Optimal. Leaf size=100 \[ -\frac{\sqrt{2 x+3} (139 x+121)}{6 \left (3 x^2+5 x+2\right )^2}+\frac{\sqrt{2 x+3} (2529 x+2090)}{6 \left (3 x^2+5 x+2\right )}+966 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-1247 \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0667299, antiderivative size = 100, 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, 822, 826, 1166, 207} \[ -\frac{\sqrt{2 x+3} (139 x+121)}{6 \left (3 x^2+5 x+2\right )^2}+\frac{\sqrt{2 x+3} (2529 x+2090)}{6 \left (3 x^2+5 x+2\right )}+966 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-1247 \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 818
Rule 822
Rule 826
Rule 1166
Rule 207
Rubi steps
\begin{align*} \int \frac{(5-x) (3+2 x)^{3/2}}{\left (2+5 x+3 x^2\right )^3} \, dx &=-\frac{\sqrt{3+2 x} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac{1}{6} \int \frac{-1142-703 x}{\sqrt{3+2 x} \left (2+5 x+3 x^2\right )^2} \, dx\\ &=-\frac{\sqrt{3+2 x} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac{\sqrt{3+2 x} (2090+2529 x)}{6 \left (2+5 x+3 x^2\right )}-\frac{1}{30} \int \frac{-27135-12645 x}{\sqrt{3+2 x} \left (2+5 x+3 x^2\right )} \, dx\\ &=-\frac{\sqrt{3+2 x} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac{\sqrt{3+2 x} (2090+2529 x)}{6 \left (2+5 x+3 x^2\right )}-\frac{1}{15} \operatorname{Subst}\left (\int \frac{-16335-12645 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt{3+2 x}\right )\\ &=-\frac{\sqrt{3+2 x} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac{\sqrt{3+2 x} (2090+2529 x)}{6 \left (2+5 x+3 x^2\right )}-2898 \operatorname{Subst}\left (\int \frac{1}{-3+3 x^2} \, dx,x,\sqrt{3+2 x}\right )+3741 \operatorname{Subst}\left (\int \frac{1}{-5+3 x^2} \, dx,x,\sqrt{3+2 x}\right )\\ &=-\frac{\sqrt{3+2 x} (121+139 x)}{6 \left (2+5 x+3 x^2\right )^2}+\frac{\sqrt{3+2 x} (2090+2529 x)}{6 \left (2+5 x+3 x^2\right )}+966 \tanh ^{-1}\left (\sqrt{3+2 x}\right )-1247 \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{3+2 x}\right )\\ \end{align*}
Mathematica [A] time = 0.083791, size = 80, normalized size = 0.8 \[ \frac{\sqrt{2 x+3} \left (2529 x^3+6305 x^2+5123 x+1353\right )}{2 \left (3 x^2+5 x+2\right )^2}+966 \tanh ^{-1}\left (\sqrt{2 x+3}\right )-1247 \sqrt{\frac{3}{5}} \tanh ^{-1}\left (\sqrt{\frac{3}{5}} \sqrt{2 x+3}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.017, size = 124, normalized size = 1.2 \begin{align*} 18\,{\frac{1}{ \left ( 6\,x+4 \right ) ^{2}} \left ({\frac{145\, \left ( 3+2\,x \right ) ^{3/2}}{2}}-{\frac{2345\,\sqrt{3+2\,x}}{18}} \right ) }-{\frac{1247\,\sqrt{15}}{5}{\it Artanh} \left ({\frac{\sqrt{15}}{5}\sqrt{3+2\,x}} \right ) }-3\, \left ( 1+\sqrt{3+2\,x} \right ) ^{-2}+68\, \left ( 1+\sqrt{3+2\,x} \right ) ^{-1}+483\,\ln \left ( 1+\sqrt{3+2\,x} \right ) +3\, \left ( -1+\sqrt{3+2\,x} \right ) ^{-2}+68\, \left ( -1+\sqrt{3+2\,x} \right ) ^{-1}-483\,\ln \left ( -1+\sqrt{3+2\,x} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.45299, size = 180, normalized size = 1.8 \begin{align*} \frac{1247}{10} \, \sqrt{15} \log \left (-\frac{\sqrt{15} - 3 \, \sqrt{2 \, x + 3}}{\sqrt{15} + 3 \, \sqrt{2 \, x + 3}}\right ) + \frac{2529 \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - 10151 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + 13115 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 5445 \, \sqrt{2 \, x + 3}}{9 \,{\left (2 \, x + 3\right )}^{4} - 48 \,{\left (2 \, x + 3\right )}^{3} + 94 \,{\left (2 \, x + 3\right )}^{2} - 160 \, x - 215} + 483 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) - 483 \, \log \left (\sqrt{2 \, x + 3} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.64838, size = 474, normalized size = 4.74 \begin{align*} \frac{1247 \, \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 ) + 4830 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\sqrt{2 \, x + 3} + 1\right ) - 4830 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (\sqrt{2 \, x + 3} - 1\right ) + 5 \,{\left (2529 \, x^{3} + 6305 \, x^{2} + 5123 \, x + 1353\right )} \sqrt{2 \, x + 3}}{10 \,{\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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.09057, size = 161, normalized size = 1.61 \begin{align*} \frac{1247}{10} \, \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{2529 \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} - 10151 \,{\left (2 \, x + 3\right )}^{\frac{5}{2}} + 13115 \,{\left (2 \, x + 3\right )}^{\frac{3}{2}} - 5445 \, \sqrt{2 \, x + 3}}{{\left (3 \,{\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}^{2}} + 483 \, \log \left (\sqrt{2 \, x + 3} + 1\right ) - 483 \, \log \left ({\left | \sqrt{2 \, x + 3} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]