Optimal. Leaf size=197 \[ \frac{(808 x+757) \left (3 x^2+5 x+2\right )^{7/2}}{1120 (2 x+3)^8}+\frac{(664 x+881) \left (3 x^2+5 x+2\right )^{5/2}}{6400 (2 x+3)^6}+\frac{(17096 x+20959) \left (3 x^2+5 x+2\right )^{3/2}}{102400 (2 x+3)^4}+\frac{3 (434104 x+559841) \sqrt{3 x^2+5 x+2}}{4096000 (2 x+3)^2}-\frac{27}{512} \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )+\frac{1673211 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{8192000 \sqrt{5}} \]
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Rubi [A] time = 0.13238, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {810, 843, 621, 206, 724} \[ \frac{(808 x+757) \left (3 x^2+5 x+2\right )^{7/2}}{1120 (2 x+3)^8}+\frac{(664 x+881) \left (3 x^2+5 x+2\right )^{5/2}}{6400 (2 x+3)^6}+\frac{(17096 x+20959) \left (3 x^2+5 x+2\right )^{3/2}}{102400 (2 x+3)^4}+\frac{3 (434104 x+559841) \sqrt{3 x^2+5 x+2}}{4096000 (2 x+3)^2}-\frac{27}{512} \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )+\frac{1673211 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{8192000 \sqrt{5}} \]
Antiderivative was successfully verified.
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Rule 810
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)^9} \, dx &=\frac{(757+808 x) \left (2+5 x+3 x^2\right )^{7/2}}{1120 (3+2 x)^8}-\frac{1}{320} \int \frac{(291+240 x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^7} \, dx\\ &=\frac{(881+664 x) \left (2+5 x+3 x^2\right )^{5/2}}{6400 (3+2 x)^6}+\frac{(757+808 x) \left (2+5 x+3 x^2\right )^{7/2}}{1120 (3+2 x)^8}+\frac{\int \frac{(-36690-43200 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx}{76800}\\ &=\frac{(20959+17096 x) \left (2+5 x+3 x^2\right )^{3/2}}{102400 (3+2 x)^4}+\frac{(881+664 x) \left (2+5 x+3 x^2\right )^{5/2}}{6400 (3+2 x)^6}+\frac{(757+808 x) \left (2+5 x+3 x^2\right )^{7/2}}{1120 (3+2 x)^8}-\frac{\int \frac{(4488660+5184000 x) \sqrt{2+5 x+3 x^2}}{(3+2 x)^3} \, dx}{12288000}\\ &=\frac{3 (559841+434104 x) \sqrt{2+5 x+3 x^2}}{4096000 (3+2 x)^2}+\frac{(20959+17096 x) \left (2+5 x+3 x^2\right )^{3/2}}{102400 (3+2 x)^4}+\frac{(881+664 x) \left (2+5 x+3 x^2\right )^{5/2}}{6400 (3+2 x)^6}+\frac{(757+808 x) \left (2+5 x+3 x^2\right )^{7/2}}{1120 (3+2 x)^8}+\frac{\int \frac{-265774680-311040000 x}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{983040000}\\ &=\frac{3 (559841+434104 x) \sqrt{2+5 x+3 x^2}}{4096000 (3+2 x)^2}+\frac{(20959+17096 x) \left (2+5 x+3 x^2\right )^{3/2}}{102400 (3+2 x)^4}+\frac{(881+664 x) \left (2+5 x+3 x^2\right )^{5/2}}{6400 (3+2 x)^6}+\frac{(757+808 x) \left (2+5 x+3 x^2\right )^{7/2}}{1120 (3+2 x)^8}-\frac{81}{512} \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx+\frac{1673211 \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx}{8192000}\\ &=\frac{3 (559841+434104 x) \sqrt{2+5 x+3 x^2}}{4096000 (3+2 x)^2}+\frac{(20959+17096 x) \left (2+5 x+3 x^2\right )^{3/2}}{102400 (3+2 x)^4}+\frac{(881+664 x) \left (2+5 x+3 x^2\right )^{5/2}}{6400 (3+2 x)^6}+\frac{(757+808 x) \left (2+5 x+3 x^2\right )^{7/2}}{1120 (3+2 x)^8}-\frac{81}{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{1673211 \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )}{4096000}\\ &=\frac{3 (559841+434104 x) \sqrt{2+5 x+3 x^2}}{4096000 (3+2 x)^2}+\frac{(20959+17096 x) \left (2+5 x+3 x^2\right )^{3/2}}{102400 (3+2 x)^4}+\frac{(881+664 x) \left (2+5 x+3 x^2\right )^{5/2}}{6400 (3+2 x)^6}+\frac{(757+808 x) \left (2+5 x+3 x^2\right )^{7/2}}{1120 (3+2 x)^8}-\frac{27}{512} \sqrt{3} \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )+\frac{1673211 \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )}{8192000 \sqrt{5}}\\ \end{align*}
Mathematica [A] time = 0.201987, size = 130, normalized size = 0.66 \[ \frac{\frac{10 \sqrt{3 x^2+5 x+2} \left (1478785536 x^7+12182619328 x^6+45214440256 x^5+97176896240 x^4+129405924160 x^3+105874603844 x^2+48950756372 x+9818427389\right )}{(2 x+3)^8}-11712477 \sqrt{5} \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )-15120000 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )}{286720000} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.027, size = 379, normalized size = 1.9 \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 = 1.9952, size = 647, normalized size = 3.28 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59802, size = 910, normalized size = 4.62 \begin{align*} \frac{15120000 \, \sqrt{3}{\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\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 ) + 11712477 \, \sqrt{5}{\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\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 (1478785536 \, x^{7} + 12182619328 \, x^{6} + 45214440256 \, x^{5} + 97176896240 \, x^{4} + 129405924160 \, x^{3} + 105874603844 \, x^{2} + 48950756372 \, x + 9818427389\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{573440000 \,{\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\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 = 1.35591, size = 737, normalized size = 3.74 \begin{align*} \frac{1673211}{40960000} \, \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{27}{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{25982914944 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{15} + 475461282240 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{14} + 12329944383680 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{13} + 66497191380480 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{12} + 747738478510240 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 2056338758898032 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{10} + 12823219634258640 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 20470141041874560 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 75774797457107080 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 72179382871515780 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 157788604924552196 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 86325470670757920 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 102935771527447390 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 28057073003987265 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 14067886443441495 \, \sqrt{3} x + 1086949713645432 \, \sqrt{3} - 14067886443441495 \, \sqrt{3 \, x^{2} + 5 \, x + 2}}{28672000 \,{\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 )}^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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