Optimal. Leaf size=99 \[ \frac{2^n 3^{2 n+1} x^{m+1} \, _2F_1\left (\frac{m+1}{2},-n;\frac{m+3}{2};\frac{4 a^2 x^2}{9}\right )}{m+1}-\frac{a 2^{n+1} 9^n x^{m+2} \, _2F_1\left (\frac{m+2}{2},-n;\frac{m+4}{2};\frac{4 a^2 x^2}{9}\right )}{m+2} \]
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Rubi [A] time = 0.0403721, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {82, 125, 364} \[ \frac{2^n 3^{2 n+1} x^{m+1} \, _2F_1\left (\frac{m+1}{2},-n;\frac{m+3}{2};\frac{4 a^2 x^2}{9}\right )}{m+1}-\frac{a 2^{n+1} 9^n x^{m+2} \, _2F_1\left (\frac{m+2}{2},-n;\frac{m+4}{2};\frac{4 a^2 x^2}{9}\right )}{m+2} \]
Antiderivative was successfully verified.
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Rule 82
Rule 125
Rule 364
Rubi steps
\begin{align*} \int x^m (3-2 a x)^{1+n} (6+4 a x)^n \, dx &=3 \int x^m (3-2 a x)^n (6+4 a x)^n \, dx-(2 a) \int x^{1+m} (3-2 a x)^n (6+4 a x)^n \, dx\\ &=3 \int x^m \left (18-8 a^2 x^2\right )^n \, dx-(2 a) \int x^{1+m} \left (18-8 a^2 x^2\right )^n \, dx\\ &=\frac{2^n 3^{1+2 n} x^{1+m} \, _2F_1\left (\frac{1+m}{2},-n;\frac{3+m}{2};\frac{4 a^2 x^2}{9}\right )}{1+m}-\frac{2^{1+n} 9^n a x^{2+m} \, _2F_1\left (\frac{2+m}{2},-n;\frac{4+m}{2};\frac{4 a^2 x^2}{9}\right )}{2+m}\\ \end{align*}
Mathematica [A] time = 0.0566317, size = 115, normalized size = 1.16 \[ \frac{x^{m+1} \left (9-4 a^2 x^2\right )^n \left (\frac{1}{2}-\frac{2 a^2 x^2}{9}\right )^{-n} \left (3 (m+2) \, _2F_1\left (\frac{m+1}{2},-n;\frac{m+3}{2};\frac{4 a^2 x^2}{9}\right )-2 a (m+1) x \, _2F_1\left (\frac{m+2}{2},-n;\frac{m+4}{2};\frac{4 a^2 x^2}{9}\right )\right )}{(m+1) (m+2)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.145, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( -2\,ax+3 \right ) ^{1+n} \left ( 4\,ax+6 \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 (4 \, a x + 6\right )}^{n}{\left (-2 \, a x + 3\right )}^{n + 1} 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 (4 \, a x + 6\right )}^{n}{\left (-2 \, a x + 3\right )}^{n + 1} 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 (4 \, a x + 6\right )}^{n}{\left (-2 \, a x + 3\right )}^{n + 1} x^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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