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

Optimal. Leaf size=170 \[ -\frac{31 e^2 \sqrt{d^2-e^2 x^2}}{8 d^2 x^2}-\frac{\sqrt{d^2-e^2 x^2}}{4 x^4}+\frac{4 e \sqrt{d^2-e^2 x^2}}{3 d x^3}+\frac{8 e^4 (d-e x)}{d^3 \sqrt{d^2-e^2 x^2}}-\frac{95 e^4 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{8 d^3}+\frac{32 e^3 \sqrt{d^2-e^2 x^2}}{3 d^3 x} \]

[Out]

(8*e^4*(d - e*x))/(d^3*Sqrt[d^2 - e^2*x^2]) - Sqrt[d^2 - e^2*x^2]/(4*x^4) + (4*e
*Sqrt[d^2 - e^2*x^2])/(3*d*x^3) - (31*e^2*Sqrt[d^2 - e^2*x^2])/(8*d^2*x^2) + (32
*e^3*Sqrt[d^2 - e^2*x^2])/(3*d^3*x) - (95*e^4*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(8
*d^3)

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Rubi [A]  time = 0.648433, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259 \[ -\frac{31 e^2 \sqrt{d^2-e^2 x^2}}{8 d^2 x^2}-\frac{\sqrt{d^2-e^2 x^2}}{4 x^4}+\frac{4 e \sqrt{d^2-e^2 x^2}}{3 d x^3}+\frac{8 e^4 (d-e x)}{d^3 \sqrt{d^2-e^2 x^2}}-\frac{95 e^4 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{8 d^3}+\frac{32 e^3 \sqrt{d^2-e^2 x^2}}{3 d^3 x} \]

Antiderivative was successfully verified.

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

[Out]

(8*e^4*(d - e*x))/(d^3*Sqrt[d^2 - e^2*x^2]) - Sqrt[d^2 - e^2*x^2]/(4*x^4) + (4*e
*Sqrt[d^2 - e^2*x^2])/(3*d*x^3) - (31*e^2*Sqrt[d^2 - e^2*x^2])/(8*d^2*x^2) + (32
*e^3*Sqrt[d^2 - e^2*x^2])/(3*d^3*x) - (95*e^4*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/(8
*d^3)

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Rubi in Sympy [A]  time = 54.6238, size = 148, normalized size = 0.87 \[ - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{4 x^{4}} + \frac{4 e \sqrt{d^{2} - e^{2} x^{2}}}{3 d x^{3}} - \frac{31 e^{2} \sqrt{d^{2} - e^{2} x^{2}}}{8 d^{2} x^{2}} - \frac{95 e^{4} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{8 d^{3}} + \frac{8 e^{4} \sqrt{d^{2} - e^{2} x^{2}}}{d^{3} \left (d + e x\right )} + \frac{32 e^{3} \sqrt{d^{2} - e^{2} x^{2}}}{3 d^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(d**2 - e**2*x**2)/(4*x**4) + 4*e*sqrt(d**2 - e**2*x**2)/(3*d*x**3) - 31*e*
*2*sqrt(d**2 - e**2*x**2)/(8*d**2*x**2) - 95*e**4*atanh(sqrt(d**2 - e**2*x**2)/d
)/(8*d**3) + 8*e**4*sqrt(d**2 - e**2*x**2)/(d**3*(d + e*x)) + 32*e**3*sqrt(d**2
- e**2*x**2)/(3*d**3*x)

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Mathematica [A]  time = 0.193004, size = 107, normalized size = 0.63 \[ \frac{-285 e^4 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+\frac{\sqrt{d^2-e^2 x^2} \left (-6 d^4+26 d^3 e x-61 d^2 e^2 x^2+163 d e^3 x^3+448 e^4 x^4\right )}{x^4 (d+e x)}+285 e^4 \log (x)}{24 d^3} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[d^2 - e^2*x^2]*(-6*d^4 + 26*d^3*e*x - 61*d^2*e^2*x^2 + 163*d*e^3*x^3 + 44
8*e^4*x^4))/(x^4*(d + e*x)) + 285*e^4*Log[x] - 285*e^4*Log[d + Sqrt[d^2 - e^2*x^
2]])/(24*d^3)

