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

Optimal. Leaf size=210 \[ -\frac{8 e^3 (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{29 e^2 \sqrt{d^2-e^2 x^2}}{3 d^6 x}-\frac{e^3 (80 d-93 e x)}{5 d^6 \sqrt{d^2-e^2 x^2}}+\frac{18 e^3 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^6}+\frac{2 e \sqrt{d^2-e^2 x^2}}{d^5 x^2}-\frac{\sqrt{d^2-e^2 x^2}}{3 d^4 x^3}-\frac{4 e^3 (5 d-6 e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{3/2}} \]

[Out]

(-8*e^3*(d - e*x))/(5*d^2*(d^2 - e^2*x^2)^(5/2)) - (4*e^3*(5*d - 6*e*x))/(5*d^4*
(d^2 - e^2*x^2)^(3/2)) - (e^3*(80*d - 93*e*x))/(5*d^6*Sqrt[d^2 - e^2*x^2]) - Sqr
t[d^2 - e^2*x^2]/(3*d^4*x^3) + (2*e*Sqrt[d^2 - e^2*x^2])/(d^5*x^2) - (29*e^2*Sqr
t[d^2 - e^2*x^2])/(3*d^6*x) + (18*e^3*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/d^6

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Rubi [A]  time = 0.795451, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259 \[ -\frac{8 e^3 (d-e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac{29 e^2 \sqrt{d^2-e^2 x^2}}{3 d^6 x}-\frac{e^3 (80 d-93 e x)}{5 d^6 \sqrt{d^2-e^2 x^2}}+\frac{18 e^3 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^6}+\frac{2 e \sqrt{d^2-e^2 x^2}}{d^5 x^2}-\frac{\sqrt{d^2-e^2 x^2}}{3 d^4 x^3}-\frac{4 e^3 (5 d-6 e x)}{5 d^4 \left (d^2-e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-8*e^3*(d - e*x))/(5*d^2*(d^2 - e^2*x^2)^(5/2)) - (4*e^3*(5*d - 6*e*x))/(5*d^4*
(d^2 - e^2*x^2)^(3/2)) - (e^3*(80*d - 93*e*x))/(5*d^6*Sqrt[d^2 - e^2*x^2]) - Sqr
t[d^2 - e^2*x^2]/(3*d^4*x^3) + (2*e*Sqrt[d^2 - e^2*x^2])/(d^5*x^2) - (29*e^2*Sqr
t[d^2 - e^2*x^2])/(3*d^6*x) + (18*e^3*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/d^6

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Rubi in Sympy [A]  time = 82.1194, size = 180, normalized size = 0.86 \[ - \frac{e^{3} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{5 d^{5} \left (d + e x\right )^{4}} + \frac{2 e \sqrt{d^{2} - e^{2} x^{2}}}{d^{5} x^{2}} + \frac{18 e^{3} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{d^{6}} - \frac{20 e^{3} \sqrt{d^{2} - e^{2} x^{2}}}{d^{6} \left (d + e x\right )} - \frac{7 e^{3} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{5 d^{6} \left (d + e x\right )^{3}} - \frac{10 e^{2} \sqrt{d^{2} - e^{2} x^{2}}}{d^{6} x} - \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{3 d^{6} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-e**3*(d**2 - e**2*x**2)**(3/2)/(5*d**5*(d + e*x)**4) + 2*e*sqrt(d**2 - e**2*x**
2)/(d**5*x**2) + 18*e**3*atanh(sqrt(d**2 - e**2*x**2)/d)/d**6 - 20*e**3*sqrt(d**
2 - e**2*x**2)/(d**6*(d + e*x)) - 7*e**3*(d**2 - e**2*x**2)**(3/2)/(5*d**6*(d +
e*x)**3) - 10*e**2*sqrt(d**2 - e**2*x**2)/(d**6*x) - (d**2 - e**2*x**2)**(3/2)/(
3*d**6*x**3)

