Optimal. Leaf size=45 \[ -\frac{\left (3 x^2+2\right ) x^2}{2 \sqrt{x^4+5}}+3 \sqrt{x^4+5}+\sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right ) \]
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Rubi [A] time = 0.0386738, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1252, 819, 641, 215} \[ -\frac{\left (3 x^2+2\right ) x^2}{2 \sqrt{x^4+5}}+3 \sqrt{x^4+5}+\sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right ) \]
Antiderivative was successfully verified.
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Rule 1252
Rule 819
Rule 641
Rule 215
Rubi steps
\begin{align*} \int \frac{x^5 \left (2+3 x^2\right )}{\left (5+x^4\right )^{3/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2 (2+3 x)}{\left (5+x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=-\frac{x^2 \left (2+3 x^2\right )}{2 \sqrt{5+x^4}}+\frac{1}{10} \operatorname{Subst}\left (\int \frac{10+30 x}{\sqrt{5+x^2}} \, dx,x,x^2\right )\\ &=-\frac{x^2 \left (2+3 x^2\right )}{2 \sqrt{5+x^4}}+3 \sqrt{5+x^4}+\operatorname{Subst}\left (\int \frac{1}{\sqrt{5+x^2}} \, dx,x,x^2\right )\\ &=-\frac{x^2 \left (2+3 x^2\right )}{2 \sqrt{5+x^4}}+3 \sqrt{5+x^4}+\sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )\\ \end{align*}
Mathematica [A] time = 0.0237748, size = 46, normalized size = 1.02 \[ \frac{3 x^4-2 x^2+2 \sqrt{x^4+5} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )+30}{2 \sqrt{x^4+5}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 37, normalized size = 0.8 \begin{align*}{\frac{3\,{x}^{4}+30}{2}{\frac{1}{\sqrt{{x}^{4}+5}}}}-{{x}^{2}{\frac{1}{\sqrt{{x}^{4}+5}}}}+{\it Arcsinh} \left ({\frac{{x}^{2}\sqrt{5}}{5}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.41993, size = 85, normalized size = 1.89 \begin{align*} -\frac{x^{2}}{\sqrt{x^{4} + 5}} + \frac{3}{2} \, \sqrt{x^{4} + 5} + \frac{15}{2 \, \sqrt{x^{4} + 5}} + \frac{1}{2} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} + 1\right ) - \frac{1}{2} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49555, size = 143, normalized size = 3.18 \begin{align*} -\frac{2 \, x^{4} + 2 \,{\left (x^{4} + 5\right )} \log \left (-x^{2} + \sqrt{x^{4} + 5}\right ) -{\left (3 \, x^{4} - 2 \, x^{2} + 30\right )} \sqrt{x^{4} + 5} + 10}{2 \,{\left (x^{4} + 5\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 10.2649, size = 48, normalized size = 1.07 \begin{align*} \frac{3 x^{4}}{2 \sqrt{x^{4} + 5}} - \frac{x^{2}}{\sqrt{x^{4} + 5}} + \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )} + \frac{15}{\sqrt{x^{4} + 5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16797, size = 53, normalized size = 1.18 \begin{align*} \frac{{\left (3 \, x^{2} - 2\right )} x^{2} + 30}{2 \, \sqrt{x^{4} + 5}} - \log \left (-x^{2} + \sqrt{x^{4} + 5}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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