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

Optimal. Leaf size=51 \[ \frac{e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d}-\frac{\sqrt{d^2-e^2 x^2}}{d x} \]

[Out]

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

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Rubi [A]  time = 0.185852, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ \frac{e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d}-\frac{\sqrt{d^2-e^2 x^2}}{d x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.3632, size = 36, normalized size = 0.71 \[ \frac{e \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{d} - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0440418, size = 53, normalized size = 1.04 \[ -\frac{\sqrt{d^2-e^2 x^2}-e x \log \left (\sqrt{d^2-e^2 x^2}+d\right )+e x \log (x)}{d x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.02, size = 222, normalized size = 4.4 \[ -{\frac{1}{{d}^{3}x} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{{e}^{2}x}{{d}^{3}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{{e}^{2}}{d}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{e}{{d}^{2}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}+{\frac{{e}^{2}}{d}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{e}{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{e\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-e^{2} x^{2} + d^{2}}}{{\left (e x + d\right )} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.284498, size = 142, normalized size = 2.78 \[ -\frac{e^{2} x^{2} - d^{2} +{\left (d e x - \sqrt{-e^{2} x^{2} + d^{2}} e x\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) + \sqrt{-e^{2} x^{2} + d^{2}} d}{d^{2} x - \sqrt{-e^{2} x^{2} + d^{2}} d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \left (- d + e x\right ) \left (d + e x\right )}}{x^{2} \left (d + e x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.299843, size = 138, normalized size = 2.71 \[ \frac{e{\rm ln}\left (\frac{{\left | -2 \, d e - 2 \, \sqrt{-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \,{\left | x \right |}}\right )}{d} + \frac{x e^{3}}{2 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} d} - \frac{{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-1\right )}}{2 \, d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*ln(1/2*abs(-2*d*e - 2*sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/abs(x))/d + 1/2*x*e^3/((d
*e + sqrt(-x^2*e^2 + d^2)*e)*d) - 1/2*(d*e + sqrt(-x^2*e^2 + d^2)*e)*e^(-1)/(d*x
)