3.2293 \(\int \frac{(d+e x)^{3/2}}{\left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=441 \[ -\frac{3 \sqrt{c} \left (-4 c e \left (-d \sqrt{b^2-4 a c}-a e+4 b d\right )+b e^2 \left (3 b-2 \sqrt{b^2-4 a c}\right )+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{2 \sqrt{2} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}+\frac{3 \sqrt{c} \left (-4 c e \left (d \sqrt{b^2-4 a c}-a e+4 b d\right )+b e^2 \left (2 \sqrt{b^2-4 a c}+3 b\right )+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{2 \sqrt{2} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{\sqrt{d+e x} (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{\sqrt{d+e x} \left (4 a c e-7 b^2 e+12 c x (2 c d-b e)+12 b c d\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )} \]

[Out]

-(Sqrt[d + e*x]*(b*d - 2*a*e + (2*c*d - b*e)*x))/(2*(b^2 - 4*a*c)*(a + b*x + c*x
^2)^2) + (Sqrt[d + e*x]*(12*b*c*d - 7*b^2*e + 4*a*c*e + 12*c*(2*c*d - b*e)*x))/(
4*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)) - (3*Sqrt[c]*(16*c^2*d^2 + b*(3*b - 2*Sqrt[
b^2 - 4*a*c])*e^2 - 4*c*e*(4*b*d - Sqrt[b^2 - 4*a*c]*d - a*e))*ArcTanh[(Sqrt[2]*
Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(2*Sqrt[2]*(b^2
- 4*a*c)^(5/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) + (3*Sqrt[c]*(16*c^2*d^
2 + b*(3*b + 2*Sqrt[b^2 - 4*a*c])*e^2 - 4*c*e*(4*b*d + Sqrt[b^2 - 4*a*c]*d - a*e
))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*
e]])/(2*Sqrt[2]*(b^2 - 4*a*c)^(5/2)*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

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Rubi [A]  time = 4.51841, antiderivative size = 441, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{3 \sqrt{c} \left (-4 c e \left (-d \sqrt{b^2-4 a c}-a e+4 b d\right )+b e^2 \left (3 b-2 \sqrt{b^2-4 a c}\right )+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{2 \sqrt{2} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}+\frac{3 \sqrt{c} \left (-4 c e \left (d \sqrt{b^2-4 a c}-a e+4 b d\right )+b e^2 \left (2 \sqrt{b^2-4 a c}+3 b\right )+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{2 \sqrt{2} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{\sqrt{d+e x} (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{\sqrt{d+e x} \left (4 a c e-7 b^2 e+12 c x (2 c d-b e)+12 b c d\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(3/2)/(a + b*x + c*x^2)^3,x]

[Out]

-(Sqrt[d + e*x]*(b*d - 2*a*e + (2*c*d - b*e)*x))/(2*(b^2 - 4*a*c)*(a + b*x + c*x
^2)^2) + (Sqrt[d + e*x]*(12*b*c*d - 7*b^2*e + 4*a*c*e + 12*c*(2*c*d - b*e)*x))/(
4*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)) - (3*Sqrt[c]*(16*c^2*d^2 + b*(3*b - 2*Sqrt[
b^2 - 4*a*c])*e^2 - 4*c*e*(4*b*d - Sqrt[b^2 - 4*a*c]*d - a*e))*ArcTanh[(Sqrt[2]*
Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(2*Sqrt[2]*(b^2
- 4*a*c)^(5/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) + (3*Sqrt[c]*(16*c^2*d^
2 + b*(3*b + 2*Sqrt[b^2 - 4*a*c])*e^2 - 4*c*e*(4*b*d + Sqrt[b^2 - 4*a*c]*d - a*e
))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*
e]])/(2*Sqrt[2]*(b^2 - 4*a*c)^(5/2)*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(3/2)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 3.77896, size = 417, normalized size = 0.95 \[ \frac{1}{4} \left (-\frac{3 \sqrt{2} \sqrt{c} \left (4 c e \left (d \sqrt{b^2-4 a c}+a e-4 b d\right )+b e^2 \left (3 b-2 \sqrt{b^2-4 a c}\right )+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right )}{\left (b^2-4 a c\right )^{5/2} \sqrt{e \left (\sqrt{b^2-4 a c}-b\right )+2 c d}}+\frac{3 \sqrt{2} \sqrt{c} \left (4 c e \left (-d \sqrt{b^2-4 a c}+a e-4 b d\right )+b e^2 \left (2 \sqrt{b^2-4 a c}+3 b\right )+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}+\frac{\sqrt{d+e x} \left (2 \left (b^2-4 a c\right ) (2 a e-b d+b e x-2 c d x)+(a+x (b+c x)) \left (4 c (a e+6 c d x)-7 b^2 e+12 b c (d-e x)\right )\right )}{\left (b^2-4 a c\right )^2 (a+x (b+c x))^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(3/2)/(a + b*x + c*x^2)^3,x]

[Out]

