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

Optimal. Leaf size=146 \[ -\frac{4 e (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}-\frac{e (15 d-19 e x)}{5 d^5 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{d^5 x}+\frac{3 e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^5}-\frac{e (5 d-7 e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{3/2}} \]

[Out]

(-4*e*(d - e*x))/(5*d*(d^2 - e^2*x^2)^(5/2)) - (e*(5*d - 7*e*x))/(5*d^3*(d^2 - e
^2*x^2)^(3/2)) - (e*(15*d - 19*e*x))/(5*d^5*Sqrt[d^2 - e^2*x^2]) - Sqrt[d^2 - e^
2*x^2]/(d^5*x) + (3*e*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/d^5

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Rubi [A]  time = 0.530599, antiderivative size = 146, 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 \[ -\frac{4 e (d-e x)}{5 d \left (d^2-e^2 x^2\right )^{5/2}}-\frac{e (15 d-19 e x)}{5 d^5 \sqrt{d^2-e^2 x^2}}-\frac{\sqrt{d^2-e^2 x^2}}{d^5 x}+\frac{3 e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^5}-\frac{e (5 d-7 e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-4*e*(d - e*x))/(5*d*(d^2 - e^2*x^2)^(5/2)) - (e*(5*d - 7*e*x))/(5*d^3*(d^2 - e
^2*x^2)^(3/2)) - (e*(15*d - 19*e*x))/(5*d^5*Sqrt[d^2 - e^2*x^2]) - Sqrt[d^2 - e^
2*x^2]/(d^5*x) + (3*e*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/d^5

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Rubi in Sympy [A]  time = 44.2998, size = 124, normalized size = 0.85 \[ - \frac{e \sqrt{d^{2} - e^{2} x^{2}}}{5 d^{3} \left (d + e x\right )^{3}} - \frac{4 e \sqrt{d^{2} - e^{2} x^{2}}}{5 d^{4} \left (d + e x\right )^{2}} + \frac{3 e \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{d^{5}} - \frac{19 e \sqrt{d^{2} - e^{2} x^{2}}}{5 d^{5} \left (d + e x\right )} - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{d^{5} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-e*sqrt(d**2 - e**2*x**2)/(5*d**3*(d + e*x)**3) - 4*e*sqrt(d**2 - e**2*x**2)/(5*
d**4*(d + e*x)**2) + 3*e*atanh(sqrt(d**2 - e**2*x**2)/d)/d**5 - 19*e*sqrt(d**2 -
e**2*x**2)/(5*d**5*(d + e*x)) - sqrt(d**2 - e**2*x**2)/(d**5*x)

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Mathematica [A]  time = 0.216613, size = 92, normalized size = 0.63 \[ -\frac{-15 e \log \left (\sqrt{d^2-e^2 x^2}+d\right )+\frac{\sqrt{d^2-e^2 x^2} \left (5 d^3+39 d^2 e x+57 d e^2 x^2+24 e^3 x^3\right )}{x (d+e x)^3}+15 e \log (x)}{5 d^5} \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[d^2 - e^2*x^2]*(5*d^3 + 39*d^2*e*x + 57*d*e^2*x^2 + 24*e^3*x^3))/(x*(d +
e*x)^3) + 15*e*Log[x] - 15*e*Log[d + Sqrt[d^2 - e^2*x^2]])/(5*d^5)

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Maple [A]  time = 0.018, size = 199, normalized size = 1.4 \[ -{\frac{1}{{d}^{5}x}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{1}{5\,{d}^{3}{e}^{2}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) } \left ( x+{\frac{d}{e}} \right ) ^{-3}}-{\frac{4}{5\,{d}^{4}e}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) } \left ( x+{\frac{d}{e}} \right ) ^{-2}}-{\frac{19}{5\,{d}^{5}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) } \left ( x+{\frac{d}{e}} \right ) ^{-1}}+3\,{\frac{e}{{d}^{4}\sqrt{{d}^{2}}}\ln \left ({\frac{2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}{x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-e^2*x^2+d^2)^(1/2)/d^5/x-1/5/d^3/e^2/(x+d/e)^3*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))
^(1/2)-4/5/d^4/e/(x+d/e)^2*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)-19/5/d^5/(x+d/e)
*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)+3*e/d^4/(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. \[ \int \frac{1}{\sqrt{-e^{2} x^{2} + d^{2}}{\left (e x + d\right )}^{3} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.292916, size = 620, normalized size = 4.25 \[ \frac{153 \, d e^{6} x^{6} + 330 \, d^{2} e^{5} x^{5} - 70 \, d^{3} e^{4} x^{4} - 525 \, d^{4} e^{3} x^{3} - 220 \, d^{5} e^{2} x^{2} + 100 \, d^{6} e x + 40 \, d^{7} - 15 \,{\left (e^{7} x^{7} - d e^{6} x^{6} - 13 \, d^{2} e^{5} x^{5} - 15 \, d^{3} e^{4} x^{4} + 8 \, d^{4} e^{3} x^{3} + 20 \, d^{5} e^{2} x^{2} + 8 \, d^{6} e x +{\left (e^{6} x^{6} + 6 \, d e^{5} x^{5} + 5 \, d^{2} e^{4} x^{4} - 12 \, d^{3} e^{3} x^{3} - 20 \, d^{4} e^{2} x^{2} - 8 \, d^{5} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (48 \, e^{6} x^{6} + 105 \, d e^{5} x^{5} - 165 \, d^{2} e^{4} x^{4} - 475 \, d^{3} e^{3} x^{3} - 200 \, d^{4} e^{2} x^{2} + 100 \, d^{5} e x + 40 \, d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{5 \,{\left (d^{5} e^{6} x^{7} - d^{6} e^{5} x^{6} - 13 \, d^{7} e^{4} x^{5} - 15 \, d^{8} e^{3} x^{4} + 8 \, d^{9} e^{2} x^{3} + 20 \, d^{10} e x^{2} + 8 \, d^{11} x +{\left (d^{5} e^{5} x^{6} + 6 \, d^{6} e^{4} x^{5} + 5 \, d^{7} e^{3} x^{4} - 12 \, d^{8} e^{2} x^{3} - 20 \, d^{9} e x^{2} - 8 \, d^{10} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*(153*d*e^6*x^6 + 330*d^2*e^5*x^5 - 70*d^3*e^4*x^4 - 525*d^4*e^3*x^3 - 220*d^
5*e^2*x^2 + 100*d^6*e*x + 40*d^7 - 15*(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) - (48*e^6*x^6 + 105*d*e^5*x^5 - 165*d
^2*e^4*x^4 - 475*d^3*e^3*x^3 - 200*d^4*e^2*x^2 + 100*d^5*e*x + 40*d^6)*sqrt(-e^2
*x^2 + d^2))/(d^5*e^6*x^7 - d^6*e^5*x^6 - 13*d^7*e^4*x^5 - 15*d^8*e^3*x^4 + 8*d^
9*e^2*x^3 + 20*d^10*e*x^2 + 8*d^11*x + (d^5*e^5*x^6 + 6*d^6*e^4*x^5 + 5*d^7*e^3*
x^4 - 12*d^8*e^2*x^3 - 20*d^9*e*x^2 - 8*d^10*x)*sqrt(-e^2*x^2 + d^2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.30971, size = 1, normalized size = 0.01 \[ +\infty \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

+Infinity