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

Optimal. Leaf size=110 \[ -\frac{4 e x}{5 d \left (d^2-e^2 x^2\right )^{3/2}}+\frac{8 d (d-e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{5 d-8 e x}{5 d^3 \sqrt{d^2-e^2 x^2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^3} \]

[Out]

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

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Rubi [A]  time = 0.424799, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259 \[ -\frac{4 e x}{5 d \left (d^2-e^2 x^2\right )^{3/2}}+\frac{8 d (d-e x)}{5 \left (d^2-e^2 x^2\right )^{5/2}}+\frac{5 d-8 e x}{5 d^3 \sqrt{d^2-e^2 x^2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 48.2365, size = 95, normalized size = 0.86 \[ \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{5 d^{2} \left (d + e x\right )^{4}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{d^{3}} + \frac{2 \sqrt{d^{2} - e^{2} x^{2}}}{d^{3} \left (d + e x\right )} + \frac{2 \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{5 d^{3} \left (d + e x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-e**2*x**2+d**2)**(1/2)/x/(e*x+d)**4,x)

[Out]

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

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Mathematica [A]  time = 0.123273, size = 76, normalized size = 0.69 \[ \frac{\frac{\sqrt{d^2-e^2 x^2} \left (13 d^2+19 d e x+8 e^2 x^2\right )}{(d+e x)^3}-5 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+5 \log (x)}{5 d^3} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[d^2 - e^2*x^2]*(13*d^2 + 19*d*e*x + 8*e^2*x^2))/(d + e*x)^3 + 5*Log[x] -
5*Log[d + Sqrt[d^2 - e^2*x^2]])/(5*d^3)

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Maple [B]  time = 0.017, size = 196, normalized size = 1.8 \[{\frac{1}{{d}^{4}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{1}{{d}^{2}}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}+{\frac{1}{{d}^{4}{e}^{2}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{d}{e}} \right ) ^{-2}}+{\frac{2}{5\,{e}^{3}{d}^{3}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{d}{e}} \right ) ^{-3}}+{\frac{1}{5\,{d}^{2}{e}^{4}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{d}{e}} \right ) ^{-4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-e^2*x^2+d^2)^(1/2)/x/(e*x+d)^4,x)

[Out]

1/d^4*(-e^2*x^2+d^2)^(1/2)-1/d^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^2/(x+d/e)^2*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)+2/5/e^3/d
^3/(x+d/e)^3*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)+1/5/d^2/e^4/(x+d/e)^4*(-(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. \[ \int \frac{\sqrt{-e^{2} x^{2} + d^{2}}}{{\left (e x + d\right )}^{4} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.287757, size = 448, normalized size = 4.07 \[ \frac{21 \, e^{5} x^{5} + 60 \, d e^{4} x^{4} + 5 \, d^{2} e^{3} x^{3} - 110 \, d^{3} e^{2} x^{2} - 80 \, d^{4} e x + 5 \,{\left (e^{5} x^{5} + 5 \, d e^{4} x^{4} + 5 \, d^{2} e^{3} x^{3} - 5 \, d^{3} e^{2} x^{2} - 10 \, d^{4} e x - 4 \, d^{5} -{\left (e^{4} x^{4} - 7 \, d^{2} e^{2} x^{2} - 10 \, d^{3} e x - 4 \, d^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) - 5 \,{\left (e^{4} x^{4} - 7 \, d e^{3} x^{3} - 22 \, d^{2} e^{2} x^{2} - 16 \, d^{3} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{5 \,{\left (d^{3} e^{5} x^{5} + 5 \, d^{4} e^{4} x^{4} + 5 \, d^{5} e^{3} x^{3} - 5 \, d^{6} e^{2} x^{2} - 10 \, d^{7} e x - 4 \, d^{8} -{\left (d^{3} e^{4} x^{4} - 7 \, d^{5} e^{2} x^{2} - 10 \, d^{6} e x - 4 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*(21*e^5*x^5 + 60*d*e^4*x^4 + 5*d^2*e^3*x^3 - 110*d^3*e^2*x^2 - 80*d^4*e*x +
5*(e^5*x^5 + 5*d*e^4*x^4 + 5*d^2*e^3*x^3 - 5*d^3*e^2*x^2 - 10*d^4*e*x - 4*d^5 -
(e^4*x^4 - 7*d^2*e^2*x^2 - 10*d^3*e*x - 4*d^4)*sqrt(-e^2*x^2 + d^2))*log(-(d - s
qrt(-e^2*x^2 + d^2))/x) - 5*(e^4*x^4 - 7*d*e^3*x^3 - 22*d^2*e^2*x^2 - 16*d^3*e*x
)*sqrt(-e^2*x^2 + d^2))/(d^3*e^5*x^5 + 5*d^4*e^4*x^4 + 5*d^5*e^3*x^3 - 5*d^6*e^2
*x^2 - 10*d^7*e*x - 4*d^8 - (d^3*e^4*x^4 - 7*d^5*e^2*x^2 - 10*d^6*e*x - 4*d^7)*s
qrt(-e^2*x^2 + d^2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e**2*x**2+d**2)**(1/2)/x/(e*x+d)**4,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.300639, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done