Optimal. Leaf size=46 \[ \frac{3 x^2+2}{10 \sqrt{x^4+5}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{x^4+5}}{\sqrt{5}}\right )}{5 \sqrt{5}} \]
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Rubi [A] time = 0.0432416, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1252, 823, 12, 266, 63, 207} \[ \frac{3 x^2+2}{10 \sqrt{x^4+5}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{x^4+5}}{\sqrt{5}}\right )}{5 \sqrt{5}} \]
Antiderivative was successfully verified.
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Rule 1252
Rule 823
Rule 12
Rule 266
Rule 63
Rule 207
Rubi steps
\begin{align*} \int \frac{2+3 x^2}{x \left (5+x^4\right )^{3/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{2+3 x}{x \left (5+x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=\frac{2+3 x^2}{10 \sqrt{5+x^4}}-\frac{1}{50} \operatorname{Subst}\left (\int -\frac{10}{x \sqrt{5+x^2}} \, dx,x,x^2\right )\\ &=\frac{2+3 x^2}{10 \sqrt{5+x^4}}+\frac{1}{5} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{5+x^2}} \, dx,x,x^2\right )\\ &=\frac{2+3 x^2}{10 \sqrt{5+x^4}}+\frac{1}{10} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{5+x}} \, dx,x,x^4\right )\\ &=\frac{2+3 x^2}{10 \sqrt{5+x^4}}+\frac{1}{5} \operatorname{Subst}\left (\int \frac{1}{-5+x^2} \, dx,x,\sqrt{5+x^4}\right )\\ &=\frac{2+3 x^2}{10 \sqrt{5+x^4}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{5+x^4}}{\sqrt{5}}\right )}{5 \sqrt{5}}\\ \end{align*}
Mathematica [A] time = 0.0388873, size = 46, normalized size = 1. \[ \frac{1}{50} \left (\frac{5 \left (3 x^2+2\right )}{\sqrt{x^4+5}}-2 \sqrt{5} \tanh ^{-1}\left (\frac{\sqrt{x^4+5}}{\sqrt{5}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 40, normalized size = 0.9 \begin{align*}{\frac{3\,{x}^{2}}{10}{\frac{1}{\sqrt{{x}^{4}+5}}}}+{\frac{1}{5}{\frac{1}{\sqrt{{x}^{4}+5}}}}-{\frac{\sqrt{5}}{25}{\it Artanh} \left ({\sqrt{5}{\frac{1}{\sqrt{{x}^{4}+5}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.42682, size = 76, normalized size = 1.65 \begin{align*} \frac{3 \, x^{2}}{10 \, \sqrt{x^{4} + 5}} + \frac{1}{50} \, \sqrt{5} \log \left (-\frac{\sqrt{5} - \sqrt{x^{4} + 5}}{\sqrt{5} + \sqrt{x^{4} + 5}}\right ) + \frac{1}{5 \, \sqrt{x^{4} + 5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49631, size = 159, normalized size = 3.46 \begin{align*} \frac{15 \, x^{4} + 2 \, \sqrt{5}{\left (x^{4} + 5\right )} \log \left (-\frac{\sqrt{5} - \sqrt{x^{4} + 5}}{x^{2}}\right ) + 5 \, \sqrt{x^{4} + 5}{\left (3 \, x^{2} + 2\right )} + 75}{50 \,{\left (x^{4} + 5\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 11.6134, size = 212, normalized size = 4.61 \begin{align*} \frac{2 x^{4} \log{\left (x^{4} \right )}}{20 \sqrt{5} x^{4} + 100 \sqrt{5}} - \frac{4 x^{4} \log{\left (\sqrt{\frac{x^{4}}{5} + 1} + 1 \right )}}{20 \sqrt{5} x^{4} + 100 \sqrt{5}} - \frac{2 x^{4} \log{\left (5 \right )}}{20 \sqrt{5} x^{4} + 100 \sqrt{5}} + \frac{3 x^{2}}{10 \sqrt{x^{4} + 5}} + \frac{4 \sqrt{5} \sqrt{x^{4} + 5}}{20 \sqrt{5} x^{4} + 100 \sqrt{5}} + \frac{10 \log{\left (x^{4} \right )}}{20 \sqrt{5} x^{4} + 100 \sqrt{5}} - \frac{20 \log{\left (\sqrt{\frac{x^{4}}{5} + 1} + 1 \right )}}{20 \sqrt{5} x^{4} + 100 \sqrt{5}} - \frac{10 \log{\left (5 \right )}}{20 \sqrt{5} x^{4} + 100 \sqrt{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16022, size = 72, normalized size = 1.57 \begin{align*} -\frac{1}{50} \, \sqrt{5} \log \left (\sqrt{5} + \sqrt{x^{4} + 5}\right ) + \frac{1}{50} \, \sqrt{5} \log \left (-\sqrt{5} + \sqrt{x^{4} + 5}\right ) + \frac{3 \, x^{2} + 2}{10 \, \sqrt{x^{4} + 5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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