Optimal. Leaf size=36 \[ -\frac {\left (a+b x^3\right )^{2/3} \, _2F_1\left (\frac {1}{3},1;\frac {2}{3};-\frac {b x^3}{a}\right )}{a x} \]
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Rubi [A] time = 0.02, antiderivative size = 49, normalized size of antiderivative = 1.36, 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 = {365, 364} \[ -\frac {\sqrt [3]{\frac {b x^3}{a}+1} \, _2F_1\left (-\frac {1}{3},\frac {1}{3};\frac {2}{3};-\frac {b x^3}{a}\right )}{x \sqrt [3]{a+b x^3}} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt [3]{a+b x^3}} \, dx &=\frac {\sqrt [3]{1+\frac {b x^3}{a}} \int \frac {1}{x^2 \sqrt [3]{1+\frac {b x^3}{a}}} \, dx}{\sqrt [3]{a+b x^3}}\\ &=-\frac {\sqrt [3]{1+\frac {b x^3}{a}} \, _2F_1\left (-\frac {1}{3},\frac {1}{3};\frac {2}{3};-\frac {b x^3}{a}\right )}{x \sqrt [3]{a+b x^3}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 49, normalized size = 1.36 \[ -\frac {\sqrt [3]{\frac {b x^3}{a}+1} \, _2F_1\left (-\frac {1}{3},\frac {1}{3};\frac {2}{3};-\frac {b x^3}{a}\right )}{x \sqrt [3]{a+b x^3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{b x^{5} + a x^{2}}, x\right ) \]
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 \[ \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.11, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {1}{3}} x^{2}}\, dx \]
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 \[ \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.22, size = 40, normalized size = 1.11 \[ -\frac {{\left (\frac {a}{b\,x^3}+1\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {2}{3};\ \frac {5}{3};\ -\frac {a}{b\,x^3}\right )}{2\,x\,{\left (b\,x^3+a\right )}^{1/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.54, size = 39, normalized size = 1.08 \[ \frac {\Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [3]{a} x \Gamma \left (\frac {2}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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