((Sqrt[d + e*x]*(2*(b^2 - 4*a*c)*(-(b*d) + 2*a*e - 2*c*d*x + b*e*x) + (a + x*(b
+ c*x))*(-7*b^2*e + 4*c*(a*e + 6*c*d*x) + 12*b*c*(d - e*x))))/((b^2 - 4*a*c)^2*(
a + x*(b + c*x))^2) - (3*Sqrt[2]*Sqrt[c]*(16*c^2*d^2 + b*(3*b - 2*Sqrt[b^2 - 4*a
*c])*e^2 + 4*c*e*(-4*b*d + Sqrt[b^2 - 4*a*c]*d + a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*
Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/((b^2 - 4*a*c)^(5/2)*Sq
rt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]) + (3*Sqrt[2]*Sqrt[c]*(16*c^2*d^2 + b*(3*
b + 2*Sqrt[b^2 - 4*a*c])*e^2 + 4*c*e*(-4*b*d - Sqrt[b^2 - 4*a*c]*d + a*e))*ArcTa
nh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/((b
^2 - 4*a*c)^(5/2)*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/4

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Maple [B]  time = 0.117, size = 15050, normalized size = 34.1 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(3/2)/(c*x^2+b*x+a)^3,x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{\frac{3}{2}}}{{\left (c x^{2} + b x + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(3/2)/(c*x^2 + b*x + a)^3,x, algorithm="maxima")

[Out]

integrate((e*x + d)^(3/2)/(c*x^2 + b*x + a)^3, x)

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Fricas [A]  time = 0.572962, size = 15023, normalized size = 34.07 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(3/2)/(c*x^2 + b*x + a)^3,x, algorithm="fricas")

[Out]

