Optimal. Leaf size=100 \[ -\frac{4 b^{3/2} \sqrt{c x} \sqrt [4]{1-\frac{a}{b x^2}} E\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 a^{3/2} c^4 \sqrt [4]{a-b x^2}}-\frac{2 \left (a-b x^2\right )^{3/4}}{5 a c (c x)^{5/2}} \]
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Rubi [A] time = 0.0358012, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {325, 317, 335, 228} \[ -\frac{4 b^{3/2} \sqrt{c x} \sqrt [4]{1-\frac{a}{b x^2}} E\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 a^{3/2} c^4 \sqrt [4]{a-b x^2}}-\frac{2 \left (a-b x^2\right )^{3/4}}{5 a c (c x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 325
Rule 317
Rule 335
Rule 228
Rubi steps
\begin{align*} \int \frac{1}{(c x)^{7/2} \sqrt [4]{a-b x^2}} \, dx &=-\frac{2 \left (a-b x^2\right )^{3/4}}{5 a c (c x)^{5/2}}+\frac{(2 b) \int \frac{1}{(c x)^{3/2} \sqrt [4]{a-b x^2}} \, dx}{5 a c^2}\\ &=-\frac{2 \left (a-b x^2\right )^{3/4}}{5 a c (c x)^{5/2}}+\frac{\left (2 b \sqrt [4]{1-\frac{a}{b x^2}} \sqrt{c x}\right ) \int \frac{1}{\sqrt [4]{1-\frac{a}{b x^2}} x^2} \, dx}{5 a c^4 \sqrt [4]{a-b x^2}}\\ &=-\frac{2 \left (a-b x^2\right )^{3/4}}{5 a c (c x)^{5/2}}-\frac{\left (2 b \sqrt [4]{1-\frac{a}{b x^2}} \sqrt{c x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{1-\frac{a x^2}{b}}} \, dx,x,\frac{1}{x}\right )}{5 a c^4 \sqrt [4]{a-b x^2}}\\ &=-\frac{2 \left (a-b x^2\right )^{3/4}}{5 a c (c x)^{5/2}}-\frac{4 b^{3/2} \sqrt [4]{1-\frac{a}{b x^2}} \sqrt{c x} E\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 a^{3/2} c^4 \sqrt [4]{a-b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0128299, size = 57, normalized size = 0.57 \[ -\frac{2 x \sqrt [4]{1-\frac{b x^2}{a}} \, _2F_1\left (-\frac{5}{4},\frac{1}{4};-\frac{1}{4};\frac{b x^2}{a}\right )}{5 (c x)^{7/2} \sqrt [4]{a-b x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.037, size = 0, normalized size = 0. \begin{align*} \int{ \left ( cx \right ) ^{-{\frac{7}{2}}}{\frac{1}{\sqrt [4]{-b{x}^{2}+a}}}}\, 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{1}{{\left (-b x^{2} + a\right )}^{\frac{1}{4}} \left (c x\right )^{\frac{7}{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{3}{4}} \sqrt{c x}}{b c^{4} x^{6} - a c^{4} x^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 98.5107, size = 39, normalized size = 0.39 \begin{align*} - \frac{i e^{- \frac{3 i \pi }{4}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{a}{b x^{2}}} \right )}}{3 \sqrt [4]{b} c^{\frac{7}{2}} x^{3}} \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{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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