Optimal. Leaf size=28 \[ -\frac{2 \left (a+b x^2\right )^{5/4}}{5 a c (c x)^{5/2}} \]
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Rubi [A] time = 0.0060003, antiderivative size = 28, 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} \[ -\frac{2 \left (a+b x^2\right )^{5/4}}{5 a c (c x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin{align*} \int \frac{\sqrt [4]{a+b x^2}}{(c x)^{7/2}} \, dx &=-\frac{2 \left (a+b x^2\right )^{5/4}}{5 a c (c x)^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.008432, size = 26, normalized size = 0.93 \[ -\frac{2 x \left (a+b x^2\right )^{5/4}}{5 a (c x)^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 21, normalized size = 0.8 \begin{align*} -{\frac{2\,x}{5\,a} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{4}}} \left ( cx \right ) ^{-{\frac{7}{2}}}} \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{{\left (b x^{2} + a\right )}^{\frac{1}{4}}}{\left (c x\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.96398, size = 62, normalized size = 2.21 \begin{align*} -\frac{2 \,{\left (b x^{2} + a\right )}^{\frac{5}{4}} \sqrt{c x}}{5 \, a c^{4} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 50.3123, size = 78, normalized size = 2.79 \begin{align*} \frac{\sqrt [4]{b} \sqrt [4]{\frac{a}{b x^{2}} + 1} \Gamma \left (- \frac{5}{4}\right )}{2 c^{\frac{7}{2}} x^{2} \Gamma \left (- \frac{1}{4}\right )} + \frac{b^{\frac{5}{4}} \sqrt [4]{\frac{a}{b x^{2}} + 1} \Gamma \left (- \frac{5}{4}\right )}{2 a c^{\frac{7}{2}} \Gamma \left (- \frac{1}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.01632, size = 58, normalized size = 2.07 \begin{align*} -\frac{2 \,{\left (b c^{4} x^{2} + a c^{4}\right )}^{\frac{1}{4}}{\left (b c^{2} + \frac{a c^{2}}{x^{2}}\right )}}{5 \, \sqrt{c x} a c^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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