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

Optimal. Leaf size=83 \[ \frac{64 c}{27 d^3 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}-\frac{22}{81 d^3 \sqrt{c+d x^3}}-\frac{32 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{243 \sqrt{c} d^3} \]

[Out]

-22/(81*d^3*Sqrt[c + d*x^3]) + (64*c)/(27*d^3*(8*c - d*x^3)*Sqrt[c + d*x^3]) - (
32*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(243*Sqrt[c]*d^3)

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Rubi [A]  time = 0.23491, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{64 c}{27 d^3 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}-\frac{22}{81 d^3 \sqrt{c+d x^3}}-\frac{32 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{243 \sqrt{c} d^3} \]

Antiderivative was successfully verified.

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

[Out]

-22/(81*d^3*Sqrt[c + d*x^3]) + (64*c)/(27*d^3*(8*c - d*x^3)*Sqrt[c + d*x^3]) - (
32*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(243*Sqrt[c]*d^3)

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Rubi in Sympy [A]  time = 26.5674, size = 73, normalized size = 0.88 \[ \frac{64 c}{27 d^{3} \sqrt{c + d x^{3}} \left (8 c - d x^{3}\right )} - \frac{22}{81 d^{3} \sqrt{c + d x^{3}}} - \frac{32 \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{243 \sqrt{c} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

64*c/(27*d**3*sqrt(c + d*x**3)*(8*c - d*x**3)) - 22/(81*d**3*sqrt(c + d*x**3)) -
32*atanh(sqrt(c + d*x**3)/(3*sqrt(c)))/(243*sqrt(c)*d**3)

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Mathematica [A]  time = 0.201241, size = 71, normalized size = 0.86 \[ \frac{2 \left (\frac{3 \left (8 c+11 d x^3\right )}{\left (8 c-d x^3\right ) \sqrt{c+d x^3}}-\frac{16 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{\sqrt{c}}\right )}{243 d^3} \]

Antiderivative was successfully verified.

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

[Out]

(2*((3*(8*c + 11*d*x^3))/((8*c - d*x^3)*Sqrt[c + d*x^3]) - (16*ArcTanh[Sqrt[c +
d*x^3]/(3*Sqrt[c])])/Sqrt[c]))/(243*d^3)

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Maple [C]  time = 0.022, size = 926, normalized size = 11.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3/d^3/(d*x^3+c)^(1/2)+64*c^2/d^2*(-1/243/d/c^2*(d*x^3+c)^(1/2)/(d*x^3-8*c)-2/
243/d/c^2/((x^3+c/d)*d)^(1/2)-1/1458*I/d^3/c^3*2^(1/2)*sum((-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)))+16*c/d^2*(2/27/d/c/((x^3+c/d)*d)^(1/2)+1/243*I/d^3/c^2*2^(1/2)*sum((-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. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223688, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (8 \, \sqrt{d x^{3} + c}{\left (d x^{3} - 8 \, c\right )} \log \left (\frac{{\left (d x^{3} + 10 \, c\right )} \sqrt{c} - 6 \, \sqrt{d x^{3} + c} c}{d x^{3} - 8 \, c}\right ) - 3 \,{\left (11 \, d x^{3} + 8 \, c\right )} \sqrt{c}\right )}}{243 \,{\left (d^{4} x^{3} - 8 \, c d^{3}\right )} \sqrt{d x^{3} + c} \sqrt{c}}, \frac{2 \,{\left (16 \, \sqrt{d x^{3} + c}{\left (d x^{3} - 8 \, c\right )} \arctan \left (\frac{3 \, c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) - 3 \,{\left (11 \, d x^{3} + 8 \, c\right )} \sqrt{-c}\right )}}{243 \,{\left (d^{4} x^{3} - 8 \, c d^{3}\right )} \sqrt{d x^{3} + c} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[2/243*(8*sqrt(d*x^3 + c)*(d*x^3 - 8*c)*log(((d*x^3 + 10*c)*sqrt(c) - 6*sqrt(d*x
^3 + c)*c)/(d*x^3 - 8*c)) - 3*(11*d*x^3 + 8*c)*sqrt(c))/((d^4*x^3 - 8*c*d^3)*sqr
t(d*x^3 + c)*sqrt(c)), 2/243*(16*sqrt(d*x^3 + c)*(d*x^3 - 8*c)*arctan(3*c/(sqrt(
d*x^3 + c)*sqrt(-c))) - 3*(11*d*x^3 + 8*c)*sqrt(-c))/((d^4*x^3 - 8*c*d^3)*sqrt(d
*x^3 + c)*sqrt(-c))]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.224799, size = 90, normalized size = 1.08 \[ \frac{32 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{243 \, \sqrt{-c} d^{3}} - \frac{2 \,{\left (11 \, d x^{3} + 8 \, c\right )}}{81 \,{\left ({\left (d x^{3} + c\right )}^{\frac{3}{2}} - 9 \, \sqrt{d x^{3} + c} c\right )} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

32/243*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*d^3) - 2/81*(11*d*x^3 + 8*
c)/(((d*x^3 + c)^(3/2) - 9*sqrt(d*x^3 + c)*c)*d^3)