Optimal. Leaf size=33 \[ \frac{x \sqrt [3]{a+b x^3} \, _2F_1\left (\frac{2}{3},1;\frac{4}{3};-\frac{b x^3}{a}\right )}{a} \]
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Rubi [A] time = 0.0249702, antiderivative size = 46, normalized size of antiderivative = 1.39, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{x \left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{\left (a+b x^3\right )^{2/3}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^3)^(-2/3),x]
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Rubi in Sympy [A] time = 3.65654, size = 39, normalized size = 1.18 \[ \frac{x \sqrt [3]{a + b x^{3}}{{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{a \sqrt [3]{1 + \frac{b x^{3}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x**3+a)**(2/3),x)
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Mathematica [C] time = 0.395106, size = 177, normalized size = 5.36 \[ \frac{3 \sqrt [3]{2} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\frac{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}\right )^{2/3} \sqrt [3]{\frac{i \left (\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}+1\right )}{\sqrt{3}+3 i}} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};\frac{\left (1-i \sqrt{3}\right ) \sqrt [3]{b} x+\left (1+i \sqrt{3}\right ) \sqrt [3]{a}}{2 \left (\sqrt [3]{b} x+\sqrt [3]{a}\right )}\right )}{\sqrt [3]{b} \left (a+b x^3\right )^{2/3}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(a + b*x^3)^(-2/3),x]
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Maple [F] time = 0.039, size = 0, normalized size = 0. \[ \int \left ( b{x}^{3}+a \right ) ^{-{\frac{2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x^3+a)^(2/3),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a)^(-2/3),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a)^(-2/3),x, algorithm="fricas")
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Sympy [A] time = 2.13136, size = 36, normalized size = 1.09 \[ \frac{x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{2}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac{2}{3}} \Gamma \left (\frac{4}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x**3+a)**(2/3),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a)^(-2/3),x, algorithm="giac")
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