Optimal. Leaf size=101 \[ \frac{(c x)^{3/2}}{3 a c \sqrt{3 a-2 a x^2}}+\frac{\sqrt{3-2 x^2} \sqrt{c x} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-\sqrt{6} x}}{\sqrt{6}}\right )\right |2\right )}{6^{3/4} a \sqrt{x} \sqrt{3 a-2 a x^2}} \]
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Rubi [A] time = 0.141017, antiderivative size = 101, 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 \[ \frac{(c x)^{3/2}}{3 a c \sqrt{3 a-2 a x^2}}+\frac{\sqrt{3-2 x^2} \sqrt{c x} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-\sqrt{6} x}}{\sqrt{6}}\right )\right |2\right )}{6^{3/4} a \sqrt{x} \sqrt{3 a-2 a x^2}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[c*x]/(3*a - 2*a*x^2)^(3/2),x]
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Rubi in Sympy [A] time = 51.4448, size = 170, normalized size = 1.68 \[ - \frac{\sqrt [4]{2} \cdot 3^{\frac{3}{4}} \sqrt{c} \sqrt{- \frac{2 x^{2}}{3} + 1} E\left (\operatorname{asin}{\left (\frac{\sqrt [4]{2} \cdot 3^{\frac{3}{4}} \sqrt{c x}}{3 \sqrt{c}} \right )}\middle | -1\right )}{6 a \sqrt{- 2 a x^{2} + 3 a}} + \frac{\sqrt [4]{2} \cdot 3^{\frac{3}{4}} \sqrt{c} \sqrt{- \frac{2 x^{2}}{3} + 1} F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{2} \cdot 3^{\frac{3}{4}} \sqrt{c x}}{3 \sqrt{c}} \right )}\middle | -1\right )}{6 a \sqrt{- 2 a x^{2} + 3 a}} + \frac{\left (c x\right )^{\frac{3}{2}}}{3 a c \sqrt{- 2 a x^{2} + 3 a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x)**(1/2)/(-2*a*x**2+3*a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0797538, size = 107, normalized size = 1.06 \[ \frac{\sqrt{c x} \left (2 x^{3/2}+\sqrt [4]{6} \sqrt{3-2 x^2} F\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac{2}{3}} \sqrt{x}\right )\right |-1\right )-\sqrt [4]{6} \sqrt{3-2 x^2} E\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac{2}{3}} \sqrt{x}\right )\right |-1\right )\right )}{6 a \sqrt{x} \sqrt{a \left (3-2 x^2\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[c*x]/(3*a - 2*a*x^2)^(3/2),x]
[Out]
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Maple [B] time = 0.023, size = 227, normalized size = 2.3 \[ -{\frac{1}{72\,{a}^{2}x \left ( 2\,{x}^{2}-3 \right ) }\sqrt{cx}\sqrt{-a \left ( 2\,{x}^{2}-3 \right ) } \left ( 2\,\sqrt{ \left ( -2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}\sqrt{3}\sqrt{-x\sqrt{3}\sqrt{2}}{\it EllipticE} \left ( 1/6\,\sqrt{3}\sqrt{2}\sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}},1/2\,\sqrt{2} \right ) \sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}\sqrt{2}-\sqrt{ \left ( -2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}\sqrt{3}\sqrt{-x\sqrt{3}\sqrt{2}}{\it EllipticF} \left ({\frac{\sqrt{3}\sqrt{2}}{6}\sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}\sqrt{2}+24\,{x}^{2} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x)^(1/2)/(-2*a*x^2+3*a)^(3/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x}}{{\left (-2 \, a x^{2} + 3 \, a\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x)/(-2*a*x^2 + 3*a)^(3/2),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{\sqrt{c x}}{{\left (2 \, a x^{2} - 3 \, a\right )} \sqrt{-2 \, a x^{2} + 3 \, a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x)/(-2*a*x^2 + 3*a)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.1292, size = 51, normalized size = 0.5 \[ \frac{\sqrt{3} \sqrt{c} x^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{3}{2} \\ \frac{7}{4} \end{matrix}\middle |{\frac{2 x^{2} e^{2 i \pi }}{3}} \right )}}{18 a^{\frac{3}{2}} \Gamma \left (\frac{7}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x)**(1/2)/(-2*a*x**2+3*a)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x}}{{\left (-2 \, a x^{2} + 3 \, a\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x)/(-2*a*x^2 + 3*a)^(3/2),x, algorithm="giac")
[Out]