3.10.41 \(\int \frac {A+B x}{x^{9/2} (a+b x+c x^2)} \, dx\)

Optimal. Leaf size=381 \[ -\frac {2 \left (a B \left (b^2-a c\right )-A \left (b^3-2 a b c\right )\right )}{a^4 \sqrt {x}}-\frac {2 \left (-a A c-a b B+A b^2\right )}{3 a^3 x^{3/2}}+\frac {2 (A b-a B)}{5 a^2 x^{5/2}}-\frac {\sqrt {2} \sqrt {c} \left (\frac {a b B \left (b^2-3 a c\right )-A \left (2 a^2 c^2-4 a b^2 c+b^4\right )}{\sqrt {b^2-4 a c}}-A \left (b^3-2 a b c\right )+a B \left (b^2-a c\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{a^4 \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {c} \left (-\frac {a b B \left (b^2-3 a c\right )-A \left (2 a^2 c^2-4 a b^2 c+b^4\right )}{\sqrt {b^2-4 a c}}-A \left (b^3-2 a b c\right )+a B \left (b^2-a c\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{a^4 \sqrt {\sqrt {b^2-4 a c}+b}}-\frac {2 A}{7 a x^{7/2}} \]

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Rubi [A]  time = 1.67, antiderivative size = 381, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {828, 826, 1166, 205} \begin {gather*} -\frac {\sqrt {2} \sqrt {c} \left (\frac {a b B \left (b^2-3 a c\right )-A \left (2 a^2 c^2-4 a b^2 c+b^4\right )}{\sqrt {b^2-4 a c}}-A \left (b^3-2 a b c\right )+a B \left (b^2-a c\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{a^4 \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {c} \left (-\frac {a b B \left (b^2-3 a c\right )-A \left (2 a^2 c^2-4 a b^2 c+b^4\right )}{\sqrt {b^2-4 a c}}-A \left (b^3-2 a b c\right )+a B \left (b^2-a c\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{a^4 \sqrt {\sqrt {b^2-4 a c}+b}}-\frac {2 \left (-a A c-a b B+A b^2\right )}{3 a^3 x^{3/2}}-\frac {2 \left (a B \left (b^2-a c\right )-A \left (b^3-2 a b c\right )\right )}{a^4 \sqrt {x}}+\frac {2 (A b-a B)}{5 a^2 x^{5/2}}-\frac {2 A}{7 a x^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x^(9/2)*(a + b*x + c*x^2)),x]

[Out]

(-2*A)/(7*a*x^(7/2)) + (2*(A*b - a*B))/(5*a^2*x^(5/2)) - (2*(A*b^2 - a*b*B - a*A*c))/(3*a^3*x^(3/2)) - (2*(a*B
*(b^2 - a*c) - A*(b^3 - 2*a*b*c)))/(a^4*Sqrt[x]) - (Sqrt[2]*Sqrt[c]*(a*B*(b^2 - a*c) - A*(b^3 - 2*a*b*c) + (a*
b*B*(b^2 - 3*a*c) - A*(b^4 - 4*a*b^2*c + 2*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]]])/(a^4*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[2]*Sqrt[c]*(a*B*(b^2 - a*c) - A*(b^3 - 2*a*
b*c) - (a*b*B*(b^2 - 3*a*c) - A*(b^4 - 4*a*b^2*c + 2*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]]])/(a^4*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 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^{9/2} \left (a+b x+c x^2\right )} \, dx &=-\frac {2 A}{7 a x^{7/2}}+\frac {\int \frac {-A b+a B-A c x}{x^{7/2} \left (a+b x+c x^2\right )} \, dx}{a}\\ &=-\frac {2 A}{7 a x^{7/2}}+\frac {2 (A b-a B)}{5 a^2 x^{5/2}}+\frac {\int \frac {-a b B+A \left (b^2-a c\right )+(A b-a B) c x}{x^{5/2} \left (a+b x+c x^2\right )} \, dx}{a^2}\\ &=-\frac {2 A}{7 a x^{7/2}}+\frac {2 (A b-a B)}{5 a^2 x^{5/2}}-\frac {2 \left (A b^2-a b B-a A c\right )}{3 a^3 x^{3/2}}+\frac {\int \frac {a B \left (b^2-a c\right )-A \left (b^3-2 a b c\right )-c \left (A b^2-a b B-a A c\right ) x}{x^{3/2} \left (a+b x+c x^2\right )} \, dx}{a^3}\\ &=-\frac {2 A}{7 a x^{7/2}}+\frac {2 (A b-a B)}{5 a^2 x^{5/2}}-\frac {2 \left (A b^2-a b B-a A c\right )}{3 a^3 x^{3/2}}-\frac {2 \left (a B \left (b^2-a c\right )-A \left (b^3-2 a b c\right )\right )}{a^4 \sqrt {x}}+\frac {\int \frac {-a b B \left (b^2-2 a c\right )+A \left (b^4-3 a b^2 c+a^2 c^2\right )-c \left (a B \left (b^2-a c\right )-A \left (b^3-2 a b c\right )\right ) x}{\sqrt {x} \left (a+b x+c x^2\right )} \, dx}{a^4}\\ &=-\frac {2 A}{7 a x^{7/2}}+\frac {2 (A b-a B)}{5 a^2 x^{5/2}}-\frac {2 \left (A b^2-a b B-a A c\right )}{3 a^3 x^{3/2}}-\frac {2 \left (a B \left (b^2-a c\right )-A \left (b^3-2 a b c\right )\right )}{a^4 \sqrt {x}}+\frac {2 \operatorname {Subst}\left (\int \frac {-a b B \left (b^2-2 a c\right )+A \left (b^4-3 a b^2 c+a^2 c^2\right )-c \left (a B \left (b^2-a c\right )-A \left (b^3-2 a b c\right )\right ) x^2}{a+b x^2+c x^4} \, dx,x,\sqrt {x}\right )}{a^4}\\ &=-\frac {2 A}{7 a x^{7/2}}+\frac {2 (A b-a B)}{5 a^2 x^{5/2}}-\frac {2 \left (A b^2-a b B-a A c\right )}{3 a^3 x^{3/2}}-\frac {2 \left (a B \left (b^2-a c\right )-A \left (b^3-2 a b c\right )\right )}{a^4 \sqrt {x}}-\frac {\left (c \left (a B \left (b^2-a c\right )-A \left (b^3-2 a b c\right )-\frac {a b B \left (b^2-3 a c\right )-A \left (b^4-4 a b^2 c+2 a^2 c^2\right )}{\sqrt {b^2-4 a c}}\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 )}{a^4}-\frac {\left (c \left (a B \left (b^2-a c\right )-A \left (b^3-2 a b c\right )+\frac {a b B \left (b^2-3 a c\right )-A \left (b^4-4 a b^2 c+2 a^2 c^2\right )}{\sqrt {b^2-4 a c}}\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 )}{a^4}\\ &=-\frac {2 A}{7 a x^{7/2}}+\frac {2 (A b-a B)}{5 a^2 x^{5/2}}-\frac {2 \left (A b^2-a b B-a A c\right )}{3 a^3 x^{3/2}}-\frac {2 \left (a B \left (b^2-a c\right )-A \left (b^3-2 a b c\right )\right )}{a^4 \sqrt {x}}-\frac {\sqrt {2} \sqrt {c} \left (a B \left (b^2-a c\right )-A \left (b^3-2 a b c\right )+\frac {a b B \left (b^2-3 a c\right )-A \left (b^4-4 a b^2 c+2 a^2 c^2\right )}{\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 )}{a^4 \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {c} \left (a B \left (b^2-a c\right )-A \left (b^3-2 a b c\right )-\frac {a b B \left (b^2-3 a c\right )-A \left (b^4-4 a b^2 c+2 a^2 c^2\right )}{\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 )}{a^4 \sqrt {b+\sqrt {b^2-4 a c}}}\\ \end {align*}

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Mathematica [A]  time = 1.07, size = 430, normalized size = 1.13 \begin {gather*} \frac {-\frac {30 a^3 A}{x^{7/2}}+\frac {105 \sqrt {2} \sqrt {c} \left (\frac {\left (A \left (2 a^2 c^2-4 a b^2 c-2 a b c \sqrt {b^2-4 a c}+b^3 \sqrt {b^2-4 a c}+b^4\right )+a B \left (-b^2 \sqrt {b^2-4 a c}+a c \sqrt {b^2-4 a c}+3 a b c-b^3\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 (-2 a^2 c^2+4 a b^2 c-2 a b c \sqrt {b^2-4 a c}+b^3 \sqrt {b^2-4 a c}-b^4\right )+a B \left (-b^2 \sqrt {b^2-4 a c}+a c \sqrt {b^2-4 a c}-3 a b c+b^3\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 {b^2-4 a c}}+\frac {42 a^2 (A b-a B)}{x^{5/2}}+\frac {70 a \left (a A c+a b B-A b^2\right )}{x^{3/2}}+\frac {210 \left (A \left (b^3-2 a b c\right )+a B \left (a c-b^2\right )\right )}{\sqrt {x}}}{105 a^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(x^(9/2)*(a + b*x + c*x^2)),x]

[Out]

((-30*a^3*A)/x^(7/2) + (42*a^2*(A*b - a*B))/x^(5/2) + (70*a*(-(A*b^2) + a*b*B + a*A*c))/x^(3/2) + (210*(a*B*(-
b^2 + a*c) + A*(b^3 - 2*a*b*c)))/Sqrt[x] + (105*Sqrt[2]*Sqrt[c]*(((a*B*(-b^3 + 3*a*b*c - b^2*Sqrt[b^2 - 4*a*c]
+ a*c*Sqrt[b^2 - 4*a*c]) + A*(b^4 - 4*a*b^2*c + 2*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 2*a*b*c*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^3 - 3
*a*b*c - b^2*Sqrt[b^2 - 4*a*c] + a*c*Sqrt[b^2 - 4*a*c]) + A*(-b^4 + 4*a*b^2*c - 2*a^2*c^2 + b^3*Sqrt[b^2 - 4*a
*c] - 2*a*b*c*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]]))/Sqrt[b^2 - 4*a*c])/(105*a^4)

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IntegrateAlgebraic [A]  time = 1.59, size = 631, normalized size = 1.66 \begin {gather*} \frac {\left (2 \sqrt {2} a^2 A c^{5/2}+\sqrt {2} a^2 B c^{3/2} \sqrt {b^2-4 a c}+3 \sqrt {2} a^2 b B c^{3/2}-4 \sqrt {2} a A b^2 c^{3/2}-2 \sqrt {2} a A b c^{3/2} \sqrt {b^2-4 a c}+\sqrt {2} A b^3 \sqrt {c} \sqrt {b^2-4 a c}-\sqrt {2} a b^3 B \sqrt {c}-\sqrt {2} a b^2 B \sqrt {c} \sqrt {b^2-4 a c}+\sqrt {2} A b^4 \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{a^4 \sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (-2 \sqrt {2} a^2 A c^{5/2}+\sqrt {2} a^2 B c^{3/2} \sqrt {b^2-4 a c}-3 \sqrt {2} a^2 b B c^{3/2}+4 \sqrt {2} a A b^2 c^{3/2}-2 \sqrt {2} a A b c^{3/2} \sqrt {b^2-4 a c}+\sqrt {2} A b^3 \sqrt {c} \sqrt {b^2-4 a c}+\sqrt {2} a b^3 B \sqrt {c}-\sqrt {2} a b^2 B \sqrt {c} \sqrt {b^2-4 a c}-\sqrt {2} A b^4 \sqrt {c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {x}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{a^4 \sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b}}-\frac {2 \left (15 a^3 A+21 a^3 B x-21 a^2 A b x-35 a^2 A c x^2-35 a^2 b B x^2-105 a^2 B c x^3+35 a A b^2 x^2+210 a A b c x^3+105 a b^2 B x^3-105 A b^3 x^3\right )}{105 a^4 x^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(A + B*x)/(x^(9/2)*(a + b*x + c*x^2)),x]

[Out]

(-2*(15*a^3*A - 21*a^2*A*b*x + 21*a^3*B*x + 35*a*A*b^2*x^2 - 35*a^2*b*B*x^2 - 35*a^2*A*c*x^2 - 105*A*b^3*x^3 +
105*a*b^2*B*x^3 + 210*a*A*b*c*x^3 - 105*a^2*B*c*x^3))/(105*a^4*x^(7/2)) + ((Sqrt[2]*A*b^4*Sqrt[c] - Sqrt[2]*a
*b^3*B*Sqrt[c] - 4*Sqrt[2]*a*A*b^2*c^(3/2) + 3*Sqrt[2]*a^2*b*B*c^(3/2) + 2*Sqrt[2]*a^2*A*c^(5/2) + Sqrt[2]*A*b
^3*Sqrt[c]*Sqrt[b^2 - 4*a*c] - Sqrt[2]*a*b^2*B*Sqrt[c]*Sqrt[b^2 - 4*a*c] - 2*Sqrt[2]*a*A*b*c^(3/2)*Sqrt[b^2 -
4*a*c] + Sqrt[2]*a^2*B*c^(3/2)*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]
])/(a^4*Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + ((-(Sqrt[2]*A*b^4*Sqrt[c]) + Sqrt[2]*a*b^3*B*Sqrt[c]
+ 4*Sqrt[2]*a*A*b^2*c^(3/2) - 3*Sqrt[2]*a^2*b*B*c^(3/2) - 2*Sqrt[2]*a^2*A*c^(5/2) + Sqrt[2]*A*b^3*Sqrt[c]*Sqrt
[b^2 - 4*a*c] - Sqrt[2]*a*b^2*B*Sqrt[c]*Sqrt[b^2 - 4*a*c] - 2*Sqrt[2]*a*A*b*c^(3/2)*Sqrt[b^2 - 4*a*c] + Sqrt[2
]*a^2*B*c^(3/2)*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(a^4*Sqrt[b^
2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]])

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fricas [B]  time = 5.38, size = 10514, normalized size = 27.60

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(9/2)/(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

