Optimal. Leaf size=33 \[ \frac {3}{5} 2^{-1+n} (1+x)^{10/3} \, _2F_1\left (\frac {10}{3},-n;\frac {13}{3};\frac {1+x}{2}\right ) \]
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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 = {71}
\begin {gather*} \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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 71
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 = 0.07, size = 33, normalized size = 1.00 \begin {gather*} \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 {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 5 in
optimal.
time = 52.35, size = 27, normalized size = 0.82 \begin {gather*} \frac {3 2^n \left (1+x\right )^{\frac {10}{3}} \text {hyper}\left [\left \{\frac {10}{3},-n\right \},\left \{\frac {13}{3}\right \},\frac {\left (1+x\right ) \text {exp\_polar}\left [2 I \text {Pi}\right ]}{2}\right ]}{10} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \left (1-x \right )^{n} \left (1+x \right )^{\frac {7}{3}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.36, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 77.09, size = 37, normalized size = 1.12 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] N/A
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (1-x\right )}^n\,{\left (x+1\right )}^{7/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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