3.169 \(\int \frac{\left (d^2-e^2 x^2\right )^{5/2}}{x^7 (d+e x)^2} \, dx\)

Optimal. Leaf size=169 \[ -\frac{\left (d^2-e^2 x^2\right )^{3/2}}{6 x^6}+\frac{2 e \left (d^2-e^2 x^2\right )^{3/2}}{5 d x^5}-\frac{3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{8 d^2 x^4}-\frac{3 e^4 \sqrt{d^2-e^2 x^2}}{16 d^2 x^2}+\frac{3 e^6 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{16 d^3}+\frac{4 e^3 \left (d^2-e^2 x^2\right )^{3/2}}{15 d^3 x^3} \]

[Out]

(-3*e^4*Sqrt[d^2 - e^2*x^2])/(16*d^2*x^2) - (d^2 - e^2*x^2)^(3/2)/(6*x^6) + (2*e
*(d^2 - e^2*x^2)^(3/2))/(5*d*x^5) - (3*e^2*(d^2 - e^2*x^2)^(3/2))/(8*d^2*x^4) +
(4*e^3*(d^2 - e^2*x^2)^(3/2))/(15*d^3*x^3) + (3*e^6*ArcTanh[Sqrt[d^2 - e^2*x^2]/
d])/(16*d^3)

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Rubi [A]  time = 0.499429, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296 \[ -\frac{\left (d^2-e^2 x^2\right )^{3/2}}{6 x^6}+\frac{2 e \left (d^2-e^2 x^2\right )^{3/2}}{5 d x^5}-\frac{3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}{8 d^2 x^4}-\frac{3 e^4 \sqrt{d^2-e^2 x^2}}{16 d^2 x^2}+\frac{3 e^6 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{16 d^3}+\frac{4 e^3 \left (d^2-e^2 x^2\right )^{3/2}}{15 d^3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(d^2 - e^2*x^2)^(5/2)/(x^7*(d + e*x)^2),x]

[Out]

(-3*e^4*Sqrt[d^2 - e^2*x^2])/(16*d^2*x^2) - (d^2 - e^2*x^2)^(3/2)/(6*x^6) + (2*e
*(d^2 - e^2*x^2)^(3/2))/(5*d*x^5) - (3*e^2*(d^2 - e^2*x^2)^(3/2))/(8*d^2*x^4) +
(4*e^3*(d^2 - e^2*x^2)^(3/2))/(15*d^3*x^3) + (3*e^6*ArcTanh[Sqrt[d^2 - e^2*x^2]/
d])/(16*d^3)

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Rubi in Sympy [A]  time = 53.5247, size = 150, normalized size = 0.89 \[ - \frac{d^{2} \sqrt{d^{2} - e^{2} x^{2}}}{6 x^{6}} - \frac{5 e^{2} \sqrt{d^{2} - e^{2} x^{2}}}{24 x^{4}} + \frac{2 e \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{5 d x^{5}} + \frac{3 e^{4} \sqrt{d^{2} - e^{2} x^{2}}}{16 d^{2} x^{2}} + \frac{3 e^{6} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{16 d^{3}} + \frac{4 e^{3} \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{15 d^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-e**2*x**2+d**2)**(5/2)/x**7/(e*x+d)**2,x)

[Out]

-d**2*sqrt(d**2 - e**2*x**2)/(6*x**6) - 5*e**2*sqrt(d**2 - e**2*x**2)/(24*x**4)
+ 2*e*(d**2 - e**2*x**2)**(3/2)/(5*d*x**5) + 3*e**4*sqrt(d**2 - e**2*x**2)/(16*d
**2*x**2) + 3*e**6*atanh(sqrt(d**2 - e**2*x**2)/d)/(16*d**3) + 4*e**3*(d**2 - e*
*2*x**2)**(3/2)/(15*d**3*x**3)

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Mathematica [A]  time = 0.167923, size = 117, normalized size = 0.69 \[ -\frac{-45 e^6 x^6 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+\sqrt{d^2-e^2 x^2} \left (40 d^5-96 d^4 e x+50 d^3 e^2 x^2+32 d^2 e^3 x^3-45 d e^4 x^4+64 e^5 x^5\right )+45 e^6 x^6 \log (x)}{240 d^3 x^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(d^2 - e^2*x^2)^(5/2)/(x^7*(d + e*x)^2),x]

[Out]

-(Sqrt[d^2 - e^2*x^2]*(40*d^5 - 96*d^4*e*x + 50*d^3*e^2*x^2 + 32*d^2*e^3*x^3 - 4
5*d*e^4*x^4 + 64*e^5*x^5) + 45*e^6*x^6*Log[x] - 45*e^6*x^6*Log[d + Sqrt[d^2 - e^
2*x^2]])/(240*d^3*x^6)

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Maple [B]  time = 0.025, size = 566, normalized size = 3.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-e^2*x^2+d^2)^(5/2)/x^7/(e*x+d)^2,x)

[Out]

