Optimal. Leaf size=33 \[ \frac {3}{5} 2^{n-1} (x+1)^{10/3} \, _2F_1\left (\frac {10}{3},-n;\frac {13}{3};\frac {x+1}{2}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, 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} \[ \frac {3}{5} 2^{n-1} (x+1)^{10/3} \, _2F_1\left (\frac {10}{3},-n;\frac {13}{3};\frac {x+1}{2}\right ) \]
Antiderivative was successfully verified.
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Rule 69
Rubi steps
\begin {align*} \int (1-x)^n (1+x)^{7/3} \, 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*}
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Mathematica [A] time = 0.02, size = 33, normalized size = 1.00 \[ \frac {3}{5} 2^{n-1} (x+1)^{10/3} \, _2F_1\left (\frac {10}{3},-n;\frac {13}{3};\frac {x+1}{2}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (x^{2} + 2 \, x + 1\right )} {\left (x + 1\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{n}, x\right ) \]
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 \[ \int {\left (x + 1\right )}^{\frac {7}{3}} {\left (-x + 1\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \left (x +1\right )^{\frac {7}{3}} \left (-x +1\right )^{n}\, dx \]
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 \[ \int {\left (x + 1\right )}^{\frac {7}{3}} {\left (-x + 1\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int {\left (1-x\right )}^n\,{\left (x+1\right )}^{7/3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 89.94, size = 37, normalized size = 1.12 \[ \frac {2^{n} \left (x + 1\right )^{\frac {10}{3}} \Gamma \left (\frac {10}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {10}{3}, - n \\ \frac {13}{3} \end {matrix}\middle | {\frac {\left (x + 1\right ) e^{2 i \pi }}{2}} \right )}}{\Gamma \left (\frac {13}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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