Optimal. Leaf size=27 \[ -\frac {2 \sqrt [4]{a-b x^2}}{a c \sqrt {c x}} \]
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Rubi [A] time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, 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*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 0.93 \[ -\frac {2 x \sqrt [4]{a-b x^2}}{a (c x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 26, normalized size = 0.96 \[ -\frac {2 \, {\left (-b x^{2} + a\right )}^{\frac {1}{4}} \sqrt {c x}}{a c^{2} x} \]
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 \[ \int \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {3}{4}} \left (c x\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 22, normalized size = 0.81 \[ -\frac {2 \left (-b \,x^{2}+a \right )^{\frac {1}{4}} x}{\left (c x \right )^{\frac {3}{2}} a} \]
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 \[ \int \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {3}{4}} \left (c x\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.08, size = 23, normalized size = 0.85 \[ -\frac {2\,{\left (a-b\,x^2\right )}^{1/4}}{a\,c\,\sqrt {c\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.00, size = 90, normalized size = 3.33 \[ \begin {cases} \frac {\sqrt [4]{b} \sqrt [4]{\frac {a}{b x^{2}} - 1} \Gamma \left (- \frac {1}{4}\right )}{2 a c^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right )} & \text {for}\: \left |{\frac {a}{b x^{2}}}\right | > 1 \\- \frac {\sqrt [4]{b} \sqrt [4]{- \frac {a}{b x^{2}} + 1} e^{- \frac {3 i \pi }{4}} \Gamma \left (- \frac {1}{4}\right )}{2 a c^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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