2/5/d^5*e/x^5*(-e^2*x^2+d^2)^(7/2)+16/15/d^7*e^3/x^3*(-e^2*x^2+d^2)^(7/2)-1/3/d^
8*e^4/(x+d/e)^2*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(7/2)+26/15/d^9*e^5/x*(-e^2*x^2+d
^2)^(7/2)+26/15/d^9*e^7*x*(-e^2*x^2+d^2)^(5/2)+13/6/d^7*e^7*x*(-e^2*x^2+d^2)^(3/
2)+13/4/d^5*e^7*x*(-e^2*x^2+d^2)^(1/2)+13/4/d^3*e^7/(e^2)^(1/2)*arctan((e^2)^(1/
2)*x/(-e^2*x^2+d^2)^(1/2))-13/6/d^7*e^7*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)*x-1
3/4/d^5*e^7*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)*x-13/4/d^3*e^7/(e^2)^(1/2)*arct
an((e^2)^(1/2)*x/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2))-17/24/d^6*e^2/x^4*(-e^2*x
^2+d^2)^(7/2)-23/16/d^8*e^4/x^2*(-e^2*x^2+d^2)^(7/2)+3/16/d^2*e^6/(d^2)^(1/2)*ln
((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)-26/15/d^8*e^6*(-(x+d/e)^2*e^2+2*d
*e*(x+d/e))^(5/2)-1/6/d^4/x^6*(-e^2*x^2+d^2)^(7/2)-3/80/d^8*e^6*(-e^2*x^2+d^2)^(
5/2)-1/16/d^6*e^6*(-e^2*x^2+d^2)^(3/2)-3/16/d^4*e^6*(-e^2*x^2+d^2)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(5/2)/((e*x + d)^2*x^7),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.360245, size = 624, normalized size = 3.69 \[ \frac{384 \, d e^{11} x^{11} - 270 \, d^{2} e^{10} x^{10} - 2240 \, d^{3} e^{9} x^{9} + 2010 \, d^{4} e^{8} x^{8} + 2304 \, d^{5} e^{7} x^{7} - 4540 \, d^{6} e^{6} x^{6} + 3648 \, d^{7} e^{5} x^{5} + 3120 \, d^{8} e^{4} x^{4} - 7168 \, d^{9} e^{3} x^{3} + 960 \, d^{10} e^{2} x^{2} + 3072 \, d^{11} e x - 1280 \, d^{12} - 45 \,{\left (e^{12} x^{12} - 18 \, d^{2} e^{10} x^{10} + 48 \, d^{4} e^{8} x^{8} - 32 \, d^{6} e^{6} x^{6} + 2 \,{\left (3 \, d e^{10} x^{10} - 16 \, d^{3} e^{8} x^{8} + 16 \, d^{5} e^{6} x^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (64 \, e^{11} x^{11} - 45 \, d e^{10} x^{10} - 1120 \, d^{2} e^{9} x^{9} + 860 \, d^{3} e^{8} x^{8} + 2400 \, d^{4} e^{7} x^{7} - 3020 \, d^{5} e^{6} x^{6} + 1216 \, d^{6} e^{5} x^{5} + 3120 \, d^{7} e^{4} x^{4} - 5632 \, d^{8} e^{3} x^{3} + 320 \, d^{9} e^{2} x^{2} + 3072 \, d^{10} e x - 1280 \, d^{11}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{240 \,{\left (d^{3} e^{6} x^{12} - 18 \, d^{5} e^{4} x^{10} + 48 \, d^{7} e^{2} x^{8} - 32 \, d^{9} x^{6} + 2 \,{\left (3 \, d^{4} e^{4} x^{10} - 16 \, d^{6} e^{2} x^{8} + 16 \, d^{8} x^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(5/2)/((e*x + d)^2*x^7),x, algorithm="fricas")

[Out]

1/240*(384*d*e^11*x^11 - 270*d^2*e^10*x^10 - 2240*d^3*e^9*x^9 + 2010*d^4*e^8*x^8
+ 2304*d^5*e^7*x^7 - 4540*d^6*e^6*x^6 + 3648*d^7*e^5*x^5 + 3120*d^8*e^4*x^4 - 7
168*d^9*e^3*x^3 + 960*d^10*e^2*x^2 + 3072*d^11*e*x - 1280*d^12 - 45*(e^12*x^12 -
18*d^2*e^10*x^10 + 48*d^4*e^8*x^8 - 32*d^6*e^6*x^6 + 2*(3*d*e^10*x^10 - 16*d^3*
e^8*x^8 + 16*d^5*e^6*x^6)*sqrt(-e^2*x^2 + d^2))*log(-(d - sqrt(-e^2*x^2 + d^2))/
x) - (64*e^11*x^11 - 45*d*e^10*x^10 - 1120*d^2*e^9*x^9 + 860*d^3*e^8*x^8 + 2400*
d^4*e^7*x^7 - 3020*d^5*e^6*x^6 + 1216*d^6*e^5*x^5 + 3120*d^7*e^4*x^4 - 5632*d^8*
e^3*x^3 + 320*d^9*e^2*x^2 + 3072*d^10*e*x - 1280*d^11)*sqrt(-e^2*x^2 + d^2))/(d^
3*e^6*x^12 - 18*d^5*e^4*x^10 + 48*d^7*e^2*x^8 - 32*d^9*x^6 + 2*(3*d^4*e^4*x^10 -
16*d^6*e^2*x^8 + 16*d^8*x^6)*sqrt(-e^2*x^2 + d^2))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e**2*x**2+d**2)**(5/2)/x**7/(e*x+d)**2,x)

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(5/2)/((e*x + d)^2*x^7),x, algorithm="giac")

[Out]

Exception raised: TypeError