3.99 \(\int \frac{\sqrt{d^2-e^2 x^2}}{x^3 (d+e x)} \, dx\)

Optimal. Leaf size=82 \[ \frac{e \sqrt{d^2-e^2 x^2}}{d^2 x}-\frac{\sqrt{d^2-e^2 x^2}}{2 d x^2}-\frac{e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^2} \]

[Out]

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

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Rubi [A]  time = 0.264977, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{e \sqrt{d^2-e^2 x^2}}{d^2 x}-\frac{\sqrt{d^2-e^2 x^2}}{2 d x^2}-\frac{e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[d^2 - e^2*x^2]/(x^3*(d + e*x)),x]

[Out]

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

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Rubi in Sympy [A]  time = 25.9517, size = 65, normalized size = 0.79 \[ - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{2 d x^{2}} - \frac{e^{2} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{2 d^{2}} + \frac{e \sqrt{d^{2} - e^{2} x^{2}}}{d^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-e**2*x**2+d**2)**(1/2)/x**3/(e*x+d),x)

[Out]

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

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Mathematica [A]  time = 0.0811528, size = 70, normalized size = 0.85 \[ -\frac{(d-2 e x) \sqrt{d^2-e^2 x^2}+e^2 x^2 \log \left (\sqrt{d^2-e^2 x^2}+d\right )-e^2 x^2 \log (x)}{2 d^2 x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[d^2 - e^2*x^2]/(x^3*(d + e*x)),x]

[Out]

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

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Maple [B]  time = 0.016, size = 254, normalized size = 3.1 \[ -{\frac{1}{2\,{d}^{3}{x}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{{e}^{2}}{2\,{d}^{3}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{{e}^{2}}{2\,d}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}-{\frac{{e}^{2}}{{d}^{3}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}-{\frac{{e}^{3}}{{d}^{2}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{e}{{d}^{4}x} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{{e}^{3}x}{{d}^{4}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{{e}^{3}}{{d}^{2}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-e^2*x^2+d^2)^(1/2)/x^3/(e*x+d),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-e^{2} x^{2} + d^{2}}}{{\left (e x + d\right )} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-e^2*x^2 + d^2)/((e*x + d)*x^3),x, algorithm="maxima")

[Out]

integrate(sqrt(-e^2*x^2 + d^2)/((e*x + d)*x^3), x)

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Fricas [A]  time = 0.279977, size = 259, normalized size = 3.16 \[ -\frac{4 \, d e^{3} x^{3} - 2 \, d^{2} e^{2} x^{2} - 4 \, d^{3} e x + 2 \, d^{4} -{\left (e^{4} x^{4} - 2 \, d^{2} e^{2} x^{2} + 2 \, \sqrt{-e^{2} x^{2} + d^{2}} d e^{2} x^{2}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (2 \, e^{3} x^{3} - d e^{2} x^{2} - 4 \, d^{2} e x + 2 \, d^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{2 \,{\left (d^{2} e^{2} x^{4} - 2 \, d^{4} x^{2} + 2 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{3} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-e^2*x^2 + d^2)/((e*x + d)*x^3),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \left (- d + e x\right ) \left (d + e x\right )}}{x^{3} \left (d + e x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e**2*x**2+d**2)**(1/2)/x**3/(e*x+d),x)

[Out]

Integral(sqrt(-(-d + e*x)*(d + e*x))/(x**3*(d + e*x)), x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-e^2*x^2 + d^2)/((e*x + d)*x^3),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError