Optimal. Leaf size=100 \[ -\frac{(139 x+121) (2 x+3)^{5/2}}{6 \left (3 x^2+5 x+2\right )^2}+\frac{7 (619 x+546) \sqrt{2 x+3}}{6 \left (3 x^2+5 x+2\right )}+1582 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-1225 \sqrt{\frac{5}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]
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Rubi [A] time = 0.0632812, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {818, 826, 1166, 207} \[ -\frac{(139 x+121) (2 x+3)^{5/2}}{6 \left (3 x^2+5 x+2\right )^2}+\frac{7 (619 x+546) \sqrt{2 x+3}}{6 \left (3 x^2+5 x+2\right )}+1582 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-1225 \sqrt{\frac{5}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]
Antiderivative was successfully verified.
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Rule 818
Rule 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 )^3} \, dx &=-\frac{(3+2 x)^{5/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac{1}{6} \int \frac{(-658-147 x) (3+2 x)^{3/2}}{\left (2+5 x+3 x^2\right )^2} \, dx\\ &=-\frac{(3+2 x)^{5/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac{7 \sqrt{3+2 x} (546+619 x)}{6 \left (2+5 x+3 x^2\right )}+\frac{1}{18} \int \frac{26649+12411 x}{\sqrt{3+2 x} \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac{(3+2 x)^{5/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac{7 \sqrt{3+2 x} (546+619 x)}{6 \left (2+5 x+3 x^2\right )}+\frac{1}{9} \operatorname{Subst}\left (\int \frac{16065+12411 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt{3+2 x}\right )\\ &=-\frac{(3+2 x)^{5/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac{7 \sqrt{3+2 x} (546+619 x)}{6 \left (2+5 x+3 x^2\right )}-4746 \operatorname{Subst}\left (\int \frac{1}{-3+3 x^2} \, dx,x,\sqrt{3+2 x}\right )+6125 \operatorname{Subst}\left (\int \frac{1}{-5+3 x^2} \, dx,x,\sqrt{3+2 x}\right )\\ &=-\frac{(3+2 x)^{5/2} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac{7 \sqrt{3+2 x} (546+619 x)}{6 \left (2+5 x+3 x^2\right )}+1582 \tanh ^{-1}\left (\sqrt{3+2 x}\right )-1225 \sqrt{\frac{5}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{3+2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.0904416, size = 80, normalized size = 0.8 \[ \frac{\sqrt{2 x+3} \left (12443 x^3+30979 x^2+25073 x+6555\right )}{6 \left (3 x^2+5 x+2\right )^2}+1582 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-1225 \sqrt{\frac{5}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 124, normalized size = 1.2 \begin{align*} 450\,{\frac{1}{ \left ( 6\,x+4 \right ) ^{2}} \left ({\frac{299\, \left ( 3+2\,x \right ) ^{3/2}}{54}}-{\frac{185\,\sqrt{3+2\,x}}{18}} \right ) }-{\frac{1225\,\sqrt{15}}{3}{\it Artanh} \left ({\frac{\sqrt{15}}{5}\sqrt{3+2\,x}} \right ) }-3\, \left ( 1+\sqrt{3+2\,x} \right ) ^{-2}+92\, \left ( 1+\sqrt{3+2\,x} \right ) ^{-1}+791\,\ln \left ( 1+\sqrt{3+2\,x} \right ) +3\, \left ( -1+\sqrt{3+2\,x} \right ) ^{-2}+92\, \left ( -1+\sqrt{3+2\,x} \right ) ^{-1}-791\,\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 = 1.43916, size = 181, normalized size = 1.81 \begin{align*} \frac{1225}{6} \, \sqrt{15} \log \left (-\frac{\sqrt{15} - 3 \, \sqrt{2 \, x + 3}}{\sqrt{15} + 3 \, \sqrt{2 \, x + 3}}\right ) + \frac{12443 \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - 50029 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + 64505 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 26775 \, \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 )}} + 791 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) - 791 \, \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 = 1.59164, size = 474, normalized size = 4.74 \begin{align*} \frac{1225 \, \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 ) + 4746 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\sqrt{2 \, x + 3} + 1\right ) - 4746 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\sqrt{2 \, x + 3} - 1\right ) +{\left (12443 \, x^{3} + 30979 \, x^{2} + 25073 \, x + 6555\right )} \sqrt{2 \, x + 3}}{6 \,{\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 = 1.09047, size = 162, normalized size = 1.62 \begin{align*} \frac{1225}{6} \, \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{12443 \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - 50029 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + 64505 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 26775 \, \sqrt{2 \, x + 3}}{3 \,{\left (3 \,{\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}^{2}} + 791 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) - 791 \, \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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