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

Optimal. Leaf size=85 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{243 c^{3/2} d^2}+\frac{8}{27 d^2 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}-\frac{2}{81 c d^2 \sqrt{c+d x^3}} \]

[Out]

-2/(81*c*d^2*Sqrt[c + d*x^3]) + 8/(27*d^2*(8*c - d*x^3)*Sqrt[c + d*x^3]) + (2*Ar
cTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(243*c^(3/2)*d^2)

_______________________________________________________________________________________

Rubi [A]  time = 0.198891, antiderivative size = 85, 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 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{243 c^{3/2} d^2}+\frac{8}{27 d^2 \left (8 c-d x^3\right ) \sqrt{c+d x^3}}-\frac{2}{81 c d^2 \sqrt{c+d x^3}} \]

Antiderivative was successfully verified.

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

[Out]

-2/(81*c*d^2*Sqrt[c + d*x^3]) + 8/(27*d^2*(8*c - d*x^3)*Sqrt[c + d*x^3]) + (2*Ar
cTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(243*c^(3/2)*d^2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 22.1705, size = 73, normalized size = 0.86 \[ \frac{8}{27 d^{2} \sqrt{c + d x^{3}} \left (8 c - d x^{3}\right )} - \frac{2}{81 c d^{2} \sqrt{c + d x^{3}}} + \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{243 c^{\frac{3}{2}} d^{2}} \]

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)**(3/2),x)

[Out]

8/(27*d**2*sqrt(c + d*x**3)*(8*c - d*x**3)) - 2/(81*c*d**2*sqrt(c + d*x**3)) + 2
*atanh(sqrt(c + d*x**3)/(3*sqrt(c)))/(243*c**(3/2)*d**2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.209956, size = 73, normalized size = 0.86 \[ \frac{2 \left (\frac{3 \sqrt{c} \left (4 c+d x^3\right )}{\left (8 c-d x^3\right ) \sqrt{c+d x^3}}+\tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )\right )}{243 c^{3/2} d^2} \]

Antiderivative was successfully verified.

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

[Out]

(2*((3*Sqrt[c]*(4*c + d*x^3))/((8*c - d*x^3)*Sqrt[c + d*x^3]) + ArcTanh[Sqrt[c +
d*x^3]/(3*Sqrt[c])]))/(243*c^(3/2)*d^2)

_______________________________________________________________________________________

Maple [C]  time = 0.017, size = 908, normalized size = 10.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/d*(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)))+8*c/d*(-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)))

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.223638, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{d x^{3} + c}{\left (d x^{3} - 8 \, c\right )} \log \left (\frac{{\left (d x^{3} + 10 \, c\right )} \sqrt{c} + 6 \, \sqrt{d x^{3} + c} c}{d x^{3} - 8 \, c}\right ) - 6 \,{\left (d x^{3} + 4 \, c\right )} \sqrt{c}}{243 \,{\left (c d^{3} x^{3} - 8 \, c^{2} d^{2}\right )} \sqrt{d x^{3} + c} \sqrt{c}}, -\frac{2 \,{\left (\sqrt{d x^{3} + c}{\left (d x^{3} - 8 \, c\right )} \arctan \left (\frac{3 \, c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) + 3 \,{\left (d x^{3} + 4 \, c\right )} \sqrt{-c}\right )}}{243 \,{\left (c d^{3} x^{3} - 8 \, c^{2} d^{2}\right )} \sqrt{d x^{3} + c} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/243*(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)) - 6*(d*x^3 + 4*c)*sqrt(c))/((c*d^3*x^3 - 8*c^2*d^2)*sqrt
(d*x^3 + c)*sqrt(c)), -2/243*(sqrt(d*x^3 + c)*(d*x^3 - 8*c)*arctan(3*c/(sqrt(d*x
^3 + c)*sqrt(-c))) + 3*(d*x^3 + 4*c)*sqrt(-c))/((c*d^3*x^3 - 8*c^2*d^2)*sqrt(d*x
^3 + c)*sqrt(-c))]

_______________________________________________________________________________________

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)**(3/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.22691, size = 103, normalized size = 1.21 \[ -\frac{2 \,{\left (\frac{\arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} c d} + \frac{3 \,{\left (d x^{3} + 4 \, c\right )}}{{\left ({\left (d x^{3} + c\right )}^{\frac{3}{2}} - 9 \, \sqrt{d x^{3} + c} c\right )} c d}\right )}}{243 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/243*(arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*c*d) + 3*(d*x^3 + 4*c)/((
(d*x^3 + c)^(3/2) - 9*sqrt(d*x^3 + c)*c)*c*d))/d