Optimal. Leaf size=111 \[ -\frac{2 e \sqrt{-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{3/2}}{5 d}-\frac{6 e^{5/2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{5 d}+\frac{6 e^{5/2} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{5 d} \]
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Rubi [A] time = 0.281521, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.162 \[ -\frac{2 e \sqrt{-c^2-2 c d x-d^2 x^2+1} (c e+d e x)^{3/2}}{5 d}-\frac{6 e^{5/2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{5 d}+\frac{6 e^{5/2} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{5 d} \]
Antiderivative was successfully verified.
[In] Int[(c*e + d*e*x)^(5/2)/Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2],x]
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Rubi in Sympy [A] time = 59.1936, size = 104, normalized size = 0.94 \[ \frac{6 e^{\frac{5}{2}} E\left (\operatorname{asin}{\left (\frac{\sqrt{c e + d e x}}{\sqrt{e}} \right )}\middle | -1\right )}{5 d} - \frac{6 e^{\frac{5}{2}} F\left (\operatorname{asin}{\left (\frac{\sqrt{c e + d e x}}{\sqrt{e}} \right )}\middle | -1\right )}{5 d} - \frac{2 e \left (c e + d e x\right )^{\frac{3}{2}} \sqrt{- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{5 d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*e*x+c*e)**(5/2)/(-d**2*x**2-2*c*d*x-c**2+1)**(1/2),x)
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Mathematica [A] time = 0.197324, size = 90, normalized size = 0.81 \[ -\frac{2 (e (c+d x))^{5/2} \left (\sqrt{-c^2-2 c d x-d^2 x^2+1} (c+d x)^{3/2}+3 F\left (\left .\sin ^{-1}\left (\sqrt{c+d x}\right )\right |-1\right )-3 E\left (\left .\sin ^{-1}\left (\sqrt{c+d x}\right )\right |-1\right )\right )}{5 d (c+d x)^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(c*e + d*e*x)^(5/2)/Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2],x]
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Maple [B] time = 0.026, size = 234, normalized size = 2.1 \[ -{\frac{{e}^{2}}{5\,d \left ({x}^{3}{d}^{3}+3\,c{d}^{2}{x}^{2}+3\,x{c}^{2}d+{c}^{3}-dx-c \right ) }\sqrt{e \left ( dx+c \right ) }\sqrt{-{d}^{2}{x}^{2}-2\,cdx-{c}^{2}+1} \left ( 2\,{d}^{4}{x}^{4}+8\,{x}^{3}c{d}^{3}+12\,{x}^{2}{c}^{2}{d}^{2}+8\,{c}^{3}dx+2\,\sqrt{-2\,dx-2\,c+2}\sqrt{dx+c}\sqrt{2\,dx+2\,c+2}{\it EllipticE} \left ( 1/2\,\sqrt{-2\,dx-2\,c+2},\sqrt{2} \right ) -5\,\sqrt{-2\,dx-2\,c+2}\sqrt{2\,dx+2\,c+2}\sqrt{-dx-c}{\it EllipticE} \left ( 1/2\,\sqrt{2\,dx+2\,c+2},\sqrt{2} \right ) -2\,{d}^{2}{x}^{2}+2\,{c}^{4}-4\,cdx-2\,{c}^{2} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*e*x+c*e)^(5/2)/(-d^2*x^2-2*c*d*x-c^2+1)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d e x + c e\right )}^{\frac{5}{2}}}{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*e*x + c*e)^(5/2)/sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (d^{2} e^{2} x^{2} + 2 \, c d e^{2} x + c^{2} e^{2}\right )} \sqrt{d e x + c e}}{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*e*x + c*e)^(5/2)/sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (e \left (c + d x\right )\right )^{\frac{5}{2}}}{\sqrt{- \left (c + d x - 1\right ) \left (c + d x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*e*x+c*e)**(5/2)/(-d**2*x**2-2*c*d*x-c**2+1)**(1/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d e x + c e\right )}^{\frac{5}{2}}}{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*e*x + c*e)^(5/2)/sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1),x, algorithm="giac")
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