3.418 \(\int \frac{x^7 \left (c+d x^3\right )^{3/2}}{\left (8 c-d x^3\right )^2} \, dx\)

Optimal. Leaf size=681 \[ \frac{108 \sqrt{3} c^{13/6} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{d^{8/3}}-\frac{108 c^{13/6} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt{c+d x^3}}\right )}{d^{8/3}}+\frac{108 c^{13/6} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{d^{8/3}}+\frac{5906 \sqrt{2} c^{7/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{13 \sqrt [4]{3} d^{8/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{2953 \sqrt [4]{3} \sqrt{2-\sqrt{3}} c^{7/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{13 d^{8/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}+\frac{5906 c^2 \sqrt{c+d x^3}}{13 d^{8/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac{103 c x^2 \sqrt{c+d x^3}}{13 d^2}+\frac{x^5 \left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )}+\frac{19 x^5 \sqrt{c+d x^3}}{39 d} \]

[Out]

(103*c*x^2*Sqrt[c + d*x^3])/(13*d^2) + (19*x^5*Sqrt[c + d*x^3])/(39*d) + (5906*c
^2*Sqrt[c + d*x^3])/(13*d^(8/3)*((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)) + (x^5*(c +
d*x^3)^(3/2))/(3*d*(8*c - d*x^3)) + (108*Sqrt[3]*c^(13/6)*ArcTan[(Sqrt[3]*c^(1/
6)*(c^(1/3) + d^(1/3)*x))/Sqrt[c + d*x^3]])/d^(8/3) - (108*c^(13/6)*ArcTanh[(c^(
1/3) + d^(1/3)*x)^2/(3*c^(1/6)*Sqrt[c + d*x^3])])/d^(8/3) + (108*c^(13/6)*ArcTan
h[Sqrt[c + d*x^3]/(3*Sqrt[c])])/d^(8/3) - (2953*3^(1/4)*Sqrt[2 - Sqrt[3]]*c^(7/3
)*(c^(1/3) + d^(1/3)*x)*Sqrt[(c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2)/((1 + S
qrt[3])*c^(1/3) + d^(1/3)*x)^2]*EllipticE[ArcSin[((1 - Sqrt[3])*c^(1/3) + d^(1/3
)*x)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)], -7 - 4*Sqrt[3]])/(13*d^(8/3)*Sqrt[(c^
(1/3)*(c^(1/3) + d^(1/3)*x))/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*Sqrt[c + d*x
^3]) + (5906*Sqrt[2]*c^(7/3)*(c^(1/3) + d^(1/3)*x)*Sqrt[(c^(2/3) - c^(1/3)*d^(1/
3)*x + d^(2/3)*x^2)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*EllipticF[ArcSin[((1
- Sqrt[3])*c^(1/3) + d^(1/3)*x)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)], -7 - 4*Sqr
t[3]])/(13*3^(1/4)*d^(8/3)*Sqrt[(c^(1/3)*(c^(1/3) + d^(1/3)*x))/((1 + Sqrt[3])*c
^(1/3) + d^(1/3)*x)^2]*Sqrt[c + d*x^3])

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Rubi [A]  time = 2.0212, antiderivative size = 681, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 15, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556 \[ \frac{108 \sqrt{3} c^{13/6} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt{c+d x^3}}\right )}{d^{8/3}}-\frac{108 c^{13/6} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2}{3 \sqrt [6]{c} \sqrt{c+d x^3}}\right )}{d^{8/3}}+\frac{108 c^{13/6} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{d^{8/3}}+\frac{5906 \sqrt{2} c^{7/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{13 \sqrt [4]{3} d^{8/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}-\frac{2953 \sqrt [4]{3} \sqrt{2-\sqrt{3}} c^{7/3} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt{\frac{c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{d} x+\left (1-\sqrt{3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt{3}\right ) \sqrt [3]{c}}\right )|-7-4 \sqrt{3}\right )}{13 d^{8/3} \sqrt{\frac{\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt{c+d x^3}}+\frac{5906 c^2 \sqrt{c+d x^3}}{13 d^{8/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )}+\frac{103 c x^2 \sqrt{c+d x^3}}{13 d^2}+\frac{x^5 \left (c+d x^3\right )^{3/2}}{3 d \left (8 c-d x^3\right )}+\frac{19 x^5 \sqrt{c+d x^3}}{39 d} \]

Antiderivative was successfully verified.