1/210*(105*sqrt(2)*a^4*x^4*sqrt(-(B^2*a^2*b^7 - 2*A*B*a*b^8 + A^2*b^9 - (4*A*B*a^5 - 9*A^2*a^4*b)*c^4 - (7*B^2
*a^5*b - 32*A*B*a^4*b^2 + 30*A^2*a^3*b^3)*c^3 + (14*B^2*a^4*b^3 - 40*A*B*a^3*b^4 + 27*A^2*a^2*b^5)*c^2 - (7*B^
2*a^3*b^5 - 16*A*B*a^2*b^6 + 9*A^2*a*b^7)*c + (a^9*b^2 - 4*a^10*c)*sqrt((B^4*a^4*b^12 - 4*A*B^3*a^3*b^13 + 6*A
^2*B^2*a^2*b^14 - 4*A^3*B*a*b^15 + A^4*b^16 + A^4*a^8*c^8 - 2*(A^2*B^2*a^9 - 8*A^3*B*a^8*b + 10*A^4*a^7*b^2)*c
^7 + (B^4*a^10 - 16*A*B^3*a^9*b + 96*A^2*B^2*a^8*b^2 - 200*A^3*B*a^7*b^3 + 130*A^4*a^6*b^4)*c^6 - 2*(6*B^4*a^9
*b^2 - 68*A*B^3*a^8*b^3 + 240*A^2*B^2*a^7*b^4 - 332*A^3*B*a^6*b^5 + 157*A^4*a^5*b^6)*c^5 + (46*B^4*a^8*b^4 - 3
44*A*B^3*a^7*b^5 + 888*A^2*B^2*a^6*b^6 - 956*A^3*B*a^5*b^7 + 367*A^4*a^4*b^8)*c^4 - 2*(31*B^4*a^7*b^6 - 182*A*
B^3*a^6*b^7 + 384*A^2*B^2*a^5*b^8 - 348*A^3*B*a^4*b^9 + 115*A^4*a^3*b^10)*c^3 + (37*B^4*a^6*b^8 - 184*A*B^3*a^
5*b^9 + 336*A^2*B^2*a^4*b^10 - 268*A^3*B*a^3*b^11 + 79*A^4*a^2*b^12)*c^2 - 2*(5*B^4*a^5*b^10 - 22*A*B^3*a^4*b^
11 + 36*A^2*B^2*a^3*b^12 - 26*A^3*B*a^2*b^13 + 7*A^4*a*b^14)*c)/(a^18*b^2 - 4*a^19*c)))/(a^9*b^2 - 4*a^10*c))*
log(sqrt(2)*(B^3*a^3*b^11 - 3*A*B^2*a^2*b^12 + 3*A^2*B*a*b^13 - A^3*b^14 + 4*A^3*a^7*c^7 - (4*A*B^2*a^8 - 40*A
^2*B*a^7*b + 53*A^3*a^6*b^2)*c^6 - (8*B^3*a^8*b - 101*A*B^2*a^7*b^2 + 270*A^2*B*a^6*b^3 - 197*A^3*a^5*b^4)*c^5
+ (54*B^3*a^7*b^3 - 313*A*B^2*a^6*b^4 + 545*A^2*B*a^5*b^5 - 294*A^3*a^4*b^6)*c^4 - (77*B^3*a^6*b^5 - 336*A*B^
2*a^5*b^6 + 468*A^2*B*a^4*b^7 - 210*A^3*a^3*b^8)*c^3 + (44*B^3*a^5*b^7 - 162*A*B^2*a^4*b^8 + 195*A^2*B*a^3*b^9
- 77*A^3*a^2*b^10)*c^2 - (11*B^3*a^4*b^9 - 36*A*B^2*a^3*b^10 + 39*A^2*B*a^2*b^11 - 14*A^3*a*b^12)*c - (B*a^10
*b^6 - A*a^9*b^7 - 4*(2*B*a^13 - 5*A*a^12*b)*c^3 + (18*B*a^12*b^2 - 25*A*a^11*b^3)*c^2 - (8*B*a^11*b^4 - 9*A*a
^10*b^5)*c)*sqrt((B^4*a^4*b^12 - 4*A*B^3*a^3*b^13 + 6*A^2*B^2*a^2*b^14 - 4*A^3*B*a*b^15 + A^4*b^16 + A^4*a^8*c
^8 - 2*(A^2*B^2*a^9 - 8*A^3*B*a^8*b + 10*A^4*a^7*b^2)*c^7 + (B^4*a^10 - 16*A*B^3*a^9*b + 96*A^2*B^2*a^8*b^2 -
200*A^3*B*a^7*b^3 + 130*A^4*a^6*b^4)*c^6 - 2*(6*B^4*a^9*b^2 - 68*A*B^3*a^8*b^3 + 240*A^2*B^2*a^7*b^4 - 332*A^3
*B*a^6*b^5 + 157*A^4*a^5*b^6)*c^5 + (46*B^4*a^8*b^4 - 344*A*B^3*a^7*b^5 + 888*A^2*B^2*a^6*b^6 - 956*A^3*B*a^5*
b^7 + 367*A^4*a^4*b^8)*c^4 - 2*(31*B^4*a^7*b^6 - 182*A*B^3*a^6*b^7 + 384*A^2*B^2*a^5*b^8 - 348*A^3*B*a^4*b^9 +
115*A^4*a^3*b^10)*c^3 + (37*B^4*a^6*b^8 - 184*A*B^3*a^5*b^9 + 336*A^2*B^2*a^4*b^10 - 268*A^3*B*a^3*b^11 + 79*
A^4*a^2*b^12)*c^2 - 2*(5*B^4*a^5*b^10 - 22*A*B^3*a^4*b^11 + 36*A^2*B^2*a^3*b^12 - 26*A^3*B*a^2*b^13 + 7*A^4*a*
b^14)*c)/(a^18*b^2 - 4*a^19*c)))*sqrt(-(B^2*a^2*b^7 - 2*A*B*a*b^8 + A^2*b^9 - (4*A*B*a^5 - 9*A^2*a^4*b)*c^4 -
(7*B^2*a^5*b - 32*A*B*a^4*b^2 + 30*A^2*a^3*b^3)*c^3 + (14*B^2*a^4*b^3 - 40*A*B*a^3*b^4 + 27*A^2*a^2*b^5)*c^2 -
(7*B^2*a^3*b^5 - 16*A*B*a^2*b^6 + 9*A^2*a*b^7)*c + (a^9*b^2 - 4*a^10*c)*sqrt((B^4*a^4*b^12 - 4*A*B^3*a^3*b^13
+ 6*A^2*B^2*a^2*b^14 - 4*A^3*B*a*b^15 + A^4*b^16 + A^4*a^8*c^8 - 2*(A^2*B^2*a^9 - 8*A^3*B*a^8*b + 10*A^4*a^7*
b^2)*c^7 + (B^4*a^10 - 16*A*B^3*a^9*b + 96*A^2*B^2*a^8*b^2 - 200*A^3*B*a^7*b^3 + 130*A^4*a^6*b^4)*c^6 - 2*(6*B
^4*a^9*b^2 - 68*A*B^3*a^8*b^3 + 240*A^2*B^2*a^7*b^4 - 332*A^3*B*a^6*b^5 + 157*A^4*a^5*b^6)*c^5 + (46*B^4*a^8*b
^4 - 344*A*B^3*a^7*b^5 + 888*A^2*B^2*a^6*b^6 - 956*A^3*B*a^5*b^7 + 367*A^4*a^4*b^8)*c^4 - 2*(31*B^4*a^7*b^6 -
182*A*B^3*a^6*b^7 + 384*A^2*B^2*a^5*b^8 - 348*A^3*B*a^4*b^9 + 115*A^4*a^3*b^10)*c^3 + (37*B^4*a^6*b^8 - 184*A*
B^3*a^5*b^9 + 336*A^2*B^2*a^4*b^10 - 268*A^3*B*a^3*b^11 + 79*A^4*a^2*b^12)*c^2 - 2*(5*B^4*a^5*b^10 - 22*A*B^3*
a^4*b^11 + 36*A^2*B^2*a^3*b^12 - 26*A^3*B*a^2*b^13 + 7*A^4*a*b^14)*c)/(a^18*b^2 - 4*a^19*c)))/(a^9*b^2 - 4*a^1
0*c)) + 4*(A^4*a^4*c^9 + (7*A^3*B*a^4*b - 10*A^4*a^3*b^2)*c^8 - (B^4*a^6 - 9*A*B^3*a^5*b + 12*A^2*B^2*a^4*b^2
+ 10*A^3*B*a^3*b^3 - 15*A^4*a^2*b^4)*c^7 + (6*B^4*a^5*b^2 - 26*A*B^3*a^4*b^3 + 30*A^2*B^2*a^3*b^4 - 3*A^3*B*a^
2*b^5 - 7*A^4*a*b^6)*c^6 - (5*B^4*a^4*b^4 - 17*A*B^3*a^3*b^5 + 18*A^2*B^2*a^2*b^6 - 5*A^3*B*a*b^7 - A^4*b^8)*c
^5 + (B^4*a^3*b^6 - 3*A*B^3*a^2*b^7 + 3*A^2*B^2*a*b^8 - A^3*B*b^9)*c^4)*sqrt(x)) - 105*sqrt(2)*a^4*x^4*sqrt(-(
B^2*a^2*b^7 - 2*A*B*a*b^8 + A^2*b^9 - (4*A*B*a^5 - 9*A^2*a^4*b)*c^4 - (7*B^2*a^5*b - 32*A*B*a^4*b^2 + 30*A^2*a
^3*b^3)*c^3 + (14*B^2*a^4*b^3 - 40*A*B*a^3*b^4 + 27*A^2*a^2*b^5)*c^2 - (7*B^2*a^3*b^5 - 16*A*B*a^2*b^6 + 9*A^2
*a*b^7)*c + (a^9*b^2 - 4*a^10*c)*sqrt((B^4*a^4*b^12 - 4*A*B^3*a^3*b^13 + 6*A^2*B^2*a^2*b^14 - 4*A^3*B*a*b^15 +
A^4*b^16 + A^4*a^8*c^8 - 2*(A^2*B^2*a^9 - 8*A^3*B*a^8*b + 10*A^4*a^7*b^2)*c^7 + (B^4*a^10 - 16*A*B^3*a^9*b +
96*A^2*B^2*a^8*b^2 - 200*A^3*B*a^7*b^3 + 130*A^4*a^6*b^4)*c^6 - 2*(6*B^4*a^9*b^2 - 68*A*B^3*a^8*b^3 + 240*A^2*
B^2*a^7*b^4 - 332*A^3*B*a^6*b^5 + 157*A^4*a^5*b^6)*c^5 + (46*B^4*a^8*b^4 - 344*A*B^3*a^7*b^5 + 888*A^2*B^2*a^6
*b^6 - 956*A^3*B*a^5*b^7 + 367*A^4*a^4*b^8)*c^4 - 2*(31*B^4*a^7*b^6 - 182*A*B^3*a^6*b^7 + 384*A^2*B^2*a^5*b^8
- 348*A^3*B*a^4*b^9 + 115*A^4*a^3*b^10)*c^3 + (37*B^4*a^6*b^8 - 184*A*B^3*a^5*b^9 + 336*A^2*B^2*a^4*b^10 - 268
*A^3*B*a^3*b^11 + 79*A^4*a^2*b^12)*c^2 - 2*(5*B^4*a^5*b^10 - 22*A*B^3*a^4*b^11 + 36*A^2*B^2*a^3*b^12 - 26*A^3*
B*a^2*b^13 + 7*A^4*a*b^14)*c)/(a^18*b^2 - 4*a^19*c)))/(a^9*b^2 - 4*a^10*c))*log(-sqrt(2)*(B^3*a^3*b^11 - 3*A*B
^2*a^2*b^12 + 3*A^2*B*a*b^13 - A^3*b^14 + 4*A^3*a^7*c^7 - (4*A*B^2*a^8 - 40*A^2*B*a^7*b + 53*A^3*a^6*b^2)*c^6
- (8*B^3*a^8*b - 101*A*B^2*a^7*b^2 + 270*A^2*B*a^6*b^3 - 197*A^3*a^5*b^4)*c^5 + (54*B^3*a^7*b^3 - 313*A*B^2*a^
6*b^4 + 545*A^2*B*a^5*b^5 - 294*A^3*a^4*b^6)*c^4 - (77*B^3*a^6*b^5 - 336*A*B^2*a^5*b^6 + 468*A^2*B*a^4*b^7 - 2
10*A^3*a^3*b^8)*c^3 + (44*B^3*a^5*b^7 - 162*A*B^2*a^4*b^8 + 195*A^2*B*a^3*b^9 - 77*A^3*a^2*b^10)*c^2 - (11*B^3
*a^4*b^9 - 36*A*B^2*a^3*b^10 + 39*A^2*B*a^2*b^11 - 14*A^3*a*b^12)*c - (B*a^10*b^6 - A*a^9*b^7 - 4*(2*B*a^13 -
5*A*a^12*b)*c^3 + (18*B*a^12*b^2 - 25*A*a^11*b^3)*c^2 - (8*B*a^11*b^4 - 9*A*a^10*b^5)*c)*sqrt((B^4*a^4*b^12 -
4*A*B^3*a^3*b^13 + 6*A^2*B^2*a^2*b^14 - 4*A^3*B*a*b^15 + A^4*b^16 + A^4*a^8*c^8 - 2*(A^2*B^2*a^9 - 8*A^3*B*a^8
*b + 10*A^4*a^7*b^2)*c^7 + (B^4*a^10 - 16*A*B^3*a^9*b + 96*A^2*B^2*a^8*b^2 - 200*A^3*B*a^7*b^3 + 130*A^4*a^6*b
^4)*c^6 - 2*(6*B^4*a^9*b^2 - 68*A*B^3*a^8*b^3 + 240*A^2*B^2*a^7*b^4 - 332*A^3*B*a^6*b^5 + 157*A^4*a^5*b^6)*c^5
+ (46*B^4*a^8*b^4 - 344*A*B^3*a^7*b^5 + 888*A^2*B^2*a^6*b^6 - 956*A^3*B*a^5*b^7 + 367*A^4*a^4*b^8)*c^4 - 2*(3
1*B^4*a^7*b^6 - 182*A*B^3*a^6*b^7 + 384*A^2*B^2*a^5*b^8 - 348*A^3*B*a^4*b^9 + 115*A^4*a^3*b^10)*c^3 + (37*B^4*
a^6*b^8 - 184*A*B^3*a^5*b^9 + 336*A^2*B^2*a^4*b^10 - 268*A^3*B*a^3*b^11 + 79*A^4*a^2*b^12)*c^2 - 2*(5*B^4*a^5*
b^10 - 22*A*B^3*a^4*b^11 + 36*A^2*B^2*a^3*b^12 - 26*A^3*B*a^2*b^13 + 7*A^4*a*b^14)*c)/(a^18*b^2 - 4*a^19*c)))*
sqrt(-(B^2*a^2*b^7 - 2*A*B*a*b^8 + A^2*b^9 - (4*A*B*a^5 - 9*A^2*a^4*b)*c^4 - (7*B^2*a^5*b - 32*A*B*a^4*b^2 + 3
0*A^2*a^3*b^3)*c^3 + (14*B^2*a^4*b^3 - 40*A*B*a^3*b^4 + 27*A^2*a^2*b^5)*c^2 - (7*B^2*a^3*b^5 - 16*A*B*a^2*b^6
+ 9*A^2*a*b^7)*c + (a^9*b^2 - 4*a^10*c)*sqrt((B^4*a^4*b^12 - 4*A*B^3*a^3*b^13 + 6*A^2*B^2*a^2*b^14 - 4*A^3*B*a
*b^15 + A^4*b^16 + A^4*a^8*c^8 - 2*(A^2*B^2*a^9 - 8*A^3*B*a^8*b + 10*A^4*a^7*b^2)*c^7 + (B^4*a^10 - 16*A*B^3*a
^9*b + 96*A^2*B^2*a^8*b^2 - 200*A^3*B*a^7*b^3 + 130*A^4*a^6*b^4)*c^6 - 2*(6*B^4*a^9*b^2 - 68*A*B^3*a^8*b^3 + 2
40*A^2*B^2*a^7*b^4 - 332*A^3*B*a^6*b^5 + 157*A^4*a^5*b^6)*c^5 + (46*B^4*a^8*b^4 - 344*A*B^3*a^7*b^5 + 888*A^2*
B^2*a^6*b^6 - 956*A^3*B*a^5*b^7 + 367*A^4*a^4*b^8)*c^4 - 2*(31*B^4*a^7*b^6 - 182*A*B^3*a^6*b^7 + 384*A^2*B^2*a
^5*b^8 - 348*A^3*B*a^4*b^9 + 115*A^4*a^3*b^10)*c^3 + (37*B^4*a^6*b^8 - 184*A*B^3*a^5*b^9 + 336*A^2*B^2*a^4*b^1
0 - 268*A^3*B*a^3*b^11 + 79*A^4*a^2*b^12)*c^2 - 2*(5*B^4*a^5*b^10 - 22*A*B^3*a^4*b^11 + 36*A^2*B^2*a^3*b^12 -
26*A^3*B*a^2*b^13 + 7*A^4*a*b^14)*c)/(a^18*b^2 - 4*a^19*c)))/(a^9*b^2 - 4*a^10*c)) + 4*(A^4*a^4*c^9 + (7*A^3*B
*a^4*b - 10*A^4*a^3*b^2)*c^8 - (B^4*a^6 - 9*A*B^3*a^5*b + 12*A^2*B^2*a^4*b^2 + 10*A^3*B*a^3*b^3 - 15*A^4*a^2*b
^4)*c^7 + (6*B^4*a^5*b^2 - 26*A*B^3*a^4*b^3 + 30*A^2*B^2*a^3*b^4 - 3*A^3*B*a^2*b^5 - 7*A^4*a*b^6)*c^6 - (5*B^4
*a^4*b^4 - 17*A*B^3*a^3*b^5 + 18*A^2*B^2*a^2*b^6 - 5*A^3*B*a*b^7 - A^4*b^8)*c^5 + (B^4*a^3*b^6 - 3*A*B^3*a^2*b
^7 + 3*A^2*B^2*a*b^8 - A^3*B*b^9)*c^4)*sqrt(x)) + 105*sqrt(2)*a^4*x^4*sqrt(-(B^2*a^2*b^7 - 2*A*B*a*b^8 + A^2*b
