Optimal. Leaf size=174 \[ -\frac{2 (47 x+37)}{5 (2 x+3)^3 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{9696 \sqrt{3 x^2+5 x+2}}{625 (2 x+3)}+\frac{1048 \sqrt{3 x^2+5 x+2}}{15 (2 x+3)^2}+\frac{47552 \sqrt{3 x^2+5 x+2}}{375 (2 x+3)^3}+\frac{12 (638 x+603)}{25 (2 x+3)^3 \sqrt{3 x^2+5 x+2}}+\frac{46108 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{625 \sqrt{5}} \]
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Rubi [A] time = 0.115879, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {822, 834, 806, 724, 206} \[ -\frac{2 (47 x+37)}{5 (2 x+3)^3 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{9696 \sqrt{3 x^2+5 x+2}}{625 (2 x+3)}+\frac{1048 \sqrt{3 x^2+5 x+2}}{15 (2 x+3)^2}+\frac{47552 \sqrt{3 x^2+5 x+2}}{375 (2 x+3)^3}+\frac{12 (638 x+603)}{25 (2 x+3)^3 \sqrt{3 x^2+5 x+2}}+\frac{46108 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{625 \sqrt{5}} \]
Antiderivative was successfully verified.
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Rule 822
Rule 834
Rule 806
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{5-x}{(3+2 x)^4 \left (2+5 x+3 x^2\right )^{5/2}} \, dx &=-\frac{2 (37+47 x)}{5 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}}-\frac{2}{15} \int \frac{1473+1410 x}{(3+2 x)^4 \left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac{2 (37+47 x)}{5 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{12 (603+638 x)}{25 (3+2 x)^3 \sqrt{2+5 x+3 x^2}}+\frac{4}{75} \int \frac{33846+34452 x}{(3+2 x)^4 \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 (37+47 x)}{5 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{12 (603+638 x)}{25 (3+2 x)^3 \sqrt{2+5 x+3 x^2}}+\frac{47552 \sqrt{2+5 x+3 x^2}}{375 (3+2 x)^3}-\frac{4 \int \frac{-222726-213984 x}{(3+2 x)^3 \sqrt{2+5 x+3 x^2}} \, dx}{1125}\\ &=-\frac{2 (37+47 x)}{5 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{12 (603+638 x)}{25 (3+2 x)^3 \sqrt{2+5 x+3 x^2}}+\frac{47552 \sqrt{2+5 x+3 x^2}}{375 (3+2 x)^3}+\frac{1048 \sqrt{2+5 x+3 x^2}}{15 (3+2 x)^2}+\frac{2 \int \frac{775170+589500 x}{(3+2 x)^2 \sqrt{2+5 x+3 x^2}} \, dx}{5625}\\ &=-\frac{2 (37+47 x)}{5 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{12 (603+638 x)}{25 (3+2 x)^3 \sqrt{2+5 x+3 x^2}}+\frac{47552 \sqrt{2+5 x+3 x^2}}{375 (3+2 x)^3}+\frac{1048 \sqrt{2+5 x+3 x^2}}{15 (3+2 x)^2}+\frac{9696 \sqrt{2+5 x+3 x^2}}{625 (3+2 x)}+\frac{46108}{625} \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 (37+47 x)}{5 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{12 (603+638 x)}{25 (3+2 x)^3 \sqrt{2+5 x+3 x^2}}+\frac{47552 \sqrt{2+5 x+3 x^2}}{375 (3+2 x)^3}+\frac{1048 \sqrt{2+5 x+3 x^2}}{15 (3+2 x)^2}+\frac{9696 \sqrt{2+5 x+3 x^2}}{625 (3+2 x)}-\frac{92216}{625} \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )\\ &=-\frac{2 (37+47 x)}{5 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{12 (603+638 x)}{25 (3+2 x)^3 \sqrt{2+5 x+3 x^2}}+\frac{47552 \sqrt{2+5 x+3 x^2}}{375 (3+2 x)^3}+\frac{1048 \sqrt{2+5 x+3 x^2}}{15 (3+2 x)^2}+\frac{9696 \sqrt{2+5 x+3 x^2}}{625 (3+2 x)}+\frac{46108 \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )}{625 \sqrt{5}}\\ \end{align*}
Mathematica [A] time = 0.121467, size = 150, normalized size = 0.86 \[ \frac{2 \left (594400 \left (3 x^2+5 x+2\right )^2+2250 (638 x+603) \left (3 x^2+5 x+2\right )+2 (2 x+3) \left (3 x^2+5 x+2\right )^{3/2} \left (10 (7272 x+27283) \sqrt{3 x^2+5 x+2}-34581 \sqrt{5} (2 x+3)^2 \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )\right )-1875 (47 x+37)\right )}{9375 (2 x+3)^3 \left (3 x^2+5 x+2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 169, normalized size = 1. \begin{align*} -{\frac{151}{200} \left ( x+{\frac{3}{2}} \right ) ^{-2} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}-{\frac{862}{125} \left ( x+{\frac{3}{2}} \right ) ^{-1} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}+{\frac{11527}{750} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}-{\frac{11830+14196\,x}{375} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}+{\frac{12120+14544\,x}{625}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}+{\frac{23054}{625}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}-{\frac{46108\,\sqrt{5}}{3125}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) }-{\frac{13}{120} \left ( x+{\frac{3}{2}} \right ) ^{-3} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.64724, size = 343, normalized size = 1.97 \begin{align*} -\frac{46108}{3125} \, \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{14544 \, x}{625 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} + \frac{35174}{625 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{4732 \, x}{125 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} - \frac{13}{15 \,{\left (8 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x^{3} + 36 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x^{2} + 54 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + 27 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}\right )}} - \frac{151}{50 \,{\left (4 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x^{2} + 12 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + 9 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}\right )}} - \frac{1724}{125 \,{\left (2 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + 3 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}\right )}} - \frac{12133}{750 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.2739, size = 527, normalized size = 3.03 \begin{align*} \frac{2 \,{\left (34581 \, \sqrt{5}{\left (72 \, x^{7} + 564 \, x^{6} + 1862 \, x^{5} + 3355 \, x^{4} + 3560 \, x^{3} + 2223 \, x^{2} + 756 \, x + 108\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 ) + 5 \,{\left (523584 \, x^{6} + 4495032 \, x^{5} + 15334836 \, x^{4} + 26717636 \, x^{3} + 25105026 \, x^{2} + 12060957 \, x + 2313929\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}}{9375 \,{\left (72 \, x^{7} + 564 \, x^{6} + 1862 \, x^{5} + 3355 \, x^{4} + 3560 \, x^{3} + 2223 \, x^{2} + 756 \, x + 108\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{x}{144 x^{8} \sqrt{3 x^{2} + 5 x + 2} + 1344 x^{7} \sqrt{3 x^{2} + 5 x + 2} + 5416 x^{6} \sqrt{3 x^{2} + 5 x + 2} + 12296 x^{5} \sqrt{3 x^{2} + 5 x + 2} + 17185 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 15126 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 8181 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 2484 x \sqrt{3 x^{2} + 5 x + 2} + 324 \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int - \frac{5}{144 x^{8} \sqrt{3 x^{2} + 5 x + 2} + 1344 x^{7} \sqrt{3 x^{2} + 5 x + 2} + 5416 x^{6} \sqrt{3 x^{2} + 5 x + 2} + 12296 x^{5} \sqrt{3 x^{2} + 5 x + 2} + 17185 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 15126 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 8181 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 2484 x \sqrt{3 x^{2} + 5 x + 2} + 324 \sqrt{3 x^{2} + 5 x + 2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19415, size = 385, normalized size = 2.21 \begin{align*} \frac{46108}{3125} \, \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{2 \,{\left ({\left (12 \,{\left (19992 \, x + 58207\right )} x + 636631\right )} x + 184301\right )}}{3125 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} - \frac{8 \,{\left (296724 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 2103870 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 16891990 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 21246975 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 38063715 \, \sqrt{3} x + 8723544 \, \sqrt{3} - 38063715 \, \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}}{9375 \,{\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 )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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