Optimal. Leaf size=93 \[ \frac{4 \sqrt{b} \sqrt{c x} \sqrt [4]{\frac{a}{b x^2}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{a^{3/2} c^2 \sqrt [4]{a+b x^2}}-\frac{2}{a c \sqrt{c x} \sqrt [4]{a+b x^2}} \]
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Rubi [A] time = 0.0346462, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {286, 284, 335, 196} \[ \frac{4 \sqrt{b} \sqrt{c x} \sqrt [4]{\frac{a}{b x^2}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{a^{3/2} c^2 \sqrt [4]{a+b x^2}}-\frac{2}{a c \sqrt{c x} \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 286
Rule 284
Rule 335
Rule 196
Rubi steps
\begin{align*} \int \frac{1}{(c x)^{3/2} \left (a+b x^2\right )^{5/4}} \, dx &=-\frac{2}{a c \sqrt{c x} \sqrt [4]{a+b x^2}}-\frac{(2 b) \int \frac{\sqrt{c x}}{\left (a+b x^2\right )^{5/4}} \, dx}{a c^2}\\ &=-\frac{2}{a c \sqrt{c x} \sqrt [4]{a+b x^2}}-\frac{\left (2 \sqrt [4]{1+\frac{a}{b x^2}} \sqrt{c x}\right ) \int \frac{1}{\left (1+\frac{a}{b x^2}\right )^{5/4} x^2} \, dx}{a c^2 \sqrt [4]{a+b x^2}}\\ &=-\frac{2}{a c \sqrt{c x} \sqrt [4]{a+b x^2}}+\frac{\left (2 \sqrt [4]{1+\frac{a}{b x^2}} \sqrt{c x}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{a x^2}{b}\right )^{5/4}} \, dx,x,\frac{1}{x}\right )}{a c^2 \sqrt [4]{a+b x^2}}\\ &=-\frac{2}{a c \sqrt{c x} \sqrt [4]{a+b x^2}}+\frac{4 \sqrt{b} \sqrt [4]{1+\frac{a}{b x^2}} \sqrt{c x} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{a^{3/2} c^2 \sqrt [4]{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0116247, size = 57, normalized size = 0.61 \[ -\frac{2 x \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (-\frac{1}{4},\frac{5}{4};\frac{3}{4};-\frac{b x^2}{a}\right )}{a (c x)^{3/2} \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.055, size = 0, normalized size = 0. \begin{align*} \int{ \left ( cx \right ) ^{-{\frac{3}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{4}}}}\, 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{5}{4}} \left (c x\right )^{\frac{3}{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^{2} c^{2} x^{6} + 2 \, a b c^{2} x^{4} + a^{2} c^{2} x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 27.3635, size = 48, normalized size = 0.52 \begin{align*} \frac{\Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{5}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{5}{4}} c^{\frac{3}{2}} \sqrt{x} \Gamma \left (\frac{3}{4}\right )} \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{5}{4}} \left (c x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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