Optimal. Leaf size=42 \[ \frac {4^n x^{m+1} \, _2F_1\left (\frac {m+1}{2},-n;\frac {m+3}{2};\frac {a^2 x^2}{4}\right )}{m+1} \]
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Rubi [A] time = 0.01, antiderivative size = 42, 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 = {125, 364} \[ \frac {4^n x^{m+1} \, _2F_1\left (\frac {m+1}{2},-n;\frac {m+3}{2};\frac {a^2 x^2}{4}\right )}{m+1} \]
Antiderivative was successfully verified.
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Rule 125
Rule 364
Rubi steps
\begin {align*} \int x^m (2-a x)^n (2+a x)^n \, dx &=\int x^m \left (4-a^2 x^2\right )^n \, dx\\ &=\frac {4^n x^{1+m} \, _2F_1\left (\frac {1+m}{2},-n;\frac {3+m}{2};\frac {a^2 x^2}{4}\right )}{1+m}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 44, normalized size = 1.05 \[ \frac {4^n x^{m+1} \, _2F_1\left (\frac {m+1}{2},-n;\frac {m+1}{2}+1;\frac {a^2 x^2}{4}\right )}{m+1} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a x + 2\right )}^{n} {\left (-a x + 2\right )}^{n} x^{m}, x\right ) \]
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 \[ \int {\left (a x + 2\right )}^{n} {\left (-a x + 2\right )}^{n} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int x^{m} \left (-a x +2\right )^{n} \left (a x +2\right )^{n}\, dx \]
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 \[ \int {\left (a x + 2\right )}^{n} {\left (-a x + 2\right )}^{n} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x^m\,{\left (2-a\,x\right )}^n\,{\left (a\,x+2\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 32.16, size = 221, normalized size = 5.26 \[ \frac {2^{m} 2^{2 n} a^{- m} {G_{6, 6}^{5, 3}\left (\begin {matrix} - \frac {m}{2} - \frac {n}{2}, - \frac {m}{2} - \frac {n}{2} + \frac {1}{2}, 1 & \frac {1}{2} - \frac {m}{2}, - \frac {m}{2} - n, - \frac {m}{2} - n + \frac {1}{2} \\- \frac {m}{2} - n - \frac {1}{2}, - \frac {m}{2} - n, - \frac {m}{2} - \frac {n}{2}, - \frac {m}{2} - n + \frac {1}{2}, - \frac {m}{2} - \frac {n}{2} + \frac {1}{2} & 0 \end {matrix} \middle | {\frac {4 e^{- 2 i \pi }}{a^{2} x^{2}}} \right )} e^{- i \pi m} e^{- i \pi n}}{2 \pi a \Gamma \left (- n\right )} - \frac {2^{m} 2^{2 n} a^{- m} {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {m}{2} - \frac {1}{2}, - \frac {m}{2}, \frac {1}{2} - \frac {m}{2}, - \frac {m}{2} - \frac {n}{2} - \frac {1}{2}, - \frac {m}{2} - \frac {n}{2}, 1 & \\- \frac {m}{2} - \frac {n}{2} - \frac {1}{2}, - \frac {m}{2} - \frac {n}{2} & - \frac {m}{2} - \frac {1}{2}, - \frac {m}{2}, - \frac {m}{2} - n - \frac {1}{2}, 0 \end {matrix} \middle | {\frac {4}{a^{2} x^{2}}} \right )}}{2 \pi a \Gamma \left (- n\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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