Integrand size = 15, antiderivative size = 33 \[ \int (1-x)^n (1+x)^{7/3} \, dx=\frac {3}{5} 2^{-1+n} (1+x)^{10/3} \operatorname {Hypergeometric2F1}\left (\frac {10}{3},-n,\frac {13}{3},\frac {1+x}{2}\right ) \]
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Time = 0.00 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {71} \[ \int (1-x)^n (1+x)^{7/3} \, dx=\frac {3}{5} 2^{n-1} (x+1)^{10/3} \operatorname {Hypergeometric2F1}\left (\frac {10}{3},-n,\frac {13}{3},\frac {x+1}{2}\right ) \]
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Rule 71
Rubi steps \begin{align*} \text {integral}& = \frac {3}{5} 2^{-1+n} (1+x)^{10/3} \, _2F_1\left (\frac {10}{3},-n;\frac {13}{3};\frac {1+x}{2}\right ) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int (1-x)^n (1+x)^{7/3} \, dx=\frac {3}{5} 2^{-1+n} (1+x)^{10/3} \operatorname {Hypergeometric2F1}\left (\frac {10}{3},-n,\frac {13}{3},\frac {1+x}{2}\right ) \]
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\[\int \left (1-x \right )^{n} \left (1+x \right )^{\frac {7}{3}}d x\]
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\[ \int (1-x)^n (1+x)^{7/3} \, dx=\int { {\left (x + 1\right )}^{\frac {7}{3}} {\left (-x + 1\right )}^{n} \,d x } \]
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Result contains complex when optimal does not.
Time = 46.82 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int (1-x)^n (1+x)^{7/3} \, dx=\frac {2^{n} \left (x + 1\right )^{\frac {10}{3}} \Gamma \left (\frac {10}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {10}{3}, - n \\ \frac {13}{3} \end {matrix}\middle | {\frac {\left (x + 1\right ) e^{2 i \pi }}{2}} \right )}}{\Gamma \left (\frac {13}{3}\right )} \]
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\[ \int (1-x)^n (1+x)^{7/3} \, dx=\int { {\left (x + 1\right )}^{\frac {7}{3}} {\left (-x + 1\right )}^{n} \,d x } \]
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\[ \int (1-x)^n (1+x)^{7/3} \, dx=\int { {\left (x + 1\right )}^{\frac {7}{3}} {\left (-x + 1\right )}^{n} \,d x } \]
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Timed out. \[ \int (1-x)^n (1+x)^{7/3} \, dx=\int {\left (1-x\right )}^n\,{\left (x+1\right )}^{7/3} \,d x \]
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