Optimal. Leaf size=26 \[ -\frac {2 \sqrt [4]{a+b x^2}}{a c \sqrt {c x}} \]
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Rubi [A] time = 0.01, antiderivative size = 26, 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} \[ -\frac {2 \sqrt [4]{a+b x^2}}{a c \sqrt {c x}} \]
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 = 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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fricas [A] time = 0.97, size = 25, 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 = 21, 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 = 4.90, size = 22, normalized size = 0.85 \[ -\frac {2\,{\left (b\,x^2+a\right )}^{1/4}}{a\,c\,\sqrt {c\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.92, size = 36, normalized size = 1.38 \[ \frac {\sqrt [4]{b} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {1}{4}\right )}{2 a c^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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