Optimal. Leaf size=132 \[ -\frac{5 \sqrt{3 a-2 a x^2}}{27 a^2 c (c x)^{3/2}}+\frac{5\ 2^{3/4} \sqrt{3-2 x^2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{\frac{2}{3}} \sqrt{c x}}{\sqrt{c}}\right )\right |-1\right )}{27 \sqrt [4]{3} a c^{5/2} \sqrt{a \left (3-2 x^2\right )}}+\frac{1}{3 a c \sqrt{3 a-2 a x^2} (c x)^{3/2}} \]
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Rubi [A] time = 0.179135, antiderivative size = 132, 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 \[ -\frac{5 \sqrt{3 a-2 a x^2}}{27 a^2 c (c x)^{3/2}}+\frac{5\ 2^{3/4} \sqrt{3-2 x^2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{\frac{2}{3}} \sqrt{c x}}{\sqrt{c}}\right )\right |-1\right )}{27 \sqrt [4]{3} a c^{5/2} \sqrt{a \left (3-2 x^2\right )}}+\frac{1}{3 a c \sqrt{3 a-2 a x^2} (c x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[1/((c*x)^(5/2)*(3*a - 2*a*x^2)^(3/2)),x]
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Rubi in Sympy [A] time = 16.7497, size = 129, normalized size = 0.98 \[ \frac{1}{3 a c \left (c x\right )^{\frac{3}{2}} \sqrt{- 2 a x^{2} + 3 a}} + \frac{5 \cdot 2^{\frac{3}{4}} \sqrt [4]{3} \sqrt{- \frac{2 x^{2}}{3} + 1} F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{2} \cdot 3^{\frac{3}{4}} \sqrt{c x}}{3 \sqrt{c}} \right )}\middle | -1\right )}{27 a c^{\frac{5}{2}} \sqrt{- 2 a x^{2} + 3 a}} - \frac{5 \sqrt{- 2 a x^{2} + 3 a}}{27 a^{2} c \left (c x\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(c*x)**(5/2)/(-2*a*x**2+3*a)**(3/2),x)
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Mathematica [A] time = 0.144872, size = 95, normalized size = 0.72 \[ \frac{x \left (6 \sqrt{2-\frac{3}{x^2}} \left (5 x^2-3\right )-5\ 6^{3/4} \sqrt{x} \left (2 x^2-3\right ) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{\frac{3}{2}}}{\sqrt{x}}\right )\right |-1\right )\right )}{81 a \sqrt{2-\frac{3}{x^2}} \sqrt{a \left (3-2 x^2\right )} (c x)^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((c*x)^(5/2)*(3*a - 2*a*x^2)^(3/2)),x]
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Maple [A] time = 0.029, size = 133, normalized size = 1. \[ -{\frac{1}{162\,{a}^{2}x{c}^{2} \left ( 2\,{x}^{2}-3 \right ) }\sqrt{-a \left ( 2\,{x}^{2}-3 \right ) } \left ( 5\,\sqrt{ \left ( -2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}\sqrt{-x\sqrt{3}\sqrt{2}}{\it EllipticF} \left ( 1/6\,\sqrt{3}\sqrt{2}\sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}},1/2\,\sqrt{2} \right ) \sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}x+60\,{x}^{2}-36 \right ){\frac{1}{\sqrt{cx}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(c*x)^(5/2)/(-2*a*x^2+3*a)^(3/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-2 \, a x^{2} + 3 \, a\right )}^{\frac{3}{2}} \left (c x\right )^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((-2*a*x^2 + 3*a)^(3/2)*(c*x)^(5/2)),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{1}{{\left (2 \, a c^{2} x^{4} - 3 \, a c^{2} x^{2}\right )} \sqrt{-2 \, a x^{2} + 3 \, a} \sqrt{c x}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((-2*a*x^2 + 3*a)^(3/2)*(c*x)^(5/2)),x, algorithm="fricas")
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(c*x)**(5/2)/(-2*a*x**2+3*a)**(3/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-2 \, a x^{2} + 3 \, a\right )}^{\frac{3}{2}} \left (c x\right )^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((-2*a*x^2 + 3*a)^(3/2)*(c*x)^(5/2)),x, algorithm="giac")
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