Optimal. Leaf size=89 \[ \frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b} x^{n/3}}{\sqrt [3]{a+b x^n}}+1}{\sqrt {3}}\right )}{\sqrt [3]{b} n}-\frac {3 \log \left (\sqrt [3]{b} x^{n/3}-\sqrt [3]{a+b x^n}\right )}{2 \sqrt [3]{b} n} \]
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Rubi [A] time = 0.04, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {345, 239} \[ \frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b} x^{n/3}}{\sqrt [3]{a+b x^n}}+1}{\sqrt {3}}\right )}{\sqrt [3]{b} n}-\frac {3 \log \left (\sqrt [3]{b} x^{n/3}-\sqrt [3]{a+b x^n}\right )}{2 \sqrt [3]{b} n} \]
Antiderivative was successfully verified.
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Rule 239
Rule 345
Rubi steps
\begin {align*} \int \frac {x^{-1+\frac {n}{3}}}{\sqrt [3]{a+b x^n}} \, dx &=\frac {3 \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a+b x^3}} \, dx,x,x^{n/3}\right )}{n}\\ &=\frac {\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x^{n/3}}{\sqrt [3]{a+b x^n}}}{\sqrt {3}}\right )}{\sqrt [3]{b} n}-\frac {3 \log \left (\sqrt [3]{b} x^{n/3}-\sqrt [3]{a+b x^n}\right )}{2 \sqrt [3]{b} n}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 56, normalized size = 0.63 \[ \frac {3 x^{n/3} \sqrt [3]{\frac {b x^n}{a}+1} \, _2F_1\left (\frac {1}{3},\frac {1}{3};\frac {4}{3};-\frac {b x^n}{a}\right )}{n \sqrt [3]{a+b x^n}} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {x^{\frac {1}{3} \, n - 1}}{{\left (b x^{n} + a\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.21, size = 0, normalized size = 0.00 \[ \int \frac {x^{\frac {n}{3}-1}}{\left (b \,x^{n}+a \right )^{\frac {1}{3}}}\, dx \]
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 \[ \int \frac {x^{\frac {1}{3} \, n - 1}}{{\left (b x^{n} + a\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^{\frac {n}{3}-1}}{{\left (a+b\,x^n\right )}^{1/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 7.81, size = 39, normalized size = 0.44 \[ \frac {x^{\frac {n}{3}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{\sqrt [3]{a} n \Gamma \left (\frac {4}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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