3.195 \(\int \frac{\sqrt{d^2-e^2 x^2}}{x^2 (d+e x)^4} \, dx\)

Optimal. Leaf size=143 \[ -\frac{4 e (5 d-8 e x)}{15 d^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{8 e (d-e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{e (60 d-79 e x)}{15 d^4 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{d^4 x}+\frac{4 e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^4} \]

[Out]

(-8*e*(d - e*x))/(5*(d^2 - e^2*x^2)^(5/2)) - (4*e*(5*d - 8*e*x))/(15*d^2*(d^2 -
e^2*x^2)^(3/2)) - (e*(60*d - 79*e*x))/(15*d^4*Sqrt[d^2 - e^2*x^2]) - Sqrt[d^2 -
e^2*x^2]/(d^4*x) + (4*e*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/d^4

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Rubi [A]  time = 0.526751, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{4 e (5 d-8 e x)}{15 d^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac{8 e (d-e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{e (60 d-79 e x)}{15 d^4 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{d^4 x}+\frac{4 e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^4} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[d^2 - e^2*x^2]/(x^2*(d + e*x)^4),x]

[Out]

(-8*e*(d - e*x))/(5*(d^2 - e^2*x^2)^(5/2)) - (4*e*(5*d - 8*e*x))/(15*d^2*(d^2 -
e^2*x^2)^(3/2)) - (e*(60*d - 79*e*x))/(15*d^4*Sqrt[d^2 - e^2*x^2]) - Sqrt[d^2 -
e^2*x^2]/(d^4*x) + (4*e*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/d^4

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Rubi in Sympy [A]  time = 60.2056, size = 122, normalized size = 0.85 \[ - \frac{e \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{5 d^{3} \left (d + e x\right )^{4}} + \frac{4 e \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{d^{4}} - \frac{6 e \sqrt{d^{2} - e^{2} x^{2}}}{d^{4} \left (d + e x\right )} - \frac{11 e \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{15 d^{4} \left (d + e x\right )^{3}} - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{d^{4} x} \]

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)**4,x)

[Out]

-e*(d**2 - e**2*x**2)**(3/2)/(5*d**3*(d + e*x)**4) + 4*e*atanh(sqrt(d**2 - e**2*
x**2)/d)/d**4 - 6*e*sqrt(d**2 - e**2*x**2)/(d**4*(d + e*x)) - 11*e*(d**2 - e**2*
x**2)**(3/2)/(15*d**4*(d + e*x)**3) - sqrt(d**2 - e**2*x**2)/(d**4*x)

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Mathematica [A]  time = 0.215231, size = 92, normalized size = 0.64 \[ -\frac{-60 e \log \left (\sqrt{d^2-e^2 x^2}+d\right )+\frac{\sqrt{d^2-e^2 x^2} \left (15 d^3+149 d^2 e x+222 d e^2 x^2+94 e^3 x^3\right )}{x (d+e x)^3}+60 e \log (x)}{15 d^4} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[d^2 - e^2*x^2]/(x^2*(d + e*x)^4),x]

[Out]

-((Sqrt[d^2 - e^2*x^2]*(15*d^3 + 149*d^2*e*x + 222*d*e^2*x^2 + 94*e^3*x^3))/(x*(
d + e*x)^3) + 60*e*Log[x] - 60*e*Log[d + Sqrt[d^2 - e^2*x^2]])/(15*d^4)

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Maple [B]  time = 0.018, size = 361, normalized size = 2.5 \[ -{\frac{1}{{d}^{6}x} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{{e}^{2}x}{{d}^{6}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{{e}^{2}}{{d}^{4}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{1}{5\,{d}^{3}{e}^{3}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{d}{e}} \right ) ^{-4}}-{\frac{11}{15\,{d}^{4}{e}^{2}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{d}{e}} \right ) ^{-3}}-4\,{\frac{e\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}{{d}^{5}}}+4\,{\frac{e}{{d}^{3}\sqrt{{d}^{2}}}\ln \left ({\frac{2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}{x}} \right ) }+{\frac{e}{{d}^{5}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}+{\frac{{e}^{2}}{{d}^{4}}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-3\,{\frac{1}{{d}^{5}e} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{3/2} \left ( x+{\frac{d}{e}} \right ) ^{-2}} \]

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)^4,x)

[Out]

-1/d^6/x*(-e^2*x^2+d^2)^(3/2)-1/d^6*e^2*x*(-e^2*x^2+d^2)^(1/2)-1/d^4*e^2/(e^2)^(
1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-1/5/d^3/e^3/(x+d/e)^4*(-(x+d/e)^
2*e^2+2*d*e*(x+d/e))^(3/2)-11/15/d^4/e^2/(x+d/e)^3*(-(x+d/e)^2*e^2+2*d*e*(x+d/e)
)^(3/2)-4/d^5*e*(-e^2*x^2+d^2)^(1/2)+4/d^3*e/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)
*(-e^2*x^2+d^2)^(1/2))/x)+1/d^5*e*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)+1/d^4*e^2
/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2))-3/d^5/e/
(x+d/e)^2*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-e^{2} x^{2} + d^{2}}}{{\left (e x + d\right )}^{4} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-e^2*x^2 + d^2)/((e*x + d)^4*x^2),x, algorithm="maxima")

