3.176 \(\int \frac{1}{x (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=118 \[ \frac{2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac{15 d-16 e x}{15 d^5 \sqrt{d^2-e^2 x^2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^5}+\frac{5 d-8 e x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}} \]

[Out]

(2*(d - e*x))/(5*d*(d^2 - e^2*x^2)^(5/2)) + (5*d - 8*e*x)/(15*d^3*(d^2 - e^2*x^2
)^(3/2)) + (15*d - 16*e*x)/(15*d^5*Sqrt[d^2 - e^2*x^2]) - ArcTanh[Sqrt[d^2 - e^2
*x^2]/d]/d^5

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Rubi [A]  time = 0.42476, antiderivative size = 118, 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 \[ \frac{2 (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}+\frac{15 d-16 e x}{15 d^5 \sqrt{d^2-e^2 x^2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^5}+\frac{5 d-8 e x}{15 d^3 \left (d^2-e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(d - e*x))/(5*d*(d^2 - e^2*x^2)^(5/2)) + (5*d - 8*e*x)/(15*d^3*(d^2 - e^2*x^2
)^(3/2)) + (15*d - 16*e*x)/(15*d^5*Sqrt[d^2 - e^2*x^2]) - ArcTanh[Sqrt[d^2 - e^2
*x^2]/d]/d^5

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Rubi in Sympy [A]  time = 33.4342, size = 114, normalized size = 0.97 \[ \frac{1}{5 d^{2} \left (d + e x\right )^{2} \sqrt{d^{2} - e^{2} x^{2}}} + \frac{8}{15 d^{3} \left (d + e x\right ) \sqrt{d^{2} - e^{2} x^{2}}} + \frac{1}{d^{4} \sqrt{d^{2} - e^{2} x^{2}}} - \frac{16 e x}{15 d^{5} \sqrt{d^{2} - e^{2} x^{2}}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(5*d**2*(d + e*x)**2*sqrt(d**2 - e**2*x**2)) + 8/(15*d**3*(d + e*x)*sqrt(d**2
- e**2*x**2)) + 1/(d**4*sqrt(d**2 - e**2*x**2)) - 16*e*x/(15*d**5*sqrt(d**2 - e*
*2*x**2)) - atanh(sqrt(d**2 - e**2*x**2)/d)/d**5

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Mathematica [A]  time = 0.0898883, size = 95, normalized size = 0.81 \[ \frac{-15 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+\frac{\sqrt{d^2-e^2 x^2} \left (26 d^3+22 d^2 e x-17 d e^2 x^2-16 e^3 x^3\right )}{(d-e x) (d+e x)^3}+15 \log (x)}{15 d^5} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[d^2 - e^2*x^2]*(26*d^3 + 22*d^2*e*x - 17*d*e^2*x^2 - 16*e^3*x^3))/((d - e
*x)*(d + e*x)^3) + 15*Log[x] - 15*Log[d + Sqrt[d^2 - e^2*x^2]])/(15*d^5)

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Maple [A]  time = 0.017, size = 187, normalized size = 1.6 \[{\frac{1}{{d}^{4}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}}-{\frac{1}{{d}^{4}}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}+{\frac{8}{15\,{d}^{3}e} \left ( x+{\frac{d}{e}} \right ) ^{-1}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}}-{\frac{16\,ex}{15\,{d}^{5}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}}+{\frac{1}{5\,{d}^{2}{e}^{2}} \left ( x+{\frac{d}{e}} \right ) ^{-2}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/d^4/(-e^2*x^2+d^2)^(1/2)-1/d^4/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d
^2)^(1/2))/x)+8/15/d^3/e/(x+d/e)/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)-16/15/d^5*
e/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)*x+1/5/d^2/e^2/(x+d/e)^2/(-(x+d/e)^2*e^2+2
*d*e*(x+d/e))^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.290636, size = 539, normalized size = 4.57 \[ \frac{26 \, e^{6} x^{6} + 4 \, d e^{5} x^{5} - 155 \, d^{2} e^{4} x^{4} - 130 \, d^{3} e^{3} x^{3} + 120 \, d^{4} e^{2} x^{2} + 120 \, d^{5} e x + 15 \,{\left (e^{6} x^{6} + 2 \, d e^{5} x^{5} - 4 \, d^{2} e^{4} x^{4} - 10 \, d^{3} e^{3} x^{3} - d^{4} e^{2} x^{2} + 8 \, d^{5} e x + 4 \, d^{6} +{\left (3 \, d e^{4} x^{4} + 6 \, d^{2} e^{3} x^{3} - d^{3} e^{2} x^{2} - 8 \, d^{4} e x - 4 \, d^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) +{\left (16 \, e^{5} x^{5} + 95 \, d e^{4} x^{4} + 70 \, d^{2} e^{3} x^{3} - 120 \, d^{3} e^{2} x^{2} - 120 \, d^{4} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (d^{5} e^{6} x^{6} + 2 \, d^{6} e^{5} x^{5} - 4 \, d^{7} e^{4} x^{4} - 10 \, d^{8} e^{3} x^{3} - d^{9} e^{2} x^{2} + 8 \, d^{10} e x + 4 \, d^{11} +{\left (3 \, d^{6} e^{4} x^{4} + 6 \, d^{7} e^{3} x^{3} - d^{8} e^{2} x^{2} - 8 \, d^{9} e x - 4 \, d^{10}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(26*e^6*x^6 + 4*d*e^5*x^5 - 155*d^2*e^4*x^4 - 130*d^3*e^3*x^3 + 120*d^4*e^2
*x^2 + 120*d^5*e*x + 15*(e^6*x^6 + 2*d*e^5*x^5 - 4*d^2*e^4*x^4 - 10*d^3*e^3*x^3
- d^4*e^2*x^2 + 8*d^5*e*x + 4*d^6 + (3*d*e^4*x^4 + 6*d^2*e^3*x^3 - d^3*e^2*x^2 -
8*d^4*e*x - 4*d^5)*sqrt(-e^2*x^2 + d^2))*log(-(d - sqrt(-e^2*x^2 + d^2))/x) + (
16*e^5*x^5 + 95*d*e^4*x^4 + 70*d^2*e^3*x^3 - 120*d^3*e^2*x^2 - 120*d^4*e*x)*sqrt
(-e^2*x^2 + d^2))/(d^5*e^6*x^6 + 2*d^6*e^5*x^5 - 4*d^7*e^4*x^4 - 10*d^8*e^3*x^3
- d^9*e^2*x^2 + 8*d^10*e*x + 4*d^11 + (3*d^6*e^4*x^4 + 6*d^7*e^3*x^3 - d^8*e^2*x
^2 - 8*d^9*e*x - 4*d^10)*sqrt(-e^2*x^2 + d^2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x*(-(-d + e*x)*(d + e*x))**(3/2)*(d + e*x)**2), x)

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GIAC/XCAS [A]  time = 0.678814, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x