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 = 0.01, antiderivative size = 47, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.02, size = 47, normalized size = 1.00 \[ \frac {2^{-m-1} (2 x+1)^{1-m} \, _2F_1(1-m,-m;2-m;-3 (2 x+1))}{1-m} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (3 \, x + 2\right )}^{m}}{{\left (2 \, x + 1\right )}^{m}}, x\right ) \]
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 \[ \int \frac {{\left (3 \, x + 2\right )}^{m}}{{\left (2 \, x + 1\right )}^{m}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \left (2 x +1\right )^{-m} \left (3 x +2\right )^{m}\, dx \]
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 \[ \int \frac {{\left (3 \, x + 2\right )}^{m}}{{\left (2 \, x + 1\right )}^{m}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (3\,x+2\right )}^m}{{\left (2\,x+1\right )}^m} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 25.03, size = 42, normalized size = 0.89 \[ \frac {3^{2 m} \left (x + \frac {2}{3}\right ) \left (x + \frac {2}{3}\right )^{m} e^{- i \pi m} \Gamma \left (m + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} m, m + 1 \\ m + 2 \end {matrix}\middle | {6 x + 4} \right )}}{\Gamma \left (m + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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