^9 - (4*A*B*a^5 - 9*A^2*a^4*b)*c^4 - (7*B^2*a^5*b - 32*A*B*a^4*b^2 + 30*A^2*a^3*b^3)*c^3 + (14*B^2*a^4*b^3 - 4
0*A*B*a^3*b^4 + 27*A^2*a^2*b^5)*c^2 - (7*B^2*a^3*b^5 - 16*A*B*a^2*b^6 + 9*A^2*a*b^7)*c - (a^9*b^2 - 4*a^10*c)*
sqrt((B^4*a^4*b^12 - 4*A*B^3*a^3*b^13 + 6*A^2*B^2*a^2*b^14 - 4*A^3*B*a*b^15 + A^4*b^16 + A^4*a^8*c^8 - 2*(A^2*
B^2*a^9 - 8*A^3*B*a^8*b + 10*A^4*a^7*b^2)*c^7 + (B^4*a^10 - 16*A*B^3*a^9*b + 96*A^2*B^2*a^8*b^2 - 200*A^3*B*a^
7*b^3 + 130*A^4*a^6*b^4)*c^6 - 2*(6*B^4*a^9*b^2 - 68*A*B^3*a^8*b^3 + 240*A^2*B^2*a^7*b^4 - 332*A^3*B*a^6*b^5 +
157*A^4*a^5*b^6)*c^5 + (46*B^4*a^8*b^4 - 344*A*B^3*a^7*b^5 + 888*A^2*B^2*a^6*b^6 - 956*A^3*B*a^5*b^7 + 367*A^
4*a^4*b^8)*c^4 - 2*(31*B^4*a^7*b^6 - 182*A*B^3*a^6*b^7 + 384*A^2*B^2*a^5*b^8 - 348*A^3*B*a^4*b^9 + 115*A^4*a^3
*b^10)*c^3 + (37*B^4*a^6*b^8 - 184*A*B^3*a^5*b^9 + 336*A^2*B^2*a^4*b^10 - 268*A^3*B*a^3*b^11 + 79*A^4*a^2*b^12
)*c^2 - 2*(5*B^4*a^5*b^10 - 22*A*B^3*a^4*b^11 + 36*A^2*B^2*a^3*b^12 - 26*A^3*B*a^2*b^13 + 7*A^4*a*b^14)*c)/(a^
18*b^2 - 4*a^19*c)))/(a^9*b^2 - 4*a^10*c))*log(sqrt(2)*(B^3*a^3*b^11 - 3*A*B^2*a^2*b^12 + 3*A^2*B*a*b^13 - A^3
*b^14 + 4*A^3*a^7*c^7 - (4*A*B^2*a^8 - 40*A^2*B*a^7*b + 53*A^3*a^6*b^2)*c^6 - (8*B^3*a^8*b - 101*A*B^2*a^7*b^2
+ 270*A^2*B*a^6*b^3 - 197*A^3*a^5*b^4)*c^5 + (54*B^3*a^7*b^3 - 313*A*B^2*a^6*b^4 + 545*A^2*B*a^5*b^5 - 294*A^
3*a^4*b^6)*c^4 - (77*B^3*a^6*b^5 - 336*A*B^2*a^5*b^6 + 468*A^2*B*a^4*b^7 - 210*A^3*a^3*b^8)*c^3 + (44*B^3*a^5*
b^7 - 162*A*B^2*a^4*b^8 + 195*A^2*B*a^3*b^9 - 77*A^3*a^2*b^10)*c^2 - (11*B^3*a^4*b^9 - 36*A*B^2*a^3*b^10 + 39*
A^2*B*a^2*b^11 - 14*A^3*a*b^12)*c + (B*a^10*b^6 - A*a^9*b^7 - 4*(2*B*a^13 - 5*A*a^12*b)*c^3 + (18*B*a^12*b^2 -
25*A*a^11*b^3)*c^2 - (8*B*a^11*b^4 - 9*A*a^10*b^5)*c)*sqrt((B^4*a^4*b^12 - 4*A*B^3*a^3*b^13 + 6*A^2*B^2*a^2*b
^14 - 4*A^3*B*a*b^15 + A^4*b^16 + A^4*a^8*c^8 - 2*(A^2*B^2*a^9 - 8*A^3*B*a^8*b + 10*A^4*a^7*b^2)*c^7 + (B^4*a^
10 - 16*A*B^3*a^9*b + 96*A^2*B^2*a^8*b^2 - 200*A^3*B*a^7*b^3 + 130*A^4*a^6*b^4)*c^6 - 2*(6*B^4*a^9*b^2 - 68*A*
B^3*a^8*b^3 + 240*A^2*B^2*a^7*b^4 - 332*A^3*B*a^6*b^5 + 157*A^4*a^5*b^6)*c^5 + (46*B^4*a^8*b^4 - 344*A*B^3*a^7
*b^5 + 888*A^2*B^2*a^6*b^6 - 956*A^3*B*a^5*b^7 + 367*A^4*a^4*b^8)*c^4 - 2*(31*B^4*a^7*b^6 - 182*A*B^3*a^6*b^7
+ 384*A^2*B^2*a^5*b^8 - 348*A^3*B*a^4*b^9 + 115*A^4*a^3*b^10)*c^3 + (37*B^4*a^6*b^8 - 184*A*B^3*a^5*b^9 + 336*
A^2*B^2*a^4*b^10 - 268*A^3*B*a^3*b^11 + 79*A^4*a^2*b^12)*c^2 - 2*(5*B^4*a^5*b^10 - 22*A*B^3*a^4*b^11 + 36*A^2*
B^2*a^3*b^12 - 26*A^3*B*a^2*b^13 + 7*A^4*a*b^14)*c)/(a^18*b^2 - 4*a^19*c)))*sqrt(-(B^2*a^2*b^7 - 2*A*B*a*b^8 +
A^2*b^9 - (4*A*B*a^5 - 9*A^2*a^4*b)*c^4 - (7*B^2*a^5*b - 32*A*B*a^4*b^2 + 30*A^2*a^3*b^3)*c^3 + (14*B^2*a^4*b
^3 - 40*A*B*a^3*b^4 + 27*A^2*a^2*b^5)*c^2 - (7*B^2*a^3*b^5 - 16*A*B*a^2*b^6 + 9*A^2*a*b^7)*c - (a^9*b^2 - 4*a^
10*c)*sqrt((B^4*a^4*b^12 - 4*A*B^3*a^3*b^13 + 6*A^2*B^2*a^2*b^14 - 4*A^3*B*a*b^15 + A^4*b^16 + A^4*a^8*c^8 - 2
*(A^2*B^2*a^9 - 8*A^3*B*a^8*b + 10*A^4*a^7*b^2)*c^7 + (B^4*a^10 - 16*A*B^3*a^9*b + 96*A^2*B^2*a^8*b^2 - 200*A^
3*B*a^7*b^3 + 130*A^4*a^6*b^4)*c^6 - 2*(6*B^4*a^9*b^2 - 68*A*B^3*a^8*b^3 + 240*A^2*B^2*a^7*b^4 - 332*A^3*B*a^6
*b^5 + 157*A^4*a^5*b^6)*c^5 + (46*B^4*a^8*b^4 - 344*A*B^3*a^7*b^5 + 888*A^2*B^2*a^6*b^6 - 956*A^3*B*a^5*b^7 +
367*A^4*a^4*b^8)*c^4 - 2*(31*B^4*a^7*b^6 - 182*A*B^3*a^6*b^7 + 384*A^2*B^2*a^5*b^8 - 348*A^3*B*a^4*b^9 + 115*A
^4*a^3*b^10)*c^3 + (37*B^4*a^6*b^8 - 184*A*B^3*a^5*b^9 + 336*A^2*B^2*a^4*b^10 - 268*A^3*B*a^3*b^11 + 79*A^4*a^
2*b^12)*c^2 - 2*(5*B^4*a^5*b^10 - 22*A*B^3*a^4*b^11 + 36*A^2*B^2*a^3*b^12 - 26*A^3*B*a^2*b^13 + 7*A^4*a*b^14)*
c)/(a^18*b^2 - 4*a^19*c)))/(a^9*b^2 - 4*a^10*c)) + 4*(A^4*a^4*c^9 + (7*A^3*B*a^4*b - 10*A^4*a^3*b^2)*c^8 - (B^
4*a^6 - 9*A*B^3*a^5*b + 12*A^2*B^2*a^4*b^2 + 10*A^3*B*a^3*b^3 - 15*A^4*a^2*b^4)*c^7 + (6*B^4*a^5*b^2 - 26*A*B^
3*a^4*b^3 + 30*A^2*B^2*a^3*b^4 - 3*A^3*B*a^2*b^5 - 7*A^4*a*b^6)*c^6 - (5*B^4*a^4*b^4 - 17*A*B^3*a^3*b^5 + 18*A
^2*B^2*a^2*b^6 - 5*A^3*B*a*b^7 - A^4*b^8)*c^5 + (B^4*a^3*b^6 - 3*A*B^3*a^2*b^7 + 3*A^2*B^2*a*b^8 - A^3*B*b^9)*
c^4)*sqrt(x)) - 105*sqrt(2)*a^4*x^4*sqrt(-(B^2*a^2*b^7 - 2*A*B*a*b^8 + A^2*b^9 - (4*A*B*a^5 - 9*A^2*a^4*b)*c^4
- (7*B^2*a^5*b - 32*A*B*a^4*b^2 + 30*A^2*a^3*b^3)*c^3 + (14*B^2*a^4*b^3 - 40*A*B*a^3*b^4 + 27*A^2*a^2*b^5)*c^
2 - (7*B^2*a^3*b^5 - 16*A*B*a^2*b^6 + 9*A^2*a*b^7)*c - (a^9*b^2 - 4*a^10*c)*sqrt((B^4*a^4*b^12 - 4*A*B^3*a^3*b
^13 + 6*A^2*B^2*a^2*b^14 - 4*A^3*B*a*b^15 + A^4*b^16 + A^4*a^8*c^8 - 2*(A^2*B^2*a^9 - 8*A^3*B*a^8*b + 10*A^4*a
^7*b^2)*c^7 + (B^4*a^10 - 16*A*B^3*a^9*b + 96*A^2*B^2*a^8*b^2 - 200*A^3*B*a^7*b^3 + 130*A^4*a^6*b^4)*c^6 - 2*(
6*B^4*a^9*b^2 - 68*A*B^3*a^8*b^3 + 240*A^2*B^2*a^7*b^4 - 332*A^3*B*a^6*b^5 + 157*A^4*a^5*b^6)*c^5 + (46*B^4*a^
8*b^4 - 344*A*B^3*a^7*b^5 + 888*A^2*B^2*a^6*b^6 - 956*A^3*B*a^5*b^7 + 367*A^4*a^4*b^8)*c^4 - 2*(31*B^4*a^7*b^6
- 182*A*B^3*a^6*b^7 + 384*A^2*B^2*a^5*b^8 - 348*A^3*B*a^4*b^9 + 115*A^4*a^3*b^10)*c^3 + (37*B^4*a^6*b^8 - 184
*A*B^3*a^5*b^9 + 336*A^2*B^2*a^4*b^10 - 268*A^3*B*a^3*b^11 + 79*A^4*a^2*b^12)*c^2 - 2*(5*B^4*a^5*b^10 - 22*A*B
^3*a^4*b^11 + 36*A^2*B^2*a^3*b^12 - 26*A^3*B*a^2*b^13 + 7*A^4*a*b^14)*c)/(a^18*b^2 - 4*a^19*c)))/(a^9*b^2 - 4*
a^10*c))*log(-sqrt(2)*(B^3*a^3*b^11 - 3*A*B^2*a^2*b^12 + 3*A^2*B*a*b^13 - A^3*b^14 + 4*A^3*a^7*c^7 - (4*A*B^2*
a^8 - 40*A^2*B*a^7*b + 53*A^3*a^6*b^2)*c^6 - (8*B^3*a^8*b - 101*A*B^2*a^7*b^2 + 270*A^2*B*a^6*b^3 - 197*A^3*a^
5*b^4)*c^5 + (54*B^3*a^7*b^3 - 313*A*B^2*a^6*b^4 + 545*A^2*B*a^5*b^5 - 294*A^3*a^4*b^6)*c^4 - (77*B^3*a^6*b^5
- 336*A*B^2*a^5*b^6 + 468*A^2*B*a^4*b^7 - 210*A^3*a^3*b^8)*c^3 + (44*B^3*a^5*b^7 - 162*A*B^2*a^4*b^8 + 195*A^2
*B*a^3*b^9 - 77*A^3*a^2*b^10)*c^2 - (11*B^3*a^4*b^9 - 36*A*B^2*a^3*b^10 + 39*A^2*B*a^2*b^11 - 14*A^3*a*b^12)*c
+ (B*a^10*b^6 - A*a^9*b^7 - 4*(2*B*a^13 - 5*A*a^12*b)*c^3 + (18*B*a^12*b^2 - 25*A*a^11*b^3)*c^2 - (8*B*a^11*b
^4 - 9*A*a^10*b^5)*c)*sqrt((B^4*a^4*b^12 - 4*A*B^3*a^3*b^13 + 6*A^2*B^2*a^2*b^14 - 4*A^3*B*a*b^15 + A^4*b^16 +
A^4*a^8*c^8 - 2*(A^2*B^2*a^9 - 8*A^3*B*a^8*b + 10*A^4*a^7*b^2)*c^7 + (B^4*a^10 - 16*A*B^3*a^9*b + 96*A^2*B^2*
a^8*b^2 - 200*A^3*B*a^7*b^3 + 130*A^4*a^6*b^4)*c^6 - 2*(6*B^4*a^9*b^2 - 68*A*B^3*a^8*b^3 + 240*A^2*B^2*a^7*b^4
- 332*A^3*B*a^6*b^5 + 157*A^4*a^5*b^6)*c^5 + (46*B^4*a^8*b^4 - 344*A*B^3*a^7*b^5 + 888*A^2*B^2*a^6*b^6 - 956*
A^3*B*a^5*b^7 + 367*A^4*a^4*b^8)*c^4 - 2*(31*B^4*a^7*b^6 - 182*A*B^3*a^6*b^7 + 384*A^2*B^2*a^5*b^8 - 348*A^3*B
*a^4*b^9 + 115*A^4*a^3*b^10)*c^3 + (37*B^4*a^6*b^8 - 184*A*B^3*a^5*b^9 + 336*A^2*B^2*a^4*b^10 - 268*A^3*B*a^3*
b^11 + 79*A^4*a^2*b^12)*c^2 - 2*(5*B^4*a^5*b^10 - 22*A*B^3*a^4*b^11 + 36*A^2*B^2*a^3*b^12 - 26*A^3*B*a^2*b^13
+ 7*A^4*a*b^14)*c)/(a^18*b^2 - 4*a^19*c)))*sqrt(-(B^2*a^2*b^7 - 2*A*B*a*b^8 + A^2*b^9 - (4*A*B*a^5 - 9*A^2*a^4
*b)*c^4 - (7*B^2*a^5*b - 32*A*B*a^4*b^2 + 30*A^2*a^3*b^3)*c^3 + (14*B^2*a^4*b^3 - 40*A*B*a^3*b^4 + 27*A^2*a^2*
b^5)*c^2 - (7*B^2*a^3*b^5 - 16*A*B*a^2*b^6 + 9*A^2*a*b^7)*c - (a^9*b^2 - 4*a^10*c)*sqrt((B^4*a^4*b^12 - 4*A*B^
3*a^3*b^13 + 6*A^2*B^2*a^2*b^14 - 4*A^3*B*a*b^15 + A^4*b^16 + A^4*a^8*c^8 - 2*(A^2*B^2*a^9 - 8*A^3*B*a^8*b + 1
0*A^4*a^7*b^2)*c^7 + (B^4*a^10 - 16*A*B^3*a^9*b + 96*A^2*B^2*a^8*b^2 - 200*A^3*B*a^7*b^3 + 130*A^4*a^6*b^4)*c^
6 - 2*(6*B^4*a^9*b^2 - 68*A*B^3*a^8*b^3 + 240*A^2*B^2*a^7*b^4 - 332*A^3*B*a^6*b^5 + 157*A^4*a^5*b^6)*c^5 + (46
*B^4*a^8*b^4 - 344*A*B^3*a^7*b^5 + 888*A^2*B^2*a^6*b^6 - 956*A^3*B*a^5*b^7 + 367*A^4*a^4*b^8)*c^4 - 2*(31*B^4*
a^7*b^6 - 182*A*B^3*a^6*b^7 + 384*A^2*B^2*a^5*b^8 - 348*A^3*B*a^4*b^9 + 115*A^4*a^3*b^10)*c^3 + (37*B^4*a^6*b^
8 - 184*A*B^3*a^5*b^9 + 336*A^2*B^2*a^4*b^10 - 268*A^3*B*a^3*b^11 + 79*A^4*a^2*b^12)*c^2 - 2*(5*B^4*a^5*b^10 -
22*A*B^3*a^4*b^11 + 36*A^2*B^2*a^3*b^12 - 26*A^3*B*a^2*b^13 + 7*A^4*a*b^14)*c)/(a^18*b^2 - 4*a^19*c)))/(a^9*b
^2 - 4*a^10*c)) + 4*(A^4*a^4*c^9 + (7*A^3*B*a^4*b - 10*A^4*a^3*b^2)*c^8 - (B^4*a^6 - 9*A*B^3*a^5*b + 12*A^2*B^
2*a^4*b^2 + 10*A^3*B*a^3*b^3 - 15*A^4*a^2*b^4)*c^7 + (6*B^4*a^5*b^2 - 26*A*B^3*a^4*b^3 + 30*A^2*B^2*a^3*b^4 -
3*A^3*B*a^2*b^5 - 7*A^4*a*b^6)*c^6 - (5*B^4*a^4*b^4 - 17*A*B^3*a^3*b^5 + 18*A^2*B^2*a^2*b^6 - 5*A^3*B*a*b^7 -
A^4*b^8)*c^5 + (B^4*a^3*b^6 - 3*A*B^3*a^2*b^7 + 3*A^2*B^2*a*b^8 - A^3*B*b^9)*c^4)*sqrt(x)) - 4*(15*A*a^3 + 105
*(B*a*b^2 - A*b^3 - (B*a^2 - 2*A*a*b)*c)*x^3 - 35*(B*a^2*b - A*a*b^2 + A*a^2*c)*x^2 + 21*(B*a^3 - A*a^2*b)*x)*
sqrt(x))/(a^4*x^4)