[In]  Int[(x^7*(c + d*x^3)^(3/2))/(8*c - d*x^3)^2,x]

[Out]

(103*c*x^2*Sqrt[c + d*x^3])/(13*d^2) + (19*x^5*Sqrt[c + d*x^3])/(39*d) + (5906*c
^2*Sqrt[c + d*x^3])/(13*d^(8/3)*((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)) + (x^5*(c +
d*x^3)^(3/2))/(3*d*(8*c - d*x^3)) + (108*Sqrt[3]*c^(13/6)*ArcTan[(Sqrt[3]*c^(1/
6)*(c^(1/3) + d^(1/3)*x))/Sqrt[c + d*x^3]])/d^(8/3) - (108*c^(13/6)*ArcTanh[(c^(
1/3) + d^(1/3)*x)^2/(3*c^(1/6)*Sqrt[c + d*x^3])])/d^(8/3) + (108*c^(13/6)*ArcTan
h[Sqrt[c + d*x^3]/(3*Sqrt[c])])/d^(8/3) - (2953*3^(1/4)*Sqrt[2 - Sqrt[3]]*c^(7/3
)*(c^(1/3) + d^(1/3)*x)*Sqrt[(c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2)/((1 + S
qrt[3])*c^(1/3) + d^(1/3)*x)^2]*EllipticE[ArcSin[((1 - Sqrt[3])*c^(1/3) + d^(1/3
)*x)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)], -7 - 4*Sqrt[3]])/(13*d^(8/3)*Sqrt[(c^
(1/3)*(c^(1/3) + d^(1/3)*x))/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*Sqrt[c + d*x
^3]) + (5906*Sqrt[2]*c^(7/3)*(c^(1/3) + d^(1/3)*x)*Sqrt[(c^(2/3) - c^(1/3)*d^(1/
3)*x + d^(2/3)*x^2)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)^2]*EllipticF[ArcSin[((1
- Sqrt[3])*c^(1/3) + d^(1/3)*x)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)], -7 - 4*Sqr
t[3]])/(13*3^(1/4)*d^(8/3)*Sqrt[(c^(1/3)*(c^(1/3) + d^(1/3)*x))/((1 + Sqrt[3])*c
^(1/3) + d^(1/3)*x)^2]*Sqrt[c + d*x^3])

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Rubi in Sympy [A]  time = 24.1229, size = 51, normalized size = 0.07 \[ \frac{x^{8} \sqrt{c + d x^{3}} \operatorname{appellf_{1}}{\left (\frac{8}{3},- \frac{3}{2},2,\frac{11}{3},- \frac{d x^{3}}{c},\frac{d x^{3}}{8 c} \right )}}{512 c \sqrt{1 + \frac{d x^{3}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7*(d*x**3+c)**(3/2)/(-d*x**3+8*c)**2,x)

[Out]

x**8*sqrt(c + d*x**3)*appellf1(8/3, -3/2, 2, 11/3, -d*x**3/c, d*x**3/(8*c))/(512
*c*sqrt(1 + d*x**3/c))

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Mathematica [C]  time = 0.362557, size = 357, normalized size = 0.52 \[ \frac{2 \left (\frac{82400 c^4 x^2 F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )}{3 d x^3 \left (F_1\left (\frac{5}{3};\frac{1}{2},2;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )-4 F_1\left (\frac{5}{3};\frac{3}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )+40 c F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )}+\frac{94496 c^3 d x^5 F_1\left (\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )}{3 d x^3 \left (F_1\left (\frac{8}{3};\frac{1}{2},2;\frac{11}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )-4 F_1\left (\frac{8}{3};\frac{3}{2},1;\frac{11}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )+64 c F_1\left (\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )}+5 \left (c+d x^3\right ) \left (-412 c^2 x^2+24 c d x^5+d^2 x^8\right )\right )}{65 d^2 \left (d x^3-8 c\right ) \sqrt{c+d x^3}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(x^7*(c + d*x^3)^(3/2))/(8*c - d*x^3)^2,x]

[Out]

(2*(5*(c + d*x^3)*(-412*c^2*x^2 + 24*c*d*x^5 + d^2*x^8) + (82400*c^4*x^2*AppellF
1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(40*c*AppellF1[2/3, 1/2, 1, 5/
3, -((d*x^3)/c), (d*x^3)/(8*c)] + 3*d*x^3*(AppellF1[5/3, 1/2, 2, 8/3, -((d*x^3)/
c), (d*x^3)/(8*c)] - 4*AppellF1[5/3, 3/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)]))
+ (94496*c^3*d*x^5*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)])/(64
*c*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), (d*x^3)/(8*c)] + 3*d*x^3*(AppellF1[8
/3, 1/2, 2, 11/3, -((d*x^3)/c), (d*x^3)/(8*c)] - 4*AppellF1[8/3, 3/2, 1, 11/3, -
((d*x^3)/c), (d*x^3)/(8*c)]))))/(65*d^2*(-8*c + d*x^3)*Sqrt[c + d*x^3])

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Maple [C]  time = 0.055, size = 2223, normalized size = 3.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7*(d*x^3+c)^(3/2)/(-d*x^3+8*c)^2,x)

[Out]

