3.576 \(\int \frac{1}{x^6 \left (a+b x^3\right )^{2/3}} \, dx\)

Optimal. Leaf size=38 \[ -\frac{\sqrt [3]{a+b x^3} \, _2F_1\left (-\frac{4}{3},1;-\frac{2}{3};-\frac{b x^3}{a}\right )}{5 a x^5} \]

[Out]

-((a + b*x^3)^(1/3)*Hypergeometric2F1[-4/3, 1, -2/3, -((b*x^3)/a)])/(5*a*x^5)

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Rubi [A]  time = 0.0528049, antiderivative size = 51, normalized size of antiderivative = 1.34, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{\left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (-\frac{5}{3},\frac{2}{3};-\frac{2}{3};-\frac{b x^3}{a}\right )}{5 x^5 \left (a+b x^3\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^6*(a + b*x^3)^(2/3)),x]

[Out]

-((1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[-5/3, 2/3, -2/3, -((b*x^3)/a)])/(5*x^5
*(a + b*x^3)^(2/3))

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Rubi in Sympy [A]  time = 6.12112, size = 48, normalized size = 1.26 \[ - \frac{\sqrt [3]{a + b x^{3}}{{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, - \frac{5}{3} \\ - \frac{2}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{5 a x^{5} \sqrt [3]{1 + \frac{b x^{3}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**6/(b*x**3+a)**(2/3),x)

[Out]

-(a + b*x**3)**(1/3)*hyper((2/3, -5/3), (-2/3,), -b*x**3/a)/(5*a*x**5*(1 + b*x**
3/a)**(1/3))

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Mathematica [B]  time = 0.0607139, size = 82, normalized size = 2.16 \[ \frac{-a^2+2 b^2 x^6 \left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right )+a b x^3+2 b^2 x^6}{5 a^2 x^5 \left (a+b x^3\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^6*(a + b*x^3)^(2/3)),x]

[Out]

(-a^2 + a*b*x^3 + 2*b^2*x^6 + 2*b^2*x^6*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[
1/3, 2/3, 4/3, -((b*x^3)/a)])/(5*a^2*x^5*(a + b*x^3)^(2/3))

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Maple [F]  time = 0.046, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{6}} \left ( b{x}^{3}+a \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(b*x^3+a)^(2/3),x)

[Out]

int(1/x^6/(b*x^3+a)^(2/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{3} + a\right )}^{\frac{2}{3}} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)^(2/3)*x^6),x, algorithm="maxima")

[Out]

integrate(1/((b*x^3 + a)^(2/3)*x^6), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{3} + a\right )}^{\frac{2}{3}} x^{6}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)^(2/3)*x^6),x, algorithm="fricas")

[Out]

integral(1/((b*x^3 + a)^(2/3)*x^6), x)

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Sympy [A]  time = 4.01704, size = 44, normalized size = 1.16 \[ \frac{\Gamma \left (- \frac{5}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{3}, \frac{2}{3} \\ - \frac{2}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac{2}{3}} x^{5} \Gamma \left (- \frac{2}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**6/(b*x**3+a)**(2/3),x)

[Out]

gamma(-5/3)*hyper((-5/3, 2/3), (-2/3,), b*x**3*exp_polar(I*pi)/a)/(3*a**(2/3)*x*
*5*gamma(-2/3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{3} + a\right )}^{\frac{2}{3}} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)^(2/3)*x^6),x, algorithm="giac")

[Out]

integrate(1/((b*x^3 + a)^(2/3)*x^6), x)