3.126 \(\int \frac{1}{x^3 (d+e x) \sqrt{d^2-e^2 x^2}} \, dx\)

Optimal. Leaf size=113 \[ \frac{\sqrt{d^2-e^2 x^2}}{d^2 x^2 (d+e x)}+\frac{2 e \sqrt{d^2-e^2 x^2}}{d^4 x}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^4}-\frac{3 \sqrt{d^2-e^2 x^2}}{2 d^3 x^2} \]

[Out]

(-3*Sqrt[d^2 - e^2*x^2])/(2*d^3*x^2) + (2*e*Sqrt[d^2 - e^2*x^2])/(d^4*x) + Sqrt[
d^2 - e^2*x^2]/(d^2*x^2*(d + e*x)) - (3*e^2*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(2*d
^4)

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Rubi [A]  time = 0.305292, antiderivative size = 113, 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 \[ \frac{\sqrt{d^2-e^2 x^2}}{d^2 x^2 (d+e x)}+\frac{2 e \sqrt{d^2-e^2 x^2}}{d^4 x}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^4}-\frac{3 \sqrt{d^2-e^2 x^2}}{2 d^3 x^2} \]

Antiderivative was successfully verified.

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

[Out]

(-3*Sqrt[d^2 - e^2*x^2])/(2*d^3*x^2) + (2*e*Sqrt[d^2 - e^2*x^2])/(d^4*x) + Sqrt[
d^2 - e^2*x^2]/(d^2*x^2*(d + e*x)) - (3*e^2*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(2*d
^4)

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Rubi in Sympy [A]  time = 33.8534, size = 97, normalized size = 0.86 \[ \frac{d - e x}{d^{2} x^{2} \sqrt{d^{2} - e^{2} x^{2}}} - \frac{3 \sqrt{d^{2} - e^{2} x^{2}}}{2 d^{3} x^{2}} - \frac{3 e^{2} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{2 d^{4}} + \frac{2 e \sqrt{d^{2} - e^{2} x^{2}}}{d^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d - e*x)/(d**2*x**2*sqrt(d**2 - e**2*x**2)) - 3*sqrt(d**2 - e**2*x**2)/(2*d**3*
x**2) - 3*e**2*atanh(sqrt(d**2 - e**2*x**2)/d)/(2*d**4) + 2*e*sqrt(d**2 - e**2*x
**2)/(d**4*x)

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Mathematica [A]  time = 0.14159, size = 84, normalized size = 0.74 \[ \frac{\frac{\sqrt{d^2-e^2 x^2} \left (-d^2+d e x+4 e^2 x^2\right )}{x^2 (d+e x)}-3 e^2 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+3 e^2 \log (x)}{2 d^4} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[d^2 - e^2*x^2]*(-d^2 + d*e*x + 4*e^2*x^2))/(x^2*(d + e*x)) + 3*e^2*Log[x]
- 3*e^2*Log[d + Sqrt[d^2 - e^2*x^2]])/(2*d^4)

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Maple [A]  time = 0.017, size = 133, normalized size = 1.2 \[ -{\frac{1}{2\,{d}^{3}{x}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{3\,{e}^{2}}{2\,{d}^{3}}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}+{\frac{e}{{d}^{4}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) } \left ( x+{\frac{d}{e}} \right ) ^{-1}}+{\frac{e}{{d}^{4}x}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-e^{2} x^{2} + d^{2}}{\left (e x + d\right )} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-e^2*x^2 + d^2)*(e*x + d)*x^3),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-e^2*x^2 + d^2)*(e*x + d)*x^3), x)

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Fricas [A]  time = 0.286798, size = 410, normalized size = 3.63 \[ \frac{6 \, e^{5} x^{5} - 5 \, d e^{4} x^{4} - 16 \, d^{2} e^{3} x^{3} + 9 \, d^{3} e^{2} x^{2} + 6 \, d^{4} e x - 4 \, d^{5} + 3 \,{\left (e^{5} x^{5} + 3 \, d e^{4} x^{4} - 2 \, d^{2} e^{3} x^{3} - 4 \, d^{3} e^{2} x^{2} -{\left (e^{4} x^{4} - 2 \, d e^{3} x^{3} - 4 \, d^{2} e^{2} x^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) +{\left (2 \, e^{4} x^{4} + 13 \, d e^{3} x^{3} - 7 \, d^{2} e^{2} x^{2} - 6 \, d^{3} e x + 4 \, d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{2 \,{\left (d^{4} e^{3} x^{5} + 3 \, d^{5} e^{2} x^{4} - 2 \, d^{6} e x^{3} - 4 \, d^{7} x^{2} -{\left (d^{4} e^{2} x^{4} - 2 \, d^{5} e x^{3} - 4 \, d^{6} x^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-e^2*x^2 + d^2)*(e*x + d)*x^3),x, algorithm="fricas")

[Out]

1/2*(6*e^5*x^5 - 5*d*e^4*x^4 - 16*d^2*e^3*x^3 + 9*d^3*e^2*x^2 + 6*d^4*e*x - 4*d^
5 + 3*(e^5*x^5 + 3*d*e^4*x^4 - 2*d^2*e^3*x^3 - 4*d^3*e^2*x^2 - (e^4*x^4 - 2*d*e^
3*x^3 - 4*d^2*e^2*x^2)*sqrt(-e^2*x^2 + d^2))*log(-(d - sqrt(-e^2*x^2 + d^2))/x)
+ (2*e^4*x^4 + 13*d*e^3*x^3 - 7*d^2*e^2*x^2 - 6*d^3*e*x + 4*d^4)*sqrt(-e^2*x^2 +
d^2))/(d^4*e^3*x^5 + 3*d^5*e^2*x^4 - 2*d^6*e*x^3 - 4*d^7*x^2 - (d^4*e^2*x^4 - 2
*d^5*e*x^3 - 4*d^6*x^2)*sqrt(-e^2*x^2 + d^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \sqrt{- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**3*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^3),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError