Optimal. Leaf size=54 \[ \frac{11 (3 x+2)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{2}{7} (3 x+2)\right )}{14 (m+1)}-\frac{5 (3 x+2)^{m+1}}{6 (m+1)} \]
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Rubi [A] time = 0.0112327, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {80, 68} \[ \frac{11 (3 x+2)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{2}{7} (3 x+2)\right )}{14 (m+1)}-\frac{5 (3 x+2)^{m+1}}{6 (m+1)} \]
Antiderivative was successfully verified.
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Rule 80
Rule 68
Rubi steps
\begin{align*} \int \frac{(2+3 x)^m (3+5 x)}{1-2 x} \, dx &=-\frac{5 (2+3 x)^{1+m}}{6 (1+m)}+\frac{11}{2} \int \frac{(2+3 x)^m}{1-2 x} \, dx\\ &=-\frac{5 (2+3 x)^{1+m}}{6 (1+m)}+\frac{11 (2+3 x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac{2}{7} (2+3 x)\right )}{14 (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0070367, size = 39, normalized size = 0.72 \[ \frac{(3 x+2)^{m+1} \left (33 \, _2F_1\left (1,m+1;m+2;\frac{2}{7} (3 x+2)\right )-35\right )}{42 (m+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.037, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( 3+5\,x \right ) \left ( 2+3\,x \right ) ^{m}}{1-2\,x}}\, 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 (5 \, x + 3\right )}}{2 \, x - 1}\,{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 (5 \, x + 3\right )}}{2 \, x - 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{3 \left (3 x + 2\right )^{m}}{2 x - 1}\, dx - \int \frac{5 x \left (3 x + 2\right )^{m}}{2 x - 1}\, dx \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 (5 \, x + 3\right )}}{2 \, x - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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