-1/8*(3*sqrt(1/2)*(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^4*c^2 - 8*a*b^2*c^3 +
16*a^2*c^4)*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^3 + (b^6 - 6*a*b^4*c
+ 32*a^3*c^3)*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x)*sqrt((512*c^5*d^5
- 1280*b*c^4*d^4*e + 160*(7*b^2*c^3 + 4*a*c^4)*d^3*e^2 - 80*(5*b^3*c^2 + 12*a*b
*c^3)*d^2*e^3 + 10*(5*b^4*c + 40*a*b^2*c^2 + 16*a^2*c^3)*d*e^4 - (b^5 + 40*a*b^3
*c + 80*a^2*b*c^2)*e^5 + sqrt(e^10/((b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 -
640*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a^5*c^7)*d^4 - 2*(b^11*c - 20*a*b^9*c
^2 + 160*a^2*b^7*c^3 - 640*a^3*b^5*c^4 + 1280*a^4*b^3*c^5 - 1024*a^5*b*c^6)*d^3*
e + (b^12 - 18*a*b^10*c + 120*a^2*b^8*c^2 - 320*a^3*b^6*c^3 + 1536*a^5*b^2*c^5 -
2048*a^6*c^6)*d^2*e^2 - 2*(a*b^11 - 20*a^2*b^9*c + 160*a^3*b^7*c^2 - 640*a^4*b^
5*c^3 + 1280*a^5*b^3*c^4 - 1024*a^6*b*c^5)*d*e^3 + (a^2*b^10 - 20*a^3*b^8*c + 16
0*a^4*b^6*c^2 - 640*a^5*b^4*c^3 + 1280*a^6*b^2*c^4 - 1024*a^7*c^5)*e^4))*((b^10*
c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a
^5*c^6)*d^2 - (b^11 - 20*a*b^9*c + 160*a^2*b^7*c^2 - 640*a^3*b^5*c^3 + 1280*a^4*
b^3*c^4 - 1024*a^5*b*c^5)*d*e + (a*b^10 - 20*a^2*b^8*c + 160*a^3*b^6*c^2 - 640*a
^4*b^4*c^3 + 1280*a^5*b^2*c^4 - 1024*a^6*c^5)*e^2))/((b^10*c - 20*a*b^8*c^2 + 16
0*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6)*d^2 - (b^11 -
20*a*b^9*c + 160*a^2*b^7*c^2 - 640*a^3*b^5*c^3 + 1280*a^4*b^3*c^4 - 1024*a^5*b*
c^5)*d*e + (a*b^10 - 20*a^2*b^8*c + 160*a^3*b^6*c^2 - 640*a^4*b^4*c^3 + 1280*a^5
*b^2*c^4 - 1024*a^6*c^5)*e^2))*log(27/2*sqrt(1/2)*(8*(b^6*c^2 - 12*a*b^4*c^3 + 4
8*a^2*b^2*c^4 - 64*a^3*c^5)*d^2*e^6 - 8*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 -
64*a^3*b*c^4)*d*e^7 + (b^8 - 8*a*b^6*c + 128*a^3*b^2*c^3 - 256*a^4*c^4)*e^8 - s
qrt(e^10/((b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^3*b^4*c^5 + 1280*a^
4*b^2*c^6 - 1024*a^5*c^7)*d^4 - 2*(b^11*c - 20*a*b^9*c^2 + 160*a^2*b^7*c^3 - 640
*a^3*b^5*c^4 + 1280*a^4*b^3*c^5 - 1024*a^5*b*c^6)*d^3*e + (b^12 - 18*a*b^10*c +
120*a^2*b^8*c^2 - 320*a^3*b^6*c^3 + 1536*a^5*b^2*c^5 - 2048*a^6*c^6)*d^2*e^2 - 2
*(a*b^11 - 20*a^2*b^9*c + 160*a^3*b^7*c^2 - 640*a^4*b^5*c^3 + 1280*a^5*b^3*c^4 -
1024*a^6*b*c^5)*d*e^3 + (a^2*b^10 - 20*a^3*b^8*c + 160*a^4*b^6*c^2 - 640*a^5*b^
4*c^3 + 1280*a^6*b^2*c^4 - 1024*a^7*c^5)*e^4))*(32*(b^10*c^4 - 20*a*b^8*c^5 + 16
0*a^2*b^6*c^6 - 640*a^3*b^4*c^7 + 1280*a^4*b^2*c^8 - 1024*a^5*c^9)*d^5 - 80*(b^1
1*c^3 - 20*a*b^9*c^4 + 160*a^2*b^7*c^5 - 640*a^3*b^5*c^6 + 1280*a^4*b^3*c^7 - 10
24*a^5*b*c^8)*d^4*e + 2*(33*b^12*c^2 - 632*a*b^10*c^3 + 4720*a^2*b^8*c^4 - 16640
*a^3*b^6*c^5 + 24320*a^4*b^4*c^6 + 2048*a^5*b^2*c^7 - 28672*a^6*c^8)*d^3*e^2 - (
19*b^13*c - 296*a*b^11*c^2 + 1360*a^2*b^9*c^3 + 1280*a^3*b^7*c^4 - 29440*a^4*b^5
*c^5 + 88064*a^5*b^3*c^6 - 86016*a^6*b*c^7)*d^2*e^3 + (b^14 + 10*a*b^12*c - 416*
a^2*b^10*c^2 + 3680*a^3*b^8*c^3 - 14080*a^4*b^6*c^4 + 22016*a^5*b^4*c^5 - 24576*
a^7*c^7)*d*e^4 - (a*b^13 - 8*a^2*b^11*c - 80*a^3*b^9*c^2 + 1280*a^4*b^7*c^3 - 64
00*a^5*b^5*c^4 + 14336*a^6*b^3*c^5 - 12288*a^7*b*c^6)*e^5))*sqrt((512*c^5*d^5 -
