Optimal. Leaf size=181 \[ \frac {5 (302-35 x)}{64009 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}+\frac {15 (7140435 x+2618306)}{14886061058 \left (5 x^2+3 x+2\right )}-\frac {5 (77020 x+223707)}{87308276 \left (5 x^2+3 x+2\right )^2}+\frac {13-6 x}{1012 \left (2 x^2-x+3\right )^2 \left (5 x^2+3 x+2\right )^2}+\frac {405 \log \left (2 x^2-x+3\right )}{1288408}-\frac {405 \log \left (5 x^2+3 x+2\right )}{1288408}-\frac {880575 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{340783916 \sqrt {23}}+\frac {2768835 \tan ^{-1}\left (\frac {10 x+3}{\sqrt {31}}\right )}{619080044 \sqrt {31}} \]
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Rubi [A] time = 0.20, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {974, 1060, 1072, 634, 618, 204, 628} \begin {gather*} \frac {5 (302-35 x)}{64009 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}+\frac {15 (7140435 x+2618306)}{14886061058 \left (5 x^2+3 x+2\right )}-\frac {5 (77020 x+223707)}{87308276 \left (5 x^2+3 x+2\right )^2}+\frac {13-6 x}{1012 \left (2 x^2-x+3\right )^2 \left (5 x^2+3 x+2\right )^2}+\frac {405 \log \left (2 x^2-x+3\right )}{1288408}-\frac {405 \log \left (5 x^2+3 x+2\right )}{1288408}-\frac {880575 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{340783916 \sqrt {23}}+\frac {2768835 \tan ^{-1}\left (\frac {10 x+3}{\sqrt {31}}\right )}{619080044 \sqrt {31}} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 974
Rule 1060
Rule 1072
Rubi steps
\begin {align*} \int \frac {1}{\left (3-x+2 x^2\right )^3 \left (2+3 x+5 x^2\right )^3} \, dx &=\frac {13-6 x}{1012 \left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )^2}-\frac {\int \frac {-4510-4400 x+2310 x^2}{\left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )^3} \, dx}{11132}\\ &=\frac {13-6 x}{1012 \left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )^2}+\frac {5 (302-35 x)}{64009 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^2}-\frac {\int \frac {-16501980-41902300 x+4235000 x^2}{\left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^3} \, dx}{61960712}\\ &=-\frac {5 (223707+77020 x)}{87308276 \left (2+3 x+5 x^2\right )^2}+\frac {13-6 x}{1012 \left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )^2}+\frac {5 (302-35 x)}{64009 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^2}-\frac {\int \frac {-28908042240+73138343520 x+24603268800 x^2}{\left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^2} \, dx}{929658522848}\\ &=-\frac {5 (223707+77020 x)}{87308276 \left (2+3 x+5 x^2\right )^2}+\frac {13-6 x}{1012 \left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )^2}+\frac {5 (302-35 x)}{64009 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^2}+\frac {15 (2618306+7140435 x)}{14886061058 \left (2+3 x+5 x^2\right )}-\frac {\int \frac {-40694764915200+36795056089440 x-100361384481600 x^2}{\left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )} \, dx}{6974298238405696}\\ &=-\frac {5 (223707+77020 x)}{87308276 \left (2+3 x+5 x^2\right )^2}+\frac {13-6 x}{1012 \left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )^2}+\frac {5 (302-35 x)}{64009 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^2}+\frac {15 (2618306+7140435 x)}{14886061058 \left (2+3 x+5 x^2\right )}-\frac {\int \frac {-1650046422874080-2122156864428480 x}{3-x+2 x^2} \, dx}{1687780173694178432}-\frac {\int \frac {-2182680087910080+5305392161071200 x}{2+3 x+5 x^2} \, dx}{1687780173694178432}\\ &=-\frac {5 (223707+77020 x)}{87308276 \left (2+3 x+5 x^2\right )^2}+\frac {13-6 x}{1012 \left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )^2}+\frac {5 (302-35 x)}{64009 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^2}+\frac {15 (2618306+7140435 x)}{14886061058 \left (2+3 x+5 x^2\right )}+\frac {405 \int \frac {-1+4 x}{3-x+2 x^2} \, dx}{1288408}-\frac {405 \int \frac {3+10 x}{2+3 x+5 x^2} \, dx}{1288408}+\frac {880575 \int \frac {1}{3-x+2 x^2} \, dx}{681567832}+\frac {2768835 \int \frac {1}{2+3 x+5 x^2} \, dx}{1238160088}\\ &=-\frac {5 (223707+77020 x)}{87308276 \left (2+3 x+5 x^2\right )^2}+\frac {13-6 x}{1012 \left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )^2}+\frac {5 (302-35 x)}{64009 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^2}+\frac {15 (2618306+7140435 x)}{14886061058 \left (2+3 x+5 x^2\right )}+\frac {405 \log \left (3-x+2 x^2\right )}{1288408}-\frac {405 \log \left (2+3 x+5 x^2\right )}{1288408}-\frac {880575 \operatorname {Subst}\left (\int \frac {1}{-23-x^2} \, dx,x,-1+4 x\right )}{340783916}-\frac {2768835 \operatorname {Subst}\left (\int \frac {1}{-31-x^2} \, dx,x,3+10 x\right )}{619080044}\\ &=-\frac {5 (223707+77020 x)}{87308276 \left (2+3 x+5 x^2\right )^2}+\frac {13-6 x}{1012 \left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )^2}+\frac {5 (302-35 x)}{64009 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^2}+\frac {15 (2618306+7140435 x)}{14886061058 \left (2+3 x+5 x^2\right )}-\frac {880575 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{340783916 \sqrt {23}}+\frac {2768835 \tan ^{-1}\left (\frac {3+10 x}{\sqrt {31}}\right )}{619080044 \sqrt {31}}+\frac {405 \log \left (3-x+2 x^2\right )}{1288408}-\frac {405 \log \left (2+3 x+5 x^2\right )}{1288408}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 151, normalized size = 0.83 \begin {gather*} \frac {405 \log \left (2 x^2-x+3\right )}{1288408}-\frac {405 \log \left (5 x^2+3 x+2\right )}{1288408}+\frac {6850 x^3-9275 x^2+11154 x-4342}{345092 \left (10 x^4+x^3+16 x^2+7 x+6\right )^2}+\frac {5 \left (42842610 x^3-5711469 x^2+51156233 x+14085977\right )}{14886061058 \left (10 x^4+x^3+16 x^2+7 x+6\right )}+\frac {880575 \tan ^{-1}\left (\frac {4 x-1}{\sqrt {23}}\right )}{340783916 \sqrt {23}}+\frac {2768835 \tan ^{-1}\left (\frac {10 x+3}{\sqrt {31}}\right )}{619080044 \sqrt {31}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (3-x+2 x^2\right )^3 \left (2+3 x+5 x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.43, size = 297, normalized size = 1.64 \begin {gather*} \frac {67202918046000 \, x^{7} - 2238718468800 \, x^{6} + 186872434930060 \, x^{5} + 62827256425340 \, x^{4} + 173919793526820 \, x^{3} + 67376830890 \, \sqrt {31} {\left (100 \, x^{8} + 20 \, x^{7} + 321 \, x^{6} + 172 \, x^{5} + 390 \, x^{4} + 236 \, x^{3} + 241 \, x^{2} + 84 \, x + 36\right )} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + 52466419650 \, \sqrt {23} {\left (100 \, x^{8} + 20 \, x^{7} + 321 \, x^{6} + 172 \, x^{5} + 390 \, x^{4} + 236 \, x^{3} + 241 \, x^{2} + 84 \, x + 36\right )} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + 73595926401690 \, x^{2} - 146799174285 \, {\left (100 \, x^{8} + 20 \, x^{7} + 321 \, x^{6} + 172 \, x^{5} + 390 \, x^{4} + 236 \, x^{3} + 241 \, x^{2} + 84 \, x + 36\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) + 146799174285 \, {\left (100 \, x^{8} + 20 \, x^{7} + 321 \, x^{6} + 172 \, x^{5} + 390 \, x^{4} + 236 \, x^{3} + 241 \, x^{2} + 84 \, x + 36\right )} \log \left (2 \, x^{2} - x + 3\right ) + 78707350628632 \, x + 7381223830244}{467005507511576 \, {\left (100 \, x^{8} + 20 \, x^{7} + 321 \, x^{6} + 172 \, x^{5} + 390 \, x^{4} + 236 \, x^{3} + 241 \, x^{2} + 84 \, x + 36\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 116, normalized size = 0.64 \begin {gather*} \frac {2768835}{19191481364} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {880575}{7838030068} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {4284261000 \, x^{7} - 142720800 \, x^{6} + 11913326210 \, x^{5} + 4005307690 \, x^{4} + 11087580870 \, x^{3} + 4691822415 \, x^{2} + 5017681412 \, x + 470561254}{29772122116 \, {\left (10 \, x^{4} + x^{3} + 16 \, x^{2} + 7 \, x + 6\right )}^{2}} - \frac {405}{1288408} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) + \frac {405}{1288408} \, \log \left (2 \, x^{2} - x + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 118, normalized size = 0.65 \begin {gather*} \frac {2768835 \sqrt {31}\, \arctan \left (\frac {\left (10 x +3\right ) \sqrt {31}}{31}\right )}{19191481364}+\frac {880575 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{7838030068}+\frac {405 \ln \left (2 x^{2}-x +3\right )}{1288408}-\frac {405 \ln \left (5 x^{2}+3 x +2\right )}{1288408}-\frac {25 \left (-\frac {3013197}{961} x^{3}-\frac {14516062}{4805} x^{2}-\frac {51193868}{24025} x -\frac {5423968}{24025}\right )}{2576816 \left (5 x^{2}+3 x +2\right )^{2}}+\frac {\frac {302907}{529} x^{3}-\frac {368291}{529} x^{2}+\frac {2501587}{2116} x -\frac {665819}{1058}}{644204 \left (2 x^{2}-x +3\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 138, normalized size = 0.76 \begin {gather*} \frac {2768835}{19191481364} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {880575}{7838030068} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {4284261000 \, x^{7} - 142720800 \, x^{6} + 11913326210 \, x^{5} + 4005307690 \, x^{4} + 11087580870 \, x^{3} + 4691822415 \, x^{2} + 5017681412 \, x + 470561254}{29772122116 \, {\left (100 \, x^{8} + 20 \, x^{7} + 321 \, x^{6} + 172 \, x^{5} + 390 \, x^{4} + 236 \, x^{3} + 241 \, x^{2} + 84 \, x + 36\right )}} - \frac {405}{1288408} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) + \frac {405}{1288408} \, \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 = 3.59, size = 155, normalized size = 0.86 \begin {gather*} \frac {\frac {21421305\,x^7}{14886061058}-\frac {356802\,x^6}{7443030529}+\frac {1191332621\,x^5}{297721221160}+\frac {400530769\,x^4}{297721221160}+\frac {1108758087\,x^3}{297721221160}+\frac {938364483\,x^2}{595442442320}+\frac {1254420353\,x}{744303052900}+\frac {235280627}{1488606105800}}{x^8+\frac {x^7}{5}+\frac {321\,x^6}{100}+\frac {43\,x^5}{25}+\frac {39\,x^4}{10}+\frac {59\,x^3}{25}+\frac {241\,x^2}{100}+\frac {21\,x}{25}+\frac {9}{25}}+\ln \left (x-\frac {1}{4}+\frac {\sqrt {23}\,1{}\mathrm {i}}{4}\right )\,\left (\frac {405}{1288408}+\frac {\sqrt {23}\,880575{}\mathrm {i}}{15676060136}\right )-\ln \left (x+\frac {3}{10}-\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (\frac {405}{1288408}+\frac {\sqrt {31}\,2768835{}\mathrm {i}}{38382962728}\right )+\ln \left (x+\frac {3}{10}+\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (-\frac {405}{1288408}+\frac {\sqrt {31}\,2768835{}\mathrm {i}}{38382962728}\right )-\ln \left (x-\frac {1}{4}-\frac {\sqrt {23}\,1{}\mathrm {i}}{4}\right )\,\left (-\frac {405}{1288408}+\frac {\sqrt {23}\,880575{}\mathrm {i}}{15676060136}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.47, size = 163, normalized size = 0.90 \begin {gather*} \frac {4284261000 x^{7} - 142720800 x^{6} + 11913326210 x^{5} + 4005307690 x^{4} + 11087580870 x^{3} + 4691822415 x^{2} + 5017681412 x + 470561254}{2977212211600 x^{8} + 595442442320 x^{7} + 9556851199236 x^{6} + 5120805003952 x^{5} + 11611127625240 x^{4} + 7026220819376 x^{3} + 7175081429956 x^{2} + 2500858257744 x + 1071796396176} + \frac {405 \log {\left (x^{2} - \frac {x}{2} + \frac {3}{2} \right )}}{1288408} - \frac {405 \log {\left (x^{2} + \frac {3 x}{5} + \frac {2}{5} \right )}}{1288408} + \frac {880575 \sqrt {23} \operatorname {atan}{\left (\frac {4 \sqrt {23} x}{23} - \frac {\sqrt {23}}{23} \right )}}{7838030068} + \frac {2768835 \sqrt {31} \operatorname {atan}{\left (\frac {10 \sqrt {31} x}{31} + \frac {3 \sqrt {31}}{31} \right )}}{19191481364} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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