Optimal. Leaf size=85 \[ -\frac{2 (47 x+37)}{5 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{12 (836 x+701)}{25 \sqrt{3 x^2+5 x+2}}+\frac{104 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{25 \sqrt{5}} \]
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Rubi [A] time = 0.0545099, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {822, 12, 724, 206} \[ -\frac{2 (47 x+37)}{5 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{12 (836 x+701)}{25 \sqrt{3 x^2+5 x+2}}+\frac{104 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{25 \sqrt{5}} \]
Antiderivative was successfully verified.
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Rule 822
Rule 12
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{5-x}{(3+2 x) \left (2+5 x+3 x^2\right )^{5/2}} \, dx &=-\frac{2 (37+47 x)}{5 \left (2+5 x+3 x^2\right )^{3/2}}-\frac{2}{15} \int \frac{807+564 x}{(3+2 x) \left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac{2 (37+47 x)}{5 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{12 (701+836 x)}{25 \sqrt{2+5 x+3 x^2}}+\frac{4}{75} \int \frac{78}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 (37+47 x)}{5 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{12 (701+836 x)}{25 \sqrt{2+5 x+3 x^2}}+\frac{104}{25} \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{2 (37+47 x)}{5 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{12 (701+836 x)}{25 \sqrt{2+5 x+3 x^2}}-\frac{208}{25} \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 \left (2+5 x+3 x^2\right )^{3/2}}+\frac{12 (701+836 x)}{25 \sqrt{2+5 x+3 x^2}}+\frac{104 \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )}{25 \sqrt{5}}\\ \end{align*}
Mathematica [A] time = 0.0579312, size = 72, normalized size = 0.85 \[ \frac{2}{125} \left (\frac{5 \left (15048 x^3+37698 x^2+30827 x+8227\right )}{\left (3 x^2+5 x+2\right )^{3/2}}-52 \sqrt{5} \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 144, normalized size = 1.7 \begin{align*}{\frac{5+6\,x}{3} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{-{\frac{3}{2}}}}-8\,{\frac{5+6\,x}{\sqrt{3\,{x}^{2}+5\,x+2}}}+{\frac{13}{15} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}-{\frac{260+312\,x}{15} \left ( 3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}} \right ) ^{-{\frac{3}{2}}}}+{\frac{9360+11232\,x}{25}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}+{\frac{52}{25}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}-{\frac{104\,\sqrt{5}}{125}{\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 = 1.46865, size = 136, normalized size = 1.6 \begin{align*} -\frac{104}{125} \, \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{10032 \, x}{25 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} + \frac{8412}{25 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{94 \, x}{5 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}} - \frac{74}{5 \,{\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 = 1.81027, size = 340, normalized size = 4. \begin{align*} \frac{2 \,{\left (26 \, \sqrt{5}{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\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 (15048 \, x^{3} + 37698 \, x^{2} + 30827 \, x + 8227\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}}{125 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\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}{18 x^{5} \sqrt{3 x^{2} + 5 x + 2} + 87 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 164 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 151 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 68 x \sqrt{3 x^{2} + 5 x + 2} + 12 \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int - \frac{5}{18 x^{5} \sqrt{3 x^{2} + 5 x + 2} + 87 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 164 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 151 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 68 x \sqrt{3 x^{2} + 5 x + 2} + 12 \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 = 1.18078, size = 138, normalized size = 1.62 \begin{align*} \frac{104}{125} \, \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 (2508 \, x + 6283\right )} x + 30827\right )} x + 8227\right )}}{25 \,{\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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