1280*b*c^4*d^4*e + 160*(7*b^2*c^3 + 4*a*c^4)*d^3*e^2 - 80*(5*b^3*c^2 + 12*a*b*c^
3)*d^2*e^3 + 10*(5*b^4*c + 40*a*b^2*c^2 + 16*a^2*c^3)*d*e^4 - (b^5 + 40*a*b^3*c
+ 80*a^2*b*c^2)*e^5 + sqrt(e^10/((b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 64
0*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a^5*c^7)*d^4 - 2*(b^11*c - 20*a*b^9*c^2
+ 160*a^2*b^7*c^3 - 640*a^3*b^5*c^4 + 1280*a^4*b^3*c^5 - 1024*a^5*b*c^6)*d^3*e +
(b^12 - 18*a*b^10*c + 120*a^2*b^8*c^2 - 320*a^3*b^6*c^3 + 1536*a^5*b^2*c^5 - 20
48*a^6*c^6)*d^2*e^2 - 2*(a*b^11 - 20*a^2*b^9*c + 160*a^3*b^7*c^2 - 640*a^4*b^5*c
^3 + 1280*a^5*b^3*c^4 - 1024*a^6*b*c^5)*d*e^3 + (a^2*b^10 - 20*a^3*b^8*c + 160*a
^4*b^6*c^2 - 640*a^5*b^4*c^3 + 1280*a^6*b^2*c^4 - 1024*a^7*c^5)*e^4))*((b^10*c -
20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*
c^6)*d^2 - (b^11 - 20*a*b^9*c + 160*a^2*b^7*c^2 - 640*a^3*b^5*c^3 + 1280*a^4*b^3
*c^4 - 1024*a^5*b*c^5)*d*e + (a*b^10 - 20*a^2*b^8*c + 160*a^3*b^6*c^2 - 640*a^4*
b^4*c^3 + 1280*a^5*b^2*c^4 - 1024*a^6*c^5)*e^2))/((b^10*c - 20*a*b^8*c^2 + 160*a
^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6)*d^2 - (b^11 - 20
*a*b^9*c + 160*a^2*b^7*c^2 - 640*a^3*b^5*c^3 + 1280*a^4*b^3*c^4 - 1024*a^5*b*c^5
)*d*e + (a*b^10 - 20*a^2*b^8*c + 160*a^3*b^6*c^2 - 640*a^4*b^4*c^3 + 1280*a^5*b^
2*c^4 - 1024*a^6*c^5)*e^2)) + 27*(256*c^5*d^4*e^5 - 512*b*c^4*d^3*e^6 + 48*(7*b^
2*c^3 + 4*a*c^4)*d^2*e^7 - 16*(5*b^3*c^2 + 12*a*b*c^3)*d*e^8 + (5*b^4*c + 40*a*b
^2*c^2 + 16*a^2*c^3)*e^9)*sqrt(e*x + d)) - 3*sqrt(1/2)*(a^2*b^4 - 8*a^3*b^2*c +
16*a^4*c^2 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^4 + 2*(b^5*c - 8*a*b^3*c^2 +
16*a^2*b*c^3)*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^2 + 2*(a*b^5 - 8*a^2*b^3*c
+ 16*a^3*b*c^2)*x)*sqrt((512*c^5*d^5 - 1280*b*c^4*d^4*e + 160*(7*b^2*c^3 + 4*a*
c^4)*d^3*e^2 - 80*(5*b^3*c^2 + 12*a*b*c^3)*d^2*e^3 + 10*(5*b^4*c + 40*a*b^2*c^2
+ 16*a^2*c^3)*d*e^4 - (b^5 + 40*a*b^3*c + 80*a^2*b*c^2)*e^5 + sqrt(e^10/((b^10*c
^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*
a^5*c^7)*d^4 - 2*(b^11*c - 20*a*b^9*c^2 + 160*a^2*b^7*c^3 - 640*a^3*b^5*c^4 + 12
80*a^4*b^3*c^5 - 1024*a^5*b*c^6)*d^3*e + (b^12 - 18*a*b^10*c + 120*a^2*b^8*c^2 -
320*a^3*b^6*c^3 + 1536*a^5*b^2*c^5 - 2048*a^6*c^6)*d^2*e^2 - 2*(a*b^11 - 20*a^2
*b^9*c + 160*a^3*b^7*c^2 - 640*a^4*b^5*c^3 + 1280*a^5*b^3*c^4 - 1024*a^6*b*c^5)*
d*e^3 + (a^2*b^10 - 20*a^3*b^8*c + 160*a^4*b^6*c^2 - 640*a^5*b^4*c^3 + 1280*a^6*
b^2*c^4 - 1024*a^7*c^5)*e^4))*((b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^
3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6)*d^2 - (b^11 - 20*a*b^9*c + 160*a^2*
b^7*c^2 - 640*a^3*b^5*c^3 + 1280*a^4*b^3*c^4 - 1024*a^5*b*c^5)*d*e + (a*b^10 - 2
0*a^2*b^8*c + 160*a^3*b^6*c^2 - 640*a^4*b^4*c^3 + 1280*a^5*b^2*c^4 - 1024*a^6*c^
5)*e^2))/((b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*
b^2*c^5 - 1024*a^5*c^6)*d^2 - (b^11 - 20*a*b^9*c + 160*a^2*b^7*c^2 - 640*a^3*b^5
*c^3 + 1280*a^4*b^3*c^4 - 1024*a^5*b*c^5)*d*e + (a*b^10 - 20*a^2*b^8*c + 160*a^3
*b^6*c^2 - 640*a^4*b^4*c^3 + 1280*a^5*b^2*c^4 - 1024*a^6*c^5)*e^2))*log(-27/2*sq
rt(1/2)*(8*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^2*e^6 - 8*(b
