Optimal. Leaf size=98 \[ \frac{4 \sqrt [4]{6} a \sqrt{3-2 x^2} \sqrt{c x} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-\sqrt{6} x}}{\sqrt{6}}\right )\right |2\right )}{c^2 \sqrt{x} \sqrt{3 a-2 a x^2}}-\frac{2 \sqrt{3 a-2 a x^2}}{c \sqrt{c x}} \]
[Out]
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Rubi [A] time = 0.141477, antiderivative size = 98, 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 \[ \frac{4 \sqrt [4]{6} a \sqrt{3-2 x^2} \sqrt{c x} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-\sqrt{6} x}}{\sqrt{6}}\right )\right |2\right )}{c^2 \sqrt{x} \sqrt{3 a-2 a x^2}}-\frac{2 \sqrt{3 a-2 a x^2}}{c \sqrt{c x}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[3*a - 2*a*x^2]/(c*x)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 50.7487, size = 168, normalized size = 1.71 \[ - \frac{4 \sqrt [4]{2} \cdot 3^{\frac{3}{4}} a \sqrt{- \frac{2 x^{2}}{3} + 1} E\left (\operatorname{asin}{\left (\frac{\sqrt [4]{2} \cdot 3^{\frac{3}{4}} \sqrt{c x}}{3 \sqrt{c}} \right )}\middle | -1\right )}{c^{\frac{3}{2}} \sqrt{- 2 a x^{2} + 3 a}} + \frac{4 \sqrt [4]{2} \cdot 3^{\frac{3}{4}} a \sqrt{- \frac{2 x^{2}}{3} + 1} F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{2} \cdot 3^{\frac{3}{4}} \sqrt{c x}}{3 \sqrt{c}} \right )}\middle | -1\right )}{c^{\frac{3}{2}} \sqrt{- 2 a x^{2} + 3 a}} - \frac{2 \sqrt{- 2 a x^{2} + 3 a}}{c \sqrt{c x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-2*a*x**2+3*a)**(1/2)/(c*x)**(3/2),x)
[Out]
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Mathematica [A] time = 0.13006, size = 83, normalized size = 0.85 \[ \frac{2 x \sqrt{a \left (3-2 x^2\right )} \left (-\frac{2 \sqrt [4]{6} \sqrt{x} \left (E\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac{2}{3}} \sqrt{x}\right )\right |-1\right )-F\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac{2}{3}} \sqrt{x}\right )\right |-1\right )\right )}{\sqrt{3-2 x^2}}-1\right )}{(c x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[3*a - 2*a*x^2]/(c*x)^(3/2),x]
[Out]
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Maple [B] time = 0.049, size = 225, normalized size = 2.3 \[ -{\frac{1}{3\,c \left ( 2\,{x}^{2}-3 \right ) }\sqrt{-a \left ( 2\,{x}^{2}-3 \right ) } \left ( 2\,\sqrt{ \left ( -2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}\sqrt{3}\sqrt{-x\sqrt{3}\sqrt{2}}{\it EllipticE} \left ( 1/6\,\sqrt{3}\sqrt{2}\sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}},1/2\,\sqrt{2} \right ) \sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}\sqrt{2}-\sqrt{ \left ( -2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}\sqrt{3}\sqrt{-x\sqrt{3}\sqrt{2}}{\it EllipticF} \left ({\frac{\sqrt{3}\sqrt{2}}{6}\sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}\sqrt{2}+12\,{x}^{2}-18 \right ){\frac{1}{\sqrt{cx}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-2*a*x^2+3*a)^(1/2)/(c*x)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-2 \, a x^{2} + 3 \, a}}{\left (c x\right )^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-2*a*x^2 + 3*a)/(c*x)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{-2 \, a x^{2} + 3 \, a}}{\sqrt{c x} c x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-2*a*x^2 + 3*a)/(c*x)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.07206, size = 56, normalized size = 0.57 \[ \frac{\sqrt{3} \sqrt{a} \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{2 x^{2} e^{2 i \pi }}{3}} \right )}}{2 c^{\frac{3}{2}} \sqrt{x} \Gamma \left (\frac{3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-2*a*x**2+3*a)**(1/2)/(c*x)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-2 \, a x^{2} + 3 \, a}}{\left (c x\right )^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-2*a*x^2 + 3*a)/(c*x)^(3/2),x, algorithm="giac")
[Out]