Optimal. Leaf size=128 \[ -\frac{3 \sqrt [4]{6} a c^2 \sqrt{3-2 x^2} \sqrt{c x} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-\sqrt{6} x}}{\sqrt{6}}\right )\right |2\right )}{5 \sqrt{x} \sqrt{3 a-2 a x^2}}+\frac{2 \sqrt{3 a-2 a x^2} (c x)^{7/2}}{9 c}-\frac{2}{15} c \sqrt{3 a-2 a x^2} (c x)^{3/2} \]
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Rubi [A] time = 0.194163, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{3 \sqrt [4]{6} a c^2 \sqrt{3-2 x^2} \sqrt{c x} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-\sqrt{6} x}}{\sqrt{6}}\right )\right |2\right )}{5 \sqrt{x} \sqrt{3 a-2 a x^2}}+\frac{2 \sqrt{3 a-2 a x^2} (c x)^{7/2}}{9 c}-\frac{2}{15} c \sqrt{3 a-2 a x^2} (c x)^{3/2} \]
Antiderivative was successfully verified.
[In] Int[(c*x)^(5/2)*Sqrt[3*a - 2*a*x^2],x]
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Rubi in Sympy [A] time = 54.6855, size = 199, normalized size = 1.55 \[ \frac{3 \sqrt [4]{2} \cdot 3^{\frac{3}{4}} a c^{\frac{5}{2}} \sqrt{- \frac{2 x^{2}}{3} + 1} E\left (\operatorname{asin}{\left (\frac{\sqrt [4]{2} \cdot 3^{\frac{3}{4}} \sqrt{c x}}{3 \sqrt{c}} \right )}\middle | -1\right )}{5 \sqrt{- 2 a x^{2} + 3 a}} - \frac{3 \sqrt [4]{2} \cdot 3^{\frac{3}{4}} a c^{\frac{5}{2}} \sqrt{- \frac{2 x^{2}}{3} + 1} F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{2} \cdot 3^{\frac{3}{4}} \sqrt{c x}}{3 \sqrt{c}} \right )}\middle | -1\right )}{5 \sqrt{- 2 a x^{2} + 3 a}} - \frac{2 c \left (c x\right )^{\frac{3}{2}} \sqrt{- 2 a x^{2} + 3 a}}{15} + \frac{2 \left (c x\right )^{\frac{7}{2}} \sqrt{- 2 a x^{2} + 3 a}}{9 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x)**(5/2)*(-2*a*x**2+3*a)**(1/2),x)
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Mathematica [A] time = 0.187564, size = 112, normalized size = 0.88 \[ \frac{\sqrt{a \left (3-2 x^2\right )} (c x)^{5/2} \left (2 \sqrt{3-2 x^2} \left (5 x^2-3\right ) x^{3/2}-27 \sqrt [4]{6} F\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac{2}{3}} \sqrt{x}\right )\right |-1\right )+27 \sqrt [4]{6} E\left (\left .\sin ^{-1}\left (\sqrt [4]{\frac{2}{3}} \sqrt{x}\right )\right |-1\right )\right )}{45 x^{5/2} \sqrt{3-2 x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(c*x)^(5/2)*Sqrt[3*a - 2*a*x^2],x]
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Maple [B] time = 0.059, size = 237, normalized size = 1.9 \[{\frac{{c}^{2}}{180\,x \left ( 2\,{x}^{2}-3 \right ) }\sqrt{cx}\sqrt{-a \left ( 2\,{x}^{2}-3 \right ) } \left ( 80\,{x}^{6}+18\,\sqrt{ \left ( -2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}\sqrt{3}\sqrt{-x\sqrt{3}\sqrt{2}}{\it EllipticE} \left ( 1/6\,\sqrt{3}\sqrt{2}\sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}},1/2\,\sqrt{2} \right ) \sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}\sqrt{2}-9\,\sqrt{ \left ( -2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}\sqrt{3}\sqrt{-x\sqrt{3}\sqrt{2}}{\it EllipticF} \left ( 1/6\,\sqrt{3}\sqrt{2}\sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}},1/2\,\sqrt{2} \right ) \sqrt{ \left ( 2\,x+\sqrt{3}\sqrt{2} \right ) \sqrt{3}\sqrt{2}}\sqrt{2}-168\,{x}^{4}+72\,{x}^{2} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x)^(5/2)*(-2*a*x^2+3*a)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{-2 \, a x^{2} + 3 \, a} \left (c x\right )^{\frac{5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-2*a*x^2 + 3*a)*(c*x)^(5/2),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{-2 \, a x^{2} + 3 \, a} \sqrt{c x} c^{2} x^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-2*a*x^2 + 3*a)*(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((c*x)**(5/2)*(-2*a*x**2+3*a)**(1/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{-2 \, a x^{2} + 3 \, a} \left (c x\right )^{\frac{5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-2*a*x^2 + 3*a)*(c*x)^(5/2),x, algorithm="giac")
[Out]