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

Optimal. Leaf size=62 \[ -\frac{(2 d-e x) \sqrt{d^2-e^2 x^2}}{2 e^2}-\frac{d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^2} \]

[Out]

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

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Rubi [A]  time = 0.135751, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ -\frac{(2 d-e x) \sqrt{d^2-e^2 x^2}}{2 e^2}-\frac{d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.8796, size = 71, normalized size = 1.15 \[ - \frac{d^{2} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{2 e^{2}} - \frac{d \sqrt{d^{2} - e^{2} x^{2}}}{2 e^{2}} - \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{2 e^{2} \left (d + e x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d**2*atan(e*x/sqrt(d**2 - e**2*x**2))/(2*e**2) - d*sqrt(d**2 - e**2*x**2)/(2*e*
*2) - (d**2 - e**2*x**2)**(3/2)/(2*e**2*(d + e*x))

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Mathematica [A]  time = 0.0528263, size = 57, normalized size = 0.92 \[ \frac{(e x-2 d) \sqrt{d^2-e^2 x^2}-d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.011, size = 140, normalized size = 2.3 \[{\frac{x}{2\,e}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{{d}^{2}}{2\,e}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{d}{{e}^{2}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}-{\frac{{d}^{2}}{e}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.289408, size = 232, normalized size = 3.74 \[ -\frac{2 \, d e^{3} x^{3} - 2 \, d^{2} e^{2} x^{2} - 2 \, d^{3} e x - 2 \,{\left (d^{2} e^{2} x^{2} - 2 \, d^{4} + 2 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{3}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (e^{3} x^{3} - 2 \, d e^{2} x^{2} - 2 \, d^{2} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{2 \,{\left (e^{4} x^{2} - 2 \, d^{2} e^{2} + 2 \, \sqrt{-e^{2} x^{2} + d^{2}} d e^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(2*d*e^3*x^3 - 2*d^2*e^2*x^2 - 2*d^3*e*x - 2*(d^2*e^2*x^2 - 2*d^4 + 2*sqrt(
-e^2*x^2 + d^2)*d^3)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - (e^3*x^3 - 2*d*
e^2*x^2 - 2*d^2*e*x)*sqrt(-e^2*x^2 + d^2))/(e^4*x^2 - 2*d^2*e^2 + 2*sqrt(-e^2*x^
2 + d^2)*d*e^2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.293226, size = 58, normalized size = 0.94 \[ -\frac{1}{2} \, d^{2} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-2\right )}{\rm sign}\left (d\right ) + \frac{1}{2} \, \sqrt{-x^{2} e^{2} + d^{2}}{\left (x e^{\left (-1\right )} - 2 \, d e^{\left (-2\right )}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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