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 = 0.01, 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, Rules used = {252, 251}
\begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 251
Rule 252
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^3\right )^{2/3}} \, dx &=\frac {\left (1+\frac {b x^3}{a}\right )^{2/3} \int \frac {1}{\left (1+\frac {b x^3}{a}\right )^{2/3}} \, dx}{\left (a+b x^3\right )^{2/3}}\\ &=\frac {x \left (1+\frac {b x^3}{a}\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}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.16, size = 177, normalized size = 5.36 \begin {gather*} \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 (1+\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 i+\sqrt {3}}} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{a}+\left (1-i \sqrt {3}\right ) \sqrt [3]{b} x}{2 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}\right )}{\sqrt [3]{b} \left (a+b x^3\right )^{2/3}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {2}{3}}}\, dx\]
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 = 0.37, size = 11, normalized size = 0.33 \begin {gather*} {\rm integral}\left (\frac {1}{{\left (b x^{3} + a\right )}^{\frac {2}{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 = 0.37, size = 36, normalized size = 1.09 \begin {gather*} \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 )} \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 [B]
time = 1.02, size = 37, normalized size = 1.12 \begin {gather*} \frac {x\,{\left (\frac {b\,x^3}{a}+1\right )}^{2/3}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {2}{3};\ \frac {4}{3};\ -\frac {b\,x^3}{a}\right )}{{\left (b\,x^3+a\right )}^{2/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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