Optimal. Leaf size=36 \[ \frac{x^{m+1} \, _2F_1\left (\frac{m+1}{2},-n;\frac{m+3}{2};a^2 x^2\right )}{m+1} \]
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Rubi [A] time = 0.0125551, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {125, 364} \[ \frac{x^{m+1} \, _2F_1\left (\frac{m+1}{2},-n;\frac{m+3}{2};a^2 x^2\right )}{m+1} \]
Antiderivative was successfully verified.
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Rule 125
Rule 364
Rubi steps
\begin{align*} \int x^m (1-a x)^n (1+a x)^n \, dx &=\int x^m \left (1-a^2 x^2\right )^n \, dx\\ &=\frac{x^{1+m} \, _2F_1\left (\frac{1+m}{2},-n;\frac{3+m}{2};a^2 x^2\right )}{1+m}\\ \end{align*}
Mathematica [A] time = 0.0085413, size = 38, normalized size = 1.06 \[ \frac{x^{m+1} \, _2F_1\left (\frac{m+1}{2},-n;\frac{m+1}{2}+1;a^2 x^2\right )}{m+1} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.125, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( -ax+1 \right ) ^{n} \left ( ax+1 \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 (a x + 1\right )}^{n}{\left (-a x + 1\right )}^{n} x^{m}\,{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 (a x + 1\right )}^{n}{\left (-a x + 1\right )}^{n} x^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 156.858, size = 202, normalized size = 5.61 \begin{align*} \frac{a^{- m}{G_{6, 6}^{5, 3}\left (\begin{matrix} - \frac{m}{2} - \frac{n}{2}, - \frac{m}{2} - \frac{n}{2} + \frac{1}{2}, 1 & \frac{1}{2} - \frac{m}{2}, - \frac{m}{2} - n, - \frac{m}{2} - n + \frac{1}{2} \\- \frac{m}{2} - n - \frac{1}{2}, - \frac{m}{2} - n, - \frac{m}{2} - \frac{n}{2}, - \frac{m}{2} - n + \frac{1}{2}, - \frac{m}{2} - \frac{n}{2} + \frac{1}{2} & 0 \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{a^{2} x^{2}}} \right )} e^{- i \pi m} e^{- i \pi n}}{4 \pi a \Gamma \left (- n\right )} - \frac{a^{- m}{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{m}{2} - \frac{1}{2}, - \frac{m}{2}, \frac{1}{2} - \frac{m}{2}, - \frac{m}{2} - \frac{n}{2} - \frac{1}{2}, - \frac{m}{2} - \frac{n}{2}, 1 & \\- \frac{m}{2} - \frac{n}{2} - \frac{1}{2}, - \frac{m}{2} - \frac{n}{2} & - \frac{m}{2} - \frac{1}{2}, - \frac{m}{2}, - \frac{m}{2} - n - \frac{1}{2}, 0 \end{matrix} \middle |{\frac{1}{a^{2} x^{2}}} \right )}}{4 \pi a \Gamma \left (- n\right )} \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 (a x + 1\right )}^{n}{\left (-a x + 1\right )}^{n} x^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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