Optimal. Leaf size=97 \[ \frac {\tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b} x^{m+1}}{\sqrt [3]{a+b x^{3 (m+1)}}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b} (m+1)}-\frac {\log \left (\sqrt [3]{b} x^{m+1}-\sqrt [3]{a+b x^{3 (m+1)}}\right )}{2 \sqrt [3]{b} (m+1)} \]
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Rubi [A] time = 0.05, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {345, 239} \[ \frac {\tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b} x^{m+1}}{\sqrt [3]{a+b x^{3 (m+1)}}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b} (m+1)}-\frac {\log \left (\sqrt [3]{b} x^{m+1}-\sqrt [3]{a+b x^{3 (m+1)}}\right )}{2 \sqrt [3]{b} (m+1)} \]
Antiderivative was successfully verified.
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Rule 239
Rule 345
Rubi steps
\begin {align*} \int \frac {x^m}{\sqrt [3]{a+b x^{3 (1+m)}}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a+b x^3}} \, dx,x,x^{1+m}\right )}{1+m}\\ &=\frac {\tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x^{1+m}}{\sqrt [3]{a+b x^{3 (1+m)}}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b} (1+m)}-\frac {\log \left (\sqrt [3]{b} x^{1+m}-\sqrt [3]{a+b x^{3 (1+m)}}\right )}{2 \sqrt [3]{b} (1+m)}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 67, normalized size = 0.69 \[ \frac {x^{m+1} \sqrt [3]{\frac {b x^{3 m+3}}{a}+1} \, _2F_1\left (\frac {1}{3},\frac {1}{3};\frac {4}{3};-\frac {b x^{3 m+3}}{a}\right )}{(m+1) \sqrt [3]{a+b x^{3 m+3}}} \]
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 \[ \int \frac {x^{m}}{{\left (b x^{3 \, m + 3} + a\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.22, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\left (b \,x^{3 m +3}+a \right )^{\frac {1}{3}}}\, dx \]
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 \[ \int \frac {x^{m}}{{\left (b x^{3 \, m + 3} + a\right )}^{\frac {1}{3}}}\,{d x} \]
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 \[ \int \frac {x^m}{{\left (a+b\,x^{3\,m+3}\right )}^{1/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.48, size = 117, normalized size = 1.21 \[ \frac {x x^{m} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{3} \\ \frac {m}{3 m + 3} + 1 + \frac {1}{3 m + 3} \end {matrix}\middle | {\frac {b x^{3} x^{3 m} e^{i \pi }}{a}} \right )}}{3 a^{\frac {m}{3 m + 3}} a^{\frac {1}{3 m + 3}} m \Gamma \left (\frac {m}{3 m + 3} + 1 + \frac {1}{3 m + 3}\right ) + 3 a^{\frac {m}{3 m + 3}} a^{\frac {1}{3 m + 3}} \Gamma \left (\frac {m}{3 m + 3} + 1 + \frac {1}{3 m + 3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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