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

Optimal. Leaf size=47 \[ \frac{d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e}-\frac{\sqrt{d^2-e^2 x^2}}{e} \]

[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 = 0.0452245, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e}-\frac{\sqrt{d^2-e^2 x^2}}{e} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)/Sqrt[d^2 - e^2*x^2],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 = 12.1172, size = 36, normalized size = 0.77 \[ \frac{d \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{e} - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)/(-e**2*x**2+d**2)**(1/2),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 = 0.03071, size = 47, normalized size = 1. \[ \frac{d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e}-\frac{\sqrt{d^2-e^2 x^2}}{e} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)/Sqrt[d^2 - e^2*x^2],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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Maple [A]  time = 0.007, size = 50, normalized size = 1.1 \[{d\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{1}{e}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.783295, size = 57, normalized size = 1.21 \[ \frac{d \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{\sqrt{e^{2}}} - \frac{\sqrt{-e^{2} x^{2} + d^{2}}}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.221844, size = 112, normalized size = 2.38 \[ -\frac{e^{2} x^{2} + 2 \,{\left (d^{2} - \sqrt{-e^{2} x^{2} + d^{2}} d\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right )}{d e - \sqrt{-e^{2} x^{2} + d^{2}} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/sqrt(-e^2*x^2 + d^2),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 [A]  time = 3.32923, size = 42, normalized size = 0.89 \[ \begin{cases} \frac{d \left (\begin{cases} \operatorname{asin}{\left (e x \sqrt{\frac{1}{d^{2}}} \right )} & \text{for}\: d^{2} > 0 \end{cases}\right ) - \sqrt{d^{2} - e^{2} x^{2}}}{e} & \text{for}\: e \neq 0 \\\frac{d x}{\sqrt{d^{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise(((d*Piecewise((asin(e*x*sqrt(d**(-2))), d**2 > 0)) - sqrt(d**2 - e**2*
x**2))/e, Ne(e, 0)), (d*x/sqrt(d**2), True))

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GIAC/XCAS [A]  time = 0.225764, size = 43, 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((e*x + d)/sqrt(-e^2*x^2 + d^2),x, algorithm="giac")

[Out]

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