3.556 \(\int \frac{1}{x^3 \sqrt [3]{a+b x^3}} \, dx\)

Optimal. Leaf size=21 \[ -\frac{\left (a+b x^3\right )^{2/3}}{2 a x^2} \]

[Out]

-(a + b*x^3)^(2/3)/(2*a*x^2)

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Rubi [A]  time = 0.0207077, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{\left (a+b x^3\right )^{2/3}}{2 a x^2} \]

Antiderivative was successfully verified.

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

[Out]

-(a + b*x^3)^(2/3)/(2*a*x^2)

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Rubi in Sympy [A]  time = 2.69498, size = 17, normalized size = 0.81 \[ - \frac{\left (a + b x^{3}\right )^{\frac{2}{3}}}{2 a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a + b*x**3)**(2/3)/(2*a*x**2)

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Mathematica [A]  time = 0.0174928, size = 21, normalized size = 1. \[ -\frac{\left (a+b x^3\right )^{2/3}}{2 a x^2} \]

Antiderivative was successfully verified.

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

[Out]

-(a + b*x^3)^(2/3)/(2*a*x^2)

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Maple [A]  time = 0.007, size = 18, normalized size = 0.9 \[ -{\frac{1}{2\,a{x}^{2}} \left ( b{x}^{3}+a \right ) ^{{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(b*x^3+a)^(1/3),x)

[Out]

-1/2*(b*x^3+a)^(2/3)/a/x^2

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Maxima [A]  time = 1.44638, size = 23, normalized size = 1.1 \[ -\frac{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{2 \, a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(b*x^3 + a)^(2/3)/(a*x^2)

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Fricas [A]  time = 0.236244, size = 23, normalized size = 1.1 \[ -\frac{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{2 \, a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(b*x^3 + a)^(2/3)/(a*x^2)

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Sympy [A]  time = 2.05887, size = 31, normalized size = 1.48 \[ \frac{b^{\frac{2}{3}} \left (\frac{a}{b x^{3}} + 1\right )^{\frac{2}{3}} \Gamma \left (- \frac{2}{3}\right )}{3 a \Gamma \left (\frac{1}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**(2/3)*(a/(b*x**3) + 1)**(2/3)*gamma(-2/3)/(3*a*gamma(1/3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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