Optimal. Leaf size=128 \[ -\frac{3 (47 x+37)}{10 (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^2}+\frac{10551 x+9146}{50 (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )}+\frac{6853}{125 \sqrt{2 x+3}}+\frac{7451}{75 (2 x+3)^{3/2}}+310 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{45603}{125} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]
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Rubi [A] time = 0.0961882, antiderivative size = 128, 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} \[ -\frac{3 (47 x+37)}{10 (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^2}+\frac{10551 x+9146}{50 (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )}+\frac{6853}{125 \sqrt{2 x+3}}+\frac{7451}{75 (2 x+3)^{3/2}}+310 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-\frac{45603}{125} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]
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)^{5/2} \left (2+5 x+3 x^2\right )^3} \, dx &=-\frac{3 (37+47 x)}{10 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^2}-\frac{1}{10} \int \frac{1550+1269 x}{(3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^2} \, dx\\ &=-\frac{3 (37+47 x)}{10 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^2}+\frac{9146+10551 x}{50 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )}+\frac{1}{50} \int \frac{60505+52755 x}{(3+2 x)^{5/2} \left (2+5 x+3 x^2\right )} \, dx\\ &=\frac{7451}{75 (3+2 x)^{3/2}}-\frac{3 (37+47 x)}{10 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^2}+\frac{9146+10551 x}{50 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )}+\frac{1}{250} \int \frac{150515+111765 x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )} \, dx\\ &=\frac{7451}{75 (3+2 x)^{3/2}}+\frac{6853}{125 \sqrt{3+2 x}}-\frac{3 (37+47 x)}{10 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^2}+\frac{9146+10551 x}{50 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )}+\frac{\int \frac{296545+102795 x}{\sqrt{3+2 x} \left (2+5 x+3 x^2\right )} \, dx}{1250}\\ &=\frac{7451}{75 (3+2 x)^{3/2}}+\frac{6853}{125 \sqrt{3+2 x}}-\frac{3 (37+47 x)}{10 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^2}+\frac{9146+10551 x}{50 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )}+\frac{1}{625} \operatorname{Subst}\left (\int \frac{284705+102795 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt{3+2 x}\right )\\ &=\frac{7451}{75 (3+2 x)^{3/2}}+\frac{6853}{125 \sqrt{3+2 x}}-\frac{3 (37+47 x)}{10 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^2}+\frac{9146+10551 x}{50 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )}-930 \operatorname{Subst}\left (\int \frac{1}{-3+3 x^2} \, dx,x,\sqrt{3+2 x}\right )+\frac{136809}{125} \operatorname{Subst}\left (\int \frac{1}{-5+3 x^2} \, dx,x,\sqrt{3+2 x}\right )\\ &=\frac{7451}{75 (3+2 x)^{3/2}}+\frac{6853}{125 \sqrt{3+2 x}}-\frac{3 (37+47 x)}{10 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^2}+\frac{9146+10551 x}{50 (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )}+310 \tanh ^{-1}\left (\sqrt{3+2 x}\right )-\frac{45603}{125} \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{3+2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.3294, size = 91, normalized size = 0.71 \[ \frac{\frac{5 \left (740124 x^5+4247856 x^4+9453447 x^3+10168583 x^2+5278129 x+1057511\right )}{(2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^2}-273618 \sqrt{15} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right )}{3750}+310 \tanh ^{-1}\left (\sqrt{2 x+3}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 142, normalized size = 1.1 \begin{align*} -{\frac{416}{375} \left ( 3+2\,x \right ) ^{-{\frac{3}{2}}}}-{\frac{9824}{625}{\frac{1}{\sqrt{3+2\,x}}}}+{\frac{4374}{625\, \left ( 6\,x+4 \right ) ^{2}} \left ({\frac{707}{18} \left ( 3+2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{1235}{18}\sqrt{3+2\,x}} \right ) }-{\frac{45603\,\sqrt{15}}{625}{\it Artanh} \left ({\frac{\sqrt{15}}{5}\sqrt{3+2\,x}} \right ) }-3\, \left ( 1+\sqrt{3+2\,x} \right ) ^{-2}+20\, \left ( 1+\sqrt{3+2\,x} \right ) ^{-1}+155\,\ln \left ( 1+\sqrt{3+2\,x} \right ) +3\, \left ( -1+\sqrt{3+2\,x} \right ) ^{-2}+20\, \left ( -1+\sqrt{3+2\,x} \right ) ^{-1}-155\,\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 = 1.6672, size = 205, normalized size = 1.6 \begin{align*} \frac{45603}{1250} \, \sqrt{15} \log \left (-\frac{\sqrt{15} - 3 \, \sqrt{2 \, x + 3}}{\sqrt{15} + 3 \, \sqrt{2 \, x + 3}}\right ) + \frac{185031 \,{\left (2 \, x + 3\right )}^{5} - 651537 \,{\left (2 \, x + 3\right )}^{4} + 619101 \,{\left (2 \, x + 3\right )}^{3} - 10115 \,{\left (2 \, x + 3\right )}^{2} - 228160 \, x - 352640}{375 \,{\left (9 \,{\left (2 \, x + 3\right )}^{\frac{11}{2}} - 48 \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} + 94 \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - 80 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + 25 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}}\right )}} + 155 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) - 155 \, \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 = 1.7724, size = 674, normalized size = 5.27 \begin{align*} \frac{136809 \, \sqrt{5} \sqrt{3}{\left (36 \, x^{6} + 228 \, x^{5} + 589 \, x^{4} + 794 \, x^{3} + 589 \, x^{2} + 228 \, x + 36\right )} \log \left (-\frac{\sqrt{5} \sqrt{3} \sqrt{2 \, x + 3} - 3 \, x - 7}{3 \, x + 2}\right ) + 581250 \,{\left (36 \, x^{6} + 228 \, x^{5} + 589 \, x^{4} + 794 \, x^{3} + 589 \, x^{2} + 228 \, x + 36\right )} \log \left (\sqrt{2 \, x + 3} + 1\right ) - 581250 \,{\left (36 \, x^{6} + 228 \, x^{5} + 589 \, x^{4} + 794 \, x^{3} + 589 \, x^{2} + 228 \, x + 36\right )} \log \left (\sqrt{2 \, x + 3} - 1\right ) + 5 \,{\left (740124 \, x^{5} + 4247856 \, x^{4} + 9453447 \, x^{3} + 10168583 \, x^{2} + 5278129 \, x + 1057511\right )} \sqrt{2 \, x + 3}}{3750 \,{\left (36 \, x^{6} + 228 \, x^{5} + 589 \, x^{4} + 794 \, x^{3} + 589 \, x^{2} + 228 \, x + 36\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 = 1.09736, size = 181, normalized size = 1.41 \begin{align*} \frac{45603}{1250} \, \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{64 \,{\left (921 \, x + 1414\right )}}{1875 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}}} + \frac{396801 \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - 1551207 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + 1922011 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 737605 \, \sqrt{2 \, x + 3}}{625 \,{\left (3 \,{\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}^{2}} + 155 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) - 155 \, \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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