Optimal. Leaf size=119 \[ \frac{1}{48} \left (18 x^2+61\right ) \left (x^4+5 x^2+3\right )^{3/2}+\frac{1}{128} \left (199-74 x^2\right ) \sqrt{x^4+5 x^2+3}+\frac{2401}{256} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )-3 \sqrt{3} \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right ) \]
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Rubi [A] time = 0.105921, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {1251, 814, 843, 621, 206, 724} \[ \frac{1}{48} \left (18 x^2+61\right ) \left (x^4+5 x^2+3\right )^{3/2}+\frac{1}{128} \left (199-74 x^2\right ) \sqrt{x^4+5 x^2+3}+\frac{2401}{256} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )-3 \sqrt{3} \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right ) \]
Antiderivative was successfully verified.
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Rule 1251
Rule 814
Rule 843
Rule 621
Rule 206
Rule 724
Rubi steps
\begin{align*} \int \frac{\left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{x} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(2+3 x) \left (3+5 x+x^2\right )^{3/2}}{x} \, dx,x,x^2\right )\\ &=\frac{1}{48} \left (61+18 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}-\frac{1}{16} \operatorname{Subst}\left (\int \frac{\left (-48+\frac{37 x}{2}\right ) \sqrt{3+5 x+x^2}}{x} \, dx,x,x^2\right )\\ &=\frac{1}{128} \left (199-74 x^2\right ) \sqrt{3+5 x^2+x^4}+\frac{1}{48} \left (61+18 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}+\frac{1}{64} \operatorname{Subst}\left (\int \frac{576+\frac{2401 x}{4}}{x \sqrt{3+5 x+x^2}} \, dx,x,x^2\right )\\ &=\frac{1}{128} \left (199-74 x^2\right ) \sqrt{3+5 x^2+x^4}+\frac{1}{48} \left (61+18 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}+9 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{3+5 x+x^2}} \, dx,x,x^2\right )+\frac{2401}{256} \operatorname{Subst}\left (\int \frac{1}{\sqrt{3+5 x+x^2}} \, dx,x,x^2\right )\\ &=\frac{1}{128} \left (199-74 x^2\right ) \sqrt{3+5 x^2+x^4}+\frac{1}{48} \left (61+18 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}-18 \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{6+5 x^2}{\sqrt{3+5 x^2+x^4}}\right )+\frac{2401}{128} \operatorname{Subst}\left (\int \frac{1}{4-x^2} \, dx,x,\frac{5+2 x^2}{\sqrt{3+5 x^2+x^4}}\right )\\ &=\frac{1}{128} \left (199-74 x^2\right ) \sqrt{3+5 x^2+x^4}+\frac{1}{48} \left (61+18 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}+\frac{2401}{256} \tanh ^{-1}\left (\frac{5+2 x^2}{2 \sqrt{3+5 x^2+x^4}}\right )-3 \sqrt{3} \tanh ^{-1}\left (\frac{6+5 x^2}{2 \sqrt{3} \sqrt{3+5 x^2+x^4}}\right )\\ \end{align*}
Mathematica [A] time = 0.0637445, size = 104, normalized size = 0.87 \[ \frac{1}{384} \sqrt{x^4+5 x^2+3} \left (144 x^6+1208 x^4+2650 x^2+2061\right )+\frac{2401}{256} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )-3 \sqrt{3} \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 117, normalized size = 1. \begin{align*}{\frac{3\,{x}^{6}}{8}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{151\,{x}^{4}}{48}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{1325\,{x}^{2}}{192}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{687}{128}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{2401}{256}\ln \left ({\frac{5}{2}}+{x}^{2}+\sqrt{{x}^{4}+5\,{x}^{2}+3} \right ) }-3\,{\it Artanh} \left ( 1/6\,{\frac{ \left ( 5\,{x}^{2}+6 \right ) \sqrt{3}}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}} \right ) \sqrt{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.4835, size = 162, normalized size = 1.36 \begin{align*} \frac{3}{8} \,{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}} x^{2} - \frac{37}{64} \, \sqrt{x^{4} + 5 \, x^{2} + 3} x^{2} + \frac{61}{48} \,{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}} - 3 \, \sqrt{3} \log \left (\frac{2 \, \sqrt{3} \sqrt{x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac{6}{x^{2}} + 5\right ) + \frac{199}{128} \, \sqrt{x^{4} + 5 \, x^{2} + 3} + \frac{2401}{256} \, \log \left (2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} + 5\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.29806, size = 300, normalized size = 2.52 \begin{align*} \frac{1}{384} \,{\left (144 \, x^{6} + 1208 \, x^{4} + 2650 \, x^{2} + 2061\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} + 3 \, \sqrt{3} \log \left (\frac{25 \, x^{2} - 2 \, \sqrt{3}{\left (5 \, x^{2} + 6\right )} - 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (5 \, \sqrt{3} - 6\right )} + 30}{x^{2}}\right ) - \frac{2401}{256} \, \log \left (-2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} - 5\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (3 x^{2} + 2\right ) \left (x^{4} + 5 x^{2} + 3\right )^{\frac{3}{2}}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}}{\left (3 \, x^{2} + 2\right )}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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