^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d*e^7 + (b^8 - 8*a*b^6*c +
128*a^3*b^2*c^3 - 256*a^4*c^4)*e^8 - sqrt(e^10/((b^10*c^2 - 20*a*b^8*c^3 + 160*a
^2*b^6*c^4 - 640*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a^5*c^7)*d^4 - 2*(b^11*c
- 20*a*b^9*c^2 + 160*a^2*b^7*c^3 - 640*a^3*b^5*c^4 + 1280*a^4*b^3*c^5 - 1024*a^5
*b*c^6)*d^3*e + (b^12 - 18*a*b^10*c + 120*a^2*b^8*c^2 - 320*a^3*b^6*c^3 + 1536*a
^5*b^2*c^5 - 2048*a^6*c^6)*d^2*e^2 - 2*(a*b^11 - 20*a^2*b^9*c + 160*a^3*b^7*c^2
- 640*a^4*b^5*c^3 + 1280*a^5*b^3*c^4 - 1024*a^6*b*c^5)*d*e^3 + (a^2*b^10 - 20*a^
3*b^8*c + 160*a^4*b^6*c^2 - 640*a^5*b^4*c^3 + 1280*a^6*b^2*c^4 - 1024*a^7*c^5)*e
^4))*(32*(b^10*c^4 - 20*a*b^8*c^5 + 160*a^2*b^6*c^6 - 640*a^3*b^4*c^7 + 1280*a^4
*b^2*c^8 - 1024*a^5*c^9)*d^5 - 80*(b^11*c^3 - 20*a*b^9*c^4 + 160*a^2*b^7*c^5 - 6
40*a^3*b^5*c^6 + 1280*a^4*b^3*c^7 - 1024*a^5*b*c^8)*d^4*e + 2*(33*b^12*c^2 - 632
*a*b^10*c^3 + 4720*a^2*b^8*c^4 - 16640*a^3*b^6*c^5 + 24320*a^4*b^4*c^6 + 2048*a^
5*b^2*c^7 - 28672*a^6*c^8)*d^3*e^2 - (19*b^13*c - 296*a*b^11*c^2 + 1360*a^2*b^9*
c^3 + 1280*a^3*b^7*c^4 - 29440*a^4*b^5*c^5 + 88064*a^5*b^3*c^6 - 86016*a^6*b*c^7
)*d^2*e^3 + (b^14 + 10*a*b^12*c - 416*a^2*b^10*c^2 + 3680*a^3*b^8*c^3 - 14080*a^
4*b^6*c^4 + 22016*a^5*b^4*c^5 - 24576*a^7*c^7)*d*e^4 - (a*b^13 - 8*a^2*b^11*c -
80*a^3*b^9*c^2 + 1280*a^4*b^7*c^3 - 6400*a^5*b^5*c^4 + 14336*a^6*b^3*c^5 - 12288
*a^7*b*c^6)*e^5))*sqrt((512*c^5*d^5 - 1280*b*c^4*d^4*e + 160*(7*b^2*c^3 + 4*a*c^
4)*d^3*e^2 - 80*(5*b^3*c^2 + 12*a*b*c^3)*d^2*e^3 + 10*(5*b^4*c + 40*a*b^2*c^2 +
16*a^2*c^3)*d*e^4 - (b^5 + 40*a*b^3*c + 80*a^2*b*c^2)*e^5 + sqrt(e^10/((b^10*c^2
- 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a^
5*c^7)*d^4 - 2*(b^11*c - 20*a*b^9*c^2 + 160*a^2*b^7*c^3 - 640*a^3*b^5*c^4 + 1280
*a^4*b^3*c^5 - 1024*a^5*b*c^6)*d^3*e + (b^12 - 18*a*b^10*c + 120*a^2*b^8*c^2 - 3
20*a^3*b^6*c^3 + 1536*a^5*b^2*c^5 - 2048*a^6*c^6)*d^2*e^2 - 2*(a*b^11 - 20*a^2*b
^9*c + 160*a^3*b^7*c^2 - 640*a^4*b^5*c^3 + 1280*a^5*b^3*c^4 - 1024*a^6*b*c^5)*d*
e^3 + (a^2*b^10 - 20*a^3*b^8*c + 160*a^4*b^6*c^2 - 640*a^5*b^4*c^3 + 1280*a^6*b^
2*c^4 - 1024*a^7*c^5)*e^4))*((b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*
b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6)*d^2 - (b^11 - 20*a*b^9*c + 160*a^2*b^
7*c^2 - 640*a^3*b^5*c^3 + 1280*a^4*b^3*c^4 - 1024*a^5*b*c^5)*d*e + (a*b^10 - 20*
a^2*b^8*c + 160*a^3*b^6*c^2 - 640*a^4*b^4*c^3 + 1280*a^5*b^2*c^4 - 1024*a^6*c^5)
*e^2))/((b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^
2*c^5 - 1024*a^5*c^6)*d^2 - (b^11 - 20*a*b^9*c + 160*a^2*b^7*c^2 - 640*a^3*b^5*c
^3 + 1280*a^4*b^3*c^4 - 1024*a^5*b*c^5)*d*e + (a*b^10 - 20*a^2*b^8*c + 160*a^3*b
^6*c^2 - 640*a^4*b^4*c^3 + 1280*a^5*b^2*c^4 - 1024*a^6*c^5)*e^2)) + 27*(256*c^5*
d^4*e^5 - 512*b*c^4*d^3*e^6 + 48*(7*b^2*c^3 + 4*a*c^4)*d^2*e^7 - 16*(5*b^3*c^2 +
12*a*b*c^3)*d*e^8 + (5*b^4*c + 40*a*b^2*c^2 + 16*a^2*c^3)*e^9)*sqrt(e*x + d)) +
3*sqrt(1/2)*(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a
^2*c^4)*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^3 + (b^6 - 6*a*b^4*c + 32
*a^3*c^3)*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x)*sqrt((512*c^5*d^5 - 12