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giac [B]  time = 1.86, size = 4086, normalized size = 10.72

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(9/2)/(c*x^2+b*x+a),x, algorithm="giac")

[Out]

1/2*((sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^8 - 11*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^6*c - 2*sqr
t(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^7*c - 2*b^8*c + 41*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^2
+ 14*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^5*c^2 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^6*c^2 + 22*
a*b^6*c^2 + 2*b^7*c^2 - 56*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^2*c^3 - 26*sqrt(2)*sqrt(b*c + sqrt(b^
2 - 4*a*c)*c)*a^2*b^3*c^3 - 7*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c^3 - 82*a^2*b^4*c^3 - 18*a*b^5*c^
3 + 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*c^4 + 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b*c^4 +
13*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^4 + 112*a^3*b^2*c^4 + 50*a^2*b^3*c^4 - 4*sqrt(2)*sqrt(b*
c + sqrt(b^2 - 4*a*c)*c)*a^3*c^5 - 32*a^4*c^5 - 40*a^3*b*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 -
4*a*c)*c)*b^7 + 9*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^5*c + 2*sqrt(2)*sqrt(b^2 - 4*
a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^6*c - 25*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*
b^3*c^2 - 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*s
qrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5*c^2 + 20*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b*c^
3 + 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^3 + 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqr
t(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^3 - 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^4
+ 2*(b^2 - 4*a*c)*b^6*c - 14*(b^2 - 4*a*c)*a*b^4*c^2 - 2*(b^2 - 4*a*c)*b^5*c^2 + 26*(b^2 - 4*a*c)*a^2*b^2*c^3
+ 10*(b^2 - 4*a*c)*a*b^3*c^3 - 8*(b^2 - 4*a*c)*a^3*c^4 - 10*(b^2 - 4*a*c)*a^2*b*c^4)*A - (sqrt(2)*sqrt(b*c +
sqrt(b^2 - 4*a*c)*c)*a*b^7 - 10*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^5*c - 2*sqrt(2)*sqrt(b*c + sqrt(
b^2 - 4*a*c)*c)*a*b^6*c - 2*a*b^7*c + 32*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^3*c^2 + 12*sqrt(2)*sqrt
(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^2 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^5*c^2 + 20*a^2*b^5*c^2 +
2*a*b^6*c^2 - 32*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b*c^3 - 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*
c)*a^3*b^2*c^3 - 6*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c^3 - 64*a^3*b^3*c^3 - 16*a^2*b^4*c^3 + 8*s
qrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b*c^4 + 64*a^4*b*c^4 + 36*a^3*b^2*c^4 - 16*a^4*c^5 - sqrt(2)*sqrt(b
^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^6 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c
)*a^2*b^4*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^5*c - 18*sqrt(2)*sqrt(b^2 - 4*a*
c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^2*c^2 - 8*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^2 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)
*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*c^3 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b*c
^3 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^3 - 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqr
t(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*c^4 + 2*(b^2 - 4*a*c)*a*b^5*c - 12*(b^2 - 4*a*c)*a^2*b^3*c^2 - 2*(b^2 - 4*a*c
)*a*b^4*c^2 + 16*(b^2 - 4*a*c)*a^3*b*c^3 + 8*(b^2 - 4*a*c)*a^2*b^2*c^3 - 4*(b^2 - 4*a*c)*a^3*c^4)*B)*arctan(2*
sqrt(1/2)*sqrt(x)/sqrt((a^4*b + sqrt(a^8*b^2 - 4*a^9*c))/(a^4*c)))/((a^5*b^4 - 8*a^6*b^2*c - 2*a^5*b^3*c + 16*
a^7*c^2 + 8*a^6*b*c^2 + a^5*b^2*c^2 - 4*a^6*c^3)*abs(c)) + 1/2*((sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^8 -
11*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^6*c - 2*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^7*c + 2*b^8*
c + 41*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^2 + 14*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^5*
c^2 + sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^6*c^2 - 22*a*b^6*c^2 - 2*b^7*c^2 - 56*sqrt(2)*sqrt(b*c - sqrt(
b^2 - 4*a*c)*c)*a^3*b^2*c^3 - 26*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c^3 - 7*sqrt(2)*sqrt(b*c - sq
rt(b^2 - 4*a*c)*c)*a*b^4*c^3 + 82*a^2*b^4*c^3 + 18*a*b^5*c^3 + 16*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*
c^4 + 8*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b*c^4 + 13*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^2
*c^4 - 112*a^3*b^2*c^4 - 50*a^2*b^3*c^4 - 4*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*c^5 + 32*a^4*c^5 + 40*
a^3*b*c^5 + sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^7 - 9*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b
*c - sqrt(b^2 - 4*a*c)*c)*a*b^5*c - 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^6*c + 25*sqr
t(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c^2 + 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - s
qrt(b^2 - 4*a*c)*c)*a*b^4*c^2 + sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^5*c^2 - 20*sqrt(2)
*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b*c^3 - 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^
2 - 4*a*c)*c)*a^2*b^2*c^3 - 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^3*c^3 + 5*sqrt(2)*
sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b*c^4 - 2*(b^2 - 4*a*c)*b^6*c + 14*(b^2 - 4*a*c)*a*b^4*c
^2 + 2*(b^2 - 4*a*c)*b^5*c^2 - 26*(b^2 - 4*a*c)*a^2*b^2*c^3 - 10*(b^2 - 4*a*c)*a*b^3*c^3 + 8*(b^2 - 4*a*c)*a^3
*c^4 + 10*(b^2 - 4*a*c)*a^2*b*c^4)*A - (sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^7 - 10*sqrt(2)*sqrt(b*c -
sqrt(b^2 - 4*a*c)*c)*a^2*b^5*c - 2*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^6*c + 2*a*b^7*c + 32*sqrt(2)*sq
rt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^3*c^2 + 12*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^2 + sqrt(2)*s
qrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^5*c^2 - 20*a^2*b^5*c^2 - 2*a*b^6*c^2 - 32*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a
*c)*c)*a^4*b*c^3 - 16*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^2*c^3 - 6*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*
a*c)*c)*a^2*b^3*c^3 + 64*a^3*b^3*c^3 + 16*a^2*b^4*c^3 + 8*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b*c^4 -
64*a^4*b*c^4 - 36*a^3*b^2*c^4 + 16*a^4*c^5 + sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^6 -
8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c - 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c
- sqrt(b^2 - 4*a*c)*c)*a*b^5*c + 18*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b^2*c^2 + 8*
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 - s
qrt(b^2 - 4*a*c)*c)*a*b^4*c^2 - 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^4*c^3 - 4*sqrt(2
)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*b*c^3 - 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^
2 - 4*a*c)*c)*a^2*b^2*c^3 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*c^4 - 2*(b^2 - 4*a
*c)*a*b^5*c + 12*(b^2 - 4*a*c)*a^2*b^3*c^2 + 2*(b^2 - 4*a*c)*a*b^4*c^2 - 16*(b^2 - 4*a*c)*a^3*b*c^3 - 8*(b^2 -
4*a*c)*a^2*b^2*c^3 + 4*(b^2 - 4*a*c)*a^3*c^4)*B)*arctan(2*sqrt(1/2)*sqrt(x)/sqrt((a^4*b - sqrt(a^8*b^2 - 4*a^
9*c))/(a^4*c)))/((a^5*b^4 - 8*a^6*b^2*c - 2*a^5*b^3*c + 16*a^7*c^2 + 8*a^6*b*c^2 + a^5*b^2*c^2 - 4*a^6*c^3)*ab
s(c)) - 2/105*(105*B*a*b^2*x^3 - 105*A*b^3*x^3 - 105*B*a^2*c*x^3 + 210*A*a*b*c*x^3 - 35*B*a^2*b*x^2 + 35*A*a*b
^2*x^2 - 35*A*a^2*c*x^2 + 21*B*a^3*x - 21*A*a^2*b*x + 15*A*a^3)/(a^4*x^(7/2))

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maple [B]  time = 0.10, size = 1210, normalized size = 3.18

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/x^(9/2)/(c*x^2+b*x+a),x)

[Out]

