Optimal. Leaf size=36 \[ -\frac{2 \sqrt{c d^2-c e^2 x^2}}{c e \sqrt{d+e x}} \]
[Out]
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Rubi [A] time = 0.05086, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.034 \[ -\frac{2 \sqrt{c d^2-c e^2 x^2}}{c e \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[d + e*x]/Sqrt[c*d^2 - c*e^2*x^2],x]
[Out]
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Rubi in Sympy [A] time = 6.41639, size = 31, normalized size = 0.86 \[ - \frac{2 \sqrt{c d^{2} - c e^{2} x^{2}}}{c e \sqrt{d + e x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(1/2)/(-c*e**2*x**2+c*d**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0424077, size = 35, normalized size = 0.97 \[ -\frac{2 \sqrt{c \left (d^2-e^2 x^2\right )}}{c e \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[d + e*x]/Sqrt[c*d^2 - c*e^2*x^2],x]
[Out]
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Maple [A] time = 0.005, size = 36, normalized size = 1. \[ -2\,{\frac{ \left ( -ex+d \right ) \sqrt{ex+d}}{e\sqrt{-c{e}^{2}{x}^{2}+c{d}^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(1/2)/(-c*e^2*x^2+c*d^2)^(1/2),x)
[Out]
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Maxima [A] time = 0.723829, size = 39, normalized size = 1.08 \[ \frac{2 \,{\left (\sqrt{c} e x - \sqrt{c} d\right )}}{\sqrt{-e x + d} c e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(e*x + d)/sqrt(-c*e^2*x^2 + c*d^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.216819, size = 57, normalized size = 1.58 \[ \frac{2 \,{\left (e^{2} x^{2} - d^{2}\right )}}{\sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(e*x + d)/sqrt(-c*e^2*x^2 + c*d^2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d + e x}}{\sqrt{- c \left (- d + e x\right ) \left (d + e x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**(1/2)/(-c*e**2*x**2+c*d**2)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{e x + d}}{\sqrt{-c e^{2} x^{2} + c d^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(e*x + d)/sqrt(-c*e^2*x^2 + c*d^2),x, algorithm="giac")
[Out]