Optimal. Leaf size=26 \[ -\frac{2 \sqrt [4]{a+b x^2}}{a c \sqrt{c x}} \]
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Rubi [A] time = 0.0059991, 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} \[ -\frac{2 \sqrt [4]{a+b x^2}}{a c \sqrt{c x}} \]
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin{align*} \int \frac{1}{(c x)^{3/2} \left (a+b x^2\right )^{3/4}} \, dx &=-\frac{2 \sqrt [4]{a+b x^2}}{a c \sqrt{c x}}\\ \end{align*}
Mathematica [A] time = 0.0053811, size = 24, normalized size = 0.92 \[ -\frac{2 x \sqrt [4]{a+b x^2}}{a (c x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 21, normalized size = 0.8 \begin{align*} -2\,{\frac{x\sqrt [4]{b{x}^{2}+a}}{a \left ( cx \right ) ^{3/2}}} \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{1}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \left (c x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.38803, size = 57, normalized size = 2.19 \begin{align*} -\frac{2 \,{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{c x}}{a c^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.23434, size = 36, normalized size = 1.38 \begin{align*} \frac{\sqrt [4]{b} \sqrt [4]{\frac{a}{b x^{2}} + 1} \Gamma \left (- \frac{1}{4}\right )}{2 a c^{\frac{3}{2}} \Gamma \left (\frac{3}{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{3}{4}} \left (c x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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