-3/a^2*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+1/a^3*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^3+4/a^3*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^2-1/a^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^4-3/a^2*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+1/a^3*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^3+4/a^3*c^2/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*ar
ctanh(2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*c*x^(1/2))*A*b^2-1/a^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^4-2/a^2*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+1/a^3*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^2+2/3*A/a^2*c/x^(3/2)-2/7*A/a/x^(7/2)-2/5*B/a/x^(5/2)+1/a^2*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-1/a^2*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+2*B/a^2*c/x^(1/2)-2/3/a^3/x^(3/2)*A*b^2
+2/3/a^2/x^(3/2)*B*b+2/a^4/x^(1/2)*A*b^3-2/a^3/x^(1/2)*B*b^2+1/a^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^3-2/a^2*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-1/a^3*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^2+2/a^3*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-1/a^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^3-2/a^3*
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/
5/a^2/x^(5/2)*A*b-4/a^3/x^(1/2)*A*b*c

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {2 \, {\left (\frac {15 \, A a^{4}}{x^{\frac {7}{2}}} - 105 \, {\left ({\left (b^{4} - 3 \, a b^{2} c + a^{2} c^{2}\right )} A - {\left (a b^{3} - 2 \, a^{2} b c\right )} B\right )} \sqrt {x} - \frac {105 \, {\left ({\left (a b^{3} - 2 \, a^{2} b c\right )} A - {\left (a^{2} b^{2} - a^{3} c\right )} B\right )}}{\sqrt {x}} - \frac {35 \, {\left (B a^{3} b - {\left (a^{2} b^{2} - a^{3} c\right )} A\right )}}{x^{\frac {3}{2}}} + \frac {21 \, {\left (B a^{4} - A a^{3} b\right )}}{x^{\frac {5}{2}}}\right )}}{105 \, a^{5}} - \int \frac {{\left ({\left (b^{4} c - 3 \, a b^{2} c^{2} + a^{2} c^{3}\right )} A - {\left (a b^{3} c - 2 \, a^{2} b c^{2}\right )} B\right )} x^{\frac {3}{2}} + {\left ({\left (b^{5} - 4 \, a b^{3} c + 3 \, a^{2} b c^{2}\right )} A - {\left (a b^{4} - 3 \, a^{2} b^{2} c + a^{3} c^{2}\right )} B\right )} \sqrt {x}}{a^{5} c x^{2} + a^{5} b x + a^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(9/2)/(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

-2/105*(15*A*a^4/x^(7/2) - 105*((b^4 - 3*a*b^2*c + a^2*c^2)*A - (a*b^3 - 2*a^2*b*c)*B)*sqrt(x) - 105*((a*b^3 -
2*a^2*b*c)*A - (a^2*b^2 - a^3*c)*B)/sqrt(x) - 35*(B*a^3*b - (a^2*b^2 - a^3*c)*A)/x^(3/2) + 21*(B*a^4 - A*a^3*
b)/x^(5/2))/a^5 - integrate((((b^4*c - 3*a*b^2*c^2 + a^2*c^3)*A - (a*b^3*c - 2*a^2*b*c^2)*B)*x^(3/2) + ((b^5 -
4*a*b^3*c + 3*a^2*b*c^2)*A - (a*b^4 - 3*a^2*b^2*c + a^3*c^2)*B)*sqrt(x))/(a^5*c*x^2 + a^5*b*x + a^6), x)

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mupad [B]  time = 5.15, size = 17910, normalized size = 47.01

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/(x^(9/2)*(a + b*x + c*x^2)),x)

[Out]

