Optimal. Leaf size=47 \[ \frac{2^{-m-1} (2 x+1)^{1-m} \, _2F_1(1-m,-m;2-m;-3 (2 x+1))}{1-m} \]
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Rubi [A] time = 0.0144975, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {69} \[ \frac{2^{-m-1} (2 x+1)^{1-m} \, _2F_1(1-m,-m;2-m;-3 (2 x+1))}{1-m} \]
Antiderivative was successfully verified.
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Rule 69
Rubi steps
\begin{align*} \int (1+2 x)^{-m} (2+3 x)^m \, dx &=\frac{2^{-1-m} (1+2 x)^{1-m} \, _2F_1(1-m,-m;2-m;-3 (1+2 x))}{1-m}\\ \end{align*}
Mathematica [A] time = 0.012871, size = 47, normalized size = 1. \[ \frac{2^{-m-1} (2 x+1)^{1-m} \, _2F_1(1-m,-m;2-m;-3 (2 x+1))}{1-m} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.06, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( 2+3\,x \right ) ^{m}}{ \left ( 1+2\,x \right ) ^{m}}}\, 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 \frac{{\left (3 \, x + 2\right )}^{m}}{{\left (2 \, x + 1\right )}^{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 (\frac{{\left (3 \, x + 2\right )}^{m}}{{\left (2 \, x + 1\right )}^{m}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 61.1545, size = 42, normalized size = 0.89 \begin{align*} \frac{3^{2 m} \left (x + \frac{2}{3}\right ) \left (x + \frac{2}{3}\right )^{m} e^{- i \pi m} \Gamma \left (m + 1\right ){{}_{2}F_{1}\left (\begin{matrix} m, m + 1 \\ m + 2 \end{matrix}\middle |{6 x + 4} \right )}}{\Gamma \left (m + 2\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 \frac{{\left (3 \, x + 2\right )}^{m}}{{\left (2 \, x + 1\right )}^{m}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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