80*b*c^4*d^4*e + 160*(7*b^2*c^3 + 4*a*c^4)*d^3*e^2 - 80*(5*b^3*c^2 + 12*a*b*c^3)
*d^2*e^3 + 10*(5*b^4*c + 40*a*b^2*c^2 + 16*a^2*c^3)*d*e^4 - (b^5 + 40*a*b^3*c +
80*a^2*b*c^2)*e^5 - sqrt(e^10/((b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*
a^3*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a^5*c^7)*d^4 - 2*(b^11*c - 20*a*b^9*c^2 +
160*a^2*b^7*c^3 - 640*a^3*b^5*c^4 + 1280*a^4*b^3*c^5 - 1024*a^5*b*c^6)*d^3*e + (
b^12 - 18*a*b^10*c + 120*a^2*b^8*c^2 - 320*a^3*b^6*c^3 + 1536*a^5*b^2*c^5 - 2048
*a^6*c^6)*d^2*e^2 - 2*(a*b^11 - 20*a^2*b^9*c + 160*a^3*b^7*c^2 - 640*a^4*b^5*c^3
+ 1280*a^5*b^3*c^4 - 1024*a^6*b*c^5)*d*e^3 + (a^2*b^10 - 20*a^3*b^8*c + 160*a^4
*b^6*c^2 - 640*a^5*b^4*c^3 + 1280*a^6*b^2*c^4 - 1024*a^7*c^5)*e^4))*((b^10*c - 2
0*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^
6)*d^2 - (b^11 - 20*a*b^9*c + 160*a^2*b^7*c^2 - 640*a^3*b^5*c^3 + 1280*a^4*b^3*c
^4 - 1024*a^5*b*c^5)*d*e + (a*b^10 - 20*a^2*b^8*c + 160*a^3*b^6*c^2 - 640*a^4*b^
4*c^3 + 1280*a^5*b^2*c^4 - 1024*a^6*c^5)*e^2))/((b^10*c - 20*a*b^8*c^2 + 160*a^2
*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6)*d^2 - (b^11 - 20*a
*b^9*c + 160*a^2*b^7*c^2 - 640*a^3*b^5*c^3 + 1280*a^4*b^3*c^4 - 1024*a^5*b*c^5)*
d*e + (a*b^10 - 20*a^2*b^8*c + 160*a^3*b^6*c^2 - 640*a^4*b^4*c^3 + 1280*a^5*b^2*
c^4 - 1024*a^6*c^5)*e^2))*log(27/2*sqrt(1/2)*(8*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2
*b^2*c^4 - 64*a^3*c^5)*d^2*e^6 - 8*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a
^3*b*c^4)*d*e^7 + (b^8 - 8*a*b^6*c + 128*a^3*b^2*c^3 - 256*a^4*c^4)*e^8 + sqrt(e
^10/((b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^3*b^4*c^5 + 1280*a^4*b^2
*c^6 - 1024*a^5*c^7)*d^4 - 2*(b^11*c - 20*a*b^9*c^2 + 160*a^2*b^7*c^3 - 640*a^3*
b^5*c^4 + 1280*a^4*b^3*c^5 - 1024*a^5*b*c^6)*d^3*e + (b^12 - 18*a*b^10*c + 120*a
^2*b^8*c^2 - 320*a^3*b^6*c^3 + 1536*a^5*b^2*c^5 - 2048*a^6*c^6)*d^2*e^2 - 2*(a*b
^11 - 20*a^2*b^9*c + 160*a^3*b^7*c^2 - 640*a^4*b^5*c^3 + 1280*a^5*b^3*c^4 - 1024
*a^6*b*c^5)*d*e^3 + (a^2*b^10 - 20*a^3*b^8*c + 160*a^4*b^6*c^2 - 640*a^5*b^4*c^3
+ 1280*a^6*b^2*c^4 - 1024*a^7*c^5)*e^4))*(32*(b^10*c^4 - 20*a*b^8*c^5 + 160*a^2
*b^6*c^6 - 640*a^3*b^4*c^7 + 1280*a^4*b^2*c^8 - 1024*a^5*c^9)*d^5 - 80*(b^11*c^3
- 20*a*b^9*c^4 + 160*a^2*b^7*c^5 - 640*a^3*b^5*c^6 + 1280*a^4*b^3*c^7 - 1024*a^
5*b*c^8)*d^4*e + 2*(33*b^12*c^2 - 632*a*b^10*c^3 + 4720*a^2*b^8*c^4 - 16640*a^3*
b^6*c^5 + 24320*a^4*b^4*c^6 + 2048*a^5*b^2*c^7 - 28672*a^6*c^8)*d^3*e^2 - (19*b^
13*c - 296*a*b^11*c^2 + 1360*a^2*b^9*c^3 + 1280*a^3*b^7*c^4 - 29440*a^4*b^5*c^5
+ 88064*a^5*b^3*c^6 - 86016*a^6*b*c^7)*d^2*e^3 + (b^14 + 10*a*b^12*c - 416*a^2*b
^10*c^2 + 3680*a^3*b^8*c^3 - 14080*a^4*b^6*c^4 + 22016*a^5*b^4*c^5 - 24576*a^7*c
^7)*d*e^4 - (a*b^13 - 8*a^2*b^11*c - 80*a^3*b^9*c^2 + 1280*a^4*b^7*c^3 - 6400*a^
5*b^5*c^4 + 14336*a^6*b^3*c^5 - 12288*a^7*b*c^6)*e^5))*sqrt((512*c^5*d^5 - 1280*
b*c^4*d^4*e + 160*(7*b^2*c^3 + 4*a*c^4)*d^3*e^2 - 80*(5*b^3*c^2 + 12*a*b*c^3)*d^
2*e^3 + 10*(5*b^4*c + 40*a*b^2*c^2 + 16*a^2*c^3)*d*e^4 - (b^5 + 40*a*b^3*c + 80*
a^2*b*c^2)*e^5 - sqrt(e^10/((b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^3
