Optimal. Leaf size=81 \[ -\frac{(139 x+121) (2 x+3)^{3/2}}{3 \left (3 x^2+5 x+2\right )}+30 \sqrt{2 x+3}-130 \tanh ^{-1}\left (\sqrt{2 x+3}\right )+100 \sqrt{\frac{5}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]
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Rubi [A] time = 0.0714276, antiderivative size = 81, 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 = {818, 824, 826, 1166, 207} \[ -\frac{(139 x+121) (2 x+3)^{3/2}}{3 \left (3 x^2+5 x+2\right )}+30 \sqrt{2 x+3}-130 \tanh ^{-1}\left (\sqrt{2 x+3}\right )+100 \sqrt{\frac{5}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]
Antiderivative was successfully verified.
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Rule 818
Rule 824
Rule 826
Rule 1166
Rule 207
Rubi steps
\begin{align*} \int \frac{(5-x) (3+2 x)^{5/2}}{\left (2+5 x+3 x^2\right )^2} \, dx &=-\frac{(3+2 x)^{3/2} (121+139 x)}{3 \left (2+5 x+3 x^2\right )}+\frac{1}{3} \int \frac{\sqrt{3+2 x} (-60+135 x)}{2+5 x+3 x^2} \, dx\\ &=30 \sqrt{3+2 x}-\frac{(3+2 x)^{3/2} (121+139 x)}{3 \left (2+5 x+3 x^2\right )}+\frac{1}{9} \int \frac{-1080-495 x}{\sqrt{3+2 x} \left (2+5 x+3 x^2\right )} \, dx\\ &=30 \sqrt{3+2 x}-\frac{(3+2 x)^{3/2} (121+139 x)}{3 \left (2+5 x+3 x^2\right )}+\frac{2}{9} \operatorname{Subst}\left (\int \frac{-675-495 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt{3+2 x}\right )\\ &=30 \sqrt{3+2 x}-\frac{(3+2 x)^{3/2} (121+139 x)}{3 \left (2+5 x+3 x^2\right )}+390 \operatorname{Subst}\left (\int \frac{1}{-3+3 x^2} \, dx,x,\sqrt{3+2 x}\right )-500 \operatorname{Subst}\left (\int \frac{1}{-5+3 x^2} \, dx,x,\sqrt{3+2 x}\right )\\ &=30 \sqrt{3+2 x}-\frac{(3+2 x)^{3/2} (121+139 x)}{3 \left (2+5 x+3 x^2\right )}-130 \tanh ^{-1}\left (\sqrt{3+2 x}\right )+100 \sqrt{\frac{5}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{3+2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.0519757, size = 92, normalized size = 1.14 \[ -\frac{\sqrt{2 x+3} \left (8 x^2+209 x+183\right )+390 \left (3 x^2+5 x+2\right ) \tanh ^{-1}\left (\sqrt{2 x+3}\right )-100 \sqrt{15} \left (3 x^2+5 x+2\right ) \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right )}{9 x^2+15 x+6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 95, normalized size = 1.2 \begin{align*} -{\frac{8}{9}\sqrt{3+2\,x}}-{\frac{850}{27}\sqrt{3+2\,x} \left ( 2\,x+{\frac{4}{3}} \right ) ^{-1}}+{\frac{100\,\sqrt{15}}{3}{\it Artanh} \left ({\frac{\sqrt{15}}{5}\sqrt{3+2\,x}} \right ) }-6\, \left ( 1+\sqrt{3+2\,x} \right ) ^{-1}-65\,\ln \left ( 1+\sqrt{3+2\,x} \right ) -6\, \left ( -1+\sqrt{3+2\,x} \right ) ^{-1}+65\,\ln \left ( -1+\sqrt{3+2\,x} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.45495, size = 144, normalized size = 1.78 \begin{align*} -\frac{50}{3} \, \sqrt{15} \log \left (-\frac{\sqrt{15} - 3 \, \sqrt{2 \, x + 3}}{\sqrt{15} + 3 \, \sqrt{2 \, x + 3}}\right ) - \frac{8}{9} \, \sqrt{2 \, x + 3} - \frac{2 \,{\left (587 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 695 \, \sqrt{2 \, x + 3}\right )}}{9 \,{\left (3 \,{\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}} - 65 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) + 65 \, \log \left (\sqrt{2 \, x + 3} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55986, size = 339, normalized size = 4.19 \begin{align*} \frac{50 \, \sqrt{5} \sqrt{3}{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\frac{\sqrt{5} \sqrt{3} \sqrt{2 \, x + 3} + 3 \, x + 7}{3 \, x + 2}\right ) - 195 \,{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\sqrt{2 \, x + 3} + 1\right ) + 195 \,{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\sqrt{2 \, x + 3} - 1\right ) -{\left (8 \, x^{2} + 209 \, x + 183\right )} \sqrt{2 \, x + 3}}{3 \,{\left (3 \, x^{2} + 5 \, x + 2\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 [A] time = 1.10828, size = 150, normalized size = 1.85 \begin{align*} -\frac{50}{3} \, \sqrt{15} \log \left (\frac{{\left | -2 \, \sqrt{15} + 6 \, \sqrt{2 \, x + 3} \right |}}{2 \,{\left (\sqrt{15} + 3 \, \sqrt{2 \, x + 3}\right )}}\right ) - \frac{8}{9} \, \sqrt{2 \, x + 3} - \frac{2 \,{\left (587 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 695 \, \sqrt{2 \, x + 3}\right )}}{9 \,{\left (3 \,{\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}} - 65 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) + 65 \, \log \left ({\left | \sqrt{2 \, x + 3} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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