3.209 \(\int \frac{\left (d^2-e^2 x^2\right )^{5/2}}{x^6 (d+e x)^4} \, dx\)

Optimal. Leaf size=196 \[ -\frac{\sqrt{d^2-e^2 x^2}}{5 x^5}+\frac{e \sqrt{d^2-e^2 x^2}}{d x^4}-\frac{13 e^2 \sqrt{d^2-e^2 x^2}}{5 d^2 x^3}-\frac{8 e^5 (d-e x)}{d^4 \sqrt{d^2-e^2 x^2}}+\frac{27 e^5 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^4}-\frac{66 e^4 \sqrt{d^2-e^2 x^2}}{5 d^4 x}+\frac{11 e^3 \sqrt{d^2-e^2 x^2}}{2 d^3 x^2} \]

[Out]

(-8*e^5*(d - e*x))/(d^4*Sqrt[d^2 - e^2*x^2]) - Sqrt[d^2 - e^2*x^2]/(5*x^5) + (e*
Sqrt[d^2 - e^2*x^2])/(d*x^4) - (13*e^2*Sqrt[d^2 - e^2*x^2])/(5*d^2*x^3) + (11*e^
3*Sqrt[d^2 - e^2*x^2])/(2*d^3*x^2) - (66*e^4*Sqrt[d^2 - e^2*x^2])/(5*d^4*x) + (2
7*e^5*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(2*d^4)

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Rubi [A]  time = 0.796705, antiderivative size = 196, 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 \[ -\frac{\sqrt{d^2-e^2 x^2}}{5 x^5}+\frac{e \sqrt{d^2-e^2 x^2}}{d x^4}-\frac{13 e^2 \sqrt{d^2-e^2 x^2}}{5 d^2 x^3}-\frac{8 e^5 (d-e x)}{d^4 \sqrt{d^2-e^2 x^2}}+\frac{27 e^5 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{2 d^4}-\frac{66 e^4 \sqrt{d^2-e^2 x^2}}{5 d^4 x}+\frac{11 e^3 \sqrt{d^2-e^2 x^2}}{2 d^3 x^2} \]

Antiderivative was successfully verified.

[In]  Int[(d^2 - e^2*x^2)^(5/2)/(x^6*(d + e*x)^4),x]

[Out]

(-8*e^5*(d - e*x))/(d^4*Sqrt[d^2 - e^2*x^2]) - Sqrt[d^2 - e^2*x^2]/(5*x^5) + (e*
Sqrt[d^2 - e^2*x^2])/(d*x^4) - (13*e^2*Sqrt[d^2 - e^2*x^2])/(5*d^2*x^3) + (11*e^
3*Sqrt[d^2 - e^2*x^2])/(2*d^3*x^2) - (66*e^4*Sqrt[d^2 - e^2*x^2])/(5*d^4*x) + (2
7*e^5*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(2*d^4)

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Rubi in Sympy [A]  time = 67.506, size = 172, normalized size = 0.88 \[ - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{5 x^{5}} + \frac{e \sqrt{d^{2} - e^{2} x^{2}}}{d x^{4}} - \frac{13 e^{2} \sqrt{d^{2} - e^{2} x^{2}}}{5 d^{2} x^{3}} + \frac{11 e^{3} \sqrt{d^{2} - e^{2} x^{2}}}{2 d^{3} x^{2}} + \frac{27 e^{5} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{2 d^{4}} - \frac{8 e^{5} \sqrt{d^{2} - e^{2} x^{2}}}{d^{4} \left (d + e x\right )} - \frac{66 e^{4} \sqrt{d^{2} - e^{2} x^{2}}}{5 d^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-e**2*x**2+d**2)**(5/2)/x**6/(e*x+d)**4,x)

[Out]

-sqrt(d**2 - e**2*x**2)/(5*x**5) + e*sqrt(d**2 - e**2*x**2)/(d*x**4) - 13*e**2*s
qrt(d**2 - e**2*x**2)/(5*d**2*x**3) + 11*e**3*sqrt(d**2 - e**2*x**2)/(2*d**3*x**
2) + 27*e**5*atanh(sqrt(d**2 - e**2*x**2)/d)/(2*d**4) - 8*e**5*sqrt(d**2 - e**2*
x**2)/(d**4*(d + e*x)) - 66*e**4*sqrt(d**2 - e**2*x**2)/(5*d**4*x)

