3.426 \(\int \frac{x^5}{\left (8 c-d x^3\right )^2 \sqrt{c+d x^3}} \, dx\)

Optimal. Leaf size=64 \[ \frac{8 \sqrt{c+d x^3}}{27 d^2 \left (8 c-d x^3\right )}-\frac{10 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{81 \sqrt{c} d^2} \]

[Out]

(8*Sqrt[c + d*x^3])/(27*d^2*(8*c - d*x^3)) - (10*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt
[c])])/(81*Sqrt[c]*d^2)

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Rubi [A]  time = 0.165489, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ \frac{8 \sqrt{c+d x^3}}{27 d^2 \left (8 c-d x^3\right )}-\frac{10 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{81 \sqrt{c} d^2} \]

Antiderivative was successfully verified.

[In]  Int[x^5/((8*c - d*x^3)^2*Sqrt[c + d*x^3]),x]

[Out]

(8*Sqrt[c + d*x^3])/(27*d^2*(8*c - d*x^3)) - (10*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt
[c])])/(81*Sqrt[c]*d^2)

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Rubi in Sympy [A]  time = 16.9277, size = 54, normalized size = 0.84 \[ \frac{8 \sqrt{c + d x^{3}}}{27 d^{2} \left (8 c - d x^{3}\right )} - \frac{10 \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{81 \sqrt{c} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(-d*x**3+8*c)**2/(d*x**3+c)**(1/2),x)

[Out]

8*sqrt(c + d*x**3)/(27*d**2*(8*c - d*x**3)) - 10*atanh(sqrt(c + d*x**3)/(3*sqrt(
c)))/(81*sqrt(c)*d**2)

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Mathematica [A]  time = 0.105544, size = 63, normalized size = 0.98 \[ -\frac{8 \sqrt{c+d x^3}}{27 d^2 \left (d x^3-8 c\right )}-\frac{10 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{81 \sqrt{c} d^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/((8*c - d*x^3)^2*Sqrt[c + d*x^3]),x]

[Out]

(-8*Sqrt[c + d*x^3])/(27*d^2*(-8*c + d*x^3)) - (10*ArcTanh[Sqrt[c + d*x^3]/(3*Sq
rt[c])])/(81*Sqrt[c]*d^2)

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Maple [C]  time = 0.019, size = 861, normalized size = 13.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(-d*x^3+8*c)^2/(d*x^3+c)^(1/2),x)

[Out]

1/27*I/d^4/c*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))+8*c/d*(-1/27/d/c*(d*x^3+c
)^(1/2)/(d*x^3-8*c)-1/486*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^5/(sqrt(d*x^3 + c)*(d*x^3 - 8*c)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.229844, size = 1, normalized size = 0.02 \[ \left [\frac{5 \,{\left (d x^{3} - 8 \, c\right )} \log \left (\frac{{\left (d x^{3} + 10 \, c\right )} \sqrt{c} - 6 \, \sqrt{d x^{3} + c} c}{d x^{3} - 8 \, c}\right ) - 24 \, \sqrt{d x^{3} + c} \sqrt{c}}{81 \,{\left (d^{3} x^{3} - 8 \, c d^{2}\right )} \sqrt{c}}, \frac{2 \,{\left (5 \,{\left (d x^{3} - 8 \, c\right )} \arctan \left (\frac{3 \, c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) - 12 \, \sqrt{d x^{3} + c} \sqrt{-c}\right )}}{81 \,{\left (d^{3} x^{3} - 8 \, c d^{2}\right )} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(sqrt(d*x^3 + c)*(d*x^3 - 8*c)^2),x, algorithm="fricas")

[Out]

[1/81*(5*(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)) - 24*sqrt(d*x^3 + c)*sqrt(c))/((d^3*x^3 - 8*c*d^2)*sqrt(c)), 2/81*(5*(d
*x^3 - 8*c)*arctan(3*c/(sqrt(d*x^3 + c)*sqrt(-c))) - 12*sqrt(d*x^3 + c)*sqrt(-c)
)/((d^3*x^3 - 8*c*d^2)*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**5/(-d*x**3+8*c)**2/(d*x**3+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.214045, size = 78, normalized size = 1.22 \[ \frac{2 \,{\left (\frac{5 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} d} - \frac{12 \, \sqrt{d x^{3} + c}}{{\left (d x^{3} - 8 \, c\right )} d}\right )}}{81 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(sqrt(d*x^3 + c)*(d*x^3 - 8*c)^2),x, algorithm="giac")

[Out]

2/81*(5*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*d) - 12*sqrt(d*x^3 + c)/(
(d*x^3 - 8*c)*d))/d