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Maple [B]  time = 0.023, size = 600, normalized size = 3.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/3/d^7*e/x^3*(-e^2*x^2+d^2)^(7/2)-2/d^7*e/(x+d/e)^3*(-(x+d/e)^2*e^2+2*d*e*(x+d/
e))^(7/2)-23/3/d^8*e^2/(x+d/e)^2*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(7/2)+44/3/d^9*e
^3/x*(-e^2*x^2+d^2)^(7/2)+44/3/d^9*e^5*x*(-e^2*x^2+d^2)^(5/2)+55/3/d^7*e^5*x*(-e
^2*x^2+d^2)^(3/2)+55/2/d^5*e^5*x*(-e^2*x^2+d^2)^(1/2)+55/2/d^3*e^5/(e^2)^(1/2)*a
rctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-55/3/d^7*e^5*(-(x+d/e)^2*e^2+2*d*e*(x+
d/e))^(3/2)*x-55/2/d^5*e^5*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)*x-55/2/d^3*e^5/(
e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2))-37/8/d^8*e
^2/x^2*(-e^2*x^2+d^2)^(7/2)-95/8/d^2*e^4/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e
^2*x^2+d^2)^(1/2))/x)+1/d^6/(x+d/e)^4*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(7/2)-44/3/
d^8*e^4*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(5/2)-1/4/d^6/x^4*(-e^2*x^2+d^2)^(7/2)+19
/8/d^8*e^4*(-e^2*x^2+d^2)^(5/2)+95/24/d^6*e^4*(-e^2*x^2+d^2)^(3/2)+95/8/d^4*e^4*
(-e^2*x^2+d^2)^(1/2)

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

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^5),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.285901, size = 648, normalized size = 3.81 \[ \frac{640 \, e^{9} x^{9} - 1117 \, d e^{8} x^{8} - 5996 \, d^{2} e^{7} x^{7} + 4147 \, d^{3} e^{6} x^{6} + 8732 \, d^{4} e^{5} x^{5} - 4190 \, d^{5} e^{4} x^{4} - 2528 \, d^{6} e^{3} x^{3} + 1064 \, d^{7} e^{2} x^{2} - 464 \, d^{8} e x + 96 \, d^{9} + 285 \,{\left (e^{9} x^{9} + 5 \, d e^{8} x^{8} - 8 \, d^{2} e^{7} x^{7} - 20 \, d^{3} e^{6} x^{6} + 8 \, d^{4} e^{5} x^{5} + 16 \, d^{5} e^{4} x^{4} -{\left (e^{8} x^{8} - 4 \, d e^{7} x^{7} - 12 \, d^{2} e^{6} x^{6} + 8 \, d^{3} e^{5} x^{5} + 16 \, d^{4} e^{4} x^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) +{\left (256 \, e^{8} x^{8} + 2723 \, d e^{7} x^{7} - 2481 \, d^{2} e^{6} x^{6} - 7294 \, d^{3} e^{5} x^{5} + 3622 \, d^{4} e^{4} x^{4} + 2760 \, d^{5} e^{3} x^{3} - 1112 \, d^{6} e^{2} x^{2} + 464 \, d^{7} e x - 96 \, d^{8}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{24 \,{\left (d^{3} e^{5} x^{9} + 5 \, d^{4} e^{4} x^{8} - 8 \, d^{5} e^{3} x^{7} - 20 \, d^{6} e^{2} x^{6} + 8 \, d^{7} e x^{5} + 16 \, d^{8} x^{4} -{\left (d^{3} e^{4} x^{8} - 4 \, d^{4} e^{3} x^{7} - 12 \, d^{5} e^{2} x^{6} + 8 \, d^{6} e x^{5} + 16 \, d^{7} x^{4}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

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^5),x, algorithm="fricas")

[Out]

1/24*(640*e^9*x^9 - 1117*d*e^8*x^8 - 5996*d^2*e^7*x^7 + 4147*d^3*e^6*x^6 + 8732*
d^4*e^5*x^5 - 4190*d^5*e^4*x^4 - 2528*d^6*e^3*x^3 + 1064*d^7*e^2*x^2 - 464*d^8*e
*x + 96*d^9 + 285*(e^9*x^9 + 5*d*e^8*x^8 - 8*d^2*e^7*x^7 - 20*d^3*e^6*x^6 + 8*d^
4*e^5*x^5 + 16*d^5*e^4*x^4 - (e^8*x^8 - 4*d*e^7*x^7 - 12*d^2*e^6*x^6 + 8*d^3*e^5
*x^5 + 16*d^4*e^4*x^4)*sqrt(-e^2*x^2 + d^2))*log(-(d - sqrt(-e^2*x^2 + d^2))/x)
+ (256*e^8*x^8 + 2723*d*e^7*x^7 - 2481*d^2*e^6*x^6 - 7294*d^3*e^5*x^5 + 3622*d^4
*e^4*x^4 + 2760*d^5*e^3*x^3 - 1112*d^6*e^2*x^2 + 464*d^7*e*x - 96*d^8)*sqrt(-e^2
*x^2 + d^2))/(d^3*e^5*x^9 + 5*d^4*e^4*x^8 - 8*d^5*e^3*x^7 - 20*d^6*e^2*x^6 + 8*d
^7*e*x^5 + 16*d^8*x^4 - (d^3*e^4*x^8 - 4*d^4*e^3*x^7 - 12*d^5*e^2*x^6 + 8*d^6*e*
x^5 + 16*d^7*x^4)*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**5/(e*x+d)**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.353409, 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^5),x, algorithm="giac")

[Out]

Done