Optimal. Leaf size=81 \[ -\frac{1}{48} \left (59-18 x^2\right ) \left (x^4+5 x^2+3\right )^{3/2}+\frac{259}{128} \left (2 x^2+5\right ) \sqrt{x^4+5 x^2+3}-\frac{3367}{256} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right ) \]
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Rubi [A] time = 0.0562794, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1251, 779, 612, 621, 206} \[ -\frac{1}{48} \left (59-18 x^2\right ) \left (x^4+5 x^2+3\right )^{3/2}+\frac{259}{128} \left (2 x^2+5\right ) \sqrt{x^4+5 x^2+3}-\frac{3367}{256} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right ) \]
Antiderivative was successfully verified.
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Rule 1251
Rule 779
Rule 612
Rule 621
Rule 206
Rubi steps
\begin{align*} \int x^3 \left (2+3 x^2\right ) \sqrt{3+5 x^2+x^4} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x (2+3 x) \sqrt{3+5 x+x^2} \, dx,x,x^2\right )\\ &=-\frac{1}{48} \left (59-18 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}+\frac{259}{32} \operatorname{Subst}\left (\int \sqrt{3+5 x+x^2} \, dx,x,x^2\right )\\ &=\frac{259}{128} \left (5+2 x^2\right ) \sqrt{3+5 x^2+x^4}-\frac{1}{48} \left (59-18 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}-\frac{3367}{256} \operatorname{Subst}\left (\int \frac{1}{\sqrt{3+5 x+x^2}} \, dx,x,x^2\right )\\ &=\frac{259}{128} \left (5+2 x^2\right ) \sqrt{3+5 x^2+x^4}-\frac{1}{48} \left (59-18 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}-\frac{3367}{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{259}{128} \left (5+2 x^2\right ) \sqrt{3+5 x^2+x^4}-\frac{1}{48} \left (59-18 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}-\frac{3367}{256} \tanh ^{-1}\left (\frac{5+2 x^2}{2 \sqrt{3+5 x^2+x^4}}\right )\\ \end{align*}
Mathematica [A] time = 0.0207215, size = 66, normalized size = 0.81 \[ \frac{1}{768} \left (2 \sqrt{x^4+5 x^2+3} \left (144 x^6+248 x^4-374 x^2+2469\right )-10101 \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 74, normalized size = 0.9 \begin{align*}{\frac{3\,{x}^{2}}{8} \left ({x}^{4}+5\,{x}^{2}+3 \right ) ^{{\frac{3}{2}}}}-{\frac{59}{48} \left ({x}^{4}+5\,{x}^{2}+3 \right ) ^{{\frac{3}{2}}}}+{\frac{518\,{x}^{2}+1295}{128}\sqrt{{x}^{4}+5\,{x}^{2}+3}}-{\frac{3367}{256}\ln \left ({\frac{5}{2}}+{x}^{2}+\sqrt{{x}^{4}+5\,{x}^{2}+3} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.977678, size = 117, normalized size = 1.44 \begin{align*} \frac{3}{8} \,{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}} x^{2} + \frac{259}{64} \, \sqrt{x^{4} + 5 \, x^{2} + 3} x^{2} - \frac{59}{48} \,{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}} + \frac{1295}{128} \, \sqrt{x^{4} + 5 \, x^{2} + 3} - \frac{3367}{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 = 1.63903, size = 161, normalized size = 1.99 \begin{align*} \frac{1}{384} \,{\left (144 \, x^{6} + 248 \, x^{4} - 374 \, x^{2} + 2469\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} + \frac{3367}{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 x^{3} \left (3 x^{2} + 2\right ) \sqrt{x^{4} + 5 x^{2} + 3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08371, size = 81, normalized size = 1. \begin{align*} \frac{1}{384} \, \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (2 \,{\left (4 \,{\left (18 \, x^{2} + 31\right )} x^{2} - 187\right )} x^{2} + 2469\right )} + \frac{3367}{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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