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Mathematica [A]  time = 0.24788, size = 118, normalized size = 0.56 \[ -\frac{-270 e^3 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+\frac{\sqrt{d^2-e^2 x^2} \left (5 d^5-15 d^4 e x+70 d^3 e^2 x^2+674 d^2 e^3 x^3+1002 d e^4 x^4+424 e^5 x^5\right )}{x^3 (d+e x)^3}+270 e^3 \log (x)}{15 d^6} \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[d^2 - e^2*x^2]*(5*d^5 - 15*d^4*e*x + 70*d^3*e^2*x^2 + 674*d^2*e^3*x^3 +
1002*d*e^4*x^4 + 424*e^5*x^5))/(x^3*(d + e*x)^3) + 270*e^3*Log[x] - 270*e^3*Log[
d + Sqrt[d^2 - e^2*x^2]])/(15*d^6)

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Maple [B]  time = 0.02, size = 412, normalized size = 2. \[ -{\frac{1}{3\,{d}^{6}{x}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{1}{5\,{d}^{5}e} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{d}{e}} \right ) ^{-4}}-{\frac{7}{5\,{d}^{6}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{d}{e}} \right ) ^{-3}}-18\,{\frac{{e}^{3}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}{{d}^{7}}}+18\,{\frac{{e}^{3}}{{d}^{5}\sqrt{{d}^{2}}}\ln \left ({\frac{2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}{x}} \right ) }+10\,{\frac{{e}^{3}}{{d}^{7}}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}+10\,{\frac{{e}^{4}}{{d}^{6}\sqrt{{e}^{2}}}\arctan \left ({\sqrt{{e}^{2}}x{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ) }-10\,{\frac{{e}^{2} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{3/2}}{{d}^{8}x}}-10\,{\frac{{e}^{4}x\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}{{d}^{8}}}-10\,{\frac{{e}^{4}}{{d}^{6}\sqrt{{e}^{2}}}\arctan \left ({\frac{\sqrt{{e}^{2}}x}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}} \right ) }-10\,{\frac{e}{{d}^{7}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{3/2} \left ( x+{\frac{d}{e}} \right ) ^{-2}}+2\,{\frac{e \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{3/2}}{{d}^{7}{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3/d^6/x^3*(-e^2*x^2+d^2)^(3/2)-1/5/d^5/e/(x+d/e)^4*(-(x+d/e)^2*e^2+2*d*e*(x+d
/e))^(3/2)-7/5/d^6/(x+d/e)^3*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)-18/d^7*e^3*(-e
^2*x^2+d^2)^(1/2)+18/d^5*e^3/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^
(1/2))/x)+10/d^7*e^3*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)+10/d^6*e^4/(e^2)^(1/2)
*arctan((e^2)^(1/2)*x/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2))-10/d^8*e^2/x*(-e^2*x
^2+d^2)^(3/2)-10/d^8*e^4*x*(-e^2*x^2+d^2)^(1/2)-10/d^6*e^4/(e^2)^(1/2)*arctan((e
^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-10/d^7*e/(x+d/e)^2*(-(x+d/e)^2*e^2+2*d*e*(x+d/
