Optimal. Leaf size=98 \[ \frac{2\ 2^{3/4} \sqrt{3-2 x^2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{\frac{2}{3}} \sqrt{c x}}{\sqrt{c}}\right ),-1\right )}{9 \sqrt [4]{3} c^{5/2} \sqrt{a \left (3-2 x^2\right )}}-\frac{2 \sqrt{3 a-2 a x^2}}{9 a c (c x)^{3/2}} \]
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Rubi [A] time = 0.0459691, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {325, 329, 224, 221} \[ \frac{2\ 2^{3/4} \sqrt{3-2 x^2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{\frac{2}{3}} \sqrt{c x}}{\sqrt{c}}\right )\right |-1\right )}{9 \sqrt [4]{3} c^{5/2} \sqrt{a \left (3-2 x^2\right )}}-\frac{2 \sqrt{3 a-2 a x^2}}{9 a c (c x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 325
Rule 329
Rule 224
Rule 221
Rubi steps
\begin{align*} \int \frac{1}{(c x)^{5/2} \sqrt{3 a-2 a x^2}} \, dx &=-\frac{2 \sqrt{3 a-2 a x^2}}{9 a c (c x)^{3/2}}+\frac{2 \int \frac{1}{\sqrt{c x} \sqrt{3 a-2 a x^2}} \, dx}{9 c^2}\\ &=-\frac{2 \sqrt{3 a-2 a x^2}}{9 a c (c x)^{3/2}}+\frac{4 \operatorname{Subst}\left (\int \frac{1}{\sqrt{3 a-\frac{2 a x^4}{c^2}}} \, dx,x,\sqrt{c x}\right )}{9 c^3}\\ &=-\frac{2 \sqrt{3 a-2 a x^2}}{9 a c (c x)^{3/2}}+\frac{\left (4 \sqrt{3-2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{2 x^4}{3 c^2}}} \, dx,x,\sqrt{c x}\right )}{9 \sqrt{3} c^3 \sqrt{a \left (3-2 x^2\right )}}\\ &=-\frac{2 \sqrt{3 a-2 a x^2}}{9 a c (c x)^{3/2}}+\frac{2\ 2^{3/4} \sqrt{3-2 x^2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{\frac{2}{3}} \sqrt{c x}}{\sqrt{c}}\right )\right |-1\right )}{9 \sqrt [4]{3} c^{5/2} \sqrt{a \left (3-2 x^2\right )}}\\ \end{align*}
Mathematica [C] time = 0.0146599, size = 53, normalized size = 0.54 \[ -\frac{2 x \sqrt{3-2 x^2} \, _2F_1\left (-\frac{3}{4},\frac{1}{2};\frac{1}{4};\frac{2 x^2}{3}\right )}{3 \sqrt{a \left (9-6 x^2\right )} (c x)^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 132, normalized size = 1.4 \begin{align*} -{\frac{1}{27\,ax{c}^{2} \left ( 2\,{x}^{2}-3 \right ) }\sqrt{-a \left ( 2\,{x}^{2}-3 \right ) } \left ( \sqrt{ \left ( 2\,x+\sqrt{2}\sqrt{3} \right ) \sqrt{2}\sqrt{3}}\sqrt{ \left ( -2\,x+\sqrt{2}\sqrt{3} \right ) \sqrt{2}\sqrt{3}}\sqrt{-x\sqrt{2}\sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{2}\sqrt{3}}{6}\sqrt{ \left ( 2\,x+\sqrt{2}\sqrt{3} \right ) \sqrt{2}\sqrt{3}}},{\frac{\sqrt{2}}{2}} \right ) x+12\,{x}^{2}-18 \right ){\frac{1}{\sqrt{cx}}}} \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}{\sqrt{-2 \, a x^{2} + 3 \, a} \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{\sqrt{-2 \, a x^{2} + 3 \, a} \sqrt{c x}}{2 \, a c^{3} x^{5} - 3 \, a c^{3} x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 10.4706, size = 54, normalized size = 0.55 \begin{align*} \frac{\sqrt{3} \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{1}{2} \\ \frac{1}{4} \end{matrix}\middle |{\frac{2 x^{2} e^{2 i \pi }}{3}} \right )}}{6 \sqrt{a} c^{\frac{5}{2}} x^{\frac{3}{2}} \Gamma \left (\frac{1}{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}{\sqrt{-2 \, a x^{2} + 3 \, a} \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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