Optimal. Leaf size=38 \[ -\frac {\sqrt [3]{a+b x^3} \, _2F_1\left (-\frac {1}{3},1;\frac {1}{3};-\frac {b x^3}{a}\right )}{2 a x^2} \]
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Rubi [A]
time = 0.01, 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, Rules used = {372, 371}
\begin {gather*} -\frac {\left (\frac {b x^3}{a}+1\right )^{2/3} \, _2F_1\left (-\frac {2}{3},\frac {2}{3};\frac {1}{3};-\frac {b x^3}{a}\right )}{2 x^2 \left (a+b x^3\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 372
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (a+b x^3\right )^{2/3}} \, dx &=\frac {\left (1+\frac {b x^3}{a}\right )^{2/3} \int \frac {1}{x^3 \left (1+\frac {b x^3}{a}\right )^{2/3}} \, dx}{\left (a+b x^3\right )^{2/3}}\\ &=-\frac {\left (1+\frac {b x^3}{a}\right )^{2/3} \, _2F_1\left (-\frac {2}{3},\frac {2}{3};\frac {1}{3};-\frac {b x^3}{a}\right )}{2 x^2 \left (a+b x^3\right )^{2/3}}\\ \end {align*}
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Mathematica [A]
time = 10.01, size = 51, normalized size = 1.34 \begin {gather*} -\frac {\left (1+\frac {b x^3}{a}\right )^{2/3} \, _2F_1\left (-\frac {2}{3},\frac {2}{3};\frac {1}{3};-\frac {b x^3}{a}\right )}{2 x^2 \left (a+b x^3\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{3} \left (b \,x^{3}+a \right )^{\frac {2}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.36, size = 25, normalized size = 0.66 \begin {gather*} {\rm integral}\left (\frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{b x^{6} + a x^{3}}, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.46, size = 41, normalized size = 1.08 \begin {gather*} \frac {\Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {2}{3} \\ \frac {1}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {2}{3}} x^{2} \Gamma \left (\frac {1}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {1}{x^3\,{\left (b\,x^3+a\right )}^{2/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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