Optimal. Leaf size=110 \[ -\frac{1}{27} (2 x+3)^2 \left (3 x^2+2\right )^{7/2}+\frac{1}{81} (63 x+226) \left (3 x^2+2\right )^{7/2}+\frac{133}{18} x \left (3 x^2+2\right )^{5/2}+\frac{665}{36} x \left (3 x^2+2\right )^{3/2}+\frac{665}{12} x \sqrt{3 x^2+2}+\frac{665 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{6 \sqrt{3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0431451, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {833, 780, 195, 215} \[ -\frac{1}{27} (2 x+3)^2 \left (3 x^2+2\right )^{7/2}+\frac{1}{81} (63 x+226) \left (3 x^2+2\right )^{7/2}+\frac{133}{18} x \left (3 x^2+2\right )^{5/2}+\frac{665}{36} x \left (3 x^2+2\right )^{3/2}+\frac{665}{12} x \sqrt{3 x^2+2}+\frac{665 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{6 \sqrt{3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 833
Rule 780
Rule 195
Rule 215
Rubi steps
\begin{align*} \int (5-x) (3+2 x)^2 \left (2+3 x^2\right )^{5/2} \, dx &=-\frac{1}{27} (3+2 x)^2 \left (2+3 x^2\right )^{7/2}+\frac{1}{27} \int (3+2 x) (413+252 x) \left (2+3 x^2\right )^{5/2} \, dx\\ &=-\frac{1}{27} (3+2 x)^2 \left (2+3 x^2\right )^{7/2}+\frac{1}{81} (226+63 x) \left (2+3 x^2\right )^{7/2}+\frac{133}{3} \int \left (2+3 x^2\right )^{5/2} \, dx\\ &=\frac{133}{18} x \left (2+3 x^2\right )^{5/2}-\frac{1}{27} (3+2 x)^2 \left (2+3 x^2\right )^{7/2}+\frac{1}{81} (226+63 x) \left (2+3 x^2\right )^{7/2}+\frac{665}{9} \int \left (2+3 x^2\right )^{3/2} \, dx\\ &=\frac{665}{36} x \left (2+3 x^2\right )^{3/2}+\frac{133}{18} x \left (2+3 x^2\right )^{5/2}-\frac{1}{27} (3+2 x)^2 \left (2+3 x^2\right )^{7/2}+\frac{1}{81} (226+63 x) \left (2+3 x^2\right )^{7/2}+\frac{665}{6} \int \sqrt{2+3 x^2} \, dx\\ &=\frac{665}{12} x \sqrt{2+3 x^2}+\frac{665}{36} x \left (2+3 x^2\right )^{3/2}+\frac{133}{18} x \left (2+3 x^2\right )^{5/2}-\frac{1}{27} (3+2 x)^2 \left (2+3 x^2\right )^{7/2}+\frac{1}{81} (226+63 x) \left (2+3 x^2\right )^{7/2}+\frac{665}{6} \int \frac{1}{\sqrt{2+3 x^2}} \, dx\\ &=\frac{665}{12} x \sqrt{2+3 x^2}+\frac{665}{36} x \left (2+3 x^2\right )^{3/2}+\frac{133}{18} x \left (2+3 x^2\right )^{5/2}-\frac{1}{27} (3+2 x)^2 \left (2+3 x^2\right )^{7/2}+\frac{1}{81} (226+63 x) \left (2+3 x^2\right )^{7/2}+\frac{665 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{6 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0729189, size = 75, normalized size = 0.68 \[ \frac{1}{324} \sqrt{3 x^2+2} \left (-1296 x^8+2916 x^7+18900 x^6+27378 x^5+41256 x^4+50571 x^3+28272 x^2+40365 x+6368\right )+\frac{665 \sinh ^{-1}\left (\sqrt{\frac{3}{2}} x\right )}{6 \sqrt{3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.006, size = 87, normalized size = 0.8 \begin{align*} -{\frac{4\,{x}^{2}}{27} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{7}{2}}}}+{\frac{199}{81} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{7}{2}}}}+{\frac{x}{3} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{7}{2}}}}+{\frac{133\,x}{18} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{5}{2}}}}+{\frac{665\,x}{36} \left ( 3\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}}+{\frac{665\,x}{12}\sqrt{3\,{x}^{2}+2}}+{\frac{665\,\sqrt{3}}{18}{\it Arcsinh} \left ({\frac{x\sqrt{6}}{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.48552, size = 116, normalized size = 1.05 \begin{align*} -\frac{4}{27} \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}} x^{2} + \frac{1}{3} \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}} x + \frac{199}{81} \,{\left (3 \, x^{2} + 2\right )}^{\frac{7}{2}} + \frac{133}{18} \,{\left (3 \, x^{2} + 2\right )}^{\frac{5}{2}} x + \frac{665}{36} \,{\left (3 \, x^{2} + 2\right )}^{\frac{3}{2}} x + \frac{665}{12} \, \sqrt{3 \, x^{2} + 2} x + \frac{665}{18} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{6} x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.84876, size = 248, normalized size = 2.25 \begin{align*} -\frac{1}{324} \,{\left (1296 \, x^{8} - 2916 \, x^{7} - 18900 \, x^{6} - 27378 \, x^{5} - 41256 \, x^{4} - 50571 \, x^{3} - 28272 \, x^{2} - 40365 \, x - 6368\right )} \sqrt{3 \, x^{2} + 2} + \frac{665}{36} \, \sqrt{3} \log \left (-\sqrt{3} \sqrt{3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 83.6017, size = 162, normalized size = 1.47 \begin{align*} - 4 x^{8} \sqrt{3 x^{2} + 2} + 9 x^{7} \sqrt{3 x^{2} + 2} + \frac{175 x^{6} \sqrt{3 x^{2} + 2}}{3} + \frac{169 x^{5} \sqrt{3 x^{2} + 2}}{2} + \frac{382 x^{4} \sqrt{3 x^{2} + 2}}{3} + \frac{1873 x^{3} \sqrt{3 x^{2} + 2}}{12} + \frac{2356 x^{2} \sqrt{3 x^{2} + 2}}{27} + \frac{1495 x \sqrt{3 x^{2} + 2}}{12} + \frac{1592 \sqrt{3 x^{2} + 2}}{81} + \frac{665 \sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{6} x}{2} \right )}}{18} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.23513, size = 97, normalized size = 0.88 \begin{align*} -\frac{1}{324} \,{\left (3 \,{\left ({\left (9 \,{\left (2 \,{\left ({\left (2 \,{\left (3 \,{\left (4 \, x - 9\right )} x - 175\right )} x - 507\right )} x - 764\right )} x - 1873\right )} x - 9424\right )} x - 13455\right )} x - 6368\right )} \sqrt{3 \, x^{2} + 2} - \frac{665}{18} \, \sqrt{3} \log \left (-\sqrt{3} x + \sqrt{3 \, x^{2} + 2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]