Optimal. Leaf size=59 \[ \frac{8 \left (a-b x^2\right )^{7/4}}{21 a^2 c (c x)^{7/2}}-\frac{2 \left (a-b x^2\right )^{3/4}}{3 a c (c x)^{7/2}} \]
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Rubi [A] time = 0.0153705, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {273, 264} \[ \frac{8 \left (a-b x^2\right )^{7/4}}{21 a^2 c (c x)^{7/2}}-\frac{2 \left (a-b x^2\right )^{3/4}}{3 a c (c x)^{7/2}} \]
Antiderivative was successfully verified.
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Rule 273
Rule 264
Rubi steps
\begin{align*} \int \frac{1}{(c x)^{9/2} \sqrt [4]{a-b x^2}} \, dx &=-\frac{2 \left (a-b x^2\right )^{3/4}}{3 a c (c x)^{7/2}}-\frac{4 \int \frac{\left (a-b x^2\right )^{3/4}}{(c x)^{9/2}} \, dx}{3 a}\\ &=-\frac{2 \left (a-b x^2\right )^{3/4}}{3 a c (c x)^{7/2}}+\frac{8 \left (a-b x^2\right )^{7/4}}{21 a^2 c (c x)^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0182048, size = 42, normalized size = 0.71 \[ -\frac{2 \sqrt{c x} \left (a-b x^2\right )^{3/4} \left (3 a+4 b x^2\right )}{21 a^2 c^5 x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 32, normalized size = 0.5 \begin{align*} -{\frac{2\,x \left ( 4\,b{x}^{2}+3\,a \right ) }{21\,{a}^{2}} \left ( -b{x}^{2}+a \right ) ^{{\frac{3}{4}}} \left ( cx \right ) ^{-{\frac{9}{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{1}{4}} \left (c x\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5763, size = 89, normalized size = 1.51 \begin{align*} -\frac{2 \,{\left (4 \, b x^{2} + 3 \, a\right )}{\left (-b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{c x}}{21 \, a^{2} c^{5} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{1}{4}} \left (c x\right )^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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