((2*x^3*(A*b^3 - B*a*b^2 + B*a^2*c - 2*A*a*b*c))/a^4 - (2*A)/(7*a) + (2*x^2*(A*a*c - A*b^2 + B*a*b))/(3*a^3) +
(2*x*(A*b - B*a))/(5*a^2))/x^(7/2) + atan((((-(A^2*b^11 + B^2*a^2*b^9 + A^2*b^8*(-(4*a*c - b^2)^3)^(1/2) - 2*
A*B*a*b^10 + 63*A^2*a^2*b^7*c^2 - 138*A^2*a^3*b^5*c^3 + 129*A^2*a^4*b^3*c^4 + A^2*a^4*c^4*(-(4*a*c - b^2)^3)^(
1/2) + B^2*a^2*b^6*(-(4*a*c - b^2)^3)^(1/2) + 42*B^2*a^4*b^5*c^2 - 63*B^2*a^5*b^3*c^3 - B^2*a^5*c^3*(-(4*a*c -
b^2)^3)^(1/2) + 16*A*B*a^6*c^5 - 13*A^2*a*b^9*c - 36*A^2*a^5*b*c^5 - 11*B^2*a^3*b^7*c + 28*B^2*a^6*b*c^4 + 15
*A^2*a^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2) - 10*A^2*a^3*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) + 6*B^2*a^4*b^2*c^2*(-
(4*a*c - b^2)^3)^(1/2) - 104*A*B*a^3*b^6*c^2 + 192*A*B*a^4*b^4*c^3 - 132*A*B*a^5*b^2*c^4 - 7*A^2*a*b^6*c*(-(4*
a*c - b^2)^3)^(1/2) - 5*B^2*a^3*b^4*c*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^7*(-(4*a*c - b^2)^3)^(1/2) + 24*A*B
*a^2*b^8*c + 12*A*B*a^2*b^5*c*(-(4*a*c - b^2)^3)^(1/2) + 8*A*B*a^4*b*c^3*(-(4*a*c - b^2)^3)^(1/2) - 20*A*B*a^3
*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^9*b^4 + 16*a^11*c^2 - 8*a^10*b^2*c)))^(1/2)*(32*A*a^19*c^5 + x^(1/2)*
(32*a^21*b*c^3 - 8*a^20*b^3*c^2)*(-(A^2*b^11 + B^2*a^2*b^9 + A^2*b^8*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^10 +
63*A^2*a^2*b^7*c^2 - 138*A^2*a^3*b^5*c^3 + 129*A^2*a^4*b^3*c^4 + A^2*a^4*c^4*(-(4*a*c - b^2)^3)^(1/2) + B^2*a
^2*b^6*(-(4*a*c - b^2)^3)^(1/2) + 42*B^2*a^4*b^5*c^2 - 63*B^2*a^5*b^3*c^3 - B^2*a^5*c^3*(-(4*a*c - b^2)^3)^(1/
2) + 16*A*B*a^6*c^5 - 13*A^2*a*b^9*c - 36*A^2*a^5*b*c^5 - 11*B^2*a^3*b^7*c + 28*B^2*a^6*b*c^4 + 15*A^2*a^2*b^4
*c^2*(-(4*a*c - b^2)^3)^(1/2) - 10*A^2*a^3*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) + 6*B^2*a^4*b^2*c^2*(-(4*a*c - b^2
)^3)^(1/2) - 104*A*B*a^3*b^6*c^2 + 192*A*B*a^4*b^4*c^3 - 132*A*B*a^5*b^2*c^4 - 7*A^2*a*b^6*c*(-(4*a*c - b^2)^3
)^(1/2) - 5*B^2*a^3*b^4*c*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^7*(-(4*a*c - b^2)^3)^(1/2) + 24*A*B*a^2*b^8*c +
12*A*B*a^2*b^5*c*(-(4*a*c - b^2)^3)^(1/2) + 8*A*B*a^4*b*c^3*(-(4*a*c - b^2)^3)^(1/2) - 20*A*B*a^3*b^3*c^2*(-(
4*a*c - b^2)^3)^(1/2))/(2*(a^9*b^4 + 16*a^11*c^2 - 8*a^10*b^2*c)))^(1/2) + 64*B*a^19*b*c^4 - 8*A*a^16*b^6*c^2
+ 56*A*a^17*b^4*c^3 - 104*A*a^18*b^2*c^4 + 8*B*a^17*b^5*c^2 - 48*B*a^18*b^3*c^3) - x^(1/2)*(16*A^2*a^16*c^7 -
16*B^2*a^17*c^6 + 8*A^2*a^12*b^8*c^3 - 64*A^2*a^13*b^6*c^4 + 160*A^2*a^14*b^4*c^5 - 128*A^2*a^15*b^2*c^6 + 8*B
^2*a^14*b^6*c^3 - 48*B^2*a^15*b^4*c^4 + 72*B^2*a^16*b^2*c^5 - 16*A*B*a^13*b^7*c^3 + 112*A*B*a^14*b^5*c^4 - 224
*A*B*a^15*b^3*c^5 + 112*A*B*a^16*b*c^6))*(-(A^2*b^11 + B^2*a^2*b^9 + A^2*b^8*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*
a*b^10 + 63*A^2*a^2*b^7*c^2 - 138*A^2*a^3*b^5*c^3 + 129*A^2*a^4*b^3*c^4 + A^2*a^4*c^4*(-(4*a*c - b^2)^3)^(1/2)
+ B^2*a^2*b^6*(-(4*a*c - b^2)^3)^(1/2) + 42*B^2*a^4*b^5*c^2 - 63*B^2*a^5*b^3*c^3 - B^2*a^5*c^3*(-(4*a*c - b^2
)^3)^(1/2) + 16*A*B*a^6*c^5 - 13*A^2*a*b^9*c - 36*A^2*a^5*b*c^5 - 11*B^2*a^3*b^7*c + 28*B^2*a^6*b*c^4 + 15*A^2
*a^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2) - 10*A^2*a^3*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) + 6*B^2*a^4*b^2*c^2*(-(4*a
*c - b^2)^3)^(1/2) - 104*A*B*a^3*b^6*c^2 + 192*A*B*a^4*b^4*c^3 - 132*A*B*a^5*b^2*c^4 - 7*A^2*a*b^6*c*(-(4*a*c
- b^2)^3)^(1/2) - 5*B^2*a^3*b^4*c*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^7*(-(4*a*c - b^2)^3)^(1/2) + 24*A*B*a^2
*b^8*c + 12*A*B*a^2*b^5*c*(-(4*a*c - b^2)^3)^(1/2) + 8*A*B*a^4*b*c^3*(-(4*a*c - b^2)^3)^(1/2) - 20*A*B*a^3*b^3
*c^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^9*b^4 + 16*a^11*c^2 - 8*a^10*b^2*c)))^(1/2)*1i - ((-(A^2*b^11 + B^2*a^2*b
^9 + A^2*b^8*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^10 + 63*A^2*a^2*b^7*c^2 - 138*A^2*a^3*b^5*c^3 + 129*A^2*a^4*
b^3*c^4 + A^2*a^4*c^4*(-(4*a*c - b^2)^3)^(1/2) + B^2*a^2*b^6*(-(4*a*c - b^2)^3)^(1/2) + 42*B^2*a^4*b^5*c^2 - 6
3*B^2*a^5*b^3*c^3 - B^2*a^5*c^3*(-(4*a*c - b^2)^3)^(1/2) + 16*A*B*a^6*c^5 - 13*A^2*a*b^9*c - 36*A^2*a^5*b*c^5
- 11*B^2*a^3*b^7*c + 28*B^2*a^6*b*c^4 + 15*A^2*a^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2) - 10*A^2*a^3*b^2*c^3*(-(4*
a*c - b^2)^3)^(1/2) + 6*B^2*a^4*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 104*A*B*a^3*b^6*c^2 + 192*A*B*a^4*b^4*c^3 -
132*A*B*a^5*b^2*c^4 - 7*A^2*a*b^6*c*(-(4*a*c - b^2)^3)^(1/2) - 5*B^2*a^3*b^4*c*(-(4*a*c - b^2)^3)^(1/2) - 2*A
*B*a*b^7*(-(4*a*c - b^2)^3)^(1/2) + 24*A*B*a^2*b^8*c + 12*A*B*a^2*b^5*c*(-(4*a*c - b^2)^3)^(1/2) + 8*A*B*a^4*b
*c^3*(-(4*a*c - b^2)^3)^(1/2) - 20*A*B*a^3*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^9*b^4 + 16*a^11*c^2 - 8*a^1
0*b^2*c)))^(1/2)*(32*A*a^19*c^5 - x^(1/2)*(32*a^21*b*c^3 - 8*a^20*b^3*c^2)*(-(A^2*b^11 + B^2*a^2*b^9 + A^2*b^8
*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^10 + 63*A^2*a^2*b^7*c^2 - 138*A^2*a^3*b^5*c^3 + 129*A^2*a^4*b^3*c^4 + A^
2*a^4*c^4*(-(4*a*c - b^2)^3)^(1/2) + B^2*a^2*b^6*(-(4*a*c - b^2)^3)^(1/2) + 42*B^2*a^4*b^5*c^2 - 63*B^2*a^5*b^
3*c^3 - B^2*a^5*c^3*(-(4*a*c - b^2)^3)^(1/2) + 16*A*B*a^6*c^5 - 13*A^2*a*b^9*c - 36*A^2*a^5*b*c^5 - 11*B^2*a^3
*b^7*c + 28*B^2*a^6*b*c^4 + 15*A^2*a^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2) - 10*A^2*a^3*b^2*c^3*(-(4*a*c - b^2)^3
)^(1/2) + 6*B^2*a^4*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 104*A*B*a^3*b^6*c^2 + 192*A*B*a^4*b^4*c^3 - 132*A*B*a^5
*b^2*c^4 - 7*A^2*a*b^6*c*(-(4*a*c - b^2)^3)^(1/2) - 5*B^2*a^3*b^4*c*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^7*(-(
4*a*c - b^2)^3)^(1/2) + 24*A*B*a^2*b^8*c + 12*A*B*a^2*b^5*c*(-(4*a*c - b^2)^3)^(1/2) + 8*A*B*a^4*b*c^3*(-(4*a*
c - b^2)^3)^(1/2) - 20*A*B*a^3*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^9*b^4 + 16*a^11*c^2 - 8*a^10*b^2*c)))^(
1/2) + 64*B*a^19*b*c^4 - 8*A*a^16*b^6*c^2 + 56*A*a^17*b^4*c^3 - 104*A*a^18*b^2*c^4 + 8*B*a^17*b^5*c^2 - 48*B*a
^18*b^3*c^3) + x^(1/2)*(16*A^2*a^16*c^7 - 16*B^2*a^17*c^6 + 8*A^2*a^12*b^8*c^3 - 64*A^2*a^13*b^6*c^4 + 160*A^2
*a^14*b^4*c^5 - 128*A^2*a^15*b^2*c^6 + 8*B^2*a^14*b^6*c^3 - 48*B^2*a^15*b^4*c^4 + 72*B^2*a^16*b^2*c^5 - 16*A*B
*a^13*b^7*c^3 + 112*A*B*a^14*b^5*c^4 - 224*A*B*a^15*b^3*c^5 + 112*A*B*a^16*b*c^6))*(-(A^2*b^11 + B^2*a^2*b^9 +
A^2*b^8*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^10 + 63*A^2*a^2*b^7*c^2 - 138*A^2*a^3*b^5*c^3 + 129*A^2*a^4*b^3*
c^4 + A^2*a^4*c^4*(-(4*a*c - b^2)^3)^(1/2) + B^2*a^2*b^6*(-(4*a*c - b^2)^3)^(1/2) + 42*B^2*a^4*b^5*c^2 - 63*B^
2*a^5*b^3*c^3 - B^2*a^5*c^3*(-(4*a*c - b^2)^3)^(1/2) + 16*A*B*a^6*c^5 - 13*A^2*a*b^9*c - 36*A^2*a^5*b*c^5 - 11
*B^2*a^3*b^7*c + 28*B^2*a^6*b*c^4 + 15*A^2*a^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2) - 10*A^2*a^3*b^2*c^3*(-(4*a*c
- b^2)^3)^(1/2) + 6*B^2*a^4*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 104*A*B*a^3*b^6*c^2 + 192*A*B*a^4*b^4*c^3 - 132
*A*B*a^5*b^2*c^4 - 7*A^2*a*b^6*c*(-(4*a*c - b^2)^3)^(1/2) - 5*B^2*a^3*b^4*c*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a
*b^7*(-(4*a*c - b^2)^3)^(1/2) + 24*A*B*a^2*b^8*c + 12*A*B*a^2*b^5*c*(-(4*a*c - b^2)^3)^(1/2) + 8*A*B*a^4*b*c^3
*(-(4*a*c - b^2)^3)^(1/2) - 20*A*B*a^3*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^9*b^4 + 16*a^11*c^2 - 8*a^10*b^
2*c)))^(1/2)*1i)/(((-(A^2*b^11 + B^2*a^2*b^9 + A^2*b^8*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^10 + 63*A^2*a^2*b^
7*c^2 - 138*A^2*a^3*b^5*c^3 + 129*A^2*a^4*b^3*c^4 + A^2*a^4*c^4*(-(4*a*c - b^2)^3)^(1/2) + B^2*a^2*b^6*(-(4*a*
c - b^2)^3)^(1/2) + 42*B^2*a^4*b^5*c^2 - 63*B^2*a^5*b^3*c^3 - B^2*a^5*c^3*(-(4*a*c - b^2)^3)^(1/2) + 16*A*B*a^
6*c^5 - 13*A^2*a*b^9*c - 36*A^2*a^5*b*c^5 - 11*B^2*a^3*b^7*c + 28*B^2*a^6*b*c^4 + 15*A^2*a^2*b^4*c^2*(-(4*a*c
- b^2)^3)^(1/2) - 10*A^2*a^3*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) + 6*B^2*a^4*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 1
04*A*B*a^3*b^6*c^2 + 192*A*B*a^4*b^4*c^3 - 132*A*B*a^5*b^2*c^4 - 7*A^2*a*b^6*c*(-(4*a*c - b^2)^3)^(1/2) - 5*B^
2*a^3*b^4*c*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^7*(-(4*a*c - b^2)^3)^(1/2) + 24*A*B*a^2*b^8*c + 12*A*B*a^2*b^
5*c*(-(4*a*c - b^2)^3)^(1/2) + 8*A*B*a^4*b*c^3*(-(4*a*c - b^2)^3)^(1/2) - 20*A*B*a^3*b^3*c^2*(-(4*a*c - b^2)^3
)^(1/2))/(2*(a^9*b^4 + 16*a^11*c^2 - 8*a^10*b^2*c)))^(1/2)*(32*A*a^19*c^5 + x^(1/2)*(32*a^21*b*c^3 - 8*a^20*b^
3*c^2)*(-(A^2*b^11 + B^2*a^2*b^9 + A^2*b^8*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^10 + 63*A^2*a^2*b^7*c^2 - 138*
A^2*a^3*b^5*c^3 + 129*A^2*a^4*b^3*c^4 + A^2*a^4*c^4*(-(4*a*c - b^2)^3)^(1/2) + B^2*a^2*b^6*(-(4*a*c - b^2)^3)^
(1/2) + 42*B^2*a^4*b^5*c^2 - 63*B^2*a^5*b^3*c^3 - B^2*a^5*c^3*(-(4*a*c - b^2)^3)^(1/2) + 16*A*B*a^6*c^5 - 13*A
^2*a*b^9*c - 36*A^2*a^5*b*c^5 - 11*B^2*a^3*b^7*c + 28*B^2*a^6*b*c^4 + 15*A^2*a^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1
/2) - 10*A^2*a^3*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) + 6*B^2*a^4*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 104*A*B*a^3*b
^6*c^2 + 192*A*B*a^4*b^4*c^3 - 132*A*B*a^5*b^2*c^4 - 7*A^2*a*b^6*c*(-(4*a*c - b^2)^3)^(1/2) - 5*B^2*a^3*b^4*c*
(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^7*(-(4*a*c - b^2)^3)^(1/2) + 24*A*B*a^2*b^8*c + 12*A*B*a^2*b^5*c*(-(4*a*c
- b^2)^3)^(1/2) + 8*A*B*a^4*b*c^3*(-(4*a*c - b^2)^3)^(1/2) - 20*A*B*a^3*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2))/(2*
(a^9*b^4 + 16*a^11*c^2 - 8*a^10*b^2*c)))^(1/2) + 64*B*a^19*b*c^4 - 8*A*a^16*b^6*c^2 + 56*A*a^17*b^4*c^3 - 104*
A*a^18*b^2*c^4 + 8*B*a^17*b^5*c^2 - 48*B*a^18*b^3*c^3) - x^(1/2)*(16*A^2*a^16*c^7 - 16*B^2*a^17*c^6 + 8*A^2*a^
12*b^8*c^3 - 64*A^2*a^13*b^6*c^4 + 160*A^2*a^14*b^4*c^5 - 128*A^2*a^15*b^2*c^6 + 8*B^2*a^14*b^6*c^3 - 48*B^2*a
^15*b^4*c^4 + 72*B^2*a^16*b^2*c^5 - 16*A*B*a^13*b^7*c^3 + 112*A*B*a^14*b^5*c^4 - 224*A*B*a^15*b^3*c^5 + 112*A*
B*a^16*b*c^6))*(-(A^2*b^11 + B^2*a^2*b^9 + A^2*b^8*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^10 + 63*A^2*a^2*b^7*c^
2 - 138*A^2*a^3*b^5*c^3 + 129*A^2*a^4*b^3*c^4 + A^2*a^4*c^4*(-(4*a*c - b^2)^3)^(1/2) + B^2*a^2*b^6*(-(4*a*c -
b^2)^3)^(1/2) + 42*B^2*a^4*b^5*c^2 - 63*B^2*a^5*b^3*c^3 - B^2*a^5*c^3*(-(4*a*c - b^2)^3)^(1/2) + 16*A*B*a^6*c^
5 - 13*A^2*a*b^9*c - 36*A^2*a^5*b*c^5 - 11*B^2*a^3*b^7*c + 28*B^2*a^6*b*c^4 + 15*A^2*a^2*b^4*c^2*(-(4*a*c - b^
2)^3)^(1/2) - 10*A^2*a^3*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) + 6*B^2*a^4*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 104*A
