Optimal. Leaf size=97 \[ \frac{44 x+39}{276 (1-x)^2 \left (4 x^2+5 x+3\right )}-\frac{11 \log \left (4 x^2+5 x+3\right )}{4608}-\frac{97}{4416 (1-x)}-\frac{21}{736 (1-x)^2}+\frac{11 \log (1-x)}{2304}+\frac{6023 \tan ^{-1}\left (\frac{8 x+5}{\sqrt{23}}\right )}{52992 \sqrt{23}} \]
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Rubi [A] time = 0.0713876, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {822, 800, 634, 618, 204, 628} \[ \frac{44 x+39}{276 (1-x)^2 \left (4 x^2+5 x+3\right )}-\frac{11 \log \left (4 x^2+5 x+3\right )}{4608}-\frac{97}{4416 (1-x)}-\frac{21}{736 (1-x)^2}+\frac{11 \log (1-x)}{2304}+\frac{6023 \tan ^{-1}\left (\frac{8 x+5}{\sqrt{23}}\right )}{52992 \sqrt{23}} \]
Antiderivative was successfully verified.
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Rule 822
Rule 800
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x}{(-1+x)^3 \left (3+5 x+4 x^2\right )^2} \, dx &=\frac{39+44 x}{276 (1-x)^2 \left (3+5 x+4 x^2\right )}+\frac{1}{276} \int \frac{57+132 x}{(-1+x)^3 \left (3+5 x+4 x^2\right )} \, dx\\ &=\frac{39+44 x}{276 (1-x)^2 \left (3+5 x+4 x^2\right )}+\frac{1}{276} \int \left (\frac{63}{4 (-1+x)^3}-\frac{97}{16 (-1+x)^2}+\frac{253}{192 (-1+x)}+\frac{2379-1012 x}{192 \left (3+5 x+4 x^2\right )}\right ) \, dx\\ &=-\frac{21}{736 (1-x)^2}-\frac{97}{4416 (1-x)}+\frac{39+44 x}{276 (1-x)^2 \left (3+5 x+4 x^2\right )}+\frac{11 \log (1-x)}{2304}+\frac{\int \frac{2379-1012 x}{3+5 x+4 x^2} \, dx}{52992}\\ &=-\frac{21}{736 (1-x)^2}-\frac{97}{4416 (1-x)}+\frac{39+44 x}{276 (1-x)^2 \left (3+5 x+4 x^2\right )}+\frac{11 \log (1-x)}{2304}-\frac{11 \int \frac{5+8 x}{3+5 x+4 x^2} \, dx}{4608}+\frac{6023 \int \frac{1}{3+5 x+4 x^2} \, dx}{105984}\\ &=-\frac{21}{736 (1-x)^2}-\frac{97}{4416 (1-x)}+\frac{39+44 x}{276 (1-x)^2 \left (3+5 x+4 x^2\right )}+\frac{11 \log (1-x)}{2304}-\frac{11 \log \left (3+5 x+4 x^2\right )}{4608}-\frac{6023 \operatorname{Subst}\left (\int \frac{1}{-23-x^2} \, dx,x,5+8 x\right )}{52992}\\ &=-\frac{21}{736 (1-x)^2}-\frac{97}{4416 (1-x)}+\frac{39+44 x}{276 (1-x)^2 \left (3+5 x+4 x^2\right )}+\frac{6023 \tan ^{-1}\left (\frac{5+8 x}{\sqrt{23}}\right )}{52992 \sqrt{23}}+\frac{11 \log (1-x)}{2304}-\frac{11 \log \left (3+5 x+4 x^2\right )}{4608}\\ \end{align*}
Mathematica [A] time = 0.0471454, size = 78, normalized size = 0.8 \[ \frac{\frac{184 (2204 x+975)}{4 x^2+5 x+3}-17457 \log \left (4 x^2+5 x+3\right )+\frac{59248}{x-1}-\frac{25392}{(x-1)^2}+34914 \log (1-x)+36138 \sqrt{23} \tan ^{-1}\left (\frac{8 x+5}{\sqrt{23}}\right )}{7312896} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 68, normalized size = 0.7 \begin{align*} -{\frac{1}{288\, \left ( -1+x \right ) ^{2}}}+{\frac{7}{-864+864\,x}}+{\frac{11\,\ln \left ( -1+x \right ) }{2304}}-{\frac{1}{6912} \left ( -{\frac{2204\,x}{23}}-{\frac{975}{23}} \right ) \left ({x}^{2}+{\frac{5\,x}{4}}+{\frac{3}{4}} \right ) ^{-1}}-{\frac{11\,\ln \left ( 4\,{x}^{2}+5\,x+3 \right ) }{4608}}+{\frac{6023\,\sqrt{23}}{1218816}\arctan \left ({\frac{ \left ( 5+8\,x \right ) \sqrt{23}}{23}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.46368, size = 101, normalized size = 1.04 \begin{align*} \frac{6023}{1218816} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (8 \, x + 5\right )}\right ) + \frac{388 \, x^{3} - 407 \, x^{2} - 120 \, x - 45}{4416 \,{\left (4 \, x^{4} - 3 \, x^{3} - 3 \, x^{2} - x + 3\right )}} - \frac{11}{4608} \, \log \left (4 \, x^{2} + 5 \, x + 3\right ) + \frac{11}{2304} \, \log \left (x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75029, size = 377, normalized size = 3.89 \begin{align*} \frac{214176 \, x^{3} + 12046 \, \sqrt{23}{\left (4 \, x^{4} - 3 \, x^{3} - 3 \, x^{2} - x + 3\right )} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (8 \, x + 5\right )}\right ) - 224664 \, x^{2} - 5819 \,{\left (4 \, x^{4} - 3 \, x^{3} - 3 \, x^{2} - x + 3\right )} \log \left (4 \, x^{2} + 5 \, x + 3\right ) + 11638 \,{\left (4 \, x^{4} - 3 \, x^{3} - 3 \, x^{2} - x + 3\right )} \log \left (x - 1\right ) - 66240 \, x - 24840}{2437632 \,{\left (4 \, x^{4} - 3 \, x^{3} - 3 \, x^{2} - x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.211788, size = 88, normalized size = 0.91 \begin{align*} \frac{388 x^{3} - 407 x^{2} - 120 x - 45}{17664 x^{4} - 13248 x^{3} - 13248 x^{2} - 4416 x + 13248} + \frac{11 \log{\left (x - 1 \right )}}{2304} - \frac{11 \log{\left (x^{2} + \frac{5 x}{4} + \frac{3}{4} \right )}}{4608} + \frac{6023 \sqrt{23} \operatorname{atan}{\left (\frac{8 \sqrt{23} x}{23} + \frac{5 \sqrt{23}}{23} \right )}}{1218816} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15479, size = 96, normalized size = 0.99 \begin{align*} \frac{6023}{1218816} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (8 \, x + 5\right )}\right ) + \frac{388 \, x^{3} - 407 \, x^{2} - 120 \, x - 45}{4416 \,{\left (4 \, x^{2} + 5 \, x + 3\right )}{\left (x - 1\right )}^{2}} - \frac{11}{4608} \, \log \left (4 \, x^{2} + 5 \, x + 3\right ) + \frac{11}{2304} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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