Optimal. Leaf size=69 \[ \frac {2 (3 x+2)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {2}{7} (3 x+2)\right )}{77 (m+1)}-\frac {5 (3 x+2)^{m+1} \, _2F_1(1,m+1;m+2;5 (3 x+2))}{11 (m+1)} \]
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Rubi [A] time = 0.02, antiderivative size = 69, normalized size of antiderivative = 1.00, 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 = {86, 68} \[ \frac {2 (3 x+2)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {2}{7} (3 x+2)\right )}{77 (m+1)}-\frac {5 (3 x+2)^{m+1} \, _2F_1(1,m+1;m+2;5 (3 x+2))}{11 (m+1)} \]
Antiderivative was successfully verified.
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Rule 68
Rule 86
Rubi steps
\begin {align*} \int \frac {(2+3 x)^m}{(1-2 x) (3+5 x)} \, dx &=\frac {2}{11} \int \frac {(2+3 x)^m}{1-2 x} \, dx+\frac {5}{11} \int \frac {(2+3 x)^m}{3+5 x} \, dx\\ &=\frac {2 (2+3 x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac {2}{7} (2+3 x)\right )}{77 (1+m)}-\frac {5 (2+3 x)^{1+m} \, _2F_1(1,1+m;2+m;5 (2+3 x))}{11 (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 55, normalized size = 0.80 \[ -\frac {(3 x+2)^{m+1} \left (35 \, _2F_1(1,m+1;m+2;5 (3 x+2))-2 \, _2F_1\left (1,m+1;m+2;\frac {2}{7} (3 x+2)\right )\right )}{77 (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.01, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (3 \, x + 2\right )}^{m}}{10 \, x^{2} + x - 3}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {{\left (3 \, x + 2\right )}^{m}}{{\left (5 \, x + 3\right )} {\left (2 \, x - 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.24, size = 0, normalized size = 0.00 \[ \int \frac {\left (3 x +2\right )^{m}}{\left (-2 x +1\right ) \left (5 x +3\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {{\left (3 \, x + 2\right )}^{m}}{{\left (5 \, x + 3\right )} {\left (2 \, x - 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int -\frac {{\left (3\,x+2\right )}^m}{\left (2\,x-1\right )\,\left (5\,x+3\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.65, size = 112, normalized size = 1.62 \[ - \frac {3^{m} m \left (x + \frac {2}{3}\right )^{m} \Phi \left (\frac {7}{6 \left (x + \frac {2}{3}\right )}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{11 \Gamma \left (1 - m\right )} + \frac {3^{m} m \left (x + \frac {2}{3}\right )^{m} \Phi \left (\frac {1}{15 \left (x + \frac {2}{3}\right )}, 1, 1 - m\right ) \Gamma \left (1 - m\right )}{165 \left (x + \frac {2}{3}\right ) \Gamma \left (2 - m\right )} - \frac {3^{m} \left (x + \frac {2}{3}\right )^{m} \Phi \left (\frac {1}{15 \left (x + \frac {2}{3}\right )}, 1, 1 - m\right ) \Gamma \left (1 - m\right )}{165 \left (x + \frac {2}{3}\right ) \Gamma \left (2 - m\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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