Optimal. Leaf size=140 \[ -\frac{\sqrt{3 a-2 a x^2}}{3 a^2 c \sqrt{c x}}+\frac{\sqrt [4]{2} \sqrt{3-2 x^2} \sqrt{c x} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-\sqrt{6} x}}{\sqrt{6}}\right )\right |2\right )}{3^{3/4} a c^2 \sqrt{x} \sqrt{3 a-2 a x^2}}+\frac{1}{3 a c \sqrt{3 a-2 a x^2} \sqrt{c x}} \]
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Rubi [A] time = 0.0561087, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {290, 325, 320, 319, 318, 424} \[ -\frac{\sqrt{3 a-2 a x^2}}{3 a^2 c \sqrt{c x}}+\frac{\sqrt [4]{2} \sqrt{3-2 x^2} \sqrt{c x} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-\sqrt{6} x}}{\sqrt{6}}\right )\right |2\right )}{3^{3/4} a c^2 \sqrt{x} \sqrt{3 a-2 a x^2}}+\frac{1}{3 a c \sqrt{3 a-2 a x^2} \sqrt{c x}} \]
Antiderivative was successfully verified.
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Rule 290
Rule 325
Rule 320
Rule 319
Rule 318
Rule 424
Rubi steps
\begin{align*} \int \frac{1}{(c x)^{3/2} \left (3 a-2 a x^2\right )^{3/2}} \, dx &=\frac{1}{3 a c \sqrt{c x} \sqrt{3 a-2 a x^2}}+\frac{\int \frac{1}{(c x)^{3/2} \sqrt{3 a-2 a x^2}} \, dx}{2 a}\\ &=\frac{1}{3 a c \sqrt{c x} \sqrt{3 a-2 a x^2}}-\frac{\sqrt{3 a-2 a x^2}}{3 a^2 c \sqrt{c x}}-\frac{\int \frac{\sqrt{c x}}{\sqrt{3 a-2 a x^2}} \, dx}{3 a c^2}\\ &=\frac{1}{3 a c \sqrt{c x} \sqrt{3 a-2 a x^2}}-\frac{\sqrt{3 a-2 a x^2}}{3 a^2 c \sqrt{c x}}-\frac{\sqrt{c x} \int \frac{\sqrt{x}}{\sqrt{3 a-2 a x^2}} \, dx}{3 a c^2 \sqrt{x}}\\ &=\frac{1}{3 a c \sqrt{c x} \sqrt{3 a-2 a x^2}}-\frac{\sqrt{3 a-2 a x^2}}{3 a^2 c \sqrt{c x}}-\frac{\left (\sqrt{c x} \sqrt{1-\frac{2 x^2}{3}}\right ) \int \frac{\sqrt{x}}{\sqrt{1-\frac{2 x^2}{3}}} \, dx}{3 a c^2 \sqrt{x} \sqrt{3 a-2 a x^2}}\\ &=\frac{1}{3 a c \sqrt{c x} \sqrt{3 a-2 a x^2}}-\frac{\sqrt{3 a-2 a x^2}}{3 a^2 c \sqrt{c x}}+\frac{\left (\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 )}{a c^2 \sqrt{x} \sqrt{3 a-2 a x^2}}\\ &=\frac{1}{3 a c \sqrt{c x} \sqrt{3 a-2 a x^2}}-\frac{\sqrt{3 a-2 a x^2}}{3 a^2 c \sqrt{c x}}+\frac{\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} a c^2 \sqrt{x} \sqrt{3 a-2 a x^2}}\\ \end{align*}
Mathematica [C] time = 0.020038, size = 58, normalized size = 0.41 \[ -\frac{2 x \left (3-2 x^2\right )^{3/2} \, _2F_1\left (-\frac{1}{4},\frac{3}{2};\frac{3}{4};\frac{2 x^2}{3}\right )}{3 \sqrt{3} \left (a \left (3-2 x^2\right )\right )^{3/2} (c x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.023, size = 228, normalized size = 1.6 \begin{align*} -{\frac{1}{36\,{a}^{2}c \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}-24 \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}} \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}}{4 \, a^{2} c^{2} x^{6} - 12 \, a^{2} c^{2} x^{4} + 9 \, 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 = 6.01221, size = 54, normalized size = 0.39 \begin{align*} \frac{\sqrt{3} \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{3}{2} \\ \frac{3}{4} \end{matrix}\middle |{\frac{2 x^{2} e^{2 i \pi }}{3}} \right )}}{18 a^{\frac{3}{2}} 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 (-2 \, a x^{2} + 3 \, a\right )}^{\frac{3}{2}} \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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