Optimal. Leaf size=59 \[ \frac{(2-33 m) (3 x+2)^{m+1} \, _2F_1(1,m+1;m+2;5 (3 x+2))}{5 (m+1)}-\frac{11 (3 x+2)^{m+1}}{5 (5 x+3)} \]
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Rubi [A] time = 0.018714, antiderivative size = 59, 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 = {78, 68} \[ \frac{(2-33 m) (3 x+2)^{m+1} \, _2F_1(1,m+1;m+2;5 (3 x+2))}{5 (m+1)}-\frac{11 (3 x+2)^{m+1}}{5 (5 x+3)} \]
Antiderivative was successfully verified.
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Rule 78
Rule 68
Rubi steps
\begin{align*} \int \frac{(1-2 x) (2+3 x)^m}{(3+5 x)^2} \, dx &=-\frac{11 (2+3 x)^{1+m}}{5 (3+5 x)}-\frac{1}{5} (2-33 m) \int \frac{(2+3 x)^m}{3+5 x} \, dx\\ &=-\frac{11 (2+3 x)^{1+m}}{5 (3+5 x)}+\frac{(2-33 m) (2+3 x)^{1+m} \, _2F_1(1,1+m;2+m;5 (2+3 x))}{5 (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0134181, size = 57, normalized size = 0.97 \[ -\frac{(3 x+2)^{m+1} ((33 m-2) (5 x+3) \, _2F_1(1,m+1;m+2;5 (3 x+2))+11 (m+1))}{5 (m+1) (5 x+3)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( 1-2\,x \right ) \left ( 2+3\,x \right ) ^{m}}{ \left ( 3+5\,x \right ) ^{2}}}\, 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 )}}{{\left (5 \, x + 3\right )}^{2}}\,{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 )}}{25 \, x^{2} + 30 \, x + 9}, 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{\left (3 x + 2\right )^{m}}{25 x^{2} + 30 x + 9}\, dx - \int \frac{2 x \left (3 x + 2\right )^{m}}{25 x^{2} + 30 x + 9}\, 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 (2 \, x - 1\right )}}{{\left (5 \, x + 3\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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