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Mathematica [A]  time = 0.298614, size = 118, normalized size = 0.6 \[ -\frac{-135 e^5 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+\frac{\sqrt{d^2-e^2 x^2} \left (2 d^5-8 d^4 e x+16 d^3 e^2 x^2-29 d^2 e^3 x^3+77 d e^4 x^4+212 e^5 x^5\right )}{x^5 (d+e x)}+135 e^5 \log (x)}{10 d^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(d^2 - e^2*x^2)^(5/2)/(x^6*(d + e*x)^4),x]

[Out]

-((Sqrt[d^2 - e^2*x^2]*(2*d^5 - 8*d^4*e*x + 16*d^3*e^2*x^2 - 29*d^2*e^3*x^3 + 77
*d*e^4*x^4 + 212*e^5*x^5))/(x^5*(d + e*x)) + 135*e^5*Log[x] - 135*e^5*Log[d + Sq
rt[d^2 - e^2*x^2]])/(10*d^4)

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Maple [B]  time = 0.024, size = 628, normalized size = 3.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-e^2*x^2+d^2)^(5/2)/x^6/(e*x+d)^4,x)

[Out]

1/d^7*e/x^4*(-e^2*x^2+d^2)^(7/2)+17/2/d^9*e^3/x^2*(-e^2*x^2+d^2)^(7/2)-1/d^7*e/(
x+d/e)^4*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(7/2)+3/d^8*e^2/(x+d/e)^3*(-(x+d/e)^2*e^
2+2*d*e*(x+d/e))^(7/2)+11/d^9*e^3/(x+d/e)^2*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(7/2)
+111/4/d^8*e^6*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)*x+333/8/d^6*e^6*(-(x+d/e)^2*
e^2+2*d*e*(x+d/e))^(1/2)*x+333/8/d^4*e^6/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-(x+d
/e)^2*e^2+2*d*e*(x+d/e))^(1/2))-16/5/d^8*e^2/x^3*(-e^2*x^2+d^2)^(7/2)-111/5/d^10
*e^4/x*(-e^2*x^2+d^2)^(7/2)-111/5/d^10*e^6*x*(-e^2*x^2+d^2)^(5/2)-111/4/d^8*e^6*
x*(-e^2*x^2+d^2)^(3/2)-333/8/d^6*e^6*x*(-e^2*x^2+d^2)^(1/2)-333/8/d^4*e^6/(e^2)^
(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))+27/2/d^3*e^5/(d^2)^(1/2)*ln((2*
d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)-27/10/d^9*e^5*(-e^2*x^2+d^2)^(5/2)-9/
2/d^7*e^5*(-e^2*x^2+d^2)^(3/2)-27/2/d^5*e^5*(-e^2*x^2+d^2)^(1/2)+111/5/d^9*e^5*(
-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(5/2)-1/5/d^6/x^5*(-e^2*x^2+d^2)^(7/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}}{{\left (e x + d\right )}^{4} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(5/2)/((e*x + d)^4*x^6),x, algorithm="maxima")

[Out]

integrate((-e^2*x^2 + d^2)^(5/2)/((e*x + d)^4*x^6), x)