[Out]

integrate(sqrt(-e^2*x^2 + d^2)/((e*x + d)^4*x^2), x)

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Fricas [A]  time = 0.291846, size = 629, normalized size = 4.4 \[ -\frac{10 \, e^{7} x^{7} - 608 \, d e^{6} x^{6} - 1415 \, d^{2} e^{5} x^{5} + 130 \, d^{3} e^{4} x^{4} + 2115 \, d^{4} e^{3} x^{3} + 1020 \, d^{5} e^{2} x^{2} - 300 \, d^{6} e x - 120 \, d^{7} + 60 \,{\left (e^{7} x^{7} - d e^{6} x^{6} - 13 \, d^{2} e^{5} x^{5} - 15 \, d^{3} e^{4} x^{4} + 8 \, d^{4} e^{3} x^{3} + 20 \, d^{5} e^{2} x^{2} + 8 \, d^{6} e x +{\left (e^{6} x^{6} + 6 \, d e^{5} x^{5} + 5 \, d^{2} e^{4} x^{4} - 12 \, d^{3} e^{3} x^{3} - 20 \, d^{4} e^{2} x^{2} - 8 \, d^{5} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) +{\left (198 \, e^{6} x^{6} + 470 \, d e^{5} x^{5} - 595 \, d^{2} e^{4} x^{4} - 1965 \, d^{3} e^{3} x^{3} - 960 \, d^{4} e^{2} x^{2} + 300 \, d^{5} e x + 120 \, d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (d^{4} e^{6} x^{7} - d^{5} e^{5} x^{6} - 13 \, d^{6} e^{4} x^{5} - 15 \, d^{7} e^{3} x^{4} + 8 \, d^{8} e^{2} x^{3} + 20 \, d^{9} e x^{2} + 8 \, d^{10} x +{\left (d^{4} e^{5} x^{6} + 6 \, d^{5} e^{4} x^{5} + 5 \, d^{6} e^{3} x^{4} - 12 \, d^{7} e^{2} x^{3} - 20 \, d^{8} e x^{2} - 8 \, d^{9} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-e^2*x^2 + d^2)/((e*x + d)^4*x^2),x, algorithm="fricas")

[Out]

-1/15*(10*e^7*x^7 - 608*d*e^6*x^6 - 1415*d^2*e^5*x^5 + 130*d^3*e^4*x^4 + 2115*d^
4*e^3*x^3 + 1020*d^5*e^2*x^2 - 300*d^6*e*x - 120*d^7 + 60*(e^7*x^7 - d*e^6*x^6 -
13*d^2*e^5*x^5 - 15*d^3*e^4*x^4 + 8*d^4*e^3*x^3 + 20*d^5*e^2*x^2 + 8*d^6*e*x +
(e^6*x^6 + 6*d*e^5*x^5 + 5*d^2*e^4*x^4 - 12*d^3*e^3*x^3 - 20*d^4*e^2*x^2 - 8*d^5
*e*x)*sqrt(-e^2*x^2 + d^2))*log(-(d - sqrt(-e^2*x^2 + d^2))/x) + (198*e^6*x^6 +
470*d*e^5*x^5 - 595*d^2*e^4*x^4 - 1965*d^3*e^3*x^3 - 960*d^4*e^2*x^2 + 300*d^5*e
*x + 120*d^6)*sqrt(-e^2*x^2 + d^2))/(d^4*e^6*x^7 - d^5*e^5*x^6 - 13*d^6*e^4*x^5
- 15*d^7*e^3*x^4 + 8*d^8*e^2*x^3 + 20*d^9*e*x^2 + 8*d^10*x + (d^4*e^5*x^6 + 6*d^
5*e^4*x^5 + 5*d^6*e^3*x^4 - 12*d^7*e^2*x^3 - 20*d^8*e*x^2 - 8*d^9*x)*sqrt(-e^2*x
^2 + d^2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \left (- d + e x\right ) \left (d + e x\right )}}{x^{2} \left (d + e x\right )^{4}}\, dx \]

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)**4,x)

[Out]

Integral(sqrt(-(-d + e*x)*(d + e*x))/(x**2*(d + e*x)**4), x)

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GIAC/XCAS [A]  time = 0.293035, size = 1, normalized size = 0.01 \[ +\infty \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-e^2*x^2 + d^2)/((e*x + d)^4*x^2),x, algorithm="giac")

[Out]

+Infinity