Optimal. Leaf size=107 \[ \frac{2 \sqrt [4]{2} \sqrt{3-2 x^2} \sqrt{c x} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-\sqrt{6} x}}{\sqrt{6}}\right )\right |2\right )}{3^{3/4} c^2 \sqrt{x} \sqrt{3 a-2 a x^2}}-\frac{2 \sqrt{3 a-2 a x^2}}{3 a c \sqrt{c x}} \]
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Rubi [A] time = 0.042602, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {325, 320, 319, 318, 424} \[ \frac{2 \sqrt [4]{2} \sqrt{3-2 x^2} \sqrt{c x} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-\sqrt{6} x}}{\sqrt{6}}\right )\right |2\right )}{3^{3/4} c^2 \sqrt{x} \sqrt{3 a-2 a x^2}}-\frac{2 \sqrt{3 a-2 a x^2}}{3 a c \sqrt{c x}} \]
Antiderivative was successfully verified.
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Rule 325
Rule 320
Rule 319
Rule 318
Rule 424
Rubi steps
\begin{align*} \int \frac{1}{(c x)^{3/2} \sqrt{3 a-2 a x^2}} \, dx &=-\frac{2 \sqrt{3 a-2 a x^2}}{3 a c \sqrt{c x}}-\frac{2 \int \frac{\sqrt{c x}}{\sqrt{3 a-2 a x^2}} \, dx}{3 c^2}\\ &=-\frac{2 \sqrt{3 a-2 a x^2}}{3 a c \sqrt{c x}}-\frac{\left (2 \sqrt{c x}\right ) \int \frac{\sqrt{x}}{\sqrt{3 a-2 a x^2}} \, dx}{3 c^2 \sqrt{x}}\\ &=-\frac{2 \sqrt{3 a-2 a x^2}}{3 a c \sqrt{c x}}-\frac{\left (2 \sqrt{c x} \sqrt{1-\frac{2 x^2}{3}}\right ) \int \frac{\sqrt{x}}{\sqrt{1-\frac{2 x^2}{3}}} \, dx}{3 c^2 \sqrt{x} \sqrt{3 a-2 a x^2}}\\ &=-\frac{2 \sqrt{3 a-2 a x^2}}{3 a c \sqrt{c x}}+\frac{\left (2 \sqrt [4]{\frac{2}{3}} \sqrt{c x} \sqrt{1-\frac{2 x^2}{3}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-2 x^2}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{1-\sqrt{\frac{2}{3}} x}}{\sqrt{2}}\right )}{c^2 \sqrt{x} \sqrt{3 a-2 a x^2}}\\ &=-\frac{2 \sqrt{3 a-2 a x^2}}{3 a c \sqrt{c x}}+\frac{2 \sqrt [4]{2} \sqrt{c x} \sqrt{3-2 x^2} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-\sqrt{6} x}}{\sqrt{6}}\right )\right |2\right )}{3^{3/4} c^2 \sqrt{x} \sqrt{3 a-2 a x^2}}\\ \end{align*}
Mathematica [C] time = 0.0158001, size = 51, normalized size = 0.48 \[ -\frac{2 x \sqrt{3-2 x^2} \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};\frac{2 x^2}{3}\right )}{\sqrt{a \left (9-6 x^2\right )} (c x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.021, size = 228, normalized size = 2.1 \begin{align*} -{\frac{1}{18\,ac \left ( 2\,{x}^{2}-3 \right ) }\sqrt{-a \left ( 2\,{x}^{2}-3 \right ) } \left ( 2\,\sqrt{2}\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{3}\sqrt{-x\sqrt{2}\sqrt{3}}{\it EllipticE} \left ( 1/6\,\sqrt{3}\sqrt{2}\sqrt{ \left ( 2\,x+\sqrt{2}\sqrt{3} \right ) \sqrt{2}\sqrt{3}},1/2\,\sqrt{2} \right ) -\sqrt{2}\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{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 ) +24\,{x}^{2}-36 \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{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{\sqrt{-2 \, a x^{2} + 3 \, a} \sqrt{c x}}{2 \, a c^{2} x^{4} - 3 \, a 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 = 1.94743, size = 54, normalized size = 0.5 \begin{align*} \frac{\sqrt{3} \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{2} \\ \frac{3}{4} \end{matrix}\middle |{\frac{2 x^{2} e^{2 i \pi }}{3}} \right )}}{6 \sqrt{a} 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}{\sqrt{-2 \, a x^{2} + 3 \, a} \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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