Optimal. Leaf size=111 \[ -\frac{3 (47 x+37)}{5 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )}-\frac{24626}{625 \sqrt{2 x+3}}-\frac{7042}{375 (2 x+3)^{3/2}}-\frac{2114}{125 (2 x+3)^{5/2}}+14 \tanh ^{-1}\left (\sqrt{2 x+3}\right )+\frac{15876}{625} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]
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Rubi [A] time = 0.0979208, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {822, 828, 826, 1166, 207} \[ -\frac{3 (47 x+37)}{5 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )}-\frac{24626}{625 \sqrt{2 x+3}}-\frac{7042}{375 (2 x+3)^{3/2}}-\frac{2114}{125 (2 x+3)^{5/2}}+14 \tanh ^{-1}\left (\sqrt{2 x+3}\right )+\frac{15876}{625} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]
Antiderivative was successfully verified.
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Rule 822
Rule 828
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 (37+47 x)}{5 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}-\frac{1}{5} \int \frac{952+987 x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac{2114}{125 (3+2 x)^{5/2}}-\frac{3 (37+47 x)}{5 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}-\frac{1}{25} \int \frac{2996+3171 x}{(3+2 x)^{5/2} \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac{2114}{125 (3+2 x)^{5/2}}-\frac{7042}{375 (3+2 x)^{3/2}}-\frac{3 (37+47 x)}{5 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}-\frac{1}{125} \int \frac{9688+10563 x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac{2114}{125 (3+2 x)^{5/2}}-\frac{7042}{375 (3+2 x)^{3/2}}-\frac{24626}{625 \sqrt{3+2 x}}-\frac{3 (37+47 x)}{5 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}-\frac{1}{625} \int \frac{32564+36939 x}{\sqrt{3+2 x} \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac{2114}{125 (3+2 x)^{5/2}}-\frac{7042}{375 (3+2 x)^{3/2}}-\frac{24626}{625 \sqrt{3+2 x}}-\frac{3 (37+47 x)}{5 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}-\frac{2}{625} \operatorname{Subst}\left (\int \frac{-45689+36939 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt{3+2 x}\right )\\ &=-\frac{2114}{125 (3+2 x)^{5/2}}-\frac{7042}{375 (3+2 x)^{3/2}}-\frac{24626}{625 \sqrt{3+2 x}}-\frac{3 (37+47 x)}{5 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}-42 \operatorname{Subst}\left (\int \frac{1}{-3+3 x^2} \, dx,x,\sqrt{3+2 x}\right )-\frac{47628}{625} \operatorname{Subst}\left (\int \frac{1}{-5+3 x^2} \, dx,x,\sqrt{3+2 x}\right )\\ &=-\frac{2114}{125 (3+2 x)^{5/2}}-\frac{7042}{375 (3+2 x)^{3/2}}-\frac{24626}{625 \sqrt{3+2 x}}-\frac{3 (37+47 x)}{5 (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )}+14 \tanh ^{-1}\left (\sqrt{3+2 x}\right )+\frac{15876}{625} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{3+2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.211072, size = 86, normalized size = 0.77 \[ \frac{47628 \sqrt{15} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right )-\frac{5 \left (886536 x^4+4348428 x^3+7782530 x^2+5977997 x+1646109\right )}{(2 x+3)^{5/2} \left (3 x^2+5 x+2\right )}}{9375}+14 \tanh ^{-1}\left (\sqrt{2 x+3}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 113, normalized size = 1. \begin{align*} -{\frac{104}{125} \left ( 3+2\,x \right ) ^{-{\frac{5}{2}}}}-{\frac{1624}{375} \left ( 3+2\,x \right ) ^{-{\frac{3}{2}}}}-{\frac{16208}{625}{\frac{1}{\sqrt{3+2\,x}}}}-{\frac{918}{625}\sqrt{3+2\,x} \left ( 2\,x+{\frac{4}{3}} \right ) ^{-1}}+{\frac{15876\,\sqrt{15}}{3125}{\it Artanh} \left ({\frac{\sqrt{15}}{5}\sqrt{3+2\,x}} \right ) }-6\, \left ( 1+\sqrt{3+2\,x} \right ) ^{-1}+7\,\ln \left ( 1+\sqrt{3+2\,x} \right ) -6\, \left ( -1+\sqrt{3+2\,x} \right ) ^{-1}-7\,\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 = 1.45481, size = 169, normalized size = 1.52 \begin{align*} -\frac{7938}{3125} \, \sqrt{15} \log \left (-\frac{\sqrt{15} - 3 \, \sqrt{2 \, x + 3}}{\sqrt{15} + 3 \, \sqrt{2 \, x + 3}}\right ) - \frac{2 \,{\left (110817 \,{\left (2 \, x + 3\right )}^{4} - 242697 \,{\left (2 \, x + 3\right )}^{3} + 91420 \,{\left (2 \, x + 3\right )}^{2} + 28120 \, x + 46080\right )}}{1875 \,{\left (3 \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} - 8 \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} + 5 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}}\right )}} + 7 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) - 7 \, \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 = 1.75487, size = 594, normalized size = 5.35 \begin{align*} \frac{23814 \, \sqrt{5} \sqrt{3}{\left (24 \, x^{5} + 148 \, x^{4} + 358 \, x^{3} + 423 \, x^{2} + 243 \, x + 54\right )} \log \left (\frac{\sqrt{5} \sqrt{3} \sqrt{2 \, x + 3} + 3 \, x + 7}{3 \, x + 2}\right ) + 65625 \,{\left (24 \, x^{5} + 148 \, x^{4} + 358 \, x^{3} + 423 \, x^{2} + 243 \, x + 54\right )} \log \left (\sqrt{2 \, x + 3} + 1\right ) - 65625 \,{\left (24 \, x^{5} + 148 \, x^{4} + 358 \, x^{3} + 423 \, x^{2} + 243 \, x + 54\right )} \log \left (\sqrt{2 \, x + 3} - 1\right ) - 5 \,{\left (886536 \, x^{4} + 4348428 \, x^{3} + 7782530 \, x^{2} + 5977997 \, x + 1646109\right )} \sqrt{2 \, x + 3}}{9375 \,{\left (24 \, x^{5} + 148 \, x^{4} + 358 \, x^{3} + 423 \, x^{2} + 243 \, x + 54\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 = 1.09842, size = 169, normalized size = 1.52 \begin{align*} -\frac{7938}{3125} \, \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{6 \,{\left (4209 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 6709 \, \sqrt{2 \, x + 3}\right )}}{625 \,{\left (3 \,{\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}} - \frac{16 \,{\left (3039 \,{\left (2 \, x + 3\right )}^{2} + 1015 \, x + 1620\right )}}{1875 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}}} + 7 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) - 7 \, \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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