*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a^5*c^7)*d^4 - 2*(b^11*c - 20*a*b^9*c^2 + 160
*a^2*b^7*c^3 - 640*a^3*b^5*c^4 + 1280*a^4*b^3*c^5 - 1024*a^5*b*c^6)*d^3*e + (b^1
2 - 18*a*b^10*c + 120*a^2*b^8*c^2 - 320*a^3*b^6*c^3 + 1536*a^5*b^2*c^5 - 2048*a^
6*c^6)*d^2*e^2 - 2*(a*b^11 - 20*a^2*b^9*c + 160*a^3*b^7*c^2 - 640*a^4*b^5*c^3 +
1280*a^5*b^3*c^4 - 1024*a^6*b*c^5)*d*e^3 + (a^2*b^10 - 20*a^3*b^8*c + 160*a^4*b^
6*c^2 - 640*a^5*b^4*c^3 + 1280*a^6*b^2*c^4 - 1024*a^7*c^5)*e^4))*((b^10*c - 20*a
*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6)*
d^2 - (b^11 - 20*a*b^9*c + 160*a^2*b^7*c^2 - 640*a^3*b^5*c^3 + 1280*a^4*b^3*c^4
- 1024*a^5*b*c^5)*d*e + (a*b^10 - 20*a^2*b^8*c + 160*a^3*b^6*c^2 - 640*a^4*b^4*c
^3 + 1280*a^5*b^2*c^4 - 1024*a^6*c^5)*e^2))/((b^10*c - 20*a*b^8*c^2 + 160*a^2*b^
6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6)*d^2 - (b^11 - 20*a*b^
9*c + 160*a^2*b^7*c^2 - 640*a^3*b^5*c^3 + 1280*a^4*b^3*c^4 - 1024*a^5*b*c^5)*d*e
+ (a*b^10 - 20*a^2*b^8*c + 160*a^3*b^6*c^2 - 640*a^4*b^4*c^3 + 1280*a^5*b^2*c^4
- 1024*a^6*c^5)*e^2)) + 27*(256*c^5*d^4*e^5 - 512*b*c^4*d^3*e^6 + 48*(7*b^2*c^3
+ 4*a*c^4)*d^2*e^7 - 16*(5*b^3*c^2 + 12*a*b*c^3)*d*e^8 + (5*b^4*c + 40*a*b^2*c^
2 + 16*a^2*c^3)*e^9)*sqrt(e*x + d)) - 3*sqrt(1/2)*(a^2*b^4 - 8*a^3*b^2*c + 16*a^
4*c^2 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a
^2*b*c^3)*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16
*a^3*b*c^2)*x)*sqrt((512*c^5*d^5 - 1280*b*c^4*d^4*e + 160*(7*b^2*c^3 + 4*a*c^4)*
d^3*e^2 - 80*(5*b^3*c^2 + 12*a*b*c^3)*d^2*e^3 + 10*(5*b^4*c + 40*a*b^2*c^2 + 16*
a^2*c^3)*d*e^4 - (b^5 + 40*a*b^3*c + 80*a^2*b*c^2)*e^5 - sqrt(e^10/((b^10*c^2 -
20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a^5*c
^7)*d^4 - 2*(b^11*c - 20*a*b^9*c^2 + 160*a^2*b^7*c^3 - 640*a^3*b^5*c^4 + 1280*a^
4*b^3*c^5 - 1024*a^5*b*c^6)*d^3*e + (b^12 - 18*a*b^10*c + 120*a^2*b^8*c^2 - 320*
a^3*b^6*c^3 + 1536*a^5*b^2*c^5 - 2048*a^6*c^6)*d^2*e^2 - 2*(a*b^11 - 20*a^2*b^9*
c + 160*a^3*b^7*c^2 - 640*a^4*b^5*c^3 + 1280*a^5*b^3*c^4 - 1024*a^6*b*c^5)*d*e^3
+ (a^2*b^10 - 20*a^3*b^8*c + 160*a^4*b^6*c^2 - 640*a^5*b^4*c^3 + 1280*a^6*b^2*c
^4 - 1024*a^7*c^5)*e^4))*((b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4
*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6)*d^2 - (b^11 - 20*a*b^9*c + 160*a^2*b^7*c
^2 - 640*a^3*b^5*c^3 + 1280*a^4*b^3*c^4 - 1024*a^5*b*c^5)*d*e + (a*b^10 - 20*a^2
*b^8*c + 160*a^3*b^6*c^2 - 640*a^4*b^4*c^3 + 1280*a^5*b^2*c^4 - 1024*a^6*c^5)*e^
2))/((b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c
^5 - 1024*a^5*c^6)*d^2 - (b^11 - 20*a*b^9*c + 160*a^2*b^7*c^2 - 640*a^3*b^5*c^3
+ 1280*a^4*b^3*c^4 - 1024*a^5*b*c^5)*d*e + (a*b^10 - 20*a^2*b^8*c + 160*a^3*b^6*
c^2 - 640*a^4*b^4*c^3 + 1280*a^5*b^2*c^4 - 1024*a^6*c^5)*e^2))*log(-27/2*sqrt(1/
2)*(8*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^2*e^6 - 8*(b^7*c
- 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d*e^7 + (b^8 - 8*a*b^6*c + 128*a
^3*b^2*c^3 - 256*a^4*c^4)*e^8 + sqrt(e^10/((b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^
6*c^4 - 640*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a^5*c^7)*d^4 - 2*(b^11*c - 20*
a*b^9*c^2 + 160*a^2*b^7*c^3 - 640*a^3*b^5*c^4 + 1280*a^4*b^3*c^5 - 1024*a^5*b*c^
