3.125 \(\int \frac{1}{x^2 (d+e x) \sqrt{d^2-e^2 x^2}} \, dx\)

Optimal. Leaf size=81 \[ \frac{\sqrt{d^2-e^2 x^2}}{d^2 x (d+e x)}-\frac{2 \sqrt{d^2-e^2 x^2}}{d^3 x}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^3} \]

[Out]

(-2*Sqrt[d^2 - e^2*x^2])/(d^3*x) + Sqrt[d^2 - e^2*x^2]/(d^2*x*(d + e*x)) + (e*Ar
cTanh[Sqrt[d^2 - e^2*x^2]/d])/d^3

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Rubi [A]  time = 0.215206, antiderivative size = 81, 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 \[ \frac{\sqrt{d^2-e^2 x^2}}{d^2 x (d+e x)}-\frac{2 \sqrt{d^2-e^2 x^2}}{d^3 x}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^3} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[d^2 - e^2*x^2])/(d^3*x) + Sqrt[d^2 - e^2*x^2]/(d^2*x*(d + e*x)) + (e*Ar
cTanh[Sqrt[d^2 - e^2*x^2]/d])/d^3

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Rubi in Sympy [A]  time = 21.3509, size = 65, normalized size = 0.8 \[ \frac{d - e x}{d^{2} x \sqrt{d^{2} - e^{2} x^{2}}} + \frac{e \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{d^{3}} - \frac{2 \sqrt{d^{2} - e^{2} x^{2}}}{d^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.104666, size = 65, normalized size = 0.8 \[ \frac{-\frac{\sqrt{d^2-e^2 x^2} (d+2 e x)}{x (d+e x)}+e \log \left (\sqrt{d^2-e^2 x^2}+d\right )-e \log (x)}{d^3} \]

Antiderivative was successfully verified.

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

[Out]

(-(((d + 2*e*x)*Sqrt[d^2 - e^2*x^2])/(x*(d + e*x))) - e*Log[x] + e*Log[d + Sqrt[
d^2 - e^2*x^2]])/d^3

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Maple [A]  time = 0.016, size = 108, normalized size = 1.3 \[ -{\frac{1}{{d}^{3}x}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{1}{{d}^{3}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) } \left ( x+{\frac{d}{e}} \right ) ^{-1}}+{\frac{e}{{d}^{2}}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-e^2*x^2+d^2)^(1/2)/d^3/x-1/d^3/(x+d/e)*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)+e
/d^2/(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. \[ \int \frac{1}{\sqrt{-e^{2} x^{2} + d^{2}}{\left (e x + d\right )} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-e^2*x^2 + d^2)*(e*x + d)*x^2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-e^2*x^2 + d^2)*(e*x + d)*x^2), x)

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Fricas [A]  time = 0.284395, size = 271, normalized size = 3.35 \[ \frac{e^{3} x^{3} + 4 \, d e^{2} x^{2} - d^{2} e x - 2 \, d^{3} -{\left (e^{3} x^{3} - d e^{2} x^{2} - 2 \, d^{2} e x +{\left (e^{2} x^{2} + 2 \, d e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (3 \, e^{2} x^{2} - d e x - 2 \, d^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{d^{3} e^{2} x^{3} - d^{4} e x^{2} - 2 \, d^{5} x +{\left (d^{3} e x^{2} + 2 \, d^{4} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(e^3*x^3 + 4*d*e^2*x^2 - d^2*e*x - 2*d^3 - (e^3*x^3 - d*e^2*x^2 - 2*d^2*e*x + (e
^2*x^2 + 2*d*e*x)*sqrt(-e^2*x^2 + d^2))*log(-(d - sqrt(-e^2*x^2 + d^2))/x) - (3*
e^2*x^2 - d*e*x - 2*d^2)*sqrt(-e^2*x^2 + d^2))/(d^3*e^2*x^3 - d^4*e*x^2 - 2*d^5*
x + (d^3*e*x^2 + 2*d^4*x)*sqrt(-e^2*x^2 + d^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \sqrt{- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: NotImplementedError