Optimal. Leaf size=96 \[ \frac{\sqrt{3-2 x^2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{\frac{2}{3}} \sqrt{c x}}{\sqrt{c}}\right ),-1\right )}{3 \sqrt [4]{6} a \sqrt{c} \sqrt{a \left (3-2 x^2\right )}}+\frac{\sqrt{c x}}{3 a c \sqrt{3 a-2 a x^2}} \]
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Rubi [A] time = 0.0463873, antiderivative size = 96, 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 = {290, 329, 224, 221} \[ \frac{\sqrt{c x}}{3 a c \sqrt{3 a-2 a x^2}}+\frac{\sqrt{3-2 x^2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{\frac{2}{3}} \sqrt{c x}}{\sqrt{c}}\right )\right |-1\right )}{3 \sqrt [4]{6} a \sqrt{c} \sqrt{a \left (3-2 x^2\right )}} \]
Antiderivative was successfully verified.
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Rule 290
Rule 329
Rule 224
Rule 221
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{c x} \left (3 a-2 a x^2\right )^{3/2}} \, dx &=\frac{\sqrt{c x}}{3 a c \sqrt{3 a-2 a x^2}}+\frac{\int \frac{1}{\sqrt{c x} \sqrt{3 a-2 a x^2}} \, dx}{6 a}\\ &=\frac{\sqrt{c x}}{3 a c \sqrt{3 a-2 a x^2}}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{3 a-\frac{2 a x^4}{c^2}}} \, dx,x,\sqrt{c x}\right )}{3 a c}\\ &=\frac{\sqrt{c x}}{3 a c \sqrt{3 a-2 a x^2}}+\frac{\sqrt{3-2 x^2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{2 x^4}{3 c^2}}} \, dx,x,\sqrt{c x}\right )}{3 \sqrt{3} a c \sqrt{a \left (3-2 x^2\right )}}\\ &=\frac{\sqrt{c x}}{3 a c \sqrt{3 a-2 a x^2}}+\frac{\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 )}{3 \sqrt [4]{6} a \sqrt{c} \sqrt{a \left (3-2 x^2\right )}}\\ \end{align*}
Mathematica [C] time = 0.0216386, size = 59, normalized size = 0.61 \[ \frac{x \left (\sqrt{9-6 x^2} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{2 x^2}{3}\right )+3\right )}{9 a \sqrt{a \left (3-2 x^2\right )} \sqrt{c x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 122, normalized size = 1.3 \begin{align*} -{\frac{1}{36\,{a}^{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 ) +12\,x \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}{{\left (-2 \, a x^{2} + 3 \, a\right )}^{\frac{3}{2}} \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{\sqrt{-2 \, a x^{2} + 3 \, a} \sqrt{c x}}{4 \, a^{2} c x^{5} - 12 \, a^{2} c x^{3} + 9 \, a^{2} c x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.30421, size = 51, normalized size = 0.53 \begin{align*} \frac{\sqrt{3} \sqrt{x} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{3}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{2 x^{2} e^{2 i \pi }}{3}} \right )}}{18 a^{\frac{3}{2}} \sqrt{c} \Gamma \left (\frac{5}{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 (-2 \, a x^{2} + 3 \, a\right )}^{\frac{3}{2}} \sqrt{c x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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