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 = 0.204182, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {974, 1060, 1072, 634, 618, 204, 628} \[ \frac{5 (302-35 x)}{64009 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}+\frac{15 (7140435 x+2618306)}{14886061058 \left (5 x^2+3 x+2\right )}-\frac{5 (77020 x+223707)}{87308276 \left (5 x^2+3 x+2\right )^2}+\frac{13-6 x}{1012 \left (2 x^2-x+3\right )^2 \left (5 x^2+3 x+2\right )^2}+\frac{405 \log \left (2 x^2-x+3\right )}{1288408}-\frac{405 \log \left (5 x^2+3 x+2\right )}{1288408}-\frac{880575 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{340783916 \sqrt{23}}+\frac{2768835 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{619080044 \sqrt{31}} \]
Antiderivative was successfully verified.
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Rule 974
Rule 1060
Rule 1072
Rule 634
Rule 618
Rule 204
Rule 628
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*}
Mathematica [A] time = 0.0852828, size = 151, normalized size = 0.83 \[ \frac{6850 x^3-9275 x^2+11154 x-4342}{345092 \left (10 x^4+x^3+16 x^2+7 x+6\right )^2}+\frac{5 \left (42842610 x^3-5711469 x^2+51156233 x+14085977\right )}{14886061058 \left (10 x^4+x^3+16 x^2+7 x+6\right )}+\frac{405 \log \left (2 x^2-x+3\right )}{1288408}-\frac{405 \log \left (5 x^2+3 x+2\right )}{1288408}+\frac{880575 \tan ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{340783916 \sqrt{23}}+\frac{2768835 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{619080044 \sqrt{31}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 118, normalized size = 0.7 \begin{align*} -{\frac{25}{2576816\, \left ( 5\,{x}^{2}+3\,x+2 \right ) ^{2}} \left ( -{\frac{3013197\,{x}^{3}}{961}}-{\frac{14516062\,{x}^{2}}{4805}}-{\frac{51193868\,x}{24025}}-{\frac{5423968}{24025}} \right ) }-{\frac{405\,\ln \left ( 5\,{x}^{2}+3\,x+2 \right ) }{1288408}}+{\frac{2768835\,\sqrt{31}}{19191481364}\arctan \left ({\frac{ \left ( 3+10\,x \right ) \sqrt{31}}{31}} \right ) }+{\frac{1}{644204\, \left ( 2\,{x}^{2}-x+3 \right ) ^{2}} \left ({\frac{302907\,{x}^{3}}{529}}-{\frac{368291\,{x}^{2}}{529}}+{\frac{2501587\,x}{2116}}-{\frac{665819}{1058}} \right ) }+{\frac{405\,\ln \left ( 2\,{x}^{2}-x+3 \right ) }{1288408}}+{\frac{880575\,\sqrt{23}}{7838030068}\arctan \left ({\frac{ \left ( -1+4\,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 = 1.44651, size = 186, normalized size = 1.03 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.17243, size = 1035, normalized size = 5.72 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.539186, size = 163, normalized size = 0.9 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16417, size = 157, normalized size = 0.87 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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