Optimal. Leaf size=115 \[ \frac {4+65 x}{1364 \left (2+3 x+5 x^2\right )^2}+\frac {7923+21605 x}{465124 \left (2+3 x+5 x^2\right )}-\frac {45 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{10648 \sqrt {23}}+\frac {847793 \tan ^{-1}\left (\frac {3+10 x}{\sqrt {31}}\right )}{10232728 \sqrt {31}}-\frac {\log \left (3-x+2 x^2\right )}{21296}+\frac {\log \left (2+3 x+5 x^2\right )}{21296} \]
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Rubi [A]
time = 0.08, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {988, 1074,
1086, 648, 632, 210, 642} \begin {gather*} -\frac {45 \text {ArcTan}\left (\frac {1-4 x}{\sqrt {23}}\right )}{10648 \sqrt {23}}+\frac {847793 \text {ArcTan}\left (\frac {10 x+3}{\sqrt {31}}\right )}{10232728 \sqrt {31}}+\frac {65 x+4}{1364 \left (5 x^2+3 x+2\right )^2}+\frac {21605 x+7923}{465124 \left (5 x^2+3 x+2\right )}-\frac {\log \left (2 x^2-x+3\right )}{21296}+\frac {\log \left (5 x^2+3 x+2\right )}{21296} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 988
Rule 1074
Rule 1086
Rubi steps
\begin {align*} \int \frac {1}{\left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^3} \, dx &=\frac {4+65 x}{1364 \left (2+3 x+5 x^2\right )^2}-\frac {\int \frac {-5753+3509 x-4290 x^2}{\left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^2} \, dx}{15004}\\ &=\frac {4+65 x}{1364 \left (2+3 x+5 x^2\right )^2}+\frac {7923+21605 x}{465124 \left (2+3 x+5 x^2\right )}-\frac {\int \frac {-14522420+3833038 x-10456820 x^2}{\left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )} \, dx}{112560008}\\ &=\frac {4+65 x}{1364 \left (2+3 x+5 x^2\right )^2}+\frac {7923+21605 x}{465124 \left (2+3 x+5 x^2\right )}-\frac {\int \frac {-58838186+5116364 x}{3-x+2 x^2} \, dx}{27239521936}-\frac {\int \frac {-1132249756-12790910 x}{2+3 x+5 x^2} \, dx}{27239521936}\\ &=\frac {4+65 x}{1364 \left (2+3 x+5 x^2\right )^2}+\frac {7923+21605 x}{465124 \left (2+3 x+5 x^2\right )}-\frac {\int \frac {-1+4 x}{3-x+2 x^2} \, dx}{21296}+\frac {\int \frac {3+10 x}{2+3 x+5 x^2} \, dx}{21296}+\frac {45 \int \frac {1}{3-x+2 x^2} \, dx}{21296}+\frac {847793 \int \frac {1}{2+3 x+5 x^2} \, dx}{20465456}\\ &=\frac {4+65 x}{1364 \left (2+3 x+5 x^2\right )^2}+\frac {7923+21605 x}{465124 \left (2+3 x+5 x^2\right )}-\frac {\log \left (3-x+2 x^2\right )}{21296}+\frac {\log \left (2+3 x+5 x^2\right )}{21296}-\frac {45 \text {Subst}\left (\int \frac {1}{-23-x^2} \, dx,x,-1+4 x\right )}{10648}-\frac {847793 \text {Subst}\left (\int \frac {1}{-31-x^2} \, dx,x,3+10 x\right )}{10232728}\\ &=\frac {4+65 x}{1364 \left (2+3 x+5 x^2\right )^2}+\frac {7923+21605 x}{465124 \left (2+3 x+5 x^2\right )}-\frac {45 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{10648 \sqrt {23}}+\frac {847793 \tan ^{-1}\left (\frac {3+10 x}{\sqrt {31}}\right )}{10232728 \sqrt {31}}-\frac {\log \left (3-x+2 x^2\right )}{21296}+\frac {\log \left (2+3 x+5 x^2\right )}{21296}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 104, normalized size = 0.90 \begin {gather*} \frac {45 \tan ^{-1}\left (\frac {-1+4 x}{\sqrt {23}}\right )}{10648 \sqrt {23}}+\frac {1695586 \sqrt {31} \tan ^{-1}\left (\frac {3+10 x}{\sqrt {31}}\right )+31 \left (\frac {44 \left (17210+89144 x+104430 x^2+108025 x^3\right )}{\left (2+3 x+5 x^2\right )^2}-961 \log \left (3-x+2 x^2\right )+961 \log \left (2+3 x+5 x^2\right )\right )}{634429136} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 89, normalized size = 0.77
method | result | size |
default | \(-\frac {\ln \left (2 x^{2}-x +3\right )}{21296}+\frac {45 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{244904}+\frac {\frac {108025}{465124} x^{3}+\frac {52215}{232562} x^{2}+\frac {2026}{10571} x +\frac {8605}{232562}}{\left (5 x^{2}+3 x +2\right )^{2}}+\frac {\ln \left (5 x^{2}+3 x +2\right )}{21296}+\frac {847793 \arctan \left (\frac {\left (3+10 x \right ) \sqrt {31}}{31}\right ) \sqrt {31}}{317214568}\) | \(89\) |
risch | \(\frac {\frac {108025}{465124} x^{3}+\frac {52215}{232562} x^{2}+\frac {2026}{10571} x +\frac {8605}{232562}}{\left (5 x^{2}+3 x +2\right )^{2}}+\frac {\ln \left (100 x^{2}+60 x +40\right )}{21296}+\frac {847793 \arctan \left (\frac {\left (3+10 x \right ) \sqrt {31}}{31}\right ) \sqrt {31}}{317214568}-\frac {\ln \left (16 x^{2}-8 x +24\right )}{21296}+\frac {45 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{244904}\) | \(89\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 98, normalized size = 0.85 \begin {gather*} \frac {847793}{317214568} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {45}{244904} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {108025 \, x^{3} + 104430 \, x^{2} + 89144 \, x + 17210}{465124 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} + \frac {1}{21296} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) - \frac {1}{21296} \, \log \left (2 \, x^{2} - x + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.31, size = 177, normalized size = 1.54 \begin {gather*} \frac {3388960300 \, x^{3} + 38998478 \, \sqrt {31} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + 2681190 \, \sqrt {23} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + 3276177960 \, x^{2} + 685193 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) - 685193 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \log \left (2 \, x^{2} - x + 3\right ) + 2796625568 \, x + 539912120}{14591870128 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.15, size = 119, normalized size = 1.03 \begin {gather*} \frac {108025 x^{3} + 104430 x^{2} + 89144 x + 17210}{11628100 x^{4} + 13953720 x^{3} + 13488596 x^{2} + 5581488 x + 1860496} - \frac {\log {\left (x^{2} - \frac {x}{2} + \frac {3}{2} \right )}}{21296} + \frac {\log {\left (x^{2} + \frac {3 x}{5} + \frac {2}{5} \right )}}{21296} + \frac {45 \sqrt {23} \operatorname {atan}{\left (\frac {4 \sqrt {23} x}{23} - \frac {\sqrt {23}}{23} \right )}}{244904} + \frac {847793 \sqrt {31} \operatorname {atan}{\left (\frac {10 \sqrt {31} x}{31} + \frac {3 \sqrt {31}}{31} \right )}}{317214568} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.25, size = 88, normalized size = 0.77 \begin {gather*} \frac {847793}{317214568} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {45}{244904} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {108025 \, x^{3} + 104430 \, x^{2} + 89144 \, x + 17210}{465124 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}^{2}} + \frac {1}{21296} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) - \frac {1}{21296} \, \log \left (2 \, x^{2} - x + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.18, size = 115, normalized size = 1.00 \begin {gather*} \frac {\frac {4321\,x^3}{465124}+\frac {10443\,x^2}{1162810}+\frac {2026\,x}{264275}+\frac {1721}{1162810}}{x^4+\frac {6\,x^3}{5}+\frac {29\,x^2}{25}+\frac {12\,x}{25}+\frac {4}{25}}+\ln \left (x-\frac {1}{4}+\frac {\sqrt {23}\,1{}\mathrm {i}}{4}\right )\,\left (-\frac {1}{21296}+\frac {\sqrt {23}\,45{}\mathrm {i}}{489808}\right )-\ln \left (x+\frac {3}{10}-\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (-\frac {1}{21296}+\frac {\sqrt {31}\,847793{}\mathrm {i}}{634429136}\right )+\ln \left (x+\frac {3}{10}+\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (\frac {1}{21296}+\frac {\sqrt {31}\,847793{}\mathrm {i}}{634429136}\right )-\ln \left (x-\frac {1}{4}-\frac {\sqrt {23}\,1{}\mathrm {i}}{4}\right )\,\left (\frac {1}{21296}+\frac {\sqrt {23}\,45{}\mathrm {i}}{489808}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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