Integrand size = 17, antiderivative size = 69 \[ \int (c x)^m \left (a+b x^3\right )^{4/3} \, dx=\frac {a (c x)^{1+m} \sqrt [3]{a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {4}{3},\frac {1+m}{3},\frac {4+m}{3},-\frac {b x^3}{a}\right )}{c (1+m) \sqrt [3]{1+\frac {b x^3}{a}}} \]
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Time = 0.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {372, 371} \[ \int (c x)^m \left (a+b x^3\right )^{4/3} \, dx=\frac {a \sqrt [3]{a+b x^3} (c x)^{m+1} \operatorname {Hypergeometric2F1}\left (-\frac {4}{3},\frac {m+1}{3},\frac {m+4}{3},-\frac {b x^3}{a}\right )}{c (m+1) \sqrt [3]{\frac {b x^3}{a}+1}} \]
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Rule 371
Rule 372
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a \sqrt [3]{a+b x^3}\right ) \int (c x)^m \left (1+\frac {b x^3}{a}\right )^{4/3} \, dx}{\sqrt [3]{1+\frac {b x^3}{a}}} \\ & = \frac {a (c x)^{1+m} \sqrt [3]{a+b x^3} \, _2F_1\left (-\frac {4}{3},\frac {1+m}{3};\frac {4+m}{3};-\frac {b x^3}{a}\right )}{c (1+m) \sqrt [3]{1+\frac {b x^3}{a}}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.97 \[ \int (c x)^m \left (a+b x^3\right )^{4/3} \, dx=\frac {a x (c x)^m \sqrt [3]{a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {4}{3},\frac {1+m}{3},1+\frac {1+m}{3},-\frac {b x^3}{a}\right )}{(1+m) \sqrt [3]{1+\frac {b x^3}{a}}} \]
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\[\int \left (c x \right )^{m} \left (b \,x^{3}+a \right )^{\frac {4}{3}}d x\]
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\[ \int (c x)^m \left (a+b x^3\right )^{4/3} \, dx=\int { {\left (b x^{3} + a\right )}^{\frac {4}{3}} \left (c x\right )^{m} \,d x } \]
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Result contains complex when optimal does not.
Time = 3.14 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.84 \[ \int (c x)^m \left (a+b x^3\right )^{4/3} \, dx=\frac {a^{\frac {4}{3}} c^{m} x^{m + 1} \Gamma \left (\frac {m}{3} + \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {4}{3}, \frac {m}{3} + \frac {1}{3} \\ \frac {m}{3} + \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {m}{3} + \frac {4}{3}\right )} \]
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\[ \int (c x)^m \left (a+b x^3\right )^{4/3} \, dx=\int { {\left (b x^{3} + a\right )}^{\frac {4}{3}} \left (c x\right )^{m} \,d x } \]
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\[ \int (c x)^m \left (a+b x^3\right )^{4/3} \, dx=\int { {\left (b x^{3} + a\right )}^{\frac {4}{3}} \left (c x\right )^{m} \,d x } \]
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Timed out. \[ \int (c x)^m \left (a+b x^3\right )^{4/3} \, dx=\int {\left (c\,x\right )}^m\,{\left (b\,x^3+a\right )}^{4/3} \,d x \]
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