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

Optimal. Leaf size=52 \[ \frac{2}{27 d^2 \sqrt{c+d x^3}}+\frac{16 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{81 \sqrt{c} d^2} \]

[Out]

2/(27*d^2*Sqrt[c + d*x^3]) + (16*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(81*Sqrt[
c]*d^2)

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Rubi [A]  time = 0.153006, antiderivative size = 52, 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 \[ \frac{2}{27 d^2 \sqrt{c+d x^3}}+\frac{16 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{81 \sqrt{c} d^2} \]

Antiderivative was successfully verified.

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

[Out]

2/(27*d^2*Sqrt[c + d*x^3]) + (16*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/(81*Sqrt[
c]*d^2)

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Rubi in Sympy [A]  time = 16.5167, size = 46, normalized size = 0.88 \[ \frac{2}{27 d^{2} \sqrt{c + d x^{3}}} + \frac{16 \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{3}}}{3 \sqrt{c}} \right )}}{81 \sqrt{c} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/(27*d**2*sqrt(c + d*x**3)) + 16*atanh(sqrt(c + d*x**3)/(3*sqrt(c)))/(81*sqrt(c
)*d**2)

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Mathematica [A]  time = 0.0673865, size = 49, normalized size = 0.94 \[ \frac{2 \left (\frac{3}{\sqrt{c+d x^3}}+\frac{8 \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{3 \sqrt{c}}\right )}{\sqrt{c}}\right )}{81 d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.014, size = 456, normalized size = 8.8 \[{\frac{2}{3\,{d}^{2}}{\frac{1}{\sqrt{d{x}^{3}+c}}}}-8\,{\frac{c}{d} \left ({\frac{2}{27\,cd}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{c}{d}} \right ) d}}}}+{\frac{{\frac{i}{243}}\sqrt{2}}{{d}^{3}{c}^{2}}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{3}d-8\,c \right ) }{\frac{\sqrt [3]{-c{d}^{2}} \left ( i\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,\sqrt{3}d+2\,{{\it \_alpha}}^{2}{d}^{2}-i\sqrt{3} \left ( -c{d}^{2} \right ) ^{2/3}-\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,d- \left ( -c{d}^{2} \right ) ^{2/3} \right ) }{\sqrt{d{x}^{3}+c}}\sqrt{{\frac{i/2d}{\sqrt [3]{-c{d}^{2}}} \left ( 2\,x+{\frac{-i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}}}{d}} \right ) }}\sqrt{{\frac{d}{-3\,\sqrt [3]{-c{d}^{2}}+i\sqrt{3}\sqrt [3]{-c{d}^{2}}} \left ( x-{\frac{\sqrt [3]{-c{d}^{2}}}{d}} \right ) }}\sqrt{{\frac{-i/2d}{\sqrt [3]{-c{d}^{2}}} \left ( 2\,x+{\frac{i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}}}{d}} \right ) }}{\it EllipticPi} \left ( 1/3\,\sqrt{3}\sqrt{{\frac{i\sqrt{3}d}{\sqrt [3]{-c{d}^{2}}} \left ( x+1/2\,{\frac{\sqrt [3]{-c{d}^{2}}}{d}}-{\frac{i/2\sqrt{3}\sqrt [3]{-c{d}^{2}}}{d}} \right ) }},-1/18\,{\frac{2\,i{{\it \_alpha}}^{2}\sqrt [3]{-c{d}^{2}}\sqrt{3}d-i{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}\sqrt{3}+i\sqrt{3}cd-3\,{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}-3\,cd}{cd}},\sqrt{{\frac{i\sqrt{3}\sqrt [3]{-c{d}^{2}}}{d} \left ( -3/2\,{\frac{\sqrt [3]{-c{d}^{2}}}{d}}+{\frac{i/2\sqrt{3}\sqrt [3]{-c{d}^{2}}}{d}} \right ) ^{-1}}} \right ) }} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3/d^2/(d*x^3+c)^(1/2)-8*c/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)))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235296, size = 1, normalized size = 0.02 \[ \left [\frac{2 \,{\left (4 \, \sqrt{d x^{3} + c} \log \left (\frac{{\left (d x^{3} + 10 \, c\right )} \sqrt{c} + 6 \, \sqrt{d x^{3} + c} c}{d x^{3} - 8 \, c}\right ) + 3 \, \sqrt{c}\right )}}{81 \, \sqrt{d x^{3} + c} \sqrt{c} d^{2}}, -\frac{2 \,{\left (8 \, \sqrt{d x^{3} + c} \arctan \left (\frac{3 \, c}{\sqrt{d x^{3} + c} \sqrt{-c}}\right ) - 3 \, \sqrt{-c}\right )}}{81 \, \sqrt{d x^{3} + c} \sqrt{-c} d^{2}}\right ] \]

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)),x, algorithm="fricas")

[Out]

[2/81*(4*sqrt(d*x^3 + c)*log(((d*x^3 + 10*c)*sqrt(c) + 6*sqrt(d*x^3 + c)*c)/(d*x
^3 - 8*c)) + 3*sqrt(c))/(sqrt(d*x^3 + c)*sqrt(c)*d^2), -2/81*(8*sqrt(d*x^3 + c)*
arctan(3*c/(sqrt(d*x^3 + c)*sqrt(-c))) - 3*sqrt(-c))/(sqrt(d*x^3 + c)*sqrt(-c)*d
^2)]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.21805, size = 63, normalized size = 1.21 \[ -\frac{2 \,{\left (\frac{8 \, \arctan \left (\frac{\sqrt{d x^{3} + c}}{3 \, \sqrt{-c}}\right )}{\sqrt{-c} d} - \frac{3}{\sqrt{d x^{3} + c} d}\right )}}{81 \, d} \]

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)),x, algorithm="giac")

[Out]

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