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Fricas [A]  time = 0.285192, size = 765, normalized size = 3.9 \[ \frac{132 \, e^{11} x^{11} + 1537 \, d e^{10} x^{10} - 2020 \, d^{2} e^{9} x^{9} - 7355 \, d^{3} e^{8} x^{8} + 5390 \, d^{4} e^{7} x^{7} + 8622 \, d^{5} e^{6} x^{6} - 4550 \, d^{6} e^{5} x^{5} - 2036 \, d^{7} e^{4} x^{4} + 760 \, d^{8} e^{3} x^{3} - 544 \, d^{9} e^{2} x^{2} + 288 \, d^{10} e x - 64 \, d^{11} - 135 \,{\left (e^{11} x^{11} - 5 \, d e^{10} x^{10} - 18 \, d^{2} e^{9} x^{9} + 20 \, d^{3} e^{8} x^{8} + 48 \, d^{4} e^{7} x^{7} - 16 \, d^{5} e^{6} x^{6} - 32 \, d^{6} e^{5} x^{5} +{\left (e^{10} x^{10} + 6 \, d e^{9} x^{9} - 12 \, d^{2} e^{8} x^{8} - 32 \, d^{3} e^{7} x^{7} + 16 \, d^{4} e^{6} x^{6} + 32 \, d^{5} e^{5} x^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (292 \, e^{10} x^{10} - 715 \, d e^{9} x^{9} - 3995 \, d^{2} e^{8} x^{8} + 3490 \, d^{3} e^{7} x^{7} + 7380 \, d^{4} e^{6} x^{6} - 4062 \, d^{5} e^{5} x^{5} - 2332 \, d^{6} e^{4} x^{4} + 904 \, d^{7} e^{3} x^{3} - 576 \, d^{8} e^{2} x^{2} + 288 \, d^{9} e x - 64 \, d^{10}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{10 \,{\left (d^{4} e^{6} x^{11} - 5 \, d^{5} e^{5} x^{10} - 18 \, d^{6} e^{4} x^{9} + 20 \, d^{7} e^{3} x^{8} + 48 \, d^{8} e^{2} x^{7} - 16 \, d^{9} e x^{6} - 32 \, d^{10} x^{5} +{\left (d^{4} e^{5} x^{10} + 6 \, d^{5} e^{4} x^{9} - 12 \, d^{6} e^{3} x^{8} - 32 \, d^{7} e^{2} x^{7} + 16 \, d^{8} e x^{6} + 32 \, d^{9} x^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(5/2)/((e*x + d)^4*x^6),x, algorithm="fricas")

[Out]

1/10*(132*e^11*x^11 + 1537*d*e^10*x^10 - 2020*d^2*e^9*x^9 - 7355*d^3*e^8*x^8 + 5
390*d^4*e^7*x^7 + 8622*d^5*e^6*x^6 - 4550*d^6*e^5*x^5 - 2036*d^7*e^4*x^4 + 760*d
^8*e^3*x^3 - 544*d^9*e^2*x^2 + 288*d^10*e*x - 64*d^11 - 135*(e^11*x^11 - 5*d*e^1
0*x^10 - 18*d^2*e^9*x^9 + 20*d^3*e^8*x^8 + 48*d^4*e^7*x^7 - 16*d^5*e^6*x^6 - 32*
d^6*e^5*x^5 + (e^10*x^10 + 6*d*e^9*x^9 - 12*d^2*e^8*x^8 - 32*d^3*e^7*x^7 + 16*d^
4*e^6*x^6 + 32*d^5*e^5*x^5)*sqrt(-e^2*x^2 + d^2))*log(-(d - sqrt(-e^2*x^2 + d^2)
)/x) - (292*e^10*x^10 - 715*d*e^9*x^9 - 3995*d^2*e^8*x^8 + 3490*d^3*e^7*x^7 + 73
80*d^4*e^6*x^6 - 4062*d^5*e^5*x^5 - 2332*d^6*e^4*x^4 + 904*d^7*e^3*x^3 - 576*d^8
*e^2*x^2 + 288*d^9*e*x - 64*d^10)*sqrt(-e^2*x^2 + d^2))/(d^4*e^6*x^11 - 5*d^5*e^
5*x^10 - 18*d^6*e^4*x^9 + 20*d^7*e^3*x^8 + 48*d^8*e^2*x^7 - 16*d^9*e*x^6 - 32*d^
10*x^5 + (d^4*e^5*x^10 + 6*d^5*e^4*x^9 - 12*d^6*e^3*x^8 - 32*d^7*e^2*x^7 + 16*d^
8*e*x^6 + 32*d^9*x^5)*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**6/(e*x+d)**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.34795, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(5/2)/((e*x + d)^4*x^6),x, algorithm="giac")

[Out]

Done