a b c -A \,b^{3}-2 B \,a^{2} c +B a \,b^{2}\right ) \sqrt {x}}{8 a c -2 b^{2}}}{c \,x^{2}+b x +a}+\frac {2 c \left (-\frac {\left (10 A \sqrt {-4 a c +b^{2}}\, a c -3 A \sqrt {-4 a c +b^{2}}\, b^{2}+16 A a b c -3 A \,b^{3}+a b B \sqrt {-4 a c +b^{2}}-12 B \,a^{2} c +B a \,b^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (10 A \sqrt {-4 a c +b^{2}}\, a c -3 A \sqrt {-4 a c +b^{2}}\, b^{2}-16 A a b c +3 A \,b^{3}+a b B \sqrt {-4 a c +b^{2}}+12 B \,a^{2} c -B a \,b^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 a c -b^{2}}\right )}{a^{2}}-\frac {2 A}{a^{2} \sqrt {x}}\)	\(383\)
risch	\(-\frac {2 A}{a^{2} \sqrt {x}}-\frac {\frac {\frac {2 c \left (2 A a c -A \,b^{2}+a b B \right ) x^{\frac {3}{2}}}{8 a c -2 b^{2}}+\frac {2 \left (3 A a b c -A \,b^{3}-2 B \,a^{2} c +B a \,b^{2}\right ) \sqrt {x}}{8 a c -2 b^{2}}}{c \,x^{2}+b x +a}+\frac {4 c \left (-\frac {\left (10 A \sqrt {-4 a c +b^{2}}\, a c -3 A \sqrt {-4 a c +b^{2}}\, b^{2}+16 A a b c -3 A \,b^{3}+a b B \sqrt {-4 a c +b^{2}}-12 B \,a^{2} c +B a \,b^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (10 A \sqrt {-4 a c +b^{2}}\, a c -3 A \sqrt {-4 a c +b^{2}}\, b^{2}-16 A a b c +3 A \,b^{3}+a b B \sqrt {-4 a c +b^{2}}+12 B \,a^{2} c -B a \,b^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {x}\, \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 a c -b^{2}}}{a^{2}}\)	\(384\)

[In]

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

[Out]

-2/a^2*((1/2*c*(2*A*a*c-A*b^2+B*a*b)/(4*a*c-b^2)*x^(3/2)+1/2*(3*A*a*b*c-A*b^3-2*B*a^2*c+B*a*b^2)/(4*a*c-b^2)*x
^(1/2))/(c*x^2+b*x+a)+2/(4*a*c-b^2)*c*(-1/8*(10*A*(-4*a*c+b^2)^(1/2)*a*c-3*A*(-4*a*c+b^2)^(1/2)*b^2+16*A*a*b*c
-3*A*b^3+a*b*B*(-4*a*c+b^2)^(1/2)-12*B*a^2*c+B*a*b^2)/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(
1/2)*arctanh(c*x^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))+1/8*(10*A*(-4*a*c+b^2)^(1/2)*a*c-3*A*(-4*a*c
+b^2)^(1/2)*b^2-16*A*a*b*c+3*A*b^3+a*b*B*(-4*a*c+b^2)^(1/2)+12*B*a^2*c-B*a*b^2)/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b
+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))))-2*A/a^2/x^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 7597 vs. \(2 (351) = 702\).

Time = 26.17 (sec) , antiderivative size = 7597, normalized size of antiderivative = 18.71 \[ \int \frac {A+B x}{x^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x}{x^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {A+B x}{x^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\int { \frac {B x + A}{{\left (c x^{2} + b x + a\right )}^{2} x^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

-(((3*b^3*c - 13*a*b*c^2)*A - (a*b^2*c - 6*a^2*c^2)*B)*x^(5/2) + ((3*b^4 - 10*a*b^2*c - 10*a^2*c^2)*A - (a*b^3
- 5*a^2*b*c)*B)*x^(3/2) + 2*(a^2*b^2 - 4*a^3*c)*A/sqrt(x) + 2*(3*(a*b^3 - 4*a^2*b*c)*A - (a^2*b^2 - 4*a^3*c)*
B)*sqrt(x))/(a^4*b^2 - 4*a^5*c + (a^3*b^2*c - 4*a^4*c^2)*x^2 + (a^3*b^3 - 4*a^4*b*c)*x) + integrate(1/2*(((3*b
^3*c - 13*a*b*c^2)*A - (a*b^2*c - 6*a^2*c^2)*B)*x^(3/2) + ((3*b^4 - 16*a*b^2*c + 10*a^2*c^2)*A - (a*b^3 - 7*a^
2*b*c)*B)*sqrt(x))/(a^4*b^2 - 4*a^5*c + (a^3*b^2*c - 4*a^4*c^2)*x^2 + (a^3*b^3 - 4*a^4*b*c)*x), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5405 vs. \(2 (351) = 702\).

Time = 1.36 (sec) , antiderivative size = 5405, normalized size of antiderivative = 13.31 \[ \int \frac {A+B x}{x^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(B*a*b*c*x^2 - 3*A*b^2*c*x^2 + 10*A*a*c^2*x^2 + B*a*b^2*x - 3*A*b^3*x - 2*B*a^2*c*x + 11*A*a*b*c*x - 2*A*a*b^2
+ 8*A*a^2*c)/((a^2*b^2 - 4*a^3*c)*(c*x^(5/2) + b*x^(3/2) + a*sqrt(x))) - 1/8*((6*b^4*c^2 - 44*a*b^2*c^3 + 80*
a^2*c^4 - 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4 + 22*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(
b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c + 6*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c - 40*sq
rt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^2 - 20*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt
(b^2 - 4*a*c)*c)*a*b*c^2 - 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c^2 + 10*sqrt(2)*sq
rt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*c^3 - 6*(b^2 - 4*a*c)*b^2*c^2 + 20*(b^2 - 4*a*c)*a*c^3)*(a^2
*b^2 - 4*a^3*c)^2*A - (2*a*b^3*c^2 - 8*a^2*b*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a
*b^3 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(
b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^2 - 2*(b^
2 - 4*a*c)*a*b*c^2)*(a^2*b^2 - 4*a^3*c)^2*B + 2*(3*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^7 - 37*sqrt(2
)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^5*c - 6*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^6*c - 6*a^2*b^7*
c + 152*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^3*c^2 + 50*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b
^4*c^2 + 3*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^5*c^2 + 74*a^3*b^5*c^2 - 208*sqrt(2)*sqrt(b*c + sqrt(
b^2 - 4*a*c)*c)*a^5*b*c^3 - 104*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^2*c^3 - 25*sqrt(2)*sqrt(b*c + sq
rt(b^2 - 4*a*c)*c)*a^3*b^3*c^3 - 304*a^4*b^3*c^3 + 52*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b*c^4 + 416*
a^5*b*c^4 + 6*(b^2 - 4*a*c)*a^2*b^5*c - 50*(b^2 - 4*a*c)*a^3*b^3*c^2 + 104*(b^2 - 4*a*c)*a^4*b*c^3)*A*abs(a^2*
b^2 - 4*a^3*c) - 2*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^6 - 14*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*
c)*a^4*b^4*c - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^5*c - 2*a^3*b^6*c + 64*sqrt(2)*sqrt(b*c + sqrt(
