Optimal. Leaf size=42 \[ \frac{2^n x^{m+1} \, _2F_1\left (\frac{m+1}{2},-n;\frac{m+3}{2};\frac{a^2 x^2}{4}\right )}{m+1} \]
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Rubi [A] time = 0.0115878, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {125, 364} \[ \frac{2^n x^{m+1} \, _2F_1\left (\frac{m+1}{2},-n;\frac{m+3}{2};\frac{a^2 x^2}{4}\right )}{m+1} \]
Antiderivative was successfully verified.
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Rule 125
Rule 364
Rubi steps
\begin{align*} \int x^m \left (1-\frac{a x}{2}\right )^n (2+a x)^n \, dx &=\int x^m \left (2-\frac{a^2 x^2}{2}\right )^n \, dx\\ &=\frac{2^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*}
Mathematica [A] time = 0.0183762, size = 42, normalized size = 1. \[ \frac{2^n x^{m+1} \, _2F_1\left (\frac{m+1}{2},-n;\frac{m+3}{2};\frac{a^2 x^2}{4}\right )}{m+1} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.143, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( 1-{\frac{ax}{2}} \right ) ^{n} \left ( ax+2 \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 + 2\right )}^{n}{\left (-\frac{1}{2} \, 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 + 2\right )}^{n}{\left (-\frac{1}{2} \, 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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 + 2\right )}^{n}{\left (-\frac{1}{2} \, 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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