Optimal. Leaf size=98 \[ -\frac{(139 x+121) (2 x+3)^{5/2}}{3 \left (3 x^2+5 x+2\right )}+\frac{826}{27} (2 x+3)^{3/2}+\frac{1358}{27} \sqrt{2 x+3}-154 \tanh ^{-1}\left (\sqrt{2 x+3}\right )+\frac{2800}{27} \sqrt{\frac{5}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]
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Rubi [A] time = 0.0811127, antiderivative size = 98, 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 = {818, 824, 826, 1166, 207} \[ -\frac{(139 x+121) (2 x+3)^{5/2}}{3 \left (3 x^2+5 x+2\right )}+\frac{826}{27} (2 x+3)^{3/2}+\frac{1358}{27} \sqrt{2 x+3}-154 \tanh ^{-1}\left (\sqrt{2 x+3}\right )+\frac{2800}{27} \sqrt{\frac{5}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]
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)^{7/2}}{\left (2+5 x+3 x^2\right )^2} \, dx &=-\frac{(3+2 x)^{5/2} (121+139 x)}{3 \left (2+5 x+3 x^2\right )}+\frac{1}{3} \int \frac{(3+2 x)^{3/2} (182+413 x)}{2+5 x+3 x^2} \, dx\\ &=\frac{826}{27} (3+2 x)^{3/2}-\frac{(3+2 x)^{5/2} (121+139 x)}{3 \left (2+5 x+3 x^2\right )}+\frac{1}{9} \int \frac{\sqrt{3+2 x} (-14+679 x)}{2+5 x+3 x^2} \, dx\\ &=\frac{1358}{27} \sqrt{3+2 x}+\frac{826}{27} (3+2 x)^{3/2}-\frac{(3+2 x)^{5/2} (121+139 x)}{3 \left (2+5 x+3 x^2\right )}+\frac{1}{27} \int \frac{-2842-763 x}{\sqrt{3+2 x} \left (2+5 x+3 x^2\right )} \, dx\\ &=\frac{1358}{27} \sqrt{3+2 x}+\frac{826}{27} (3+2 x)^{3/2}-\frac{(3+2 x)^{5/2} (121+139 x)}{3 \left (2+5 x+3 x^2\right )}+\frac{2}{27} \operatorname{Subst}\left (\int \frac{-3395-763 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt{3+2 x}\right )\\ &=\frac{1358}{27} \sqrt{3+2 x}+\frac{826}{27} (3+2 x)^{3/2}-\frac{(3+2 x)^{5/2} (121+139 x)}{3 \left (2+5 x+3 x^2\right )}+462 \operatorname{Subst}\left (\int \frac{1}{-3+3 x^2} \, dx,x,\sqrt{3+2 x}\right )-\frac{14000}{27} \operatorname{Subst}\left (\int \frac{1}{-5+3 x^2} \, dx,x,\sqrt{3+2 x}\right )\\ &=\frac{1358}{27} \sqrt{3+2 x}+\frac{826}{27} (3+2 x)^{3/2}-\frac{(3+2 x)^{5/2} (121+139 x)}{3 \left (2+5 x+3 x^2\right )}-154 \tanh ^{-1}\left (\sqrt{3+2 x}\right )+\frac{2800}{27} \sqrt{\frac{5}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{3+2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.0944341, size = 81, normalized size = 0.83 \[ \frac{1}{81} \left (2800 \sqrt{15} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right )-\frac{3 \sqrt{2 x+3} \left (48 x^3-400 x^2+1843 x+2129\right )}{3 x^2+5 x+2}\right )-154 \tanh ^{-1}\left (\sqrt{2 x+3}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 104, normalized size = 1.1 \begin{align*} -{\frac{8}{27} \left ( 3+2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{184}{27}\sqrt{3+2\,x}}-{\frac{4250}{81}\sqrt{3+2\,x} \left ( 2\,x+{\frac{4}{3}} \right ) ^{-1}}+{\frac{2800\,\sqrt{15}}{81}{\it Artanh} \left ({\frac{\sqrt{15}}{5}\sqrt{3+2\,x}} \right ) }-6\, \left ( 1+\sqrt{3+2\,x} \right ) ^{-1}-77\,\ln \left ( 1+\sqrt{3+2\,x} \right ) -6\, \left ( -1+\sqrt{3+2\,x} \right ) ^{-1}+77\,\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 = 1.46223, size = 157, normalized size = 1.6 \begin{align*} -\frac{8}{27} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - \frac{1400}{81} \, \sqrt{15} \log \left (-\frac{\sqrt{15} - 3 \, \sqrt{2 \, x + 3}}{\sqrt{15} + 3 \, \sqrt{2 \, x + 3}}\right ) + \frac{184}{27} \, \sqrt{2 \, x + 3} - \frac{2 \,{\left (2611 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 2935 \, \sqrt{2 \, x + 3}\right )}}{27 \,{\left (3 \,{\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}} - 77 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) + 77 \, \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 = 1.60452, size = 366, normalized size = 3.73 \begin{align*} \frac{1400 \, \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 ) - 6237 \,{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\sqrt{2 \, x + 3} + 1\right ) + 6237 \,{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\sqrt{2 \, x + 3} - 1\right ) - 3 \,{\left (48 \, x^{3} - 400 \, x^{2} + 1843 \, x + 2129\right )} \sqrt{2 \, x + 3}}{81 \,{\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 = 1.13587, size = 162, normalized size = 1.65 \begin{align*} -\frac{8}{27} \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - \frac{1400}{81} \, \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{184}{27} \, \sqrt{2 \, x + 3} - \frac{2 \,{\left (2611 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 2935 \, \sqrt{2 \, x + 3}\right )}}{27 \,{\left (3 \,{\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}} - 77 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) + 77 \, \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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