*B*a^3*b^6*c^2 + 192*A*B*a^4*b^4*c^3 - 132*A*B*a^5*b^2*c^4 - 7*A^2*a*b^6*c*(-(4*a*c - b^2)^3)^(1/2) - 5*B^2*a^
3*b^4*c*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^7*(-(4*a*c - b^2)^3)^(1/2) + 24*A*B*a^2*b^8*c + 12*A*B*a^2*b^5*c*
(-(4*a*c - b^2)^3)^(1/2) + 8*A*B*a^4*b*c^3*(-(4*a*c - b^2)^3)^(1/2) - 20*A*B*a^3*b^3*c^2*(-(4*a*c - b^2)^3)^(1
/2))/(2*(a^9*b^4 + 16*a^11*c^2 - 8*a^10*b^2*c)))^(1/2) + ((-(A^2*b^11 + B^2*a^2*b^9 + A^2*b^8*(-(4*a*c - b^2)^
3)^(1/2) - 2*A*B*a*b^10 + 63*A^2*a^2*b^7*c^2 - 138*A^2*a^3*b^5*c^3 + 129*A^2*a^4*b^3*c^4 + A^2*a^4*c^4*(-(4*a*
c - b^2)^3)^(1/2) + B^2*a^2*b^6*(-(4*a*c - b^2)^3)^(1/2) + 42*B^2*a^4*b^5*c^2 - 63*B^2*a^5*b^3*c^3 - B^2*a^5*c
^3*(-(4*a*c - b^2)^3)^(1/2) + 16*A*B*a^6*c^5 - 13*A^2*a*b^9*c - 36*A^2*a^5*b*c^5 - 11*B^2*a^3*b^7*c + 28*B^2*a
^6*b*c^4 + 15*A^2*a^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2) - 10*A^2*a^3*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) + 6*B^2*a
^4*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 104*A*B*a^3*b^6*c^2 + 192*A*B*a^4*b^4*c^3 - 132*A*B*a^5*b^2*c^4 - 7*A^2*
a*b^6*c*(-(4*a*c - b^2)^3)^(1/2) - 5*B^2*a^3*b^4*c*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^7*(-(4*a*c - b^2)^3)^(
1/2) + 24*A*B*a^2*b^8*c + 12*A*B*a^2*b^5*c*(-(4*a*c - b^2)^3)^(1/2) + 8*A*B*a^4*b*c^3*(-(4*a*c - b^2)^3)^(1/2)
- 20*A*B*a^3*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^9*b^4 + 16*a^11*c^2 - 8*a^10*b^2*c)))^(1/2)*(32*A*a^19*c
^5 - x^(1/2)*(32*a^21*b*c^3 - 8*a^20*b^3*c^2)*(-(A^2*b^11 + B^2*a^2*b^9 + A^2*b^8*(-(4*a*c - b^2)^3)^(1/2) - 2
*A*B*a*b^10 + 63*A^2*a^2*b^7*c^2 - 138*A^2*a^3*b^5*c^3 + 129*A^2*a^4*b^3*c^4 + A^2*a^4*c^4*(-(4*a*c - b^2)^3)^
(1/2) + B^2*a^2*b^6*(-(4*a*c - b^2)^3)^(1/2) + 42*B^2*a^4*b^5*c^2 - 63*B^2*a^5*b^3*c^3 - B^2*a^5*c^3*(-(4*a*c
- b^2)^3)^(1/2) + 16*A*B*a^6*c^5 - 13*A^2*a*b^9*c - 36*A^2*a^5*b*c^5 - 11*B^2*a^3*b^7*c + 28*B^2*a^6*b*c^4 + 1
5*A^2*a^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2) - 10*A^2*a^3*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) + 6*B^2*a^4*b^2*c^2*(
-(4*a*c - b^2)^3)^(1/2) - 104*A*B*a^3*b^6*c^2 + 192*A*B*a^4*b^4*c^3 - 132*A*B*a^5*b^2*c^4 - 7*A^2*a*b^6*c*(-(4
*a*c - b^2)^3)^(1/2) - 5*B^2*a^3*b^4*c*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^7*(-(4*a*c - b^2)^3)^(1/2) + 24*A*
B*a^2*b^8*c + 12*A*B*a^2*b^5*c*(-(4*a*c - b^2)^3)^(1/2) + 8*A*B*a^4*b*c^3*(-(4*a*c - b^2)^3)^(1/2) - 20*A*B*a^
3*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^9*b^4 + 16*a^11*c^2 - 8*a^10*b^2*c)))^(1/2) + 64*B*a^19*b*c^4 - 8*A*
a^16*b^6*c^2 + 56*A*a^17*b^4*c^3 - 104*A*a^18*b^2*c^4 + 8*B*a^17*b^5*c^2 - 48*B*a^18*b^3*c^3) + x^(1/2)*(16*A^
2*a^16*c^7 - 16*B^2*a^17*c^6 + 8*A^2*a^12*b^8*c^3 - 64*A^2*a^13*b^6*c^4 + 160*A^2*a^14*b^4*c^5 - 128*A^2*a^15*
b^2*c^6 + 8*B^2*a^14*b^6*c^3 - 48*B^2*a^15*b^4*c^4 + 72*B^2*a^16*b^2*c^5 - 16*A*B*a^13*b^7*c^3 + 112*A*B*a^14*
b^5*c^4 - 224*A*B*a^15*b^3*c^5 + 112*A*B*a^16*b*c^6))*(-(A^2*b^11 + B^2*a^2*b^9 + A^2*b^8*(-(4*a*c - b^2)^3)^(
1/2) - 2*A*B*a*b^10 + 63*A^2*a^2*b^7*c^2 - 138*A^2*a^3*b^5*c^3 + 129*A^2*a^4*b^3*c^4 + A^2*a^4*c^4*(-(4*a*c -
b^2)^3)^(1/2) + B^2*a^2*b^6*(-(4*a*c - b^2)^3)^(1/2) + 42*B^2*a^4*b^5*c^2 - 63*B^2*a^5*b^3*c^3 - B^2*a^5*c^3*(
-(4*a*c - b^2)^3)^(1/2) + 16*A*B*a^6*c^5 - 13*A^2*a*b^9*c - 36*A^2*a^5*b*c^5 - 11*B^2*a^3*b^7*c + 28*B^2*a^6*b
*c^4 + 15*A^2*a^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2) - 10*A^2*a^3*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) + 6*B^2*a^4*b
^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 104*A*B*a^3*b^6*c^2 + 192*A*B*a^4*b^4*c^3 - 132*A*B*a^5*b^2*c^4 - 7*A^2*a*b^
6*c*(-(4*a*c - b^2)^3)^(1/2) - 5*B^2*a^3*b^4*c*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^7*(-(4*a*c - b^2)^3)^(1/2)
+ 24*A*B*a^2*b^8*c + 12*A*B*a^2*b^5*c*(-(4*a*c - b^2)^3)^(1/2) + 8*A*B*a^4*b*c^3*(-(4*a*c - b^2)^3)^(1/2) - 2
0*A*B*a^3*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^9*b^4 + 16*a^11*c^2 - 8*a^10*b^2*c)))^(1/2) + 16*B^3*a^15*c^
7 + 16*A^3*a^12*b^3*c^7 - 16*B^3*a^14*b^2*c^6 + 16*A^2*B*a^14*c^8 - 32*A^3*a^13*b*c^8 - 48*A*B^2*a^14*b*c^7 +
32*A*B^2*a^13*b^3*c^6 - 16*A^2*B*a^12*b^4*c^6 + 16*A^2*B*a^13*b^2*c^7))*(-(A^2*b^11 + B^2*a^2*b^9 + A^2*b^8*(-
(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^10 + 63*A^2*a^2*b^7*c^2 - 138*A^2*a^3*b^5*c^3 + 129*A^2*a^4*b^3*c^4 + A^2*a
^4*c^4*(-(4*a*c - b^2)^3)^(1/2) + B^2*a^2*b^6*(-(4*a*c - b^2)^3)^(1/2) + 42*B^2*a^4*b^5*c^2 - 63*B^2*a^5*b^3*c
^3 - B^2*a^5*c^3*(-(4*a*c - b^2)^3)^(1/2) + 16*A*B*a^6*c^5 - 13*A^2*a*b^9*c - 36*A^2*a^5*b*c^5 - 11*B^2*a^3*b^
7*c + 28*B^2*a^6*b*c^4 + 15*A^2*a^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2) - 10*A^2*a^3*b^2*c^3*(-(4*a*c - b^2)^3)^(
1/2) + 6*B^2*a^4*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 104*A*B*a^3*b^6*c^2 + 192*A*B*a^4*b^4*c^3 - 132*A*B*a^5*b^
2*c^4 - 7*A^2*a*b^6*c*(-(4*a*c - b^2)^3)^(1/2) - 5*B^2*a^3*b^4*c*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^7*(-(4*a
*c - b^2)^3)^(1/2) + 24*A*B*a^2*b^8*c + 12*A*B*a^2*b^5*c*(-(4*a*c - b^2)^3)^(1/2) + 8*A*B*a^4*b*c^3*(-(4*a*c -
b^2)^3)^(1/2) - 20*A*B*a^3*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^9*b^4 + 16*a^11*c^2 - 8*a^10*b^2*c)))^(1/2
)*2i + atan((((-(A^2*b^11 + B^2*a^2*b^9 - A^2*b^8*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^10 + 63*A^2*a^2*b^7*c^2
- 138*A^2*a^3*b^5*c^3 + 129*A^2*a^4*b^3*c^4 - A^2*a^4*c^4*(-(4*a*c - b^2)^3)^(1/2) - B^2*a^2*b^6*(-(4*a*c - b
^2)^3)^(1/2) + 42*B^2*a^4*b^5*c^2 - 63*B^2*a^5*b^3*c^3 + B^2*a^5*c^3*(-(4*a*c - b^2)^3)^(1/2) + 16*A*B*a^6*c^5
- 13*A^2*a*b^9*c - 36*A^2*a^5*b*c^5 - 11*B^2*a^3*b^7*c + 28*B^2*a^6*b*c^4 - 15*A^2*a^2*b^4*c^2*(-(4*a*c - b^2
)^3)^(1/2) + 10*A^2*a^3*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) - 6*B^2*a^4*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 104*A*
B*a^3*b^6*c^2 + 192*A*B*a^4*b^4*c^3 - 132*A*B*a^5*b^2*c^4 + 7*A^2*a*b^6*c*(-(4*a*c - b^2)^3)^(1/2) + 5*B^2*a^3
*b^4*c*(-(4*a*c - b^2)^3)^(1/2) + 2*A*B*a*b^7*(-(4*a*c - b^2)^3)^(1/2) + 24*A*B*a^2*b^8*c - 12*A*B*a^2*b^5*c*(
-(4*a*c - b^2)^3)^(1/2) - 8*A*B*a^4*b*c^3*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a^3*b^3*c^2*(-(4*a*c - b^2)^3)^(1/
2))/(2*(a^9*b^4 + 16*a^11*c^2 - 8*a^10*b^2*c)))^(1/2)*(32*A*a^19*c^5 + x^(1/2)*(32*a^21*b*c^3 - 8*a^20*b^3*c^2
)*(-(A^2*b^11 + B^2*a^2*b^9 - A^2*b^8*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^10 + 63*A^2*a^2*b^7*c^2 - 138*A^2*a
^3*b^5*c^3 + 129*A^2*a^4*b^3*c^4 - A^2*a^4*c^4*(-(4*a*c - b^2)^3)^(1/2) - B^2*a^2*b^6*(-(4*a*c - b^2)^3)^(1/2)
+ 42*B^2*a^4*b^5*c^2 - 63*B^2*a^5*b^3*c^3 + B^2*a^5*c^3*(-(4*a*c - b^2)^3)^(1/2) + 16*A*B*a^6*c^5 - 13*A^2*a*
b^9*c - 36*A^2*a^5*b*c^5 - 11*B^2*a^3*b^7*c + 28*B^2*a^6*b*c^4 - 15*A^2*a^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2) +
10*A^2*a^3*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) - 6*B^2*a^4*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 104*A*B*a^3*b^6*c^
2 + 192*A*B*a^4*b^4*c^3 - 132*A*B*a^5*b^2*c^4 + 7*A^2*a*b^6*c*(-(4*a*c - b^2)^3)^(1/2) + 5*B^2*a^3*b^4*c*(-(4*
a*c - b^2)^3)^(1/2) + 2*A*B*a*b^7*(-(4*a*c - b^2)^3)^(1/2) + 24*A*B*a^2*b^8*c - 12*A*B*a^2*b^5*c*(-(4*a*c - b^
2)^3)^(1/2) - 8*A*B*a^4*b*c^3*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a^3*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^9*
b^4 + 16*a^11*c^2 - 8*a^10*b^2*c)))^(1/2) + 64*B*a^19*b*c^4 - 8*A*a^16*b^6*c^2 + 56*A*a^17*b^4*c^3 - 104*A*a^1
8*b^2*c^4 + 8*B*a^17*b^5*c^2 - 48*B*a^18*b^3*c^3) - x^(1/2)*(16*A^2*a^16*c^7 - 16*B^2*a^17*c^6 + 8*A^2*a^12*b^
8*c^3 - 64*A^2*a^13*b^6*c^4 + 160*A^2*a^14*b^4*c^5 - 128*A^2*a^15*b^2*c^6 + 8*B^2*a^14*b^6*c^3 - 48*B^2*a^15*b
^4*c^4 + 72*B^2*a^16*b^2*c^5 - 16*A*B*a^13*b^7*c^3 + 112*A*B*a^14*b^5*c^4 - 224*A*B*a^15*b^3*c^5 + 112*A*B*a^1
6*b*c^6))*(-(A^2*b^11 + B^2*a^2*b^9 - A^2*b^8*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^10 + 63*A^2*a^2*b^7*c^2 - 1
38*A^2*a^3*b^5*c^3 + 129*A^2*a^4*b^3*c^4 - A^2*a^4*c^4*(-(4*a*c - b^2)^3)^(1/2) - B^2*a^2*b^6*(-(4*a*c - b^2)^
3)^(1/2) + 42*B^2*a^4*b^5*c^2 - 63*B^2*a^5*b^3*c^3 + B^2*a^5*c^3*(-(4*a*c - b^2)^3)^(1/2) + 16*A*B*a^6*c^5 - 1
3*A^2*a*b^9*c - 36*A^2*a^5*b*c^5 - 11*B^2*a^3*b^7*c + 28*B^2*a^6*b*c^4 - 15*A^2*a^2*b^4*c^2*(-(4*a*c - b^2)^3)
^(1/2) + 10*A^2*a^3*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) - 6*B^2*a^4*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 104*A*B*a^
3*b^6*c^2 + 192*A*B*a^4*b^4*c^3 - 132*A*B*a^5*b^2*c^4 + 7*A^2*a*b^6*c*(-(4*a*c - b^2)^3)^(1/2) + 5*B^2*a^3*b^4
*c*(-(4*a*c - b^2)^3)^(1/2) + 2*A*B*a*b^7*(-(4*a*c - b^2)^3)^(1/2) + 24*A*B*a^2*b^8*c - 12*A*B*a^2*b^5*c*(-(4*
a*c - b^2)^3)^(1/2) - 8*A*B*a^4*b*c^3*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a^3*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2))/
(2*(a^9*b^4 + 16*a^11*c^2 - 8*a^10*b^2*c)))^(1/2)*1i - ((-(A^2*b^11 + B^2*a^2*b^9 - A^2*b^8*(-(4*a*c - b^2)^3)
^(1/2) - 2*A*B*a*b^10 + 63*A^2*a^2*b^7*c^2 - 138*A^2*a^3*b^5*c^3 + 129*A^2*a^4*b^3*c^4 - A^2*a^4*c^4*(-(4*a*c
- b^2)^3)^(1/2) - B^2*a^2*b^6*(-(4*a*c - b^2)^3)^(1/2) + 42*B^2*a^4*b^5*c^2 - 63*B^2*a^5*b^3*c^3 + B^2*a^5*c^3
*(-(4*a*c - b^2)^3)^(1/2) + 16*A*B*a^6*c^5 - 13*A^2*a*b^9*c - 36*A^2*a^5*b*c^5 - 11*B^2*a^3*b^7*c + 28*B^2*a^6
*b*c^4 - 15*A^2*a^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2) + 10*A^2*a^3*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) - 6*B^2*a^4
*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 104*A*B*a^3*b^6*c^2 + 192*A*B*a^4*b^4*c^3 - 132*A*B*a^5*b^2*c^4 + 7*A^2*a*
b^6*c*(-(4*a*c - b^2)^3)^(1/2) + 5*B^2*a^3*b^4*c*(-(4*a*c - b^2)^3)^(1/2) + 2*A*B*a*b^7*(-(4*a*c - b^2)^3)^(1/
2) + 24*A*B*a^2*b^8*c - 12*A*B*a^2*b^5*c*(-(4*a*c - b^2)^3)^(1/2) - 8*A*B*a^4*b*c^3*(-(4*a*c - b^2)^3)^(1/2) +
20*A*B*a^3*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^9*b^4 + 16*a^11*c^2 - 8*a^10*b^2*c)))^(1/2)*(32*A*a^19*c^5
- x^(1/2)*(32*a^21*b*c^3 - 8*a^20*b^3*c^2)*(-(A^2*b^11 + B^2*a^2*b^9 - A^2*b^8*(-(4*a*c - b^2)^3)^(1/2) - 2*A
*B*a*b^10 + 63*A^2*a^2*b^7*c^2 - 138*A^2*a^3*b^5*c^3 + 129*A^2*a^4*b^3*c^4 - A^2*a^4*c^4*(-(4*a*c - b^2)^3)^(1
/2) - B^2*a^2*b^6*(-(4*a*c - b^2)^3)^(1/2) + 42*B^2*a^4*b^5*c^2 - 63*B^2*a^5*b^3*c^3 + B^2*a^5*c^3*(-(4*a*c -
b^2)^3)^(1/2) + 16*A*B*a^6*c^5 - 13*A^2*a*b^9*c - 36*A^2*a^5*b*c^5 - 11*B^2*a^3*b^7*c + 28*B^2*a^6*b*c^4 - 15*
A^2*a^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2) + 10*A^2*a^3*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) - 6*B^2*a^4*b^2*c^2*(-(
4*a*c - b^2)^3)^(1/2) - 104*A*B*a^3*b^6*c^2 + 192*A*B*a^4*b^4*c^3 - 132*A*B*a^5*b^2*c^4 + 7*A^2*a*b^6*c*(-(4*a
*c - b^2)^3)^(1/2) + 5*B^2*a^3*b^4*c*(-(4*a*c - b^2)^3)^(1/2) + 2*A*B*a*b^7*(-(4*a*c - b^2)^3)^(1/2) + 24*A*B*
a^2*b^8*c - 12*A*B*a^2*b^5*c*(-(4*a*c - b^2)^3)^(1/2) - 8*A*B*a^4*b*c^3*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a^3*
b^3*c^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^9*b^4 + 16*a^11*c^2 - 8*a^10*b^2*c)))^(1/2) + 64*B*a^19*b*c^4 - 8*A*a^
16*b^6*c^2 + 56*A*a^17*b^4*c^3 - 104*A*a^18*b^2*c^4 + 8*B*a^17*b^5*c^2 - 48*B*a^18*b^3*c^3) + x^(1/2)*(16*A^2*
a^16*c^7 - 16*B^2*a^17*c^6 + 8*A^2*a^12*b^8*c^3 - 64*A^2*a^13*b^6*c^4 + 160*A^2*a^14*b^4*c^5 - 128*A^2*a^15*b^
2*c^6 + 8*B^2*a^14*b^6*c^3 - 48*B^2*a^15*b^4*c^4 + 72*B^2*a^16*b^2*c^5 - 16*A*B*a^13*b^7*c^3 + 112*A*B*a^14*b^
5*c^4 - 224*A*B*a^15*b^3*c^5 + 112*A*B*a^16*b*c^6))*(-(A^2*b^11 + B^2*a^2*b^9 - A^2*b^8*(-(4*a*c - b^2)^3)^(1/
2) - 2*A*B*a*b^10 + 63*A^2*a^2*b^7*c^2 - 138*A^2*a^3*b^5*c^3 + 129*A^2*a^4*b^3*c^4 - A^2*a^4*c^4*(-(4*a*c - b^
2)^3)^(1/2) - B^2*a^2*b^6*(-(4*a*c - b^2)^3)^(1/2) + 42*B^2*a^4*b^5*c^2 - 63*B^2*a^5*b^3*c^3 + B^2*a^5*c^3*(-(
4*a*c - b^2)^3)^(1/2) + 16*A*B*a^6*c^5 - 13*A^2*a*b^9*c - 36*A^2*a^5*b*c^5 - 11*B^2*a^3*b^7*c + 28*B^2*a^6*b*c
^4 - 15*A^2*a^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2) + 10*A^2*a^3*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) - 6*B^2*a^4*b^2
*c^2*(-(4*a*c - b^2)^3)^(1/2) - 104*A*B*a^3*b^6*c^2 + 192*A*B*a^4*b^4*c^3 - 132*A*B*a^5*b^2*c^4 + 7*A^2*a*b^6*
c*(-(4*a*c - b^2)^3)^(1/2) + 5*B^2*a^3*b^4*c*(-(4*a*c - b^2)^3)^(1/2) + 2*A*B*a*b^7*(-(4*a*c - b^2)^3)^(1/2) +
24*A*B*a^2*b^8*c - 12*A*B*a^2*b^5*c*(-(4*a*c - b^2)^3)^(1/2) - 8*A*B*a^4*b*c^3*(-(4*a*c - b^2)^3)^(1/2) + 20*
A*B*a^3*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^9*b^4 + 16*a^11*c^2 - 8*a^10*b^2*c)))^(1/2)*1i)/(((-(A^2*b^11
+ B^2*a^2*b^9 - A^2*b^8*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^10 + 63*A^2*a^2*b^7*c^2 - 138*A^2*a^3*b^5*c^3 + 1
29*A^2*a^4*b^3*c^4 - A^2*a^4*c^4*(-(4*a*c - b^2)^3)^(1/2) - B^2*a^2*b^6*(-(4*a*c - b^2)^3)^(1/2) + 42*B^2*a^4*
b^5*c^2 - 63*B^2*a^5*b^3*c^3 + B^2*a^5*c^3*(-(4*a*c - b^2)^3)^(1/2) + 16*A*B*a^6*c^5 - 13*A^2*a*b^9*c - 36*A^2
*a^5*b*c^5 - 11*B^2*a^3*b^7*c + 28*B^2*a^6*b*c^4 - 15*A^2*a^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2) + 10*A^2*a^3*b^
2*c^3*(-(4*a*c - b^2)^3)^(1/2) - 6*B^2*a^4*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 104*A*B*a^3*b^6*c^2 + 192*A*B*a^
4*b^4*c^3 - 132*A*B*a^5*b^2*c^4 + 7*A^2*a*b^6*c*(-(4*a*c - b^2)^3)^(1/2) + 5*B^2*a^3*b^4*c*(-(4*a*c - b^2)^3)^
(1/2) + 2*A*B*a*b^7*(-(4*a*c - b^2)^3)^(1/2) + 24*A*B*a^2*b^8*c - 12*A*B*a^2*b^5*c*(-(4*a*c - b^2)^3)^(1/2) -
8*A*B*a^4*b*c^3*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a^3*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^9*b^4 + 16*a^11*
c^2 - 8*a^10*b^2*c)))^(1/2)*(32*A*a^19*c^5 + x^(1/2)*(32*a^21*b*c^3 - 8*a^20*b^3*c^2)*(-(A^2*b^11 + B^2*a^2*b^
9 - A^2*b^8*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^10 + 63*A^2*a^2*b^7*c^2 - 138*A^2*a^3*b^5*c^3 + 129*A^2*a^4*b
^3*c^4 - A^2*a^4*c^4*(-(4*a*c - b^2)^3)^(1/2) - B^2*a^2*b^6*(-(4*a*c - b^2)^3)^(1/2) + 42*B^2*a^4*b^5*c^2 - 63
*B^2*a^5*b^3*c^3 + B^2*a^5*c^3*(-(4*a*c - b^2)^3)^(1/2) + 16*A*B*a^6*c^5 - 13*A^2*a*b^9*c - 36*A^2*a^5*b*c^5 -
11*B^2*a^3*b^7*c + 28*B^2*a^6*b*c^4 - 15*A^2*a^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2) + 10*A^2*a^3*b^2*c^3*(-(4*a
*c - b^2)^3)^(1/2) - 6*B^2*a^4*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 104*A*B*a^3*b^6*c^2 + 192*A*B*a^4*b^4*c^3 -
132*A*B*a^5*b^2*c^4 + 7*A^2*a*b^6*c*(-(4*a*c - b^2)^3)^(1/2) + 5*B^2*a^3*b^4*c*(-(4*a*c - b^2)^3)^(1/2) + 2*A*
B*a*b^7*(-(4*a*c - b^2)^3)^(1/2) + 24*A*B*a^2*b^8*c - 12*A*B*a^2*b^5*c*(-(4*a*c - b^2)^3)^(1/2) - 8*A*B*a^4*b*
c^3*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a^3*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^9*b^4 + 16*a^11*c^2 - 8*a^10
*b^2*c)))^(1/2) + 64*B*a^19*b*c^4 - 8*A*a^16*b^6*c^2 + 56*A*a^17*b^4*c^3 - 104*A*a^18*b^2*c^4 + 8*B*a^17*b^5*c
^2 - 48*B*a^18*b^3*c^3) - x^(1/2)*(16*A^2*a^16*c^7 - 16*B^2*a^17*c^6 + 8*A^2*a^12*b^8*c^3 - 64*A^2*a^13*b^6*c^
4 + 160*A^2*a^14*b^4*c^5 - 128*A^2*a^15*b^2*c^6 + 8*B^2*a^14*b^6*c^3 - 48*B^2*a^15*b^4*c^4 + 72*B^2*a^16*b^2*c
^5 - 16*A*B*a^13*b^7*c^3 + 112*A*B*a^14*b^5*c^4 - 224*A*B*a^15*b^3*c^5 + 112*A*B*a^16*b*c^6))*(-(A^2*b^11 + B^
2*a^2*b^9 - A^2*b^8*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^10 + 63*A^2*a^2*b^7*c^2 - 138*A^2*a^3*b^5*c^3 + 129*A
^2*a^4*b^3*c^4 - A^2*a^4*c^4*(-(4*a*c - b^2)^3)^(1/2) - B^2*a^2*b^6*(-(4*a*c - b^2)^3)^(1/2) + 42*B^2*a^4*b^5*
c^2 - 63*B^2*a^5*b^3*c^3 + B^2*a^5*c^3*(-(4*a*c - b^2)^3)^(1/2) + 16*A*B*a^6*c^5 - 13*A^2*a*b^9*c - 36*A^2*a^5
*b*c^5 - 11*B^2*a^3*b^7*c + 28*B^2*a^6*b*c^4 - 15*A^2*a^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2) + 10*A^2*a^3*b^2*c^
3*(-(4*a*c - b^2)^3)^(1/2) - 6*B^2*a^4*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 104*A*B*a^3*b^6*c^2 + 192*A*B*a^4*b^
4*c^3 - 132*A*B*a^5*b^2*c^4 + 7*A^2*a*b^6*c*(-(4*a*c - b^2)^3)^(1/2) + 5*B^2*a^3*b^4*c*(-(4*a*c - b^2)^3)^(1/2
) + 2*A*B*a*b^7*(-(4*a*c - b^2)^3)^(1/2) + 24*A*B*a^2*b^8*c - 12*A*B*a^2*b^5*c*(-(4*a*c - b^2)^3)^(1/2) - 8*A*
B*a^4*b*c^3*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a^3*b^3*c^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^9*b^4 + 16*a^11*c^2
- 8*a^10*b^2*c)))^(1/2) + ((-(A^2*b^11 + B^2*a^2*b^9 - A^2*b^8*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^10 + 63*A^
2*a^2*b^7*c^2 - 138*A^2*a^3*b^5*c^3 + 129*A^2*a^4*b^3*c^4 - A^2*a^4*c^4*(-(4*a*c - b^2)^3)^(1/2) - B^2*a^2*b^6
*(-(4*a*c - b^2)^3)^(1/2) + 42*B^2*a^4*b^5*c^2 - 63*B^2*a^5*b^3*c^3 + B^2*a^5*c^3*(-(4*a*c - b^2)^3)^(1/2) + 1
6*A*B*a^6*c^5 - 13*A^2*a*b^9*c - 36*A^2*a^5*b*c^5 - 11*B^2*a^3*b^7*c + 28*B^2*a^6*b*c^4 - 15*A^2*a^2*b^4*c^2*(
-(4*a*c - b^2)^3)^(1/2) + 10*A^2*a^3*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) - 6*B^2*a^4*b^2*c^2*(-(4*a*c - b^2)^3)^(
1/2) - 104*A*B*a^3*b^6*c^2 + 192*A*B*a^4*b^4*c^3 - 132*A*B*a^5*b^2*c^4 + 7*A^2*a*b^6*c*(-(4*a*c - b^2)^3)^(1/2
) + 5*B^2*a^3*b^4*c*(-(4*a*c - b^2)^3)^(1/2) + 2*A*B*a*b^7*(-(4*a*c - b^2)^3)^(1/2) + 24*A*B*a^2*b^8*c - 12*A*
B*a^2*b^5*c*(-(4*a*c - b^2)^3)^(1/2) - 8*A*B*a^4*b*c^3*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a^3*b^3*c^2*(-(4*a*c
- b^2)^3)^(1/2))/(2*(a^9*b^4 + 16*a^11*c^2 - 8*a^10*b^2*c)))^(1/2)*(32*A*a^19*c^5 - x^(1/2)*(32*a^21*b*c^3 - 8
*a^20*b^3*c^2)*(-(A^2*b^11 + B^2*a^2*b^9 - A^2*b^8*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^10 + 63*A^2*a^2*b^7*c^
2 - 138*A^2*a^3*b^5*c^3 + 129*A^2*a^4*b^3*c^4 - A^2*a^4*c^4*(-(4*a*c - b^2)^3)^(1/2) - B^2*a^2*b^6*(-(4*a*c -
b^2)^3)^(1/2) + 42*B^2*a^4*b^5*c^2 - 63*B^2*a^5*b^3*c^3 + B^2*a^5*c^3*(-(4*a*c - b^2)^3)^(1/2) + 16*A*B*a^6*c^
5 - 13*A^2*a*b^9*c - 36*A^2*a^5*b*c^5 - 11*B^2*a^3*b^7*c + 28*B^2*a^6*b*c^4 - 15*A^2*a^2*b^4*c^2*(-(4*a*c - b^
2)^3)^(1/2) + 10*A^2*a^3*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) - 6*B^2*a^4*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 104*A
*B*a^3*b^6*c^2 + 192*A*B*a^4*b^4*c^3 - 132*A*B*a^5*b^2*c^4 + 7*A^2*a*b^6*c*(-(4*a*c - b^2)^3)^(1/2) + 5*B^2*a^
3*b^4*c*(-(4*a*c - b^2)^3)^(1/2) + 2*A*B*a*b^7*(-(4*a*c - b^2)^3)^(1/2) + 24*A*B*a^2*b^8*c - 12*A*B*a^2*b^5*c*
(-(4*a*c - b^2)^3)^(1/2) - 8*A*B*a^4*b*c^3*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a^3*b^3*c^2*(-(4*a*c - b^2)^3)^(1
/2))/(2*(a^9*b^4 + 16*a^11*c^2 - 8*a^10*b^2*c)))^(1/2) + 64*B*a^19*b*c^4 - 8*A*a^16*b^6*c^2 + 56*A*a^17*b^4*c^
3 - 104*A*a^18*b^2*c^4 + 8*B*a^17*b^5*c^2 - 48*B*a^18*b^3*c^3) + x^(1/2)*(16*A^2*a^16*c^7 - 16*B^2*a^17*c^6 +
8*A^2*a^12*b^8*c^3 - 64*A^2*a^13*b^6*c^4 + 160*A^2*a^14*b^4*c^5 - 128*A^2*a^15*b^2*c^6 + 8*B^2*a^14*b^6*c^3 -
48*B^2*a^15*b^4*c^4 + 72*B^2*a^16*b^2*c^5 - 16*A*B*a^13*b^7*c^3 + 112*A*B*a^14*b^5*c^4 - 224*A*B*a^15*b^3*c^5
+ 112*A*B*a^16*b*c^6))*(-(A^2*b^11 + B^2*a^2*b^9 - A^2*b^8*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*a*b^10 + 63*A^2*a^
2*b^7*c^2 - 138*A^2*a^3*b^5*c^3 + 129*A^2*a^4*b^3*c^4 - A^2*a^4*c^4*(-(4*a*c - b^2)^3)^(1/2) - B^2*a^2*b^6*(-(
4*a*c - b^2)^3)^(1/2) + 42*B^2*a^4*b^5*c^2 - 63*B^2*a^5*b^3*c^3 + B^2*a^5*c^3*(-(4*a*c - b^2)^3)^(1/2) + 16*A*
B*a^6*c^5 - 13*A^2*a*b^9*c - 36*A^2*a^5*b*c^5 - 11*B^2*a^3*b^7*c + 28*B^2*a^6*b*c^4 - 15*A^2*a^2*b^4*c^2*(-(4*
a*c - b^2)^3)^(1/2) + 10*A^2*a^3*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) - 6*B^2*a^4*b^2*c^2*(-(4*a*c - b^2)^3)^(1/2)
- 104*A*B*a^3*b^6*c^2 + 192*A*B*a^4*b^4*c^3 - 132*A*B*a^5*b^2*c^4 + 7*A^2*a*b^6*c*(-(4*a*c - b^2)^3)^(1/2) +
5*B^2*a^3*b^4*c*(-(4*a*c - b^2)^3)^(1/2) + 2*A*B*a*b^7*(-(4*a*c - b^2)^3)^(1/2) + 24*A*B*a^2*b^8*c - 12*A*B*a^
2*b^5*c*(-(4*a*c - b^2)^3)^(1/2) - 8*A*B*a^4*b*c^3*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a^3*b^3*c^2*(-(4*a*c - b^
2)^3)^(1/2))/(2*(a^9*b^4 + 16*a^11*c^2 - 8*a^10*b^2*c)))^(1/2) + 16*B^3*a^15*c^7 + 16*A^3*a^12*b^3*c^7 - 16*B^
3*a^14*b^2*c^6 + 16*A^2*B*a^14*c^8 - 32*A^3*a^13*b*c^8 - 48*A*B^2*a^14*b*c^7 + 32*A*B^2*a^13*b^3*c^6 - 16*A^2*
B*a^12*b^4*c^6 + 16*A^2*B*a^13*b^2*c^7))*(-(A^2*b^11 + B^2*a^2*b^9 - A^2*b^8*(-(4*a*c - b^2)^3)^(1/2) - 2*A*B*
a*b^10 + 63*A^2*a^2*b^7*c^2 - 138*A^2*a^3*b^5*c^3 + 129*A^2*a^4*b^3*c^4 - A^2*a^4*c^4*(-(4*a*c - b^2)^3)^(1/2)
- B^2*a^2*b^6*(-(4*a*c - b^2)^3)^(1/2) + 42*B^2*a^4*b^5*c^2 - 63*B^2*a^5*b^3*c^3 + B^2*a^5*c^3*(-(4*a*c - b^2
)^3)^(1/2) + 16*A*B*a^6*c^5 - 13*A^2*a*b^9*c - 36*A^2*a^5*b*c^5 - 11*B^2*a^3*b^7*c + 28*B^2*a^6*b*c^4 - 15*A^2
*a^2*b^4*c^2*(-(4*a*c - b^2)^3)^(1/2) + 10*A^2*a^3*b^2*c^3*(-(4*a*c - b^2)^3)^(1/2) - 6*B^2*a^4*b^2*c^2*(-(4*a
*c - b^2)^3)^(1/2) - 104*A*B*a^3*b^6*c^2 + 192*A*B*a^4*b^4*c^3 - 132*A*B*a^5*b^2*c^4 + 7*A^2*a*b^6*c*(-(4*a*c
- b^2)^3)^(1/2) + 5*B^2*a^3*b^4*c*(-(4*a*c - b^2)^3)^(1/2) + 2*A*B*a*b^7*(-(4*a*c - b^2)^3)^(1/2) + 24*A*B*a^2
*b^8*c - 12*A*B*a^2*b^5*c*(-(4*a*c - b^2)^3)^(1/2) - 8*A*B*a^4*b*c^3*(-(4*a*c - b^2)^3)^(1/2) + 20*A*B*a^3*b^3
*c^2*(-(4*a*c - b^2)^3)^(1/2))/(2*(a^9*b^4 + 16*a^11*c^2 - 8*a^10*b^2*c)))^(1/2)*2i

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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**(9/2)/(c*x**2+b*x+a),x)

[Out]

Timed out

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