b^2 - 4*a*c)*c)*a^5*b^2*c^2 + 20*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^3*c^2 + sqrt(2)*sqrt(b*c + sqrt
(b^2 - 4*a*c)*c)*a^3*b^4*c^2 + 28*a^4*b^4*c^2 - 96*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*c^3 - 48*sqrt(2
)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b*c^3 - 10*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^2*c^3 - 128*a^5
*b^2*c^3 + 24*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*c^4 + 192*a^6*c^4 + 2*(b^2 - 4*a*c)*a^3*b^4*c - 20*(
b^2 - 4*a*c)*a^4*b^2*c^2 + 48*(b^2 - 4*a*c)*a^5*c^3)*B*abs(a^2*b^2 - 4*a^3*c) + (6*a^4*b^8*c^2 - 80*a^5*b^6*c^
3 + 352*a^6*b^4*c^4 - 512*a^7*b^2*c^5 - 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^8 +
40*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^6*c + 6*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c
+ sqrt(b^2 - 4*a*c)*c)*a^4*b^7*c - 176*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^4*c^2 -
56*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^5*c^2 - 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b
*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^6*c^2 + 256*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^2*
c^3 + 128*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^3*c^3 + 28*sqrt(2)*sqrt(b^2 - 4*a*c)
*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^4*c^3 - 64*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^
6*b^2*c^4 - 6*(b^2 - 4*a*c)*a^4*b^6*c^2 + 56*(b^2 - 4*a*c)*a^5*b^4*c^3 - 128*(b^2 - 4*a*c)*a^6*b^2*c^4)*A - (2
*a^5*b^7*c^2 - 40*a^6*b^5*c^3 + 224*a^7*b^3*c^4 - 384*a^8*b*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^
2 - 4*a*c)*c)*a^5*b^7 + 20*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^5*c + 2*sqrt(2)*sqr
t(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^6*c - 112*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 -
4*a*c)*c)*a^7*b^3*c^2 - 32*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^4*c^2 - sqrt(2)*sq
rt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^5*c^2 + 192*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^
2 - 4*a*c)*c)*a^8*b*c^3 + 96*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^2*c^3 + 16*sqrt(2
)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^3*c^3 - 48*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt
(b^2 - 4*a*c)*c)*a^7*b*c^4 - 2*(b^2 - 4*a*c)*a^5*b^5*c^2 + 32*(b^2 - 4*a*c)*a^6*b^3*c^3 - 96*(b^2 - 4*a*c)*a^7
*b*c^4)*B)*arctan(2*sqrt(1/2)*sqrt(x)/sqrt((a^2*b^3 - 4*a^3*b*c + sqrt((a^2*b^3 - 4*a^3*b*c)^2 - 4*(a^3*b^2 -
4*a^4*c)*(a^2*b^2*c - 4*a^3*c^2)))/(a^2*b^2*c - 4*a^3*c^2)))/((a^5*b^6 - 12*a^6*b^4*c - 2*a^5*b^5*c + 48*a^7*b
^2*c^2 + 16*a^6*b^3*c^2 + a^5*b^4*c^2 - 64*a^8*c^3 - 32*a^7*b*c^3 - 8*a^6*b^2*c^3 + 16*a^7*c^4)*abs(a^2*b^2 -
4*a^3*c)*abs(c)) - 1/8*((6*b^4*c^2 - 44*a*b^2*c^3 + 80*a^2*c^4 - 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b
^2 - 4*a*c)*c)*b^4 + 22*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^2*c + 6*sqrt(2)*sqrt(b^2
- 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^3*c - 40*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)
*a^2*c^2 - 20*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b*c^2 - 3*sqrt(2)*sqrt(b^2 - 4*a*c)*
sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^2*c^2 + 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*c^3 -
6*(b^2 - 4*a*c)*b^2*c^2 + 20*(b^2 - 4*a*c)*a*c^3)*(a^2*b^2 - 4*a^3*c)^2*A - (2*a*b^3*c^2 - 8*a^2*b*c^3 - sqrt
(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^3 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2
- 4*a*c)*c)*a^2*b*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^2*c - sqrt(2)*sqrt(b^2 -
4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b*c^2 - 2*(b^2 - 4*a*c)*a*b*c^2)*(a^2*b^2 - 4*a^3*c)^2*B + 2*(3*sqrt
(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^7 - 37*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^5*c - 6*sqrt(2)
*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^6*c + 6*a^2*b^7*c + 152*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b^3
*c^2 + 50*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^4*c^2 + 3*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*
b^5*c^2 - 74*a^3*b^5*c^2 - 208*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^5*b*c^3 - 104*sqrt(2)*sqrt(b*c - sqrt
(b^2 - 4*a*c)*c)*a^4*b^2*c^3 - 25*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^3*c^3 + 304*a^4*b^3*c^3 + 52*s
qrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b*c^4 - 416*a^5*b*c^4 - 6*(b^2 - 4*a*c)*a^2*b^5*c + 50*(b^2 - 4*a*c
)*a^3*b^3*c^2 - 104*(b^2 - 4*a*c)*a^4*b*c^3)*A*abs(a^2*b^2 - 4*a^3*c) - 2*(sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c
)*c)*a^3*b^6 - 14*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b^4*c - 2*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c
)*a^3*b^5*c + 2*a^3*b^6*c + 64*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^5*b^2*c^2 + 20*sqrt(2)*sqrt(b*c - sqr
t(b^2 - 4*a*c)*c)*a^4*b^3*c^2 + sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^4*c^2 - 28*a^4*b^4*c^2 - 96*sqrt
(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^6*c^3 - 48*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^5*b*c^3 - 10*sqrt(2
)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b^2*c^3 + 128*a^5*b^2*c^3 + 24*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a
^5*c^4 - 192*a^6*c^4 - 2*(b^2 - 4*a*c)*a^3*b^4*c + 20*(b^2 - 4*a*c)*a^4*b^2*c^2 - 48*(b^2 - 4*a*c)*a^5*c^3)*B*
abs(a^2*b^2 - 4*a^3*c) + (6*a^4*b^8*c^2 - 80*a^5*b^6*c^3 + 352*a^6*b^4*c^4 - 512*a^7*b^2*c^5 - 3*sqrt(2)*sqrt(
b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b^8 + 40*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*
