Optimal. Leaf size=69 \[ -\frac{(35 x+29) (2 x+3)^3}{2 \left (3 x^2+5 x+2\right )^2}+\frac{141 (8 x+7) (2 x+3)}{2 \left (3 x^2+5 x+2\right )}-705 \log (x+1)+705 \log (3 x+2) \]
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Rubi [A] time = 0.0303941, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {804, 722, 616, 31} \[ -\frac{(35 x+29) (2 x+3)^3}{2 \left (3 x^2+5 x+2\right )^2}+\frac{141 (8 x+7) (2 x+3)}{2 \left (3 x^2+5 x+2\right )}-705 \log (x+1)+705 \log (3 x+2) \]
Antiderivative was successfully verified.
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Rule 804
Rule 722
Rule 616
Rule 31
Rubi steps
\begin{align*} \int \frac{(5-x) (3+2 x)^3}{\left (2+5 x+3 x^2\right )^3} \, dx &=-\frac{(3+2 x)^3 (29+35 x)}{2 \left (2+5 x+3 x^2\right )^2}-\frac{141}{2} \int \frac{(3+2 x)^2}{\left (2+5 x+3 x^2\right )^2} \, dx\\ &=-\frac{(3+2 x)^3 (29+35 x)}{2 \left (2+5 x+3 x^2\right )^2}+\frac{141 (3+2 x) (7+8 x)}{2 \left (2+5 x+3 x^2\right )}+705 \int \frac{1}{2+5 x+3 x^2} \, dx\\ &=-\frac{(3+2 x)^3 (29+35 x)}{2 \left (2+5 x+3 x^2\right )^2}+\frac{141 (3+2 x) (7+8 x)}{2 \left (2+5 x+3 x^2\right )}+2115 \int \frac{1}{2+3 x} \, dx-2115 \int \frac{1}{3+3 x} \, dx\\ &=-\frac{(3+2 x)^3 (29+35 x)}{2 \left (2+5 x+3 x^2\right )^2}+\frac{141 (3+2 x) (7+8 x)}{2 \left (2+5 x+3 x^2\right )}-705 \log (1+x)+705 \log (2+3 x)\\ \end{align*}
Mathematica [A] time = 0.0268445, size = 59, normalized size = 0.86 \[ -\frac{2611 x+2449}{54 \left (3 x^2+5 x+2\right )^2}+\frac{38118 x+31673}{54 \left (3 x^2+5 x+2\right )}+705 \log (-6 x-4)-705 \log (-2 (x+1)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 48, normalized size = 0.7 \begin{align*} 3\, \left ( 1+x \right ) ^{-2}+89\, \left ( 1+x \right ) ^{-1}-705\,\ln \left ( 1+x \right ) -{\frac{2125}{18\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{3950}{18+27\,x}}+705\,\ln \left ( 2+3\,x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00291, size = 73, normalized size = 1.06 \begin{align*} \frac{38118 \, x^{3} + 95203 \, x^{2} + 77330 \, x + 20299}{18 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} + 705 \, \log \left (3 \, x + 2\right ) - 705 \, \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.22891, size = 269, normalized size = 3.9 \begin{align*} \frac{38118 \, x^{3} + 95203 \, x^{2} + 12690 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (3 \, x + 2\right ) - 12690 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (x + 1\right ) + 77330 \, x + 20299}{18 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.182768, size = 49, normalized size = 0.71 \begin{align*} \frac{38118 x^{3} + 95203 x^{2} + 77330 x + 20299}{162 x^{4} + 540 x^{3} + 666 x^{2} + 360 x + 72} + 705 \log{\left (x + \frac{2}{3} \right )} - 705 \log{\left (x + 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11995, size = 62, normalized size = 0.9 \begin{align*} \frac{38118 \, x^{3} + 95203 \, x^{2} + 77330 \, x + 20299}{18 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{2}} + 705 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - 705 \, \log \left ({\left | x + 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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