3.820 \(\int \frac{1}{(d+e x) \sqrt{d^2-e^2 x^2}} \, dx\)

Optimal. Leaf size=31 \[ -\frac{\sqrt{d^2-e^2 x^2}}{d e (d+e x)} \]

[Out]

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

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Rubi [A]  time = 0.0390997, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ -\frac{\sqrt{d^2-e^2 x^2}}{d e (d+e x)} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)*Sqrt[d^2 - e^2*x^2]),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.14703, size = 22, normalized size = 0.71 \[ - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{d e \left (d + e x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)/(-e**2*x**2+d**2)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0261186, size = 31, normalized size = 1. \[ -\frac{\sqrt{d^2-e^2 x^2}}{d e (d+e x)} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)*Sqrt[d^2 - e^2*x^2]),x]

[Out]

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

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Maple [A]  time = 0.009, size = 29, normalized size = 0.9 \[ -{\frac{-ex+d}{de}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)/(-e^2*x^2+d^2)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-e^2*x^2 + d^2)*(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.21421, size = 41, normalized size = 1.32 \[ -\frac{2 \, x}{d e x + d^{2} - \sqrt{-e^{2} x^{2} + d^{2}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-e^2*x^2 + d^2)*(e*x + d)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)/(-e**2*x**2+d**2)**(1/2),x)

[Out]

Integral(1/(sqrt(-(-d + e*x)*(d + e*x))*(d + e*x)), x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-e^2*x^2 + d^2)*(e*x + d)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError