Optimal. Leaf size=74 \[ \frac{1}{2} \left (x^4+5 x^2+3\right )^{3/2}-\frac{11}{16} \left (2 x^2+5\right ) \sqrt{x^4+5 x^2+3}+\frac{143}{32} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right ) \]
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Rubi [A] time = 0.0435766, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {1247, 640, 612, 621, 206} \[ \frac{1}{2} \left (x^4+5 x^2+3\right )^{3/2}-\frac{11}{16} \left (2 x^2+5\right ) \sqrt{x^4+5 x^2+3}+\frac{143}{32} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right ) \]
Antiderivative was successfully verified.
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Rule 1247
Rule 640
Rule 612
Rule 621
Rule 206
Rubi steps
\begin{align*} \int x \left (2+3 x^2\right ) \sqrt{3+5 x^2+x^4} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int (2+3 x) \sqrt{3+5 x+x^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \left (3+5 x^2+x^4\right )^{3/2}-\frac{11}{4} \operatorname{Subst}\left (\int \sqrt{3+5 x+x^2} \, dx,x,x^2\right )\\ &=-\frac{11}{16} \left (5+2 x^2\right ) \sqrt{3+5 x^2+x^4}+\frac{1}{2} \left (3+5 x^2+x^4\right )^{3/2}+\frac{143}{32} \operatorname{Subst}\left (\int \frac{1}{\sqrt{3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac{11}{16} \left (5+2 x^2\right ) \sqrt{3+5 x^2+x^4}+\frac{1}{2} \left (3+5 x^2+x^4\right )^{3/2}+\frac{143}{16} \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{11}{16} \left (5+2 x^2\right ) \sqrt{3+5 x^2+x^4}+\frac{1}{2} \left (3+5 x^2+x^4\right )^{3/2}+\frac{143}{32} \tanh ^{-1}\left (\frac{5+2 x^2}{2 \sqrt{3+5 x^2+x^4}}\right )\\ \end{align*}
Mathematica [A] time = 0.0169141, size = 61, normalized size = 0.82 \[ \frac{1}{32} \left (2 \sqrt{x^4+5 x^2+3} \left (8 x^4+18 x^2-31\right )+143 \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 57, normalized size = 0.8 \begin{align*}{\frac{1}{2} \left ({x}^{4}+5\,{x}^{2}+3 \right ) ^{{\frac{3}{2}}}}-{\frac{22\,{x}^{2}+55}{16}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{143}{32}\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 = 1.22665, size = 95, normalized size = 1.28 \begin{align*} -\frac{11}{8} \, \sqrt{x^{4} + 5 \, x^{2} + 3} x^{2} + \frac{1}{2} \,{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}} - \frac{55}{16} \, \sqrt{x^{4} + 5 \, x^{2} + 3} + \frac{143}{32} \, \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 = 1.69985, size = 136, normalized size = 1.84 \begin{align*} \frac{1}{16} \,{\left (8 \, x^{4} + 18 \, x^{2} - 31\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} - \frac{143}{32} \, \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 \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 = 1.11622, size = 72, normalized size = 0.97 \begin{align*} \frac{1}{16} \, \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (2 \,{\left (4 \, x^{2} + 9\right )} x^{2} - 31\right )} - \frac{143}{32} \, \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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