Optimal. Leaf size=72 \[ \frac{121 (3 x+2)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{2}{7} (3 x+2)\right )}{28 (m+1)}-\frac{155 (3 x+2)^{m+1}}{36 (m+1)}-\frac{25 (3 x+2)^{m+2}}{18 (m+2)} \]
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Rubi [A] time = 0.0254671, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {88, 68} \[ \frac{121 (3 x+2)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{2}{7} (3 x+2)\right )}{28 (m+1)}-\frac{155 (3 x+2)^{m+1}}{36 (m+1)}-\frac{25 (3 x+2)^{m+2}}{18 (m+2)} \]
Antiderivative was successfully verified.
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Rule 88
Rule 68
Rubi steps
\begin{align*} \int \frac{(2+3 x)^m (3+5 x)^2}{1-2 x} \, dx &=\int \left (-\frac{155}{12} (2+3 x)^m+\frac{121 (2+3 x)^m}{4 (1-2 x)}-\frac{25}{6} (2+3 x)^{1+m}\right ) \, dx\\ &=-\frac{155 (2+3 x)^{1+m}}{36 (1+m)}-\frac{25 (2+3 x)^{2+m}}{18 (2+m)}+\frac{121}{4} \int \frac{(2+3 x)^m}{1-2 x} \, dx\\ &=-\frac{155 (2+3 x)^{1+m}}{36 (1+m)}-\frac{25 (2+3 x)^{2+m}}{18 (2+m)}+\frac{121 (2+3 x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac{2}{7} (2+3 x)\right )}{28 (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0277024, size = 60, normalized size = 0.83 \[ \frac{(3 x+2)^{m+1} \left (1089 (m+2) \, _2F_1\left (1,m+1;m+2;\frac{2}{7} (3 x+2)\right )-35 (m (30 x+51)+30 x+82)\right )}{252 (m+1) (m+2)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( 3+5\,x \right ) ^{2} \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}}{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 (25 \, x^{2} + 30 \, x + 9\right )}{\left (3 \, x + 2\right )}^{m}}{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{9 \left (3 x + 2\right )^{m}}{2 x - 1}\, dx - \int \frac{30 x \left (3 x + 2\right )^{m}}{2 x - 1}\, dx - \int \frac{25 x^{2} \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}}{2 \, x - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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