c)*c)*a^5*b^6*c + 6*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b^7*c - 176*sqrt(2)*sqrt(b^2
- 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^6*b^4*c^2 - 56*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a
*c)*c)*a^5*b^5*c^2 - 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*b^6*c^2 + 256*sqrt(2)*sqr
t(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^7*b^2*c^3 + 128*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2
- 4*a*c)*c)*a^6*b^3*c^3 + 28*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^5*b^4*c^3 - 64*sqrt(
2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^6*b^2*c^4 - 6*(b^2 - 4*a*c)*a^4*b^6*c^2 + 56*(b^2 - 4*a
*c)*a^5*b^4*c^3 - 128*(b^2 - 4*a*c)*a^6*b^2*c^4)*A - (2*a^5*b^7*c^2 - 40*a^6*b^5*c^3 + 224*a^7*b^3*c^4 - 384*a
^8*b*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^5*b^7 + 20*sqrt(2)*sqrt(b^2 - 4*a*c)*sq
rt(b*c - sqrt(b^2 - 4*a*c)*c)*a^6*b^5*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^5*b^6*
c - 112*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^7*b^3*c^2 - 32*sqrt(2)*sqrt(b^2 - 4*a*c)*s
qrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^6*b^4*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^5*b^5
*c^2 + 192*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^8*b*c^3 + 96*sqrt(2)*sqrt(b^2 - 4*a*c)*
sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^7*b^2*c^3 + 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^6
*b^3*c^3 - 48*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^7*b*c^4 - 2*(b^2 - 4*a*c)*a^5*b^5*c^
2 + 32*(b^2 - 4*a*c)*a^6*b^3*c^3 - 96*(b^2 - 4*a*c)*a^7*b*c^4)*B)*arctan(2*sqrt(1/2)*sqrt(x)/sqrt((a^2*b^3 - 4
*a^3*b*c - sqrt((a^2*b^3 - 4*a^3*b*c)^2 - 4*(a^3*b^2 - 4*a^4*c)*(a^2*b^2*c - 4*a^3*c^2)))/(a^2*b^2*c - 4*a^3*c
^2)))/((a^5*b^6 - 12*a^6*b^4*c - 2*a^5*b^5*c + 48*a^7*b^2*c^2 + 16*a^6*b^3*c^2 + a^5*b^4*c^2 - 64*a^8*c^3 - 32
*a^7*b*c^3 - 8*a^6*b^2*c^3 + 16*a^7*c^4)*abs(a^2*b^2 - 4*a^3*c)*abs(c))

Mupad [B] (verification not implemented)

Time = 15.45 (sec) , antiderivative size = 17623, normalized size of antiderivative = 43.41 \[ \int \frac {A+B x}{x^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

- ((2*A)/a - (x*(3*A*b^3 - B*a*b^2 + 2*B*a^2*c - 11*A*a*b*c))/(a^2*(4*a*c - b^2)) + (c*x^2*(10*A*a*c - 3*A*b^2
+ B*a*b))/(a^2*(4*a*c - b^2)))/(a*x^(1/2) + b*x^(3/2) + c*x^(5/2)) - atan(((x^(1/2)*(25600*A^2*a^12*c^9 - 921
6*B^2*a^13*c^8 + 18*A^2*a^6*b^12*c^3 - 408*A^2*a^7*b^10*c^4 + 3764*A^2*a^8*b^8*c^5 - 17920*A^2*a^9*b^6*c^6 + 4
5696*A^2*a^10*b^4*c^7 - 57344*A^2*a^11*b^2*c^8 + 2*B^2*a^8*b^10*c^3 - 52*B^2*a^9*b^8*c^4 + 576*B^2*a^10*b^6*c^
5 - 3200*B^2*a^11*b^4*c^6 + 8704*B^2*a^12*b^2*c^7 - 12*A*B*a^7*b^11*c^3 + 292*A*B*a^8*b^9*c^4 - 2816*A*B*a^9*b
^7*c^5 + 13440*A*B*a^10*b^5*c^6 - 31744*A*B*a^11*b^3*c^7 + 29696*A*B*a^12*b*c^8) + (-(9*A^2*b^13 + B^2*a^2*b^1
1 + 9*A^2*b^4*(-(4*a*c - b^2)^9)^(1/2) - 6*A*B*a*b^12 + 2077*A^2*a^2*b^9*c^2 - 10656*A^2*a^3*b^7*c^3 + 30240*A
^2*a^4*b^5*c^4 - 44800*A^2*a^5*b^3*c^5 + 25*A^2*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + B^2*a^2*b^2*(-(4*a*c - b^2)
^9)^(1/2) + 288*B^2*a^4*b^7*c^2 - 1504*B^2*a^5*b^5*c^3 + 3840*B^2*a^6*b^3*c^4 - 15360*A*B*a^7*c^6 - 213*A^2*a*
b^11*c + 26880*A^2*a^6*b*c^6 - 27*B^2*a^3*b^9*c - 3840*B^2*a^7*b*c^5 - 9*B^2*a^3*c*(-(4*a*c - b^2)^9)^(1/2) -
1548*A*B*a^3*b^8*c^2 + 8064*A*B*a^4*b^6*c^3 - 22400*A*B*a^5*b^4*c^4 + 30720*A*B*a^6*b^2*c^5 - 51*A^2*a*b^2*c*(
-(4*a*c - b^2)^9)^(1/2) - 6*A*B*a*b^3*(-(4*a*c - b^2)^9)^(1/2) + 152*A*B*a^2*b^10*c + 44*A*B*a^2*b*c*(-(4*a*c
- b^2)^9)^(1/2))/(8*(a^5*b^12 + 4096*a^11*c^6 - 24*a^6*b^10*c + 240*a^7*b^8*c^2 - 1280*a^8*b^6*c^3 + 3840*a^9*
b^4*c^4 - 6144*a^10*b^2*c^5)))^(1/2)*(x^(1/2)*(-(9*A^2*b^13 + B^2*a^2*b^11 + 9*A^2*b^4*(-(4*a*c - b^2)^9)^(1/2
) - 6*A*B*a*b^12 + 2077*A^2*a^2*b^9*c^2 - 10656*A^2*a^3*b^7*c^3 + 30240*A^2*a^4*b^5*c^4 - 44800*A^2*a^5*b^3*c^
5 + 25*A^2*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + B^2*a^2*b^2*(-(4*a*c - b^2)^9)^(1/2) + 288*B^2*a^4*b^7*c^2 - 150
4*B^2*a^5*b^5*c^3 + 3840*B^2*a^6*b^3*c^4 - 15360*A*B*a^7*c^6 - 213*A^2*a*b^11*c + 26880*A^2*a^6*b*c^6 - 27*B^2
*a^3*b^9*c - 3840*B^2*a^7*b*c^5 - 9*B^2*a^3*c*(-(4*a*c - b^2)^9)^(1/2) - 1548*A*B*a^3*b^8*c^2 + 8064*A*B*a^4*b
^6*c^3 - 22400*A*B*a^5*b^4*c^4 + 30720*A*B*a^6*b^2*c^5 - 51*A^2*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2) - 6*A*B*a*b^3
*(-(4*a*c - b^2)^9)^(1/2) + 152*A*B*a^2*b^10*c + 44*A*B*a^2*b*c*(-(4*a*c - b^2)^9)^(1/2))/(8*(a^5*b^12 + 4096*
a^11*c^6 - 24*a^6*b^10*c + 240*a^7*b^8*c^2 - 1280*a^8*b^6*c^3 + 3840*a^9*b^4*c^4 - 6144*a^10*b^2*c^5)))^(1/2)*
(32768*a^16*b*c^8 + 8*a^10*b^13*c^2 - 192*a^11*b^11*c^3 + 1920*a^12*b^9*c^4 - 10240*a^13*b^7*c^5 + 30720*a^14*
b^5*c^6 - 49152*a^15*b^3*c^7) - 24576*B*a^15*c^8 + 53248*A*a^14*b*c^8 + 12*A*a^8*b^13*c^2 - 292*A*a^9*b^11*c^3
+ 2960*A*a^10*b^9*c^4 - 16000*A*a^11*b^7*c^5 + 48640*A*a^12*b^5*c^6 - 78848*A*a^13*b^3*c^7 - 4*B*a^9*b^12*c^2
+ 104*B*a^10*b^10*c^3 - 1120*B*a^11*b^8*c^4 + 6400*B*a^12*b^6*c^5 - 20480*B*a^13*b^4*c^6 + 34816*B*a^14*b^2*c
^7))*(-(9*A^2*b^13 + B^2*a^2*b^11 + 9*A^2*b^4*(-(4*a*c - b^2)^9)^(1/2) - 6*A*B*a*b^12 + 2077*A^2*a^2*b^9*c^2 -
10656*A^2*a^3*b^7*c^3 + 30240*A^2*a^4*b^5*c^4 - 44800*A^2*a^5*b^3*c^5 + 25*A^2*a^2*c^2*(-(4*a*c - b^2)^9)^(1/
2) + B^2*a^2*b^2*(-(4*a*c - b^2)^9)^(1/2) + 288*B^2*a^4*b^7*c^2 - 1504*B^2*a^5*b^5*c^3 + 3840*B^2*a^6*b^3*c^4
- 15360*A*B*a^7*c^6 - 213*A^2*a*b^11*c + 26880*A^2*a^6*b*c^6 - 27*B^2*a^3*b^9*c - 3840*B^2*a^7*b*c^5 - 9*B^2*a
^3*c*(-(4*a*c - b^2)^9)^(1/2) - 1548*A*B*a^3*b^8*c^2 + 8064*A*B*a^4*b^6*c^3 - 22400*A*B*a^5*b^4*c^4 + 30720*A*
B*a^6*b^2*c^5 - 51*A^2*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2) - 6*A*B*a*b^3*(-(4*a*c - b^2)^9)^(1/2) + 152*A*B*a^2*b
^10*c + 44*A*B*a^2*b*c*(-(4*a*c - b^2)^9)^(1/2))/(8*(a^5*b^12 + 4096*a^11*c^6 - 24*a^6*b^10*c + 240*a^7*b^8*c^
2 - 1280*a^8*b^6*c^3 + 3840*a^9*b^4*c^4 - 6144*a^10*b^2*c^5)))^(1/2)*1i + (x^(1/2)*(25600*A^2*a^12*c^9 - 9216*
B^2*a^13*c^8 + 18*A^2*a^6*b^12*c^3 - 408*A^2*a^7*b^10*c^4 + 3764*A^2*a^8*b^8*c^5 - 17920*A^2*a^9*b^6*c^6 + 456
96*A^2*a^10*b^4*c^7 - 57344*A^2*a^11*b^2*c^8 + 2*B^2*a^8*b^10*c^3 - 52*B^2*a^9*b^8*c^4 + 576*B^2*a^10*b^6*c^5
- 3200*B^2*a^11*b^4*c^6 + 8704*B^2*a^12*b^2*c^7 - 12*A*B*a^7*b^11*c^3 + 292*A*B*a^8*b^9*c^4 - 2816*A*B*a^9*b^7
*c^5 + 13440*A*B*a^10*b^5*c^6 - 31744*A*B*a^11*b^3*c^7 + 29696*A*B*a^12*b*c^8) + (-(9*A^2*b^13 + B^2*a^2*b^11
+ 9*A^2*b^4*(-(4*a*c - b^2)^9)^(1/2) - 6*A*B*a*b^12 + 2077*A^2*a^2*b^9*c^2 - 10656*A^2*a^3*b^7*c^3 + 30240*A^2
*a^4*b^5*c^4 - 44800*A^2*a^5*b^3*c^5 + 25*A^2*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + B^2*a^2*b^2*(-(4*a*c - b^2)^9
)^(1/2) + 288*B^2*a^4*b^7*c^2 - 1504*B^2*a^5*b^5*c^3 + 3840*B^2*a^6*b^3*c^4 - 15360*A*B*a^7*c^6 - 213*A^2*a*b^
11*c + 26880*A^2*a^6*b*c^6 - 27*B^2*a^3*b^9*c - 3840*B^2*a^7*b*c^5 - 9*B^2*a^3*c*(-(4*a*c - b^2)^9)^(1/2) - 15
48*A*B*a^3*b^8*c^2 + 8064*A*B*a^4*b^6*c^3 - 22400*A*B*a^5*b^4*c^4 + 30720*A*B*a^6*b^2*c^5 - 51*A^2*a*b^2*c*(-(
4*a*c - b^2)^9)^(1/2) - 6*A*B*a*b^3*(-(4*a*c - b^2)^9)^(1/2) + 152*A*B*a^2*b^10*c + 44*A*B*a^2*b*c*(-(4*a*c -
b^2)^9)^(1/2))/(8*(a^5*b^12 + 4096*a^11*c^6 - 24*a^6*b^10*c + 240*a^7*b^8*c^2 - 1280*a^8*b^6*c^3 + 3840*a^9*b^
4*c^4 - 6144*a^10*b^2*c^5)))^(1/2)*(24576*B*a^15*c^8 + x^(1/2)*(-(9*A^2*b^13 + B^2*a^2*b^11 + 9*A^2*b^4*(-(4*a
*c - b^2)^9)^(1/2) - 6*A*B*a*b^12 + 2077*A^2*a^2*b^9*c^2 - 10656*A^2*a^3*b^7*c^3 + 30240*A^2*a^4*b^5*c^4 - 448
00*A^2*a^5*b^3*c^5 + 25*A^2*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + B^2*a^2*b^2*(-(4*a*c - b^2)^9)^(1/2) + 288*B^2*
a^4*b^7*c^2 - 1504*B^2*a^5*b^5*c^3 + 3840*B^2*a^6*b^3*c^4 - 15360*A*B*a^7*c^6 - 213*A^2*a*b^11*c + 26880*A^2*a
^6*b*c^6 - 27*B^2*a^3*b^9*c - 3840*B^2*a^7*b*c^5 - 9*B^2*a^3*c*(-(4*a*c - b^2)^9)^(1/2) - 1548*A*B*a^3*b^8*c^2
+ 8064*A*B*a^4*b^6*c^3 - 22400*A*B*a^5*b^4*c^4 + 30720*A*B*a^6*b^2*c^5 - 51*A^2*a*b^2*c*(-(4*a*c - b^2)^9)^(1
/2) - 6*A*B*a*b^3*(-(4*a*c - b^2)^9)^(1/2) + 152*A*B*a^2*b^10*c + 44*A*B*a^2*b*c*(-(4*a*c - b^2)^9)^(1/2))/(8*
(a^5*b^12 + 4096*a^11*c^6 - 24*a^6*b^10*c + 240*a^7*b^8*c^2 - 1280*a^8*b^6*c^3 + 3840*a^9*b^4*c^4 - 6144*a^10*
b^2*c^5)))^(1/2)*(32768*a^16*b*c^8 + 8*a^10*b^13*c^2 - 192*a^11*b^11*c^3 + 1920*a^12*b^9*c^4 - 10240*a^13*b^7*
c^5 + 30720*a^14*b^5*c^6 - 49152*a^15*b^3*c^7) - 53248*A*a^14*b*c^8 - 12*A*a^8*b^13*c^2 + 292*A*a^9*b^11*c^3 -
2960*A*a^10*b^9*c^4 + 16000*A*a^11*b^7*c^5 - 48640*A*a^12*b^5*c^6 + 78848*A*a^13*b^3*c^7 + 4*B*a^9*b^12*c^2 -
104*B*a^10*b^10*c^3 + 1120*B*a^11*b^8*c^4 - 6400*B*a^12*b^6*c^5 + 20480*B*a^13*b^4*c^6 - 34816*B*a^14*b^2*c^7
))*(-(9*A^2*b^13 + B^2*a^2*b^11 + 9*A^2*b^4*(-(4*a*c - b^2)^9)^(1/2) - 6*A*B*a*b^12 + 2077*A^2*a^2*b^9*c^2 - 1
0656*A^2*a^3*b^7*c^3 + 30240*A^2*a^4*b^5*c^4 - 44800*A^2*a^5*b^3*c^5 + 25*A^2*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2)
+ B^2*a^2*b^2*(-(4*a*c - b^2)^9)^(1/2) + 288*B^2*a^4*b^7*c^2 - 1504*B^2*a^5*b^5*c^3 + 3840*B^2*a^6*b^3*c^4 -
15360*A*B*a^7*c^6 - 213*A^2*a*b^11*c + 26880*A^2*a^6*b*c^6 - 27*B^2*a^3*b^9*c - 3840*B^2*a^7*b*c^5 - 9*B^2*a^3
*c*(-(4*a*c - b^2)^9)^(1/2) - 1548*A*B*a^3*b^8*c^2 + 8064*A*B*a^4*b^6*c^3 - 22400*A*B*a^5*b^4*c^4 + 30720*A*B*
a^6*b^2*c^5 - 51*A^2*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2) - 6*A*B*a*b^3*(-(4*a*c - b^2)^9)^(1/2) + 152*A*B*a^2*b^1
0*c + 44*A*B*a^2*b*c*(-(4*a*c - b^2)^9)^(1/2))/(8*(a^5*b^12 + 4096*a^11*c^6 - 24*a^6*b^10*c + 240*a^7*b^8*c^2
- 1280*a^8*b^6*c^3 + 3840*a^9*b^4*c^4 - 6144*a^10*b^2*c^5)))^(1/2)*1i)/((x^(1/2)*(25600*A^2*a^12*c^9 - 9216*B^
2*a^13*c^8 + 18*A^2*a^6*b^12*c^3 - 408*A^2*a^7*b^10*c^4 + 3764*A^2*a^8*b^8*c^5 - 17920*A^2*a^9*b^6*c^6 + 45696
*A^2*a^10*b^4*c^7 - 57344*A^2*a^11*b^2*c^8 + 2*B^2*a^8*b^10*c^3 - 52*B^2*a^9*b^8*c^4 + 576*B^2*a^10*b^6*c^5 -
3200*B^2*a^11*b^4*c^6 + 8704*B^2*a^12*b^2*c^7 - 12*A*B*a^7*b^11*c^3 + 292*A*B*a^8*b^9*c^4 - 2816*A*B*a^9*b^7*c
^5 + 13440*A*B*a^10*b^5*c^6 - 31744*A*B*a^11*b^3*c^7 + 29696*A*B*a^12*b*c^8) + (-(9*A^2*b^13 + B^2*a^2*b^11 +
9*A^2*b^4*(-(4*a*c - b^2)^9)^(1/2) - 6*A*B*a*b^12 + 2077*A^2*a^2*b^9*c^2 - 10656*A^2*a^3*b^7*c^3 + 30240*A^2*a
^4*b^5*c^4 - 44800*A^2*a^5*b^3*c^5 + 25*A^2*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + B^2*a^2*b^2*(-(4*a*c - b^2)^9)^
(1/2) + 288*B^2*a^4*b^7*c^2 - 1504*B^2*a^5*b^5*c^3 + 3840*B^2*a^6*b^3*c^4 - 15360*A*B*a^7*c^6 - 213*A^2*a*b^11
*c + 26880*A^2*a^6*b*c^6 - 27*B^2*a^3*b^9*c - 3840*B^2*a^7*b*c^5 - 9*B^2*a^3*c*(-(4*a*c - b^2)^9)^(1/2) - 1548
*A*B*a^3*b^8*c^2 + 8064*A*B*a^4*b^6*c^3 - 22400*A*B*a^5*b^4*c^4 + 30720*A*B*a^6*b^2*c^5 - 51*A^2*a*b^2*c*(-(4*
a*c - b^2)^9)^(1/2) - 6*A*B*a*b^3*(-(4*a*c - b^2)^9)^(1/2) + 152*A*B*a^2*b^10*c + 44*A*B*a^2*b*c*(-(4*a*c - b^
2)^9)^(1/2))/(8*(a^5*b^12 + 4096*a^11*c^6 - 24*a^6*b^10*c + 240*a^7*b^8*c^2 - 1280*a^8*b^6*c^3 + 3840*a^9*b^4*
c^4 - 6144*a^10*b^2*c^5)))^(1/2)*(x^(1/2)*(-(9*A^2*b^13 + B^2*a^2*b^11 + 9*A^2*b^4*(-(4*a*c - b^2)^9)^(1/2) -
6*A*B*a*b^12 + 2077*A^2*a^2*b^9*c^2 - 10656*A^2*a^3*b^7*c^3 + 30240*A^2*a^4*b^5*c^4 - 44800*A^2*a^5*b^3*c^5 +
25*A^2*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + B^2*a^2*b^2*(-(4*a*c - b^2)^9)^(1/2) + 288*B^2*a^4*b^7*c^2 - 1504*B^
