Optimal. Leaf size=147 \[ -\frac{2 (47 x+37)}{5 (2 x+3)^2 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{11808 \sqrt{3 x^2+5 x+2}}{125 (2 x+3)}+\frac{152 \sqrt{3 x^2+5 x+2}}{(2 x+3)^2}+\frac{4 (2112 x+1907)}{25 (2 x+3)^2 \sqrt{3 x^2+5 x+2}}+\frac{4884 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{125 \sqrt{5}} \]
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Rubi [A] time = 0.094746, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ -\frac{2 (47 x+37)}{5 (2 x+3)^2 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{11808 \sqrt{3 x^2+5 x+2}}{125 (2 x+3)}+\frac{152 \sqrt{3 x^2+5 x+2}}{(2 x+3)^2}+\frac{4 (2112 x+1907)}{25 (2 x+3)^2 \sqrt{3 x^2+5 x+2}}+\frac{4884 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{125 \sqrt{5}} \]
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)^3 \left (2+5 x+3 x^2\right )^{5/2}} \, dx &=-\frac{2 (37+47 x)}{5 (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}}-\frac{2}{15} \int \frac{1251+1128 x}{(3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac{2 (37+47 x)}{5 (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 (1907+2112 x)}{25 (3+2 x)^2 \sqrt{2+5 x+3 x^2}}+\frac{4}{75} \int \frac{23766+25344 x}{(3+2 x)^3 \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 (37+47 x)}{5 (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 (1907+2112 x)}{25 (3+2 x)^2 \sqrt{2+5 x+3 x^2}}+\frac{152 \sqrt{2+5 x+3 x^2}}{(3+2 x)^2}-\frac{2}{375} \int \frac{-83970-85500 x}{(3+2 x)^2 \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 (37+47 x)}{5 (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 (1907+2112 x)}{25 (3+2 x)^2 \sqrt{2+5 x+3 x^2}}+\frac{152 \sqrt{2+5 x+3 x^2}}{(3+2 x)^2}+\frac{11808 \sqrt{2+5 x+3 x^2}}{125 (3+2 x)}+\frac{4884}{125} \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 (37+47 x)}{5 (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 (1907+2112 x)}{25 (3+2 x)^2 \sqrt{2+5 x+3 x^2}}+\frac{152 \sqrt{2+5 x+3 x^2}}{(3+2 x)^2}+\frac{11808 \sqrt{2+5 x+3 x^2}}{125 (3+2 x)}-\frac{9768}{125} \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)^2 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{4 (1907+2112 x)}{25 (3+2 x)^2 \sqrt{2+5 x+3 x^2}}+\frac{152 \sqrt{2+5 x+3 x^2}}{(3+2 x)^2}+\frac{11808 \sqrt{2+5 x+3 x^2}}{125 (3+2 x)}+\frac{4884 \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )}{125 \sqrt{5}}\\ \end{align*}
Mathematica [A] time = 0.0853277, size = 143, normalized size = 0.97 \[ \frac{2 \left (142500 \left (3 x^2+5 x+2\right )^2+50 (6336 x+5721) \left (3 x^2+5 x+2\right )+18 (2 x+3) \left (3 x^2+5 x+2\right )^{3/2} \left (4920 \sqrt{3 x^2+5 x+2}-407 \sqrt{5} (2 x+3) \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )\right )-375 (47 x+37)\right )}{1875 (2 x+3)^2 \left (3 x^2+5 x+2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 148, normalized size = 1. \begin{align*} -{\frac{13}{40} \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{177}{50} \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{407}{50} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}-{\frac{530+636\,x}{25} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}+{\frac{14760+17712\,x}{125}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}+{\frac{2442}{125}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}-{\frac{4884\,\sqrt{5}}{625}{\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 ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.51344, size = 251, normalized size = 1.71 \begin{align*} -\frac{4884}{625} \, \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{17712 \, x}{125 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} + \frac{17202}{125 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{636 \, x}{25 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} - \frac{13}{10 \,{\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{177}{25 \,{\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{653}{50 \,{\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 = 2.06037, size = 455, normalized size = 3.1 \begin{align*} \frac{2 \,{\left (1221 \, \sqrt{5}{\left (36 \, x^{6} + 228 \, x^{5} + 589 \, x^{4} + 794 \, x^{3} + 589 \, x^{2} + 228 \, x + 36\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 (106272 \, x^{5} + 599148 \, x^{4} + 1316616 \, x^{3} + 1405814 \, x^{2} + 727887 \, x + 146063\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}}{625 \,{\left (36 \, x^{6} + 228 \, x^{5} + 589 \, x^{4} + 794 \, x^{3} + 589 \, x^{2} + 228 \, x + 36\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}{72 x^{7} \sqrt{3 x^{2} + 5 x + 2} + 564 x^{6} \sqrt{3 x^{2} + 5 x + 2} + 1862 x^{5} \sqrt{3 x^{2} + 5 x + 2} + 3355 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 3560 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 2223 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 756 x \sqrt{3 x^{2} + 5 x + 2} + 108 \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int - \frac{5}{72 x^{7} \sqrt{3 x^{2} + 5 x + 2} + 564 x^{6} \sqrt{3 x^{2} + 5 x + 2} + 1862 x^{5} \sqrt{3 x^{2} + 5 x + 2} + 3355 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 3560 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 2223 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 756 x \sqrt{3 x^{2} + 5 x + 2} + 108 \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 = 1.19761, size = 316, normalized size = 2.15 \begin{align*} \frac{4884}{625} \, \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 (6 \,{\left (23826 \, x + 61591\right )} x + 309599\right )} x + 84259\right )}}{625 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} - \frac{8 \,{\left (4106 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 16447 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 57729 \, \sqrt{3} x + 20987 \, \sqrt{3} - 57729 \, \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}}{625 \,{\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 )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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