1/d^2*(2/13*x^5*d*(d*x^3+c)^(1/2)+32/91*c*x^2*(d*x^3+c)^(1/2)-18/91*I*c^2*3^(1/2
)/d*(-c*d^2)^(1/3)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^
(1/2)*d/(-c*d^2)^(1/3))^(1/2)*((x-1/d*(-c*d^2)^(1/3))/(-3/2/d*(-c*d^2)^(1/3)+1/2
*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)*(-I*(x+1/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*
(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*((-3/2/d*(-c*d^2
)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*EllipticE(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^
2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),(I*3^(1
/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/
2))+1/d*(-c*d^2)^(1/3)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^
(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),(I*3^(1/2)/d*(-c*d^2)^(1
/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2))))+16*c/d^2*(2
/7*x^2*(d*x^3+c)^(1/2)-44/7*I*c*3^(1/2)/d*(-c*d^2)^(1/3)*(I*(x+1/2/d*(-c*d^2)^(1
/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2)*((x-1/d*(-c*
d^2)^(1/3))/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)*(-I*(x
+1/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^
(1/2)/(d*x^3+c)^(1/2)*((-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*El
lipticE(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3
^(1/2)*d/(-c*d^2)^(1/3))^(1/2),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3
)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2))+1/d*(-c*d^2)^(1/3)*EllipticF(1/3*3^(1/
2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)
^(1/3))^(1/2),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d
*(-c*d^2)^(1/3)))^(1/2)))+3*I*c/d^3*2^(1/2)*sum(1/_alpha*(-c*d^2)^(1/3)*(1/2*I*d
*(2*x+1/d*(-I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)*(d*(
x-1/d*(-c*d^2)^(1/3))/(-3*(-c*d^2)^(1/3)+I*3^(1/2)*(-c*d^2)^(1/3)))^(1/2)*(-1/2*
I*d*(2*x+1/d*(I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)/(d
*x^3+c)^(1/2)*(I*(-c*d^2)^(1/3)*_alpha*3^(1/2)*d+2*_alpha^2*d^2-I*3^(1/2)*(-c*d^
2)^(2/3)-(-c*d^2)^(1/3)*_alpha*d-(-c*d^2)^(2/3))*EllipticPi(1/3*3^(1/2)*(I*(x+1/
2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/
2),-1/18/d*(2*I*_alpha^2*(-c*d^2)^(1/3)*3^(1/2)*d-I*_alpha*(-c*d^2)^(2/3)*3^(1/2
)+I*3^(1/2)*c*d-3*_alpha*(-c*d^2)^(2/3)-3*c*d)/c,(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3
/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)),_alpha=RootOf(_Z^3*d
-8*c)))+64*c^2/d^2*(-3/8*x^2/(d*x^3-8*c)*(d*x^3+c)^(1/2)-19/24*I*3^(1/2)/d*(-c*d
^2)^(1/3)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(
-c*d^2)^(1/3))^(1/2)*((x-1/d*(-c*d^2)^(1/3))/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2
)/d*(-c*d^2)^(1/3)))^(1/2)*(-I*(x+1/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^
(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*((-3/2/d*(-c*d^2)^(1/3)+1
/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*EllipticE(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-
1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),(I*3^(1/2)/d*(-c
*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2))+1/d*(
-c*d^2)^(1/3)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(
-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2
/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)))+3/8*I/d^3*2^(1/2)*sum
(1/_alpha*(-c*d^2)^(1/3)*(1/2*I*d*(2*x+1/d*(-I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(
1/3)))/(-c*d^2)^(1/3))^(1/2)*(d*(x-1/d*(-c*d^2)^(1/3))/(-3*(-c*d^2)^(1/3)+I*3^(1
/2)*(-c*d^2)^(1/3)))^(1/2)*(-1/2*I*d*(2*x+1/d*(I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)
^(1/3)))/(-c*d^2)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*(I*(-c*d^2)^(1/3)*_alpha*3^(1/2)*
d+2*_alpha^2*d^2-I*3^(1/2)*(-c*d^2)^(2/3)-(-c*d^2)^(1/3)*_alpha*d-(-c*d^2)^(2/3)
)*EllipticPi(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/
3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),-1/18/d*(2*I*_alpha^2*(-c*d^2)^(1/3)*3^(1/2)
*d-I*_alpha*(-c*d^2)^(2/3)*3^(1/2)+I*3^(1/2)*c*d-3*_alpha*(-c*d^2)^(2/3)-3*c*d)/
c,(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1
/3)))^(1/2)),_alpha=RootOf(_Z^3*d-8*c)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{3} + c\right )}^{\frac{3}{2}} x^{7}}{{\left (d x^{3} - 8 \, c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^3 + c)^(3/2)*x^7/(d*x^3 - 8*c)^2,x, algorithm="maxima")

[Out]

integrate((d*x^3 + c)^(3/2)*x^7/(d*x^3 - 8*c)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^3 + c)^(3/2)*x^7/(d*x^3 - 8*c)^2,x, algorithm="fricas")

[Out]

Timed out

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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(x**7*(d*x**3+c)**(3/2)/(-d*x**3+8*c)**2,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{3} + c\right )}^{\frac{3}{2}} x^{7}}{{\left (d x^{3} - 8 \, c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^3 + c)^(3/2)*x^7/(d*x^3 - 8*c)^2,x, algorithm="giac")

[Out]

integrate((d*x^3 + c)^(3/2)*x^7/(d*x^3 - 8*c)^2, x)