Optimal. Leaf size=97 \[ -\frac {21}{736 (1-x)^2}-\frac {97}{4416 (1-x)}+\frac {39+44 x}{276 (1-x)^2 \left (3+5 x+4 x^2\right )}+\frac {6023 \tan ^{-1}\left (\frac {5+8 x}{\sqrt {23}}\right )}{52992 \sqrt {23}}+\frac {11 \log (1-x)}{2304}-\frac {11 \log \left (3+5 x+4 x^2\right )}{4608} \]
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Rubi [A]
time = 0.05, 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 = {836, 814, 648,
632, 210, 642} \begin {gather*} \frac {6023 \text {ArcTan}\left (\frac {8 x+5}{\sqrt {23}}\right )}{52992 \sqrt {23}}+\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 814
Rule 836
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 \text {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 = 0.03, size = 78, normalized size = 0.80 \begin {gather*} \frac {-\frac {25392}{(-1+x)^2}+\frac {59248}{-1+x}+\frac {184 (975+2204 x)}{3+5 x+4 x^2}+36138 \sqrt {23} \tan ^{-1}\left (\frac {5+8 x}{\sqrt {23}}\right )+34914 \log (1-x)-17457 \log \left (3+5 x+4 x^2\right )}{7312896} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 68, normalized size = 0.70
method | result | size |
default | \(-\frac {1}{288 \left (-1+x \right )^{2}}+\frac {7}{864 \left (-1+x \right )}+\frac {11 \ln \left (-1+x \right )}{2304}-\frac {-\frac {2204 x}{23}-\frac {975}{23}}{6912 \left (x^{2}+\frac {5}{4} x +\frac {3}{4}\right )}-\frac {11 \ln \left (4 x^{2}+5 x +3\right )}{4608}+\frac {6023 \arctan \left (\frac {\left (5+8 x \right ) \sqrt {23}}{23}\right ) \sqrt {23}}{1218816}\) | \(68\) |
risch | \(\frac {\frac {97}{1104} x^{3}-\frac {407}{4416} x^{2}-\frac {5}{184} x -\frac {15}{1472}}{\left (-1+x \right )^{2} \left (4 x^{2}+5 x +3\right )}-\frac {11 \ln \left (64 x^{2}+80 x +48\right )}{4608}+\frac {6023 \arctan \left (\frac {\left (5+8 x \right ) \sqrt {23}}{23}\right ) \sqrt {23}}{1218816}+\frac {11 \ln \left (-1+x \right )}{2304}\) | \(71\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 75, normalized size = 0.77 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.14, size = 134, normalized size = 1.38 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 88, normalized size = 0.91 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.53, size = 71, normalized size = 0.73 \begin {gather*} \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 {gather*}
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 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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