Optimal. Leaf size=117 \[ \frac{6^{3/4} a c^{3/2} \sqrt{3-2 x^2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{\frac{2}{3}} \sqrt{c x}}{\sqrt{c}}\right ),-1\right )}{7 \sqrt{a \left (3-2 x^2\right )}}+\frac{2 \sqrt{3 a-2 a x^2} (c x)^{5/2}}{7 c}-\frac{2}{7} c \sqrt{3 a-2 a x^2} \sqrt{c x} \]
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Rubi [A] time = 0.0774048, antiderivative size = 117, 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 = {279, 321, 329, 224, 221} \[ \frac{6^{3/4} a c^{3/2} \sqrt{3-2 x^2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{\frac{2}{3}} \sqrt{c x}}{\sqrt{c}}\right )\right |-1\right )}{7 \sqrt{a \left (3-2 x^2\right )}}+\frac{2 \sqrt{3 a-2 a x^2} (c x)^{5/2}}{7 c}-\frac{2}{7} c \sqrt{3 a-2 a x^2} \sqrt{c x} \]
Antiderivative was successfully verified.
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Rule 279
Rule 321
Rule 329
Rule 224
Rule 221
Rubi steps
\begin{align*} \int (c x)^{3/2} \sqrt{3 a-2 a x^2} \, dx &=\frac{2 (c x)^{5/2} \sqrt{3 a-2 a x^2}}{7 c}+\frac{1}{7} (6 a) \int \frac{(c x)^{3/2}}{\sqrt{3 a-2 a x^2}} \, dx\\ &=-\frac{2}{7} c \sqrt{c x} \sqrt{3 a-2 a x^2}+\frac{2 (c x)^{5/2} \sqrt{3 a-2 a x^2}}{7 c}+\frac{1}{7} \left (3 a c^2\right ) \int \frac{1}{\sqrt{c x} \sqrt{3 a-2 a x^2}} \, dx\\ &=-\frac{2}{7} c \sqrt{c x} \sqrt{3 a-2 a x^2}+\frac{2 (c x)^{5/2} \sqrt{3 a-2 a x^2}}{7 c}+\frac{1}{7} (6 a c) \operatorname{Subst}\left (\int \frac{1}{\sqrt{3 a-\frac{2 a x^4}{c^2}}} \, dx,x,\sqrt{c x}\right )\\ &=-\frac{2}{7} c \sqrt{c x} \sqrt{3 a-2 a x^2}+\frac{2 (c x)^{5/2} \sqrt{3 a-2 a x^2}}{7 c}+\frac{\left (2 \sqrt{3} a c \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 )}{7 \sqrt{a \left (3-2 x^2\right )}}\\ &=-\frac{2}{7} c \sqrt{c x} \sqrt{3 a-2 a x^2}+\frac{2 (c x)^{5/2} \sqrt{3 a-2 a x^2}}{7 c}+\frac{6^{3/4} a c^{3/2} \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 )}{7 \sqrt{a \left (3-2 x^2\right )}}\\ \end{align*}
Mathematica [C] time = 0.0232594, size = 74, normalized size = 0.63 \[ \frac{c \sqrt{a \left (3-2 x^2\right )} \sqrt{c x} \left (3 \sqrt{3} \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{5}{4};\frac{2 x^2}{3}\right )-\left (3-2 x^2\right )^{3/2}\right )}{7 \sqrt{3-2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 133, normalized size = 1.1 \begin{align*} -{\frac{c}{14\,x \left ( 2\,{x}^{2}-3 \right ) }\sqrt{cx}\sqrt{-a \left ( 2\,{x}^{2}-3 \right ) } \left ( -8\,{x}^{5}+\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 ) +20\,{x}^{3}-12\,x \right ) } \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 \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 (\sqrt{-2 \, a x^{2} + 3 \, a} \sqrt{c x} c x, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.67944, size = 53, normalized size = 0.45 \begin{align*} \frac{\sqrt{3} \sqrt{a} c^{\frac{3}{2}} x^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{2 x^{2} e^{2 i \pi }}{3}} \right )}}{2 \Gamma \left (\frac{9}{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 \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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