3.871 \(\int \frac{\sqrt{d+e x}}{\sqrt{c d^2-c e^2 x^2}} \, dx\)

Optimal. Leaf size=36 \[ -\frac{2 \sqrt{c d^2-c e^2 x^2}}{c e \sqrt{d+e x}} \]

[Out]

(-2*Sqrt[c*d^2 - c*e^2*x^2])/(c*e*Sqrt[d + e*x])

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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]

(-2*Sqrt[c*d^2 - c*e^2*x^2])/(c*e*Sqrt[d + e*x])

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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]

-2*sqrt(c*d**2 - c*e**2*x**2)/(c*e*sqrt(d + e*x))

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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]

(-2*Sqrt[c*(d^2 - e^2*x^2)])/(c*e*Sqrt[d + e*x])

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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]

-2*(-e*x+d)*(e*x+d)^(1/2)/e/(-c*e^2*x^2+c*d^2)^(1/2)

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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]

2*(sqrt(c)*e*x - sqrt(c)*d)/(sqrt(-e*x + d)*c*e)

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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]

2*(e^2*x^2 - d^2)/(sqrt(-c*e^2*x^2 + c*d^2)*sqrt(e*x + d)*e)

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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]

Integral(sqrt(d + e*x)/sqrt(-c*(-d + e*x)*(d + e*x)), x)

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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]

integrate(sqrt(e*x + d)/sqrt(-c*e^2*x^2 + c*d^2), x)