2*a^5*b^5*c^3 + 3840*B^2*a^6*b^3*c^4 - 15360*A*B*a^7*c^6 - 213*A^2*a*b^11*c + 26880*A^2*a^6*b*c^6 - 27*B^2*a^3
*b^9*c - 3840*B^2*a^7*b*c^5 - 9*B^2*a^3*c*(-(4*a*c - b^2)^9)^(1/2) - 1548*A*B*a^3*b^8*c^2 + 8064*A*B*a^4*b^6*c
^3 - 22400*A*B*a^5*b^4*c^4 + 30720*A*B*a^6*b^2*c^5 - 51*A^2*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2) - 6*A*B*a*b^3*(-(
4*a*c - b^2)^9)^(1/2) + 152*A*B*a^2*b^10*c + 44*A*B*a^2*b*c*(-(4*a*c - b^2)^9)^(1/2))/(8*(a^5*b^12 + 4096*a^11
*c^6 - 24*a^6*b^10*c + 240*a^7*b^8*c^2 - 1280*a^8*b^6*c^3 + 3840*a^9*b^4*c^4 - 6144*a^10*b^2*c^5)))^(1/2)*(327
68*a^16*b*c^8 + 8*a^10*b^13*c^2 - 192*a^11*b^11*c^3 + 1920*a^12*b^9*c^4 - 10240*a^13*b^7*c^5 + 30720*a^14*b^5*
c^6 - 49152*a^15*b^3*c^7) - 24576*B*a^15*c^8 + 53248*A*a^14*b*c^8 + 12*A*a^8*b^13*c^2 - 292*A*a^9*b^11*c^3 + 2
960*A*a^10*b^9*c^4 - 16000*A*a^11*b^7*c^5 + 48640*A*a^12*b^5*c^6 - 78848*A*a^13*b^3*c^7 - 4*B*a^9*b^12*c^2 + 1
04*B*a^10*b^10*c^3 - 1120*B*a^11*b^8*c^4 + 6400*B*a^12*b^6*c^5 - 20480*B*a^13*b^4*c^6 + 34816*B*a^14*b^2*c^7))
*(-(9*A^2*b^13 + B^2*a^2*b^11 + 9*A^2*b^4*(-(4*a*c - b^2)^9)^(1/2) - 6*A*B*a*b^12 + 2077*A^2*a^2*b^9*c^2 - 106
56*A^2*a^3*b^7*c^3 + 30240*A^2*a^4*b^5*c^4 - 44800*A^2*a^5*b^3*c^5 + 25*A^2*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) +
B^2*a^2*b^2*(-(4*a*c - b^2)^9)^(1/2) + 288*B^2*a^4*b^7*c^2 - 1504*B^2*a^5*b^5*c^3 + 3840*B^2*a^6*b^3*c^4 - 15
360*A*B*a^7*c^6 - 213*A^2*a*b^11*c + 26880*A^2*a^6*b*c^6 - 27*B^2*a^3*b^9*c - 3840*B^2*a^7*b*c^5 - 9*B^2*a^3*c
*(-(4*a*c - b^2)^9)^(1/2) - 1548*A*B*a^3*b^8*c^2 + 8064*A*B*a^4*b^6*c^3 - 22400*A*B*a^5*b^4*c^4 + 30720*A*B*a^
6*b^2*c^5 - 51*A^2*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2) - 6*A*B*a*b^3*(-(4*a*c - b^2)^9)^(1/2) + 152*A*B*a^2*b^10*
c + 44*A*B*a^2*b*c*(-(4*a*c - b^2)^9)^(1/2))/(8*(a^5*b^12 + 4096*a^11*c^6 - 24*a^6*b^10*c + 240*a^7*b^8*c^2 -
1280*a^8*b^6*c^3 + 3840*a^9*b^4*c^4 - 6144*a^10*b^2*c^5)))^(1/2) - (x^(1/2)*(25600*A^2*a^12*c^9 - 9216*B^2*a^1
3*c^8 + 18*A^2*a^6*b^12*c^3 - 408*A^2*a^7*b^10*c^4 + 3764*A^2*a^8*b^8*c^5 - 17920*A^2*a^9*b^6*c^6 + 45696*A^2*
a^10*b^4*c^7 - 57344*A^2*a^11*b^2*c^8 + 2*B^2*a^8*b^10*c^3 - 52*B^2*a^9*b^8*c^4 + 576*B^2*a^10*b^6*c^5 - 3200*
B^2*a^11*b^4*c^6 + 8704*B^2*a^12*b^2*c^7 - 12*A*B*a^7*b^11*c^3 + 292*A*B*a^8*b^9*c^4 - 2816*A*B*a^9*b^7*c^5 +
13440*A*B*a^10*b^5*c^6 - 31744*A*B*a^11*b^3*c^7 + 29696*A*B*a^12*b*c^8) + (-(9*A^2*b^13 + B^2*a^2*b^11 + 9*A^2
*b^4*(-(4*a*c - b^2)^9)^(1/2) - 6*A*B*a*b^12 + 2077*A^2*a^2*b^9*c^2 - 10656*A^2*a^3*b^7*c^3 + 30240*A^2*a^4*b^
5*c^4 - 44800*A^2*a^5*b^3*c^5 + 25*A^2*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + B^2*a^2*b^2*(-(4*a*c - b^2)^9)^(1/2)
+ 288*B^2*a^4*b^7*c^2 - 1504*B^2*a^5*b^5*c^3 + 3840*B^2*a^6*b^3*c^4 - 15360*A*B*a^7*c^6 - 213*A^2*a*b^11*c +
26880*A^2*a^6*b*c^6 - 27*B^2*a^3*b^9*c - 3840*B^2*a^7*b*c^5 - 9*B^2*a^3*c*(-(4*a*c - b^2)^9)^(1/2) - 1548*A*B*
a^3*b^8*c^2 + 8064*A*B*a^4*b^6*c^3 - 22400*A*B*a^5*b^4*c^4 + 30720*A*B*a^6*b^2*c^5 - 51*A^2*a*b^2*c*(-(4*a*c -
b^2)^9)^(1/2) - 6*A*B*a*b^3*(-(4*a*c - b^2)^9)^(1/2) + 152*A*B*a^2*b^10*c + 44*A*B*a^2*b*c*(-(4*a*c - b^2)^9)
^(1/2))/(8*(a^5*b^12 + 4096*a^11*c^6 - 24*a^6*b^10*c + 240*a^7*b^8*c^2 - 1280*a^8*b^6*c^3 + 3840*a^9*b^4*c^4 -
6144*a^10*b^2*c^5)))^(1/2)*(24576*B*a^15*c^8 + x^(1/2)*(-(9*A^2*b^13 + B^2*a^2*b^11 + 9*A^2*b^4*(-(4*a*c - b^
2)^9)^(1/2) - 6*A*B*a*b^12 + 2077*A^2*a^2*b^9*c^2 - 10656*A^2*a^3*b^7*c^3 + 30240*A^2*a^4*b^5*c^4 - 44800*A^2*
a^5*b^3*c^5 + 25*A^2*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + B^2*a^2*b^2*(-(4*a*c - b^2)^9)^(1/2) + 288*B^2*a^4*b^7
*c^2 - 1504*B^2*a^5*b^5*c^3 + 3840*B^2*a^6*b^3*c^4 - 15360*A*B*a^7*c^6 - 213*A^2*a*b^11*c + 26880*A^2*a^6*b*c^
6 - 27*B^2*a^3*b^9*c - 3840*B^2*a^7*b*c^5 - 9*B^2*a^3*c*(-(4*a*c - b^2)^9)^(1/2) - 1548*A*B*a^3*b^8*c^2 + 8064
*A*B*a^4*b^6*c^3 - 22400*A*B*a^5*b^4*c^4 + 30720*A*B*a^6*b^2*c^5 - 51*A^2*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2) - 6
*A*B*a*b^3*(-(4*a*c - b^2)^9)^(1/2) + 152*A*B*a^2*b^10*c + 44*A*B*a^2*b*c*(-(4*a*c - b^2)^9)^(1/2))/(8*(a^5*b^
12 + 4096*a^11*c^6 - 24*a^6*b^10*c + 240*a^7*b^8*c^2 - 1280*a^8*b^6*c^3 + 3840*a^9*b^4*c^4 - 6144*a^10*b^2*c^5
)))^(1/2)*(32768*a^16*b*c^8 + 8*a^10*b^13*c^2 - 192*a^11*b^11*c^3 + 1920*a^12*b^9*c^4 - 10240*a^13*b^7*c^5 + 3
0720*a^14*b^5*c^6 - 49152*a^15*b^3*c^7) - 53248*A*a^14*b*c^8 - 12*A*a^8*b^13*c^2 + 292*A*a^9*b^11*c^3 - 2960*A
*a^10*b^9*c^4 + 16000*A*a^11*b^7*c^5 - 48640*A*a^12*b^5*c^6 + 78848*A*a^13*b^3*c^7 + 4*B*a^9*b^12*c^2 - 104*B*
a^10*b^10*c^3 + 1120*B*a^11*b^8*c^4 - 6400*B*a^12*b^6*c^5 + 20480*B*a^13*b^4*c^6 - 34816*B*a^14*b^2*c^7))*(-(9
*A^2*b^13 + B^2*a^2*b^11 + 9*A^2*b^4*(-(4*a*c - b^2)^9)^(1/2) - 6*A*B*a*b^12 + 2077*A^2*a^2*b^9*c^2 - 10656*A^
2*a^3*b^7*c^3 + 30240*A^2*a^4*b^5*c^4 - 44800*A^2*a^5*b^3*c^5 + 25*A^2*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + B^2*
a^2*b^2*(-(4*a*c - b^2)^9)^(1/2) + 288*B^2*a^4*b^7*c^2 - 1504*B^2*a^5*b^5*c^3 + 3840*B^2*a^6*b^3*c^4 - 15360*A
*B*a^7*c^6 - 213*A^2*a*b^11*c + 26880*A^2*a^6*b*c^6 - 27*B^2*a^3*b^9*c - 3840*B^2*a^7*b*c^5 - 9*B^2*a^3*c*(-(4
*a*c - b^2)^9)^(1/2) - 1548*A*B*a^3*b^8*c^2 + 8064*A*B*a^4*b^6*c^3 - 22400*A*B*a^5*b^4*c^4 + 30720*A*B*a^6*b^2
*c^5 - 51*A^2*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2) - 6*A*B*a*b^3*(-(4*a*c - b^2)^9)^(1/2) + 152*A*B*a^2*b^10*c + 4
4*A*B*a^2*b*c*(-(4*a*c - b^2)^9)^(1/2))/(8*(a^5*b^12 + 4096*a^11*c^6 - 24*a^6*b^10*c + 240*a^7*b^8*c^2 - 1280*
a^8*b^6*c^3 + 3840*a^9*b^4*c^4 - 6144*a^10*b^2*c^5)))^(1/2) + 32000*A^3*a^10*c^9 + 126*A^3*a^6*b^8*c^5 - 2028*
A^3*a^7*b^6*c^6 + 12176*A^3*a^8*b^4*c^7 - 32320*A^3*a^9*b^2*c^8 - 10*B^3*a^8*b^7*c^4 + 152*B^3*a^9*b^5*c^5 - 7
