Optimal. Leaf size=28 \[ -\frac {2 \left (a+b x^2\right )^{3/4}}{3 a c (c x)^{3/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, 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} \[ -\frac {2 \left (a+b x^2\right )^{3/4}}{3 a c (c x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin {align*} \int \frac {1}{(c x)^{5/2} \sqrt [4]{a+b x^2}} \, dx &=-\frac {2 \left (a+b x^2\right )^{3/4}}{3 a c (c x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 0.93 \[ -\frac {2 x \left (a+b x^2\right )^{3/4}}{3 a (c x)^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.34, size = 25, normalized size = 0.89 \[ -\frac {2 \, {\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {c x}}{3 \, a c^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {1}{4}} \left (c x\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 21, normalized size = 0.75 \[ -\frac {2 \left (b \,x^{2}+a \right )^{\frac {3}{4}} x}{3 \left (c x \right )^{\frac {5}{2}} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {1}{4}} \left (c x\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.95, size = 25, normalized size = 0.89 \[ -\frac {2\,{\left (b\,x^2+a\right )}^{3/4}}{3\,a\,c^2\,x\,\sqrt {c\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.58, size = 36, normalized size = 1.29 \[ \frac {b^{\frac {3}{4}} \left (\frac {a}{b x^{2}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {3}{4}\right )}{2 a c^{\frac {5}{2}} \Gamma \left (\frac {1}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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