Optimal. Leaf size=67 \[ -\frac {\sqrt [4]{6} \sqrt {3-2 x^2} \sqrt {c x} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {3-\sqrt {6} x}}{\sqrt {6}}\right )\right |2\right )}{\sqrt {x} \sqrt {3 a-2 a x^2}} \]
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Rubi [A] time = 0.03, antiderivative size = 67, 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 = {320, 319, 318, 424} \[ -\frac {\sqrt [4]{6} \sqrt {3-2 x^2} \sqrt {c x} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {3-\sqrt {6} x}}{\sqrt {6}}\right )\right |2\right )}{\sqrt {x} \sqrt {3 a-2 a x^2}} \]
Antiderivative was successfully verified.
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Rule 318
Rule 319
Rule 320
Rule 424
Rubi steps
\begin {align*} \int \frac {\sqrt {c x}}{\sqrt {3 a-2 a x^2}} \, dx &=\frac {\sqrt {c x} \int \frac {\sqrt {x}}{\sqrt {3 a-2 a x^2}} \, dx}{\sqrt {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}{\sqrt {x} \sqrt {3 a-2 a x^2}}\\ &=-\frac {\left (\sqrt [4]{2} 3^{3/4} \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 )}{\sqrt {x} \sqrt {3 a-2 a x^2}}\\ &=-\frac {\sqrt [4]{6} \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 )}{\sqrt {x} \sqrt {3 a-2 a x^2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 53, normalized size = 0.79 \[ \frac {2 x \sqrt {3-2 x^2} \sqrt {c x} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};\frac {2 x^2}{3}\right )}{3 \sqrt {a \left (9-6 x^2\right )}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.18, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-2 \, a x^{2} + 3 \, a} \sqrt {c x}}{2 \, a x^{2} - 3 \, a}, x\right ) \]
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 \[ \int \frac {\sqrt {c x}}{\sqrt {-2 \, a x^{2} + 3 \, a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 165, normalized size = 2.46 \[ \frac {\sqrt {c x}\, \sqrt {-\left (2 x^{2}-3\right ) a}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {\left (-2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}\, \sqrt {3}\, \sqrt {-\sqrt {2}\, \sqrt {3}\, x}\, \left (2 \EllipticE \left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}}{6}, \frac {\sqrt {2}}{2}\right )-\EllipticF \left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\left (2 x +\sqrt {2}\, \sqrt {3}\right ) \sqrt {2}\, \sqrt {3}}}{6}, \frac {\sqrt {2}}{2}\right )\right )}{12 \left (2 x^{2}-3\right ) a x} \]
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 \[ \int \frac {\sqrt {c x}}{\sqrt {-2 \, a x^{2} + 3 \, a}}\,{d x} \]
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 \[ \int \frac {\sqrt {c\,x}}{\sqrt {3\,a-2\,a\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.86, size = 51, normalized size = 0.76 \[ \frac {\sqrt {3} \sqrt {c} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {2 x^{2} e^{2 i \pi }}{3}} \right )}}{6 \sqrt {a} \Gamma \left (\frac {7}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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