Optimal. Leaf size=102 \[ -\frac{(139 x+121) (2 x+3)^{3/2}}{6 \left (3 x^2+5 x+2\right )^2}+\frac{25 (131 x+112) \sqrt{2 x+3}}{6 \left (3 x^2+5 x+2\right )}+1250 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{2905}{3} \sqrt{\frac{5}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]
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Rubi [A] time = 0.0594209, antiderivative size = 102, 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, 820, 826, 1166, 207} \[ -\frac{(139 x+121) (2 x+3)^{3/2}}{6 \left (3 x^2+5 x+2\right )^2}+\frac{25 (131 x+112) \sqrt{2 x+3}}{6 \left (3 x^2+5 x+2\right )}+1250 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{2905}{3} \sqrt{\frac{5}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]
Antiderivative was successfully verified.
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Rule 818
Rule 820
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 )^3} \, dx &=-\frac{(3+2 x)^{3/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac{1}{6} \int \frac{(-900-425 x) \sqrt{3+2 x}}{\left (2+5 x+3 x^2\right )^2} \, dx\\ &=-\frac{(3+2 x)^{3/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac{25 \sqrt{3+2 x} (112+131 x)}{6 \left (2+5 x+3 x^2\right )}-\frac{1}{6} \int \frac{-7025-3275 x}{\sqrt{3+2 x} \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac{(3+2 x)^{3/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac{25 \sqrt{3+2 x} (112+131 x)}{6 \left (2+5 x+3 x^2\right )}-\frac{1}{3} \operatorname{Subst}\left (\int \frac{-4225-3275 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt{3+2 x}\right )\\ &=-\frac{(3+2 x)^{3/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac{25 \sqrt{3+2 x} (112+131 x)}{6 \left (2+5 x+3 x^2\right )}-3750 \operatorname{Subst}\left (\int \frac{1}{-3+3 x^2} \, dx,x,\sqrt{3+2 x}\right )+\frac{14525}{3} \operatorname{Subst}\left (\int \frac{1}{-5+3 x^2} \, dx,x,\sqrt{3+2 x}\right )\\ &=-\frac{(3+2 x)^{3/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac{25 \sqrt{3+2 x} (112+131 x)}{6 \left (2+5 x+3 x^2\right )}+1250 \tanh ^{-1}\left (\sqrt{3+2 x}\right )-\frac{2905}{3} \sqrt{\frac{5}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{3+2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.0847627, size = 82, normalized size = 0.8 \[ \frac{\sqrt{2 x+3} \left (9825 x^3+24497 x^2+19891 x+5237\right )}{6 \left (3 x^2+5 x+2\right )^2}+1250 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{2905}{3} \sqrt{\frac{5}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 124, normalized size = 1.2 \begin{align*} 90\,{\frac{1}{ \left ( 6\,x+4 \right ) ^{2}} \left ({\frac{367\, \left ( 3+2\,x \right ) ^{3/2}}{18}}-{\frac{2005\,\sqrt{3+2\,x}}{54}} \right ) }-{\frac{2905\,\sqrt{15}}{9}{\it Artanh} \left ({\frac{\sqrt{15}}{5}\sqrt{3+2\,x}} \right ) }-3\, \left ( 1+\sqrt{3+2\,x} \right ) ^{-2}+80\, \left ( 1+\sqrt{3+2\,x} \right ) ^{-1}+625\,\ln \left ( 1+\sqrt{3+2\,x} \right ) +3\, \left ( -1+\sqrt{3+2\,x} \right ) ^{-2}+80\, \left ( -1+\sqrt{3+2\,x} \right ) ^{-1}-625\,\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 = 1.43811, size = 181, normalized size = 1.77 \begin{align*} \frac{2905}{18} \, \sqrt{15} \log \left (-\frac{\sqrt{15} - 3 \, \sqrt{2 \, x + 3}}{\sqrt{15} + 3 \, \sqrt{2 \, x + 3}}\right ) + \frac{9825 \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - 39431 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + 50875 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 21125 \, \sqrt{2 \, x + 3}}{3 \,{\left (9 \,{\left (2 \, x + 3\right )}^{4} - 48 \,{\left (2 \, x + 3\right )}^{3} + 94 \,{\left (2 \, x + 3\right )}^{2} - 160 \, x - 215\right )}} + 625 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) - 625 \, \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 [B] time = 1.63403, size = 479, normalized size = 4.7 \begin{align*} \frac{2905 \, \sqrt{5} \sqrt{3}{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (-\frac{\sqrt{5} \sqrt{3} \sqrt{2 \, x + 3} - 3 \, x - 7}{3 \, x + 2}\right ) + 11250 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\sqrt{2 \, x + 3} + 1\right ) - 11250 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\sqrt{2 \, x + 3} - 1\right ) + 3 \,{\left (9825 \, x^{3} + 24497 \, x^{2} + 19891 \, x + 5237\right )} \sqrt{2 \, x + 3}}{18 \,{\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(-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 = 1.09723, size = 162, normalized size = 1.59 \begin{align*} \frac{2905}{18} \, \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{9825 \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - 39431 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + 50875 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 21125 \, \sqrt{2 \, x + 3}}{3 \,{\left (3 \,{\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}^{2}} + 625 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) - 625 \, \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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