Optimal. Leaf size=94 \[ \frac {2\ 2^{3/4} a \sqrt {3-2 x^2} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3}} \sqrt {c x}}{\sqrt {c}}\right ),-1\right )}{\sqrt [4]{3} \sqrt {c} \sqrt {a \left (3-2 x^2\right )}}+\frac {2 \sqrt {3 a-2 a x^2} \sqrt {c x}}{3 c} \]
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Rubi [A] time = 0.05, antiderivative size = 94, normalized size of antiderivative = 1.00, 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 = {279, 329, 224, 221} \[ \frac {2 \sqrt {3 a-2 a x^2} \sqrt {c x}}{3 c}+\frac {2\ 2^{3/4} a \sqrt {3-2 x^2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3}} \sqrt {c x}}{\sqrt {c}}\right )\right |-1\right )}{\sqrt [4]{3} \sqrt {c} \sqrt {a \left (3-2 x^2\right )}} \]
Antiderivative was successfully verified.
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Rule 221
Rule 224
Rule 279
Rule 329
Rubi steps
\begin {align*} \int \frac {\sqrt {3 a-2 a x^2}}{\sqrt {c x}} \, dx &=\frac {2 \sqrt {c x} \sqrt {3 a-2 a x^2}}{3 c}+(2 a) \int \frac {1}{\sqrt {c x} \sqrt {3 a-2 a x^2}} \, dx\\ &=\frac {2 \sqrt {c x} \sqrt {3 a-2 a x^2}}{3 c}+\frac {(4 a) \operatorname {Subst}\left (\int \frac {1}{\sqrt {3 a-\frac {2 a x^4}{c^2}}} \, dx,x,\sqrt {c x}\right )}{c}\\ &=\frac {2 \sqrt {c x} \sqrt {3 a-2 a x^2}}{3 c}+\frac {\left (4 a \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 )}{\sqrt {3} c \sqrt {a \left (3-2 x^2\right )}}\\ &=\frac {2 \sqrt {c x} \sqrt {3 a-2 a x^2}}{3 c}+\frac {2\ 2^{3/4} a \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 )}{\sqrt [4]{3} \sqrt {c} \sqrt {a \left (3-2 x^2\right )}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 51, normalized size = 0.54 \[ \frac {2 x \sqrt {a \left (9-6 x^2\right )} \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {5}{4};\frac {2 x^2}{3}\right )}{\sqrt {3-2 x^2} \sqrt {c x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x}}{c x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-2 \, a x^{2} + 3 \, a}}{\sqrt {c x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 124, normalized size = 1.32 \[ -\frac {\sqrt {-\left (2 x^{2}-3\right ) a}\, \left (-4 x^{3}+6 x +\sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {\left (-2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {-\sqrt {2}\, \sqrt {3}\, x}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}}{6}, \frac {\sqrt {2}}{2}\right )\right )}{3 \sqrt {c x}\, \left (2 x^{2}-3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-2 \, a x^{2} + 3 \, a}}{\sqrt {c x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {3\,a-2\,a\,x^2}}{\sqrt {c\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.93, size = 53, normalized size = 0.56 \[ \frac {\sqrt {3} \sqrt {a} \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {2 x^{2} e^{2 i \pi }}{3}} \right )}}{2 \sqrt {c} \Gamma \left (\frac {5}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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