e))^(3/2)+2/d^7*e/x^2*(-e^2*x^2+d^2)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.310684, size = 882, normalized size = 4.2 \[ \frac{100 \, e^{11} x^{11} + 4094 \, d e^{10} x^{10} + 6800 \, d^{2} e^{9} x^{9} - 16272 \, d^{3} e^{8} x^{8} - 34165 \, d^{4} e^{7} x^{7} + 7670 \, d^{5} e^{6} x^{6} + 38895 \, d^{6} e^{5} x^{5} + 8210 \, d^{7} e^{4} x^{4} - 10900 \, d^{8} e^{3} x^{3} - 2240 \, d^{9} e^{2} x^{2} + 560 \, d^{10} e x - 160 \, d^{11} - 270 \,{\left (e^{11} x^{11} - 3 \, d e^{10} x^{10} - 27 \, d^{2} e^{9} x^{9} - 21 \, d^{3} e^{8} x^{8} + 70 \, d^{4} e^{7} x^{7} + 100 \, d^{5} e^{6} x^{6} - 16 \, d^{6} e^{5} x^{5} - 80 \, d^{7} e^{4} x^{4} - 32 \, d^{8} e^{3} x^{3} +{\left (e^{10} x^{10} + 8 \, d e^{9} x^{9} + d^{2} e^{8} x^{8} - 50 \, d^{3} e^{7} x^{7} - 60 \, d^{4} e^{6} x^{6} + 32 \, d^{5} e^{5} x^{5} + 80 \, d^{6} e^{4} x^{4} + 32 \, d^{7} e^{3} x^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (748 \, e^{10} x^{10} + 1050 \, d e^{9} x^{9} - 10102 \, d^{2} e^{8} x^{8} - 18630 \, d^{3} e^{7} x^{7} + 10885 \, d^{4} e^{6} x^{6} + 33655 \, d^{5} e^{5} x^{5} + 7030 \, d^{6} e^{4} x^{4} - 10620 \, d^{7} e^{3} x^{3} - 2320 \, d^{8} e^{2} x^{2} + 560 \, d^{9} e x - 160 \, d^{10}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{15 \,{\left (d^{6} e^{8} x^{11} - 3 \, d^{7} e^{7} x^{10} - 27 \, d^{8} e^{6} x^{9} - 21 \, d^{9} e^{5} x^{8} + 70 \, d^{10} e^{4} x^{7} + 100 \, d^{11} e^{3} x^{6} - 16 \, d^{12} e^{2} x^{5} - 80 \, d^{13} e x^{4} - 32 \, d^{14} x^{3} +{\left (d^{6} e^{7} x^{10} + 8 \, d^{7} e^{6} x^{9} + d^{8} e^{5} x^{8} - 50 \, d^{9} e^{4} x^{7} - 60 \, d^{10} e^{3} x^{6} + 32 \, d^{11} e^{2} x^{5} + 80 \, d^{12} e x^{4} + 32 \, d^{13} x^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(100*e^11*x^11 + 4094*d*e^10*x^10 + 6800*d^2*e^9*x^9 - 16272*d^3*e^8*x^8 -
34165*d^4*e^7*x^7 + 7670*d^5*e^6*x^6 + 38895*d^6*e^5*x^5 + 8210*d^7*e^4*x^4 - 10
900*d^8*e^3*x^3 - 2240*d^9*e^2*x^2 + 560*d^10*e*x - 160*d^11 - 270*(e^11*x^11 -
3*d*e^10*x^10 - 27*d^2*e^9*x^9 - 21*d^3*e^8*x^8 + 70*d^4*e^7*x^7 + 100*d^5*e^6*x
^6 - 16*d^6*e^5*x^5 - 80*d^7*e^4*x^4 - 32*d^8*e^3*x^3 + (e^10*x^10 + 8*d*e^9*x^9
+ d^2*e^8*x^8 - 50*d^3*e^7*x^7 - 60*d^4*e^6*x^6 + 32*d^5*e^5*x^5 + 80*d^6*e^4*x
^4 + 32*d^7*e^3*x^3)*sqrt(-e^2*x^2 + d^2))*log(-(d - sqrt(-e^2*x^2 + d^2))/x) -
(748*e^10*x^10 + 1050*d*e^9*x^9 - 10102*d^2*e^8*x^8 - 18630*d^3*e^7*x^7 + 10885*
d^4*e^6*x^6 + 33655*d^5*e^5*x^5 + 7030*d^6*e^4*x^4 - 10620*d^7*e^3*x^3 - 2320*d^
8*e^2*x^2 + 560*d^9*e*x - 160*d^10)*sqrt(-e^2*x^2 + d^2))/(d^6*e^8*x^11 - 3*d^7*
e^7*x^10 - 27*d^8*e^6*x^9 - 21*d^9*e^5*x^8 + 70*d^10*e^4*x^7 + 100*d^11*e^3*x^6
- 16*d^12*e^2*x^5 - 80*d^13*e*x^4 - 32*d^14*x^3 + (d^6*e^7*x^10 + 8*d^7*e^6*x^9
+ d^8*e^5*x^8 - 50*d^9*e^4*x^7 - 60*d^10*e^3*x^6 + 32*d^11*e^2*x^5 + 80*d^12*e*x
^4 + 32*d^13*x^3)*sqrt(-e^2*x^2 + d^2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.336527, size = 1, normalized size = 0. \[ +\infty \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

+Infinity