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

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 = 0.0123493, 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, Rules used = {641, 217, 203} \[ \frac{d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{e}-\frac{\sqrt{d^2-e^2 x^2}}{e} \]

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

Rule 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{d+e x}{\sqrt{d^2-e^2 x^2}} \, dx &=-\frac{\sqrt{d^2-e^2 x^2}}{e}+d \int \frac{1}{\sqrt{d^2-e^2 x^2}} \, dx\\ &=-\frac{\sqrt{d^2-e^2 x^2}}{e}+d \operatorname{Subst}\left (\int \frac{1}{1+e^2 x^2} \, dx,x,\frac{x}{\sqrt{d^2-e^2 x^2}}\right )\\ &=-\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}\\ \end{align*}

Mathematica [A]  time = 0.0152537, 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 = 0.044, size = 50, normalized size = 1.1 \begin{align*} -{\frac{1}{e}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{d\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \end{align*}

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]

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

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Maxima [A]  time = 1.96742, size = 57, normalized size = 1.21 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(-e^2*x^2+d^2)^(1/2),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 = 2.13811, size = 101, normalized size = 2.15 \begin{align*} -\frac{2 \, d \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + \sqrt{-e^{2} x^{2} + d^{2}}}{e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.52875, size = 42, normalized size = 0.89 \begin{align*} \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} \end{align*}

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 [A]  time = 1.26415, size = 43, normalized size = 0.91 \begin{align*} d \arcsin \left (\frac{x e}{d}\right ) e^{\left (-1\right )} \mathrm{sgn}\left (d\right ) - \sqrt{-x^{2} e^{2} + d^{2}} e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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