Optimal. Leaf size=197 \[ \frac{(266 x+269) \left (3 x^2+5 x+2\right )^{7/2}}{280 (2 x+3)^7}+\frac{3 (106 x+135) \left (3 x^2+5 x+2\right )^{5/2}}{640 (2 x+3)^5}+\frac{(30858 x+39767) \left (3 x^2+5 x+2\right )^{3/2}}{25600 (2 x+3)^3}-\frac{3 (61278 x+131465) \sqrt{3 x^2+5 x+2}}{102400 (2 x+3)}+\frac{603}{512} \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )-\frac{934161 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{204800 \sqrt{5}} \]
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Rubi [A] time = 0.128826, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {810, 812, 843, 621, 206, 724} \[ \frac{(266 x+269) \left (3 x^2+5 x+2\right )^{7/2}}{280 (2 x+3)^7}+\frac{3 (106 x+135) \left (3 x^2+5 x+2\right )^{5/2}}{640 (2 x+3)^5}+\frac{(30858 x+39767) \left (3 x^2+5 x+2\right )^{3/2}}{25600 (2 x+3)^3}-\frac{3 (61278 x+131465) \sqrt{3 x^2+5 x+2}}{102400 (2 x+3)}+\frac{603}{512} \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )-\frac{934161 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{204800 \sqrt{5}} \]
Antiderivative was successfully verified.
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Rule 810
Rule 812
Rule 843
Rule 621
Rule 206
Rule 724
Rubi steps
\begin{align*} \int \frac{(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^8} \, dx &=\frac{(269+266 x) \left (2+5 x+3 x^2\right )^{7/2}}{280 (3+2 x)^7}-\frac{1}{240} \int \frac{(513+522 x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^6} \, dx\\ &=\frac{3 (135+106 x) \left (2+5 x+3 x^2\right )^{5/2}}{640 (3+2 x)^5}+\frac{(269+266 x) \left (2+5 x+3 x^2\right )^{7/2}}{280 (3+2 x)^7}+\frac{\int \frac{(-78210-91260 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^4} \, dx}{38400}\\ &=\frac{(39767+30858 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600 (3+2 x)^3}+\frac{3 (135+106 x) \left (2+5 x+3 x^2\right )^{5/2}}{640 (3+2 x)^5}+\frac{(269+266 x) \left (2+5 x+3 x^2\right )^{7/2}}{280 (3+2 x)^7}-\frac{\int \frac{(9426420+11030040 x) \sqrt{2+5 x+3 x^2}}{(3+2 x)^2} \, dx}{3072000}\\ &=-\frac{3 (131465+61278 x) \sqrt{2+5 x+3 x^2}}{102400 (3+2 x)}+\frac{(39767+30858 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600 (3+2 x)^3}+\frac{3 (135+106 x) \left (2+5 x+3 x^2\right )^{5/2}}{640 (3+2 x)^5}+\frac{(269+266 x) \left (2+5 x+3 x^2\right )^{7/2}}{280 (3+2 x)^7}+\frac{\int \frac{148396680+173664000 x}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{24576000}\\ &=-\frac{3 (131465+61278 x) \sqrt{2+5 x+3 x^2}}{102400 (3+2 x)}+\frac{(39767+30858 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600 (3+2 x)^3}+\frac{3 (135+106 x) \left (2+5 x+3 x^2\right )^{5/2}}{640 (3+2 x)^5}+\frac{(269+266 x) \left (2+5 x+3 x^2\right )^{7/2}}{280 (3+2 x)^7}+\frac{1809}{512} \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx-\frac{934161 \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{204800}\\ &=-\frac{3 (131465+61278 x) \sqrt{2+5 x+3 x^2}}{102400 (3+2 x)}+\frac{(39767+30858 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600 (3+2 x)^3}+\frac{3 (135+106 x) \left (2+5 x+3 x^2\right )^{5/2}}{640 (3+2 x)^5}+\frac{(269+266 x) \left (2+5 x+3 x^2\right )^{7/2}}{280 (3+2 x)^7}+\frac{1809}{256} \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )+\frac{934161 \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )}{102400}\\ &=-\frac{3 (131465+61278 x) \sqrt{2+5 x+3 x^2}}{102400 (3+2 x)}+\frac{(39767+30858 x) \left (2+5 x+3 x^2\right )^{3/2}}{25600 (3+2 x)^3}+\frac{3 (135+106 x) \left (2+5 x+3 x^2\right )^{5/2}}{640 (3+2 x)^5}+\frac{(269+266 x) \left (2+5 x+3 x^2\right )^{7/2}}{280 (3+2 x)^7}+\frac{603}{512} \sqrt{3} \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )-\frac{934161 \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )}{204800 \sqrt{5}}\\ \end{align*}
Mathematica [A] time = 0.193334, size = 130, normalized size = 0.66 \[ \frac{-\frac{10 \sqrt{3 x^2+5 x+2} \left (9676800 x^7+338443008 x^6+2361590432 x^5+7622049520 x^4+13619671040 x^3+13975079520 x^2+7753535702 x+1810375853\right )}{(2 x+3)^7}+6539127 \sqrt{5} \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )+8442000 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )}{7168000} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.019, size = 358, normalized size = 1.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.10226, size = 571, normalized size = 2.9 \begin{align*} \frac{45801}{1000000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} - \frac{13 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{35 \,{\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac{3 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{14 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac{24 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{125 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac{4719 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{35000 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac{5147 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{43750 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac{15267 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}}}{250000 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac{142623}{500000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x + \frac{16659}{4000000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} - \frac{78423 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}}}{350000 \,{\left (2 \, x + 3\right )}} + \frac{44331}{80000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x - \frac{15847}{640000} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} + \frac{88983}{64000} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{603}{512} \, \sqrt{3} \log \left (\sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac{5}{2}\right ) + \frac{934161}{1024000} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) - \frac{340941}{512000} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.32902, size = 836, normalized size = 4.24 \begin{align*} \frac{8442000 \, \sqrt{3}{\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )} \log \left (4 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 6539127 \, \sqrt{5}{\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )} \log \left (-\frac{4 \, \sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )} - 124 \, x^{2} - 212 \, x - 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 20 \,{\left (9676800 \, x^{7} + 338443008 \, x^{6} + 2361590432 \, x^{5} + 7622049520 \, x^{4} + 13619671040 \, x^{3} + 13975079520 \, x^{2} + 7753535702 \, x + 1810375853\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{14336000 \,{\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.3866, size = 687, normalized size = 3.49 \begin{align*} -\frac{934161}{1024000} \, \sqrt{5} \log \left (\frac{{\left | -4 \, \sqrt{3} x - 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt{3} x + 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}\right ) - \frac{603}{512} \, \sqrt{3} \log \left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) - \frac{27}{256} \, \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{2310353472 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{13} + 39459777504 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{12} + 930047331808 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 4439192854544 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{10} + 42996771835920 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 98991221694624 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 500967391220544 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 626374342937616 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 1740466332835804 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 1179088946690970 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 1703610278292706 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 552456024942507 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 324453464706399 \, \sqrt{3} x + 28970271150072 \, \sqrt{3} - 324453464706399 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{716800 \,{\left (2 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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