Optimal. Leaf size=26 \[ \frac{2 \sqrt{c x}}{a c \sqrt [4]{a+b x^2}} \]
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Rubi [A] time = 0.0055901, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {264} \[ \frac{2 \sqrt{c x}}{a c \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{c x} \left (a+b x^2\right )^{5/4}} \, dx &=\frac{2 \sqrt{c x}}{a c \sqrt [4]{a+b x^2}}\\ \end{align*}
Mathematica [A] time = 0.0060322, size = 24, normalized size = 0.92 \[ \frac{2 x}{a \sqrt{c x} \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 21, normalized size = 0.8 \begin{align*} 2\,{\frac{x}{\sqrt [4]{b{x}^{2}+a}a\sqrt{cx}}} \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{1}{{\left (b x^{2} + a\right )}^{\frac{5}{4}} \sqrt{c x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50464, size = 69, normalized size = 2.65 \begin{align*} \frac{2 \,{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{c x}}{a b c x^{2} + a^{2} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.04361, size = 34, normalized size = 1.31 \begin{align*} \frac{\Gamma \left (\frac{1}{4}\right )}{2 a \sqrt [4]{b} \sqrt{c} \sqrt [4]{\frac{a}{b x^{2}} + 1} \Gamma \left (\frac{5}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{5}{4}} \sqrt{c x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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