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 = 0.07, antiderivative size = 97, normalized size of antiderivative = 1.00, 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 204
Rule 618
Rule 628
Rule 634
Rule 800
Rule 822
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*}
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Mathematica [A] time = 0.05, size = 78, normalized size = 0.80 \[ \frac {\frac {184 (2204 x+975)}{4 x^2+5 x+3}-17457 \log \left (4 x^2+5 x+3\right )+\frac {59248}{x-1}-\frac {25392}{(x-1)^2}+34914 \log (1-x)+36138 \sqrt {23} \tan ^{-1}\left (\frac {8 x+5}{\sqrt {23}}\right )}{7312896} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.01, size = 134, normalized size = 1.38 \[ \frac {214176 \, x^{3} + 12046 \, \sqrt {23} {\left (4 \, x^{4} - 3 \, x^{3} - 3 \, x^{2} - x + 3\right )} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (8 \, x + 5\right )}\right ) - 224664 \, x^{2} - 5819 \, {\left (4 \, x^{4} - 3 \, x^{3} - 3 \, x^{2} - x + 3\right )} \log \left (4 \, x^{2} + 5 \, x + 3\right ) + 11638 \, {\left (4 \, x^{4} - 3 \, x^{3} - 3 \, x^{2} - x + 3\right )} \log \left (x - 1\right ) - 66240 \, x - 24840}{2437632 \, {\left (4 \, x^{4} - 3 \, x^{3} - 3 \, x^{2} - x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 71, normalized size = 0.73 \[ \frac {6023}{1218816} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (8 \, x + 5\right )}\right ) + \frac {388 \, x^{3} - 407 \, x^{2} - 120 \, x - 45}{4416 \, {\left (4 \, x^{2} + 5 \, x + 3\right )} {\left (x - 1\right )}^{2}} - \frac {11}{4608} \, \log \left (4 \, x^{2} + 5 \, x + 3\right ) + \frac {11}{2304} \, \log \left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 68, normalized size = 0.70 \[ \frac {6023 \sqrt {23}\, \arctan \left (\frac {\left (8 x +5\right ) \sqrt {23}}{23}\right )}{1218816}+\frac {11 \ln \left (x -1\right )}{2304}-\frac {11 \ln \left (4 x^{2}+5 x +3\right )}{4608}-\frac {-\frac {2204 x}{23}-\frac {975}{23}}{6912 \left (x^{2}+\frac {5}{4} x +\frac {3}{4}\right )}-\frac {1}{288 \left (x -1\right )^{2}}+\frac {7}{864 \left (x -1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 75, normalized size = 0.77 \[ \frac {6023}{1218816} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (8 \, x + 5\right )}\right ) + \frac {388 \, x^{3} - 407 \, x^{2} - 120 \, x - 45}{4416 \, {\left (4 \, x^{4} - 3 \, x^{3} - 3 \, x^{2} - x + 3\right )}} - \frac {11}{4608} \, \log \left (4 \, x^{2} + 5 \, x + 3\right ) + \frac {11}{2304} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 84, normalized size = 0.87 \[ \frac {11\,\ln \left (x-1\right )}{2304}+\frac {-\frac {97\,x^3}{4416}+\frac {407\,x^2}{17664}+\frac {5\,x}{736}+\frac {15}{5888}}{-x^4+\frac {3\,x^3}{4}+\frac {3\,x^2}{4}+\frac {x}{4}-\frac {3}{4}}-\ln \left (x+\frac {5}{8}-\frac {\sqrt {23}\,1{}\mathrm {i}}{8}\right )\,\left (\frac {11}{4608}+\frac {\sqrt {23}\,6023{}\mathrm {i}}{2437632}\right )+\ln \left (x+\frac {5}{8}+\frac {\sqrt {23}\,1{}\mathrm {i}}{8}\right )\,\left (-\frac {11}{4608}+\frac {\sqrt {23}\,6023{}\mathrm {i}}{2437632}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.24, size = 88, normalized size = 0.91 \[ \frac {388 x^{3} - 407 x^{2} - 120 x - 45}{17664 x^{4} - 13248 x^{3} - 13248 x^{2} - 4416 x + 13248} + \frac {11 \log {\left (x - 1 \right )}}{2304} - \frac {11 \log {\left (x^{2} + \frac {5 x}{4} + \frac {3}{4} \right )}}{4608} + \frac {6023 \sqrt {23} \operatorname {atan}{\left (\frac {8 \sqrt {23} x}{23} + \frac {5 \sqrt {23}}{23} \right )}}{1218816} \]
Verification of antiderivative is not currently implemented for this CAS.
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