36*B^3*a^10*b^3*c^6 + 11520*A*B^2*a^11*c^8 + 1152*B^3*a^11*b*c^7 - 21120*A^2*B*a^10*b*c^8 + 60*A*B^2*a^7*b^8*c
^4 - 948*A*B^2*a^8*b^6*c^5 + 5424*A*B^2*a^9*b^4*c^6 - 13248*A*B^2*a^10*b^2*c^7 - 90*A^2*B*a^6*b^9*c^4 + 1434*A
^2*B*a^7*b^7*c^5 - 8472*A^2*B*a^8*b^5*c^6 + 21984*A^2*B*a^9*b^3*c^7))*(-(9*A^2*b^13 + B^2*a^2*b^11 + 9*A^2*b^4
*(-(4*a*c - b^2)^9)^(1/2) - 6*A*B*a*b^12 + 2077*A^2*a^2*b^9*c^2 - 10656*A^2*a^3*b^7*c^3 + 30240*A^2*a^4*b^5*c^
4 - 44800*A^2*a^5*b^3*c^5 + 25*A^2*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + B^2*a^2*b^2*(-(4*a*c - b^2)^9)^(1/2) + 2
88*B^2*a^4*b^7*c^2 - 1504*B^2*a^5*b^5*c^3 + 3840*B^2*a^6*b^3*c^4 - 15360*A*B*a^7*c^6 - 213*A^2*a*b^11*c + 2688
0*A^2*a^6*b*c^6 - 27*B^2*a^3*b^9*c - 3840*B^2*a^7*b*c^5 - 9*B^2*a^3*c*(-(4*a*c - b^2)^9)^(1/2) - 1548*A*B*a^3*
b^8*c^2 + 8064*A*B*a^4*b^6*c^3 - 22400*A*B*a^5*b^4*c^4 + 30720*A*B*a^6*b^2*c^5 - 51*A^2*a*b^2*c*(-(4*a*c - b^2
)^9)^(1/2) - 6*A*B*a*b^3*(-(4*a*c - b^2)^9)^(1/2) + 152*A*B*a^2*b^10*c + 44*A*B*a^2*b*c*(-(4*a*c - b^2)^9)^(1/
2))/(8*(a^5*b^12 + 4096*a^11*c^6 - 24*a^6*b^10*c + 240*a^7*b^8*c^2 - 1280*a^8*b^6*c^3 + 3840*a^9*b^4*c^4 - 614
4*a^10*b^2*c^5)))^(1/2)*2i - atan(((x^(1/2)*(25600*A^2*a^12*c^9 - 9216*B^2*a^13*c^8 + 18*A^2*a^6*b^12*c^3 - 40
8*A^2*a^7*b^10*c^4 + 3764*A^2*a^8*b^8*c^5 - 17920*A^2*a^9*b^6*c^6 + 45696*A^2*a^10*b^4*c^7 - 57344*A^2*a^11*b^
2*c^8 + 2*B^2*a^8*b^10*c^3 - 52*B^2*a^9*b^8*c^4 + 576*B^2*a^10*b^6*c^5 - 3200*B^2*a^11*b^4*c^6 + 8704*B^2*a^12
*b^2*c^7 - 12*A*B*a^7*b^11*c^3 + 292*A*B*a^8*b^9*c^4 - 2816*A*B*a^9*b^7*c^5 + 13440*A*B*a^10*b^5*c^6 - 31744*A
*B*a^11*b^3*c^7 + 29696*A*B*a^12*b*c^8) + ((9*A^2*b^4*(-(4*a*c - b^2)^9)^(1/2) - B^2*a^2*b^11 - 9*A^2*b^13 + 6
*A*B*a*b^12 - 2077*A^2*a^2*b^9*c^2 + 10656*A^2*a^3*b^7*c^3 - 30240*A^2*a^4*b^5*c^4 + 44800*A^2*a^5*b^3*c^5 + 2
5*A^2*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + B^2*a^2*b^2*(-(4*a*c - b^2)^9)^(1/2) - 288*B^2*a^4*b^7*c^2 + 1504*B^2
*a^5*b^5*c^3 - 3840*B^2*a^6*b^3*c^4 + 15360*A*B*a^7*c^6 + 213*A^2*a*b^11*c - 26880*A^2*a^6*b*c^6 + 27*B^2*a^3*
b^9*c + 3840*B^2*a^7*b*c^5 - 9*B^2*a^3*c*(-(4*a*c - b^2)^9)^(1/2) + 1548*A*B*a^3*b^8*c^2 - 8064*A*B*a^4*b^6*c^
3 + 22400*A*B*a^5*b^4*c^4 - 30720*A*B*a^6*b^2*c^5 - 51*A^2*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2) - 6*A*B*a*b^3*(-(4
*a*c - b^2)^9)^(1/2) - 152*A*B*a^2*b^10*c + 44*A*B*a^2*b*c*(-(4*a*c - b^2)^9)^(1/2))/(8*(a^5*b^12 + 4096*a^11*
c^6 - 24*a^6*b^10*c + 240*a^7*b^8*c^2 - 1280*a^8*b^6*c^3 + 3840*a^9*b^4*c^4 - 6144*a^10*b^2*c^5)))^(1/2)*(x^(1
/2)*((9*A^2*b^4*(-(4*a*c - b^2)^9)^(1/2) - B^2*a^2*b^11 - 9*A^2*b^13 + 6*A*B*a*b^12 - 2077*A^2*a^2*b^9*c^2 + 1
0656*A^2*a^3*b^7*c^3 - 30240*A^2*a^4*b^5*c^4 + 44800*A^2*a^5*b^3*c^5 + 25*A^2*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2)
+ B^2*a^2*b^2*(-(4*a*c - b^2)^9)^(1/2) - 288*B^2*a^4*b^7*c^2 + 1504*B^2*a^5*b^5*c^3 - 3840*B^2*a^6*b^3*c^4 +
15360*A*B*a^7*c^6 + 213*A^2*a*b^11*c - 26880*A^2*a^6*b*c^6 + 27*B^2*a^3*b^9*c + 3840*B^2*a^7*b*c^5 - 9*B^2*a^3
*c*(-(4*a*c - b^2)^9)^(1/2) + 1548*A*B*a^3*b^8*c^2 - 8064*A*B*a^4*b^6*c^3 + 22400*A*B*a^5*b^4*c^4 - 30720*A*B*
a^6*b^2*c^5 - 51*A^2*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2) - 6*A*B*a*b^3*(-(4*a*c - b^2)^9)^(1/2) - 152*A*B*a^2*b^1
0*c + 44*A*B*a^2*b*c*(-(4*a*c - b^2)^9)^(1/2))/(8*(a^5*b^12 + 4096*a^11*c^6 - 24*a^6*b^10*c + 240*a^7*b^8*c^2
- 1280*a^8*b^6*c^3 + 3840*a^9*b^4*c^4 - 6144*a^10*b^2*c^5)))^(1/2)*(32768*a^16*b*c^8 + 8*a^10*b^13*c^2 - 192*a
^11*b^11*c^3 + 1920*a^12*b^9*c^4 - 10240*a^13*b^7*c^5 + 30720*a^14*b^5*c^6 - 49152*a^15*b^3*c^7) - 24576*B*a^1
5*c^8 + 53248*A*a^14*b*c^8 + 12*A*a^8*b^13*c^2 - 292*A*a^9*b^11*c^3 + 2960*A*a^10*b^9*c^4 - 16000*A*a^11*b^7*c
^5 + 48640*A*a^12*b^5*c^6 - 78848*A*a^13*b^3*c^7 - 4*B*a^9*b^12*c^2 + 104*B*a^10*b^10*c^3 - 1120*B*a^11*b^8*c^
4 + 6400*B*a^12*b^6*c^5 - 20480*B*a^13*b^4*c^6 + 34816*B*a^14*b^2*c^7))*((9*A^2*b^4*(-(4*a*c - b^2)^9)^(1/2) -
B^2*a^2*b^11 - 9*A^2*b^13 + 6*A*B*a*b^12 - 2077*A^2*a^2*b^9*c^2 + 10656*A^2*a^3*b^7*c^3 - 30240*A^2*a^4*b^5*c
^4 + 44800*A^2*a^5*b^3*c^5 + 25*A^2*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + B^2*a^2*b^2*(-(4*a*c - b^2)^9)^(1/2) -
288*B^2*a^4*b^7*c^2 + 1504*B^2*a^5*b^5*c^3 - 3840*B^2*a^6*b^3*c^4 + 15360*A*B*a^7*c^6 + 213*A^2*a*b^11*c - 268
80*A^2*a^6*b*c^6 + 27*B^2*a^3*b^9*c + 3840*B^2*a^7*b*c^5 - 9*B^2*a^3*c*(-(4*a*c - b^2)^9)^(1/2) + 1548*A*B*a^3
*b^8*c^2 - 8064*A*B*a^4*b^6*c^3 + 22400*A*B*a^5*b^4*c^4 - 30720*A*B*a^6*b^2*c^5 - 51*A^2*a*b^2*c*(-(4*a*c - b^
2)^9)^(1/2) - 6*A*B*a*b^3*(-(4*a*c - b^2)^9)^(1/2) - 152*A*B*a^2*b^10*c + 44*A*B*a^2*b*c*(-(4*a*c - b^2)^9)^(1
/2))/(8*(a^5*b^12 + 4096*a^11*c^6 - 24*a^6*b^10*c + 240*a^7*b^8*c^2 - 1280*a^8*b^6*c^3 + 3840*a^9*b^4*c^4 - 61
44*a^10*b^2*c^5)))^(1/2)*1i + (x^(1/2)*(25600*A^2*a^12*c^9 - 9216*B^2*a^13*c^8 + 18*A^2*a^6*b^12*c^3 - 408*A^2
*a^7*b^10*c^4 + 3764*A^2*a^8*b^8*c^5 - 17920*A^2*a^9*b^6*c^6 + 45696*A^2*a^10*b^4*c^7 - 57344*A^2*a^11*b^2*c^8
+ 2*B^2*a^8*b^10*c^3 - 52*B^2*a^9*b^8*c^4 + 576*B^2*a^10*b^6*c^5 - 3200*B^2*a^11*b^4*c^6 + 8704*B^2*a^12*b^2*
c^7 - 12*A*B*a^7*b^11*c^3 + 292*A*B*a^8*b^9*c^4 - 2816*A*B*a^9*b^7*c^5 + 13440*A*B*a^10*b^5*c^6 - 31744*A*B*a^
11*b^3*c^7 + 29696*A*B*a^12*b*c^8) + ((9*A^2*b^4*(-(4*a*c - b^2)^9)^(1/2) - B^2*a^2*b^11 - 9*A^2*b^13 + 6*A*B*
a*b^12 - 2077*A^2*a^2*b^9*c^2 + 10656*A^2*a^3*b^7*c^3 - 30240*A^2*a^4*b^5*c^4 + 44800*A^2*a^5*b^3*c^5 + 25*A^2
*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + B^2*a^2*b^2*(-(4*a*c - b^2)^9)^(1/2) - 288*B^2*a^4*b^7*c^2 + 1504*B^2*a^5*
