Optimal. Leaf size=66 \[ -\frac{2 \sqrt{b} (c x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ),2\right )}{\sqrt{a} c^2 \left (a+b x^2\right )^{3/4}} \]
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Rubi [A] time = 0.0629385, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {329, 237, 335, 275, 231} \[ -\frac{2 \sqrt{b} (c x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{a} c^2 \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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Rule 329
Rule 237
Rule 335
Rule 275
Rule 231
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{c x} \left (a+b x^2\right )^{3/4}} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{\left (a+\frac{b x^4}{c^2}\right )^{3/4}} \, dx,x,\sqrt{c x}\right )}{c}\\ &=\frac{\left (2 \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 )}{c \left (a+b x^2\right )^{3/4}}\\ &=-\frac{\left (2 \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 )}{c \left (a+b x^2\right )^{3/4}}\\ &=-\frac{\left (\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 )}{c \left (a+b x^2\right )^{3/4}}\\ &=-\frac{2 \sqrt{b} \left (1+\frac{a}{b x^2}\right )^{3/4} (c x)^{3/2} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{a} c^2 \left (a+b x^2\right )^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.0108494, size = 54, normalized size = 0.82 \[ \frac{2 x \left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^2}{a}\right )}{\sqrt{c x} \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{{\frac{1}{\sqrt{cx}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{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{3}{4}} \sqrt{c x}}\,{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}}{b c x^{3} + a c x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.56298, size = 31, normalized size = 0.47 \begin{align*} - \frac{{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{3}{2} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{2}}} \right )}}{b^{\frac{3}{4}} \sqrt{c} 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{1}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{c x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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