Optimal. Leaf size=97 \[ \frac{2 b^{3/2} (c x)^{3/2} \left (1-\frac{a}{b x^2}\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ),2\right )}{3 \sqrt{a} c^4 \left (a-b x^2\right )^{3/4}}-\frac{2 \sqrt [4]{a-b x^2}}{3 c (c x)^{3/2}} \]
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Rubi [A] time = 0.0785771, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {277, 329, 237, 335, 275, 232} \[ \frac{2 b^{3/2} (c x)^{3/2} \left (1-\frac{a}{b x^2}\right )^{3/4} F\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 \sqrt{a} c^4 \left (a-b x^2\right )^{3/4}}-\frac{2 \sqrt [4]{a-b x^2}}{3 c (c x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 277
Rule 329
Rule 237
Rule 335
Rule 275
Rule 232
Rubi steps
\begin{align*} \int \frac{\sqrt [4]{a-b x^2}}{(c x)^{5/2}} \, dx &=-\frac{2 \sqrt [4]{a-b x^2}}{3 c (c x)^{3/2}}-\frac{b \int \frac{1}{\sqrt{c x} \left (a-b x^2\right )^{3/4}} \, dx}{3 c^2}\\ &=-\frac{2 \sqrt [4]{a-b x^2}}{3 c (c x)^{3/2}}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{1}{\left (a-\frac{b x^4}{c^2}\right )^{3/4}} \, dx,x,\sqrt{c x}\right )}{3 c^3}\\ &=-\frac{2 \sqrt [4]{a-b x^2}}{3 c (c x)^{3/2}}-\frac{\left (2 b \left (1-\frac{a}{b x^2}\right )^{3/4} (c x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{a c^2}{b x^4}\right )^{3/4} x^3} \, dx,x,\sqrt{c x}\right )}{3 c^3 \left (a-b x^2\right )^{3/4}}\\ &=-\frac{2 \sqrt [4]{a-b x^2}}{3 c (c x)^{3/2}}+\frac{\left (2 b \left (1-\frac{a}{b x^2}\right )^{3/4} (c x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{x}{\left (1-\frac{a c^2 x^4}{b}\right )^{3/4}} \, dx,x,\frac{1}{\sqrt{c x}}\right )}{3 c^3 \left (a-b x^2\right )^{3/4}}\\ &=-\frac{2 \sqrt [4]{a-b x^2}}{3 c (c x)^{3/2}}+\frac{\left (b \left (1-\frac{a}{b x^2}\right )^{3/4} (c x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{a c^2 x^2}{b}\right )^{3/4}} \, dx,x,\frac{1}{c x}\right )}{3 c^3 \left (a-b x^2\right )^{3/4}}\\ &=-\frac{2 \sqrt [4]{a-b x^2}}{3 c (c x)^{3/2}}+\frac{2 b^{3/2} \left (1-\frac{a}{b x^2}\right )^{3/4} (c x)^{3/2} F\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 \sqrt{a} c^4 \left (a-b x^2\right )^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.0153646, size = 57, normalized size = 0.59 \[ -\frac{2 x \sqrt [4]{a-b x^2} \, _2F_1\left (-\frac{3}{4},-\frac{1}{4};\frac{1}{4};\frac{b x^2}{a}\right )}{3 (c x)^{5/2} \sqrt [4]{1-\frac{b x^2}{a}}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.024, size = 0, normalized size = 0. \begin{align*} \int{\sqrt [4]{-b{x}^{2}+a} \left ( cx \right ) ^{-{\frac{5}{2}}}}\, dx \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{{\left (-b x^{2} + a\right )}^{\frac{1}{4}}}{\left (c x\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (-b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{c x}}{c^{3} x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 11.0769, size = 36, normalized size = 0.37 \begin{align*} - \frac{i \sqrt [4]{b} e^{- \frac{i \pi }{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{a}{b x^{2}}} \right )}}{c^{\frac{5}{2}} x} \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{{\left (-b x^{2} + a\right )}^{\frac{1}{4}}}{\left (c x\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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