Optimal. Leaf size=37 \[ -\frac{3}{5} 2^{n-1} (1-x)^{10/3} \, _2F_1\left (\frac{10}{3},-n;\frac{13}{3};\frac{1-x}{2}\right ) \]
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Rubi [A] time = 0.0052561, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {69} \[ -\frac{3}{5} 2^{n-1} (1-x)^{10/3} \, _2F_1\left (\frac{10}{3},-n;\frac{13}{3};\frac{1-x}{2}\right ) \]
Antiderivative was successfully verified.
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Rule 69
Rubi steps
\begin{align*} \int (1-x)^{7/3} (1+x)^n \, dx &=-\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*}
Mathematica [A] time = 0.0119005, size = 37, normalized size = 1. \[ -\frac{3}{5} 2^{n-1} (1-x)^{10/3} \, _2F_1\left (\frac{10}{3},-n;\frac{13}{3};\frac{1-x}{2}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.026, size = 0, normalized size = 0. \begin{align*} \int \left ( 1-x \right ) ^{{\frac{7}{3}}} \left ( 1+x \right ) ^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (x + 1\right )}^{n}{\left (-x + 1\right )}^{\frac{7}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (x^{2} - 2 \, x + 1\right )}{\left (x + 1\right )}^{n}{\left (-x + 1\right )}^{\frac{1}{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (x + 1\right )}^{n}{\left (-x + 1\right )}^{\frac{7}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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