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

Optimal. Leaf size=23 \[ -\frac{\sqrt [3]{a x^3+b x^6}}{a x^2} \]

[Out]

-((a*x^3 + b*x^6)^(1/3)/(a*x^2))

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Rubi [A]  time = 0.0130483, antiderivative size = 23, 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 \[ -\frac{\sqrt [3]{a x^3+b x^6}}{a x^2} \]

Antiderivative was successfully verified.

[In]  Int[(a*x^3 + b*x^6)^(-2/3),x]

[Out]

-((a*x^3 + b*x^6)^(1/3)/(a*x^2))

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Rubi in Sympy [A]  time = 1.38468, size = 19, normalized size = 0.83 \[ - \frac{\sqrt [3]{a x^{3} + b x^{6}}}{a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a*x**3 + b*x**6)**(1/3)/(a*x**2)

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Mathematica [A]  time = 0.0216424, size = 23, normalized size = 1. \[ -\frac{\sqrt [3]{x^3 \left (a+b x^3\right )}}{a x^2} \]

Antiderivative was successfully verified.

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

[Out]

-((x^3*(a + b*x^3))^(1/3)/(a*x^2))

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Maple [A]  time = 0.006, size = 27, normalized size = 1.2 \[ -{\frac{x \left ( b{x}^{3}+a \right ) }{a} \left ( b{x}^{6}+a{x}^{3} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^6+a*x^3)^(2/3),x)

[Out]

-x*(b*x^3+a)/a/(b*x^6+a*x^3)^(2/3)

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Maxima [A]  time = 0.767714, size = 23, normalized size = 1. \[ -\frac{{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(b*x^3 + a)^(1/3)/(a*x)

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Fricas [A]  time = 0.274283, size = 28, normalized size = 1.22 \[ -\frac{{\left (b x^{6} + a x^{3}\right )}^{\frac{1}{3}}}{a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(b*x^6 + a*x^3)^(1/3)/(a*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a*x**3 + b*x**6)**(-2/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^6 + a*x^3)^(-2/3), x)