Optimal. Leaf size=82 \[ \frac{\sqrt{x} \sqrt{a+b x^2+c x^4} \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{2 \sqrt{c} \sqrt{a x+b x^3+c x^5}} \]
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Rubi [A] time = 0.0618114, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1914, 1107, 621, 206} \[ \frac{\sqrt{x} \sqrt{a+b x^2+c x^4} \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{2 \sqrt{c} \sqrt{a x+b x^3+c x^5}} \]
Antiderivative was successfully verified.
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Rule 1914
Rule 1107
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{x^{3/2}}{\sqrt{a x+b x^3+c x^5}} \, dx &=\frac{\left (\sqrt{x} \sqrt{a+b x^2+c x^4}\right ) \int \frac{x}{\sqrt{a+b x^2+c x^4}} \, dx}{\sqrt{a x+b x^3+c x^5}}\\ &=\frac{\left (\sqrt{x} \sqrt{a+b x^2+c x^4}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{2 \sqrt{a x+b x^3+c x^5}}\\ &=\frac{\left (\sqrt{x} \sqrt{a+b x^2+c x^4}\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x^2}{\sqrt{a+b x^2+c x^4}}\right )}{\sqrt{a x+b x^3+c x^5}}\\ &=\frac{\sqrt{x} \sqrt{a+b x^2+c x^4} \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{2 \sqrt{c} \sqrt{a x+b x^3+c x^5}}\\ \end{align*}
Mathematica [A] time = 0.0182489, size = 82, normalized size = 1. \[ \frac{\sqrt{x} \sqrt{a+b x^2+c x^4} \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{2 \sqrt{c} \sqrt{x \left (a+b x^2+c x^4\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 72, normalized size = 0.9 \begin{align*}{\frac{1}{2}\sqrt{x \left ( c{x}^{4}+b{x}^{2}+a \right ) }\ln \left ({\frac{1}{2} \left ( 2\,c{x}^{2}+2\,\sqrt{c{x}^{4}+b{x}^{2}+a}\sqrt{c}+b \right ){\frac{1}{\sqrt{c}}}} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}{\frac{1}{\sqrt{c}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{\frac{3}{2}}}{\sqrt{c x^{5} + b x^{3} + a x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46885, size = 325, normalized size = 3.96 \begin{align*} \left [\frac{\log \left (-\frac{8 \, c^{2} x^{5} + 8 \, b c x^{3} + 4 \, \sqrt{c x^{5} + b x^{3} + a x}{\left (2 \, c x^{2} + b\right )} \sqrt{c} \sqrt{x} +{\left (b^{2} + 4 \, a c\right )} x}{x}\right )}{4 \, \sqrt{c}}, -\frac{\sqrt{-c} \arctan \left (\frac{\sqrt{c x^{5} + b x^{3} + a x}{\left (2 \, c x^{2} + b\right )} \sqrt{-c} \sqrt{x}}{2 \,{\left (c^{2} x^{5} + b c x^{3} + a c x\right )}}\right )}{2 \, c}\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{x^{\frac{3}{2}}}{\sqrt{x \left (a + b x^{2} + c x^{4}\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31361, size = 54, normalized size = 0.66 \begin{align*} -\frac{\log \left ({\left | -2 \,{\left (\sqrt{c} x^{2} - \sqrt{c x^{4} + b x^{2} + a}\right )} \sqrt{c} - b \right |}\right )}{1024 \, c^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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