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

Optimal. Leaf size=46 \[ \frac{\sqrt{d^2-e^2 x^2}}{e}+\frac{d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e} \]

[Out]

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

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Rubi [A]  time = 0.0446184, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{\sqrt{d^2-e^2 x^2}}{e}+\frac{d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.3079, size = 36, normalized size = 0.78 \[ \frac{d \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{e} + \frac{\sqrt{d^{2} - e^{2} x^{2}}}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0272258, size = 43, normalized size = 0.93 \[ \frac{\sqrt{d^2-e^2 x^2}+d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 77, normalized size = 1.7 \[{\frac{1}{e}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}+{d\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/e*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)+d/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-(x
+d/e)^2*e^2+2*d*e*(x+d/e))^(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(sqrt(-e^2*x^2 + d^2)/(e*x + d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.28337, size = 111, normalized size = 2.41 \[ \frac{e^{2} x^{2} - 2 \,{\left (d^{2} - \sqrt{-e^{2} x^{2} + d^{2}} d\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right )}{d e - \sqrt{-e^{2} x^{2} + d^{2}} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.2846, size = 42, normalized size = 0.91 \[ d \arcsin \left (\frac{x e}{d}\right ) e^{\left (-1\right )}{\rm sign}\left (d\right ) + \sqrt{-x^{2} e^{2} + d^{2}} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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