b^5*c^3 - 3840*B^2*a^6*b^3*c^4 + 15360*A*B*a^7*c^6 + 213*A^2*a*b^11*c - 26880*A^2*a^6*b*c^6 + 27*B^2*a^3*b^9*c
+ 3840*B^2*a^7*b*c^5 - 9*B^2*a^3*c*(-(4*a*c - b^2)^9)^(1/2) + 1548*A*B*a^3*b^8*c^2 - 8064*A*B*a^4*b^6*c^3 + 2
2400*A*B*a^5*b^4*c^4 - 30720*A*B*a^6*b^2*c^5 - 51*A^2*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2) - 6*A*B*a*b^3*(-(4*a*c
- b^2)^9)^(1/2) - 152*A*B*a^2*b^10*c + 44*A*B*a^2*b*c*(-(4*a*c - b^2)^9)^(1/2))/(8*(a^5*b^12 + 4096*a^11*c^6 -
24*a^6*b^10*c + 240*a^7*b^8*c^2 - 1280*a^8*b^6*c^3 + 3840*a^9*b^4*c^4 - 6144*a^10*b^2*c^5)))^(1/2)*(24576*B*a
^15*c^8 + x^(1/2)*((9*A^2*b^4*(-(4*a*c - b^2)^9)^(1/2) - B^2*a^2*b^11 - 9*A^2*b^13 + 6*A*B*a*b^12 - 2077*A^2*a
^2*b^9*c^2 + 10656*A^2*a^3*b^7*c^3 - 30240*A^2*a^4*b^5*c^4 + 44800*A^2*a^5*b^3*c^5 + 25*A^2*a^2*c^2*(-(4*a*c -
b^2)^9)^(1/2) + B^2*a^2*b^2*(-(4*a*c - b^2)^9)^(1/2) - 288*B^2*a^4*b^7*c^2 + 1504*B^2*a^5*b^5*c^3 - 3840*B^2*
a^6*b^3*c^4 + 15360*A*B*a^7*c^6 + 213*A^2*a*b^11*c - 26880*A^2*a^6*b*c^6 + 27*B^2*a^3*b^9*c + 3840*B^2*a^7*b*c
^5 - 9*B^2*a^3*c*(-(4*a*c - b^2)^9)^(1/2) + 1548*A*B*a^3*b^8*c^2 - 8064*A*B*a^4*b^6*c^3 + 22400*A*B*a^5*b^4*c^
4 - 30720*A*B*a^6*b^2*c^5 - 51*A^2*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2) - 6*A*B*a*b^3*(-(4*a*c - b^2)^9)^(1/2) - 1
52*A*B*a^2*b^10*c + 44*A*B*a^2*b*c*(-(4*a*c - b^2)^9)^(1/2))/(8*(a^5*b^12 + 4096*a^11*c^6 - 24*a^6*b^10*c + 24
0*a^7*b^8*c^2 - 1280*a^8*b^6*c^3 + 3840*a^9*b^4*c^4 - 6144*a^10*b^2*c^5)))^(1/2)*(32768*a^16*b*c^8 + 8*a^10*b^
13*c^2 - 192*a^11*b^11*c^3 + 1920*a^12*b^9*c^4 - 10240*a^13*b^7*c^5 + 30720*a^14*b^5*c^6 - 49152*a^15*b^3*c^7)
- 53248*A*a^14*b*c^8 - 12*A*a^8*b^13*c^2 + 292*A*a^9*b^11*c^3 - 2960*A*a^10*b^9*c^4 + 16000*A*a^11*b^7*c^5 -
48640*A*a^12*b^5*c^6 + 78848*A*a^13*b^3*c^7 + 4*B*a^9*b^12*c^2 - 104*B*a^10*b^10*c^3 + 1120*B*a^11*b^8*c^4 - 6
400*B*a^12*b^6*c^5 + 20480*B*a^13*b^4*c^6 - 34816*B*a^14*b^2*c^7))*((9*A^2*b^4*(-(4*a*c - b^2)^9)^(1/2) - B^2*
a^2*b^11 - 9*A^2*b^13 + 6*A*B*a*b^12 - 2077*A^2*a^2*b^9*c^2 + 10656*A^2*a^3*b^7*c^3 - 30240*A^2*a^4*b^5*c^4 +
44800*A^2*a^5*b^3*c^5 + 25*A^2*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + B^2*a^2*b^2*(-(4*a*c - b^2)^9)^(1/2) - 288*B
^2*a^4*b^7*c^2 + 1504*B^2*a^5*b^5*c^3 - 3840*B^2*a^6*b^3*c^4 + 15360*A*B*a^7*c^6 + 213*A^2*a*b^11*c - 26880*A^
2*a^6*b*c^6 + 27*B^2*a^3*b^9*c + 3840*B^2*a^7*b*c^5 - 9*B^2*a^3*c*(-(4*a*c - b^2)^9)^(1/2) + 1548*A*B*a^3*b^8*
c^2 - 8064*A*B*a^4*b^6*c^3 + 22400*A*B*a^5*b^4*c^4 - 30720*A*B*a^6*b^2*c^5 - 51*A^2*a*b^2*c*(-(4*a*c - b^2)^9)
^(1/2) - 6*A*B*a*b^3*(-(4*a*c - b^2)^9)^(1/2) - 152*A*B*a^2*b^10*c + 44*A*B*a^2*b*c*(-(4*a*c - b^2)^9)^(1/2))/
(8*(a^5*b^12 + 4096*a^11*c^6 - 24*a^6*b^10*c + 240*a^7*b^8*c^2 - 1280*a^8*b^6*c^3 + 3840*a^9*b^4*c^4 - 6144*a^
10*b^2*c^5)))^(1/2)*1i)/((x^(1/2)*(25600*A^2*a^12*c^9 - 9216*B^2*a^13*c^8 + 18*A^2*a^6*b^12*c^3 - 408*A^2*a^7*
b^10*c^4 + 3764*A^2*a^8*b^8*c^5 - 17920*A^2*a^9*b^6*c^6 + 45696*A^2*a^10*b^4*c^7 - 57344*A^2*a^11*b^2*c^8 + 2*
B^2*a^8*b^10*c^3 - 52*B^2*a^9*b^8*c^4 + 576*B^2*a^10*b^6*c^5 - 3200*B^2*a^11*b^4*c^6 + 8704*B^2*a^12*b^2*c^7 -
12*A*B*a^7*b^11*c^3 + 292*A*B*a^8*b^9*c^4 - 2816*A*B*a^9*b^7*c^5 + 13440*A*B*a^10*b^5*c^6 - 31744*A*B*a^11*b^
3*c^7 + 29696*A*B*a^12*b*c^8) + ((9*A^2*b^4*(-(4*a*c - b^2)^9)^(1/2) - B^2*a^2*b^11 - 9*A^2*b^13 + 6*A*B*a*b^1
2 - 2077*A^2*a^2*b^9*c^2 + 10656*A^2*a^3*b^7*c^3 - 30240*A^2*a^4*b^5*c^4 + 44800*A^2*a^5*b^3*c^5 + 25*A^2*a^2*
c^2*(-(4*a*c - b^2)^9)^(1/2) + B^2*a^2*b^2*(-(4*a*c - b^2)^9)^(1/2) - 288*B^2*a^4*b^7*c^2 + 1504*B^2*a^5*b^5*c
^3 - 3840*B^2*a^6*b^3*c^4 + 15360*A*B*a^7*c^6 + 213*A^2*a*b^11*c - 26880*A^2*a^6*b*c^6 + 27*B^2*a^3*b^9*c + 38
40*B^2*a^7*b*c^5 - 9*B^2*a^3*c*(-(4*a*c - b^2)^9)^(1/2) + 1548*A*B*a^3*b^8*c^2 - 8064*A*B*a^4*b^6*c^3 + 22400*
A*B*a^5*b^4*c^4 - 30720*A*B*a^6*b^2*c^5 - 51*A^2*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2) - 6*A*B*a*b^3*(-(4*a*c - b^2
)^9)^(1/2) - 152*A*B*a^2*b^10*c + 44*A*B*a^2*b*c*(-(4*a*c - b^2)^9)^(1/2))/(8*(a^5*b^12 + 4096*a^11*c^6 - 24*a
^6*b^10*c + 240*a^7*b^8*c^2 - 1280*a^8*b^6*c^3 + 3840*a^9*b^4*c^4 - 6144*a^10*b^2*c^5)))^(1/2)*(x^(1/2)*((9*A^
2*b^4*(-(4*a*c - b^2)^9)^(1/2) - B^2*a^2*b^11 - 9*A^2*b^13 + 6*A*B*a*b^12 - 2077*A^2*a^2*b^9*c^2 + 10656*A^2*a
^3*b^7*c^3 - 30240*A^2*a^4*b^5*c^4 + 44800*A^2*a^5*b^3*c^5 + 25*A^2*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + B^2*a^2
*b^2*(-(4*a*c - b^2)^9)^(1/2) - 288*B^2*a^4*b^7*c^2 + 1504*B^2*a^5*b^5*c^3 - 3840*B^2*a^6*b^3*c^4 + 15360*A*B*
a^7*c^6 + 213*A^2*a*b^11*c - 26880*A^2*a^6*b*c^6 + 27*B^2*a^3*b^9*c + 3840*B^2*a^7*b*c^5 - 9*B^2*a^3*c*(-(4*a*
c - b^2)^9)^(1/2) + 1548*A*B*a^3*b^8*c^2 - 8064*A*B*a^4*b^6*c^3 + 22400*A*B*a^5*b^4*c^4 - 30720*A*B*a^6*b^2*c^
5 - 51*A^2*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2) - 6*A*B*a*b^3*(-(4*a*c - b^2)^9)^(1/2) - 152*A*B*a^2*b^10*c + 44*A
*B*a^2*b*c*(-(4*a*c - b^2)^9)^(1/2))/(8*(a^5*b^12 + 4096*a^11*c^6 - 24*a^6*b^10*c + 240*a^7*b^8*c^2 - 1280*a^8
*b^6*c^3 + 3840*a^9*b^4*c^4 - 6144*a^10*b^2*c^5)))^(1/2)*(32768*a^16*b*c^8 + 8*a^10*b^13*c^2 - 192*a^11*b^11*c
^3 + 1920*a^12*b^9*c^4 - 10240*a^13*b^7*c^5 + 30720*a^14*b^5*c^6 - 49152*a^15*b^3*c^7) - 24576*B*a^15*c^8 + 53
248*A*a^14*b*c^8 + 12*A*a^8*b^13*c^2 - 292*A*a^9*b^11*c^3 + 2960*A*a^10*b^9*c^4 - 16000*A*a^11*b^7*c^5 + 48640
*A*a^12*b^5*c^6 - 78848*A*a^13*b^3*c^7 - 4*B*a^9*b^12*c^2 + 104*B*a^10*b^10*c^3 - 1120*B*a^11*b^8*c^4 + 6400*B
*a^12*b^6*c^5 - 20480*B*a^13*b^4*c^6 + 34816*B*a^14*b^2*c^7))*((9*A^2*b^4*(-(4*a*c - b^2)^9)^(1/2) - B^2*a^2*b
^11 - 9*A^2*b^13 + 6*A*B*a*b^12 - 2077*A^2*a^2*b^9*c^2 + 10656*A^2*a^3*b^7*c^3 - 30240*A^2*a^4*b^5*c^4 + 44800
