Optimal. Leaf size=36 \[ \frac {x^{1+m} \, _2F_1\left (\frac {1+m}{2},-n;\frac {3+m}{2};a^2 x^2\right )}{1+m} \]
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Rubi [A]
time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, 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 = {126, 371}
\begin {gather*} \frac {x^{m+1} \, _2F_1\left (\frac {m+1}{2},-n;\frac {m+3}{2};a^2 x^2\right )}{m+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 126
Rule 371
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*}
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Mathematica [A]
time = 0.06, size = 38, normalized size = 1.06 \begin {gather*} \frac {x^{1+m} \, _2F_1\left (\frac {1+m}{2},-n;1+\frac {1+m}{2};a^2 x^2\right )}{1+m} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int x^{m} \left (-a x +1\right )^{n} \left (a x +1\right )^{n}\, dx\]
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 \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 = 0.00, 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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Sympy [C] Result contains complex when optimal does not.
time = 19.04, size = 202, normalized size = 5.61 \begin {gather*} \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 {gather*}
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 \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 x^m\,{\left (1-a\,x\right )}^n\,{\left (a\,x+1\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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