6)*d^3*e + (b^12 - 18*a*b^10*c + 120*a^2*b^8*c^2 - 320*a^3*b^6*c^3 + 1536*a^5*b^
2*c^5 - 2048*a^6*c^6)*d^2*e^2 - 2*(a*b^11 - 20*a^2*b^9*c + 160*a^3*b^7*c^2 - 640
*a^4*b^5*c^3 + 1280*a^5*b^3*c^4 - 1024*a^6*b*c^5)*d*e^3 + (a^2*b^10 - 20*a^3*b^8
*c + 160*a^4*b^6*c^2 - 640*a^5*b^4*c^3 + 1280*a^6*b^2*c^4 - 1024*a^7*c^5)*e^4))*
(32*(b^10*c^4 - 20*a*b^8*c^5 + 160*a^2*b^6*c^6 - 640*a^3*b^4*c^7 + 1280*a^4*b^2*
c^8 - 1024*a^5*c^9)*d^5 - 80*(b^11*c^3 - 20*a*b^9*c^4 + 160*a^2*b^7*c^5 - 640*a^
3*b^5*c^6 + 1280*a^4*b^3*c^7 - 1024*a^5*b*c^8)*d^4*e + 2*(33*b^12*c^2 - 632*a*b^
10*c^3 + 4720*a^2*b^8*c^4 - 16640*a^3*b^6*c^5 + 24320*a^4*b^4*c^6 + 2048*a^5*b^2
*c^7 - 28672*a^6*c^8)*d^3*e^2 - (19*b^13*c - 296*a*b^11*c^2 + 1360*a^2*b^9*c^3 +
1280*a^3*b^7*c^4 - 29440*a^4*b^5*c^5 + 88064*a^5*b^3*c^6 - 86016*a^6*b*c^7)*d^2
*e^3 + (b^14 + 10*a*b^12*c - 416*a^2*b^10*c^2 + 3680*a^3*b^8*c^3 - 14080*a^4*b^6
*c^4 + 22016*a^5*b^4*c^5 - 24576*a^7*c^7)*d*e^4 - (a*b^13 - 8*a^2*b^11*c - 80*a^
3*b^9*c^2 + 1280*a^4*b^7*c^3 - 6400*a^5*b^5*c^4 + 14336*a^6*b^3*c^5 - 12288*a^7*
b*c^6)*e^5))*sqrt((512*c^5*d^5 - 1280*b*c^4*d^4*e + 160*(7*b^2*c^3 + 4*a*c^4)*d^
3*e^2 - 80*(5*b^3*c^2 + 12*a*b*c^3)*d^2*e^3 + 10*(5*b^4*c + 40*a*b^2*c^2 + 16*a^
2*c^3)*d*e^4 - (b^5 + 40*a*b^3*c + 80*a^2*b*c^2)*e^5 - sqrt(e^10/((b^10*c^2 - 20
*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a^5*c^7
)*d^4 - 2*(b^11*c - 20*a*b^9*c^2 + 160*a^2*b^7*c^3 - 640*a^3*b^5*c^4 + 1280*a^4*
b^3*c^5 - 1024*a^5*b*c^6)*d^3*e + (b^12 - 18*a*b^10*c + 120*a^2*b^8*c^2 - 320*a^
3*b^6*c^3 + 1536*a^5*b^2*c^5 - 2048*a^6*c^6)*d^2*e^2 - 2*(a*b^11 - 20*a^2*b^9*c
+ 160*a^3*b^7*c^2 - 640*a^4*b^5*c^3 + 1280*a^5*b^3*c^4 - 1024*a^6*b*c^5)*d*e^3 +
(a^2*b^10 - 20*a^3*b^8*c + 160*a^4*b^6*c^2 - 640*a^5*b^4*c^3 + 1280*a^6*b^2*c^4
- 1024*a^7*c^5)*e^4))*((b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c
^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6)*d^2 - (b^11 - 20*a*b^9*c + 160*a^2*b^7*c^2
- 640*a^3*b^5*c^3 + 1280*a^4*b^3*c^4 - 1024*a^5*b*c^5)*d*e + (a*b^10 - 20*a^2*b
^8*c + 160*a^3*b^6*c^2 - 640*a^4*b^4*c^3 + 1280*a^5*b^2*c^4 - 1024*a^6*c^5)*e^2)
)/((b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5
- 1024*a^5*c^6)*d^2 - (b^11 - 20*a*b^9*c + 160*a^2*b^7*c^2 - 640*a^3*b^5*c^3 +
1280*a^4*b^3*c^4 - 1024*a^5*b*c^5)*d*e + (a*b^10 - 20*a^2*b^8*c + 160*a^3*b^6*c^
2 - 640*a^4*b^4*c^3 + 1280*a^5*b^2*c^4 - 1024*a^6*c^5)*e^2)) + 27*(256*c^5*d^4*e
^5 - 512*b*c^4*d^3*e^6 + 48*(7*b^2*c^3 + 4*a*c^4)*d^2*e^7 - 16*(5*b^3*c^2 + 12*a
*b*c^3)*d*e^8 + (5*b^4*c + 40*a*b^2*c^2 + 16*a^2*c^3)*e^9)*sqrt(e*x + d)) - 2*(1
2*(2*c^3*d - b*c^2*e)*x^3 + (36*b*c^2*d - (19*b^2*c - 4*a*c^2)*e)*x^2 - 2*(b^3 -
10*a*b*c)*d - 3*(a*b^2 + 4*a^2*c)*e + (8*(b^2*c + 5*a*c^2)*d - (5*b^3 + 16*a*b*
c)*e)*x)*sqrt(e*x + d))/(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^4*c^2 - 8*a*b^2
*c^3 + 16*a^2*c^4)*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^3 + (b^6 - 6*a
*b^4*c + 32*a^3*c^3)*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**(3/2)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(3/2)/(c*x^2 + b*x + a)^3,x, algorithm="giac")

[Out]

Timed out