*A^2*a^5*b^3*c^5 + 25*A^2*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + B^2*a^2*b^2*(-(4*a*c - b^2)^9)^(1/2) - 288*B^2*a^
4*b^7*c^2 + 1504*B^2*a^5*b^5*c^3 - 3840*B^2*a^6*b^3*c^4 + 15360*A*B*a^7*c^6 + 213*A^2*a*b^11*c - 26880*A^2*a^6
*b*c^6 + 27*B^2*a^3*b^9*c + 3840*B^2*a^7*b*c^5 - 9*B^2*a^3*c*(-(4*a*c - b^2)^9)^(1/2) + 1548*A*B*a^3*b^8*c^2 -
8064*A*B*a^4*b^6*c^3 + 22400*A*B*a^5*b^4*c^4 - 30720*A*B*a^6*b^2*c^5 - 51*A^2*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2
) - 6*A*B*a*b^3*(-(4*a*c - b^2)^9)^(1/2) - 152*A*B*a^2*b^10*c + 44*A*B*a^2*b*c*(-(4*a*c - b^2)^9)^(1/2))/(8*(a
^5*b^12 + 4096*a^11*c^6 - 24*a^6*b^10*c + 240*a^7*b^8*c^2 - 1280*a^8*b^6*c^3 + 3840*a^9*b^4*c^4 - 6144*a^10*b^
2*c^5)))^(1/2) - (x^(1/2)*(25600*A^2*a^12*c^9 - 9216*B^2*a^13*c^8 + 18*A^2*a^6*b^12*c^3 - 408*A^2*a^7*b^10*c^4
+ 3764*A^2*a^8*b^8*c^5 - 17920*A^2*a^9*b^6*c^6 + 45696*A^2*a^10*b^4*c^7 - 57344*A^2*a^11*b^2*c^8 + 2*B^2*a^8*
b^10*c^3 - 52*B^2*a^9*b^8*c^4 + 576*B^2*a^10*b^6*c^5 - 3200*B^2*a^11*b^4*c^6 + 8704*B^2*a^12*b^2*c^7 - 12*A*B*
a^7*b^11*c^3 + 292*A*B*a^8*b^9*c^4 - 2816*A*B*a^9*b^7*c^5 + 13440*A*B*a^10*b^5*c^6 - 31744*A*B*a^11*b^3*c^7 +
29696*A*B*a^12*b*c^8) + ((9*A^2*b^4*(-(4*a*c - b^2)^9)^(1/2) - B^2*a^2*b^11 - 9*A^2*b^13 + 6*A*B*a*b^12 - 2077
*A^2*a^2*b^9*c^2 + 10656*A^2*a^3*b^7*c^3 - 30240*A^2*a^4*b^5*c^4 + 44800*A^2*a^5*b^3*c^5 + 25*A^2*a^2*c^2*(-(4
*a*c - b^2)^9)^(1/2) + B^2*a^2*b^2*(-(4*a*c - b^2)^9)^(1/2) - 288*B^2*a^4*b^7*c^2 + 1504*B^2*a^5*b^5*c^3 - 384
0*B^2*a^6*b^3*c^4 + 15360*A*B*a^7*c^6 + 213*A^2*a*b^11*c - 26880*A^2*a^6*b*c^6 + 27*B^2*a^3*b^9*c + 3840*B^2*a
^7*b*c^5 - 9*B^2*a^3*c*(-(4*a*c - b^2)^9)^(1/2) + 1548*A*B*a^3*b^8*c^2 - 8064*A*B*a^4*b^6*c^3 + 22400*A*B*a^5*
b^4*c^4 - 30720*A*B*a^6*b^2*c^5 - 51*A^2*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2) - 6*A*B*a*b^3*(-(4*a*c - b^2)^9)^(1/
2) - 152*A*B*a^2*b^10*c + 44*A*B*a^2*b*c*(-(4*a*c - b^2)^9)^(1/2))/(8*(a^5*b^12 + 4096*a^11*c^6 - 24*a^6*b^10*
c + 240*a^7*b^8*c^2 - 1280*a^8*b^6*c^3 + 3840*a^9*b^4*c^4 - 6144*a^10*b^2*c^5)))^(1/2)*(24576*B*a^15*c^8 + x^(
1/2)*((9*A^2*b^4*(-(4*a*c - b^2)^9)^(1/2) - B^2*a^2*b^11 - 9*A^2*b^13 + 6*A*B*a*b^12 - 2077*A^2*a^2*b^9*c^2 +
10656*A^2*a^3*b^7*c^3 - 30240*A^2*a^4*b^5*c^4 + 44800*A^2*a^5*b^3*c^5 + 25*A^2*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2
) + B^2*a^2*b^2*(-(4*a*c - b^2)^9)^(1/2) - 288*B^2*a^4*b^7*c^2 + 1504*B^2*a^5*b^5*c^3 - 3840*B^2*a^6*b^3*c^4 +
15360*A*B*a^7*c^6 + 213*A^2*a*b^11*c - 26880*A^2*a^6*b*c^6 + 27*B^2*a^3*b^9*c + 3840*B^2*a^7*b*c^5 - 9*B^2*a^
3*c*(-(4*a*c - b^2)^9)^(1/2) + 1548*A*B*a^3*b^8*c^2 - 8064*A*B*a^4*b^6*c^3 + 22400*A*B*a^5*b^4*c^4 - 30720*A*B
*a^6*b^2*c^5 - 51*A^2*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2) - 6*A*B*a*b^3*(-(4*a*c - b^2)^9)^(1/2) - 152*A*B*a^2*b^
10*c + 44*A*B*a^2*b*c*(-(4*a*c - b^2)^9)^(1/2))/(8*(a^5*b^12 + 4096*a^11*c^6 - 24*a^6*b^10*c + 240*a^7*b^8*c^2
- 1280*a^8*b^6*c^3 + 3840*a^9*b^4*c^4 - 6144*a^10*b^2*c^5)))^(1/2)*(32768*a^16*b*c^8 + 8*a^10*b^13*c^2 - 192*
a^11*b^11*c^3 + 1920*a^12*b^9*c^4 - 10240*a^13*b^7*c^5 + 30720*a^14*b^5*c^6 - 49152*a^15*b^3*c^7) - 53248*A*a^
14*b*c^8 - 12*A*a^8*b^13*c^2 + 292*A*a^9*b^11*c^3 - 2960*A*a^10*b^9*c^4 + 16000*A*a^11*b^7*c^5 - 48640*A*a^12*
b^5*c^6 + 78848*A*a^13*b^3*c^7 + 4*B*a^9*b^12*c^2 - 104*B*a^10*b^10*c^3 + 1120*B*a^11*b^8*c^4 - 6400*B*a^12*b^
6*c^5 + 20480*B*a^13*b^4*c^6 - 34816*B*a^14*b^2*c^7))*((9*A^2*b^4*(-(4*a*c - b^2)^9)^(1/2) - B^2*a^2*b^11 - 9*
A^2*b^13 + 6*A*B*a*b^12 - 2077*A^2*a^2*b^9*c^2 + 10656*A^2*a^3*b^7*c^3 - 30240*A^2*a^4*b^5*c^4 + 44800*A^2*a^5
*b^3*c^5 + 25*A^2*a^2*c^2*(-(4*a*c - b^2)^9)^(1/2) + B^2*a^2*b^2*(-(4*a*c - b^2)^9)^(1/2) - 288*B^2*a^4*b^7*c^
2 + 1504*B^2*a^5*b^5*c^3 - 3840*B^2*a^6*b^3*c^4 + 15360*A*B*a^7*c^6 + 213*A^2*a*b^11*c - 26880*A^2*a^6*b*c^6 +
27*B^2*a^3*b^9*c + 3840*B^2*a^7*b*c^5 - 9*B^2*a^3*c*(-(4*a*c - b^2)^9)^(1/2) + 1548*A*B*a^3*b^8*c^2 - 8064*A*
B*a^4*b^6*c^3 + 22400*A*B*a^5*b^4*c^4 - 30720*A*B*a^6*b^2*c^5 - 51*A^2*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2) - 6*A*
B*a*b^3*(-(4*a*c - b^2)^9)^(1/2) - 152*A*B*a^2*b^10*c + 44*A*B*a^2*b*c*(-(4*a*c - b^2)^9)^(1/2))/(8*(a^5*b^12
+ 4096*a^11*c^6 - 24*a^6*b^10*c + 240*a^7*b^8*c^2 - 1280*a^8*b^6*c^3 + 3840*a^9*b^4*c^4 - 6144*a^10*b^2*c^5)))
^(1/2) + 32000*A^3*a^10*c^9 + 126*A^3*a^6*b^8*c^5 - 2028*A^3*a^7*b^6*c^6 + 12176*A^3*a^8*b^4*c^7 - 32320*A^3*a
^9*b^2*c^8 - 10*B^3*a^8*b^7*c^4 + 152*B^3*a^9*b^5*c^5 - 736*B^3*a^10*b^3*c^6 + 11520*A*B^2*a^11*c^8 + 1152*B^3
*a^11*b*c^7 - 21120*A^2*B*a^10*b*c^8 + 60*A*B^2*a^7*b^8*c^4 - 948*A*B^2*a^8*b^6*c^5 + 5424*A*B^2*a^9*b^4*c^6 -
13248*A*B^2*a^10*b^2*c^7 - 90*A^2*B*a^6*b^9*c^4 + 1434*A^2*B*a^7*b^7*c^5 - 8472*A^2*B*a^8*b^5*c^6 + 21984*A^2
*B*a^9*b^3*c^7))*((9*A^2*b^4*(-(4*a*c - b^2)^9)^(1/2) - B^2*a^2*b^11 - 9*A^2*b^13 + 6*A*B*a*b^12 - 2077*A^2*a^
2*b^9*c^2 + 10656*A^2*a^3*b^7*c^3 - 30240*A^2*a^4*b^5*c^4 + 44800*A^2*a^5*b^3*c^5 + 25*A^2*a^2*c^2*(-(4*a*c -
b^2)^9)^(1/2) + B^2*a^2*b^2*(-(4*a*c - b^2)^9)^(1/2) - 288*B^2*a^4*b^7*c^2 + 1504*B^2*a^5*b^5*c^3 - 3840*B^2*a
^6*b^3*c^4 + 15360*A*B*a^7*c^6 + 213*A^2*a*b^11*c - 26880*A^2*a^6*b*c^6 + 27*B^2*a^3*b^9*c + 3840*B^2*a^7*b*c^
5 - 9*B^2*a^3*c*(-(4*a*c - b^2)^9)^(1/2) + 1548*A*B*a^3*b^8*c^2 - 8064*A*B*a^4*b^6*c^3 + 22400*A*B*a^5*b^4*c^4
- 30720*A*B*a^6*b^2*c^5 - 51*A^2*a*b^2*c*(-(4*a*c - b^2)^9)^(1/2) - 6*A*B*a*b^3*(-(4*a*c - b^2)^9)^(1/2) - 15
2*A*B*a^2*b^10*c + 44*A*B*a^2*b*c*(-(4*a*c - b^2)^9)^(1/2))/(8*(a^5*b^12 + 4096*a^11*c^6 - 24*a^6*b^10*c + 240
*a^7*b^8*c^2 - 1280*a^8*b^6*c^3 + 3840*a^9*b^4*c^4 - 6144*a^10*b^2*c^5)))^(1/2)*2i