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

Optimal. Leaf size=137 \[ -\frac{23 e^2 \sqrt{d^2-e^2 x^2}}{3 d^2 x}+\frac{2 e \sqrt{d^2-e^2 x^2}}{d x^2}-\frac{\sqrt{d^2-e^2 x^2}}{3 x^3}-\frac{8 e^3 (d-e x)}{d^2 \sqrt{d^2-e^2 x^2}}+\frac{10 e^3 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^2} \]

[Out]

(-8*e^3*(d - e*x))/(d^2*Sqrt[d^2 - e^2*x^2]) - Sqrt[d^2 - e^2*x^2]/(3*x^3) + (2*
e*Sqrt[d^2 - e^2*x^2])/(d*x^2) - (23*e^2*Sqrt[d^2 - e^2*x^2])/(3*d^2*x) + (10*e^
3*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/d^2

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Rubi [A]  time = 0.526274, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259 \[ -\frac{23 e^2 \sqrt{d^2-e^2 x^2}}{3 d^2 x}+\frac{2 e \sqrt{d^2-e^2 x^2}}{d x^2}-\frac{\sqrt{d^2-e^2 x^2}}{3 x^3}-\frac{8 e^3 (d-e x)}{d^2 \sqrt{d^2-e^2 x^2}}+\frac{10 e^3 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right )}{d^2} \]

Antiderivative was successfully verified.

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

[Out]

(-8*e^3*(d - e*x))/(d^2*Sqrt[d^2 - e^2*x^2]) - Sqrt[d^2 - e^2*x^2]/(3*x^3) + (2*
e*Sqrt[d^2 - e^2*x^2])/(d*x^2) - (23*e^2*Sqrt[d^2 - e^2*x^2])/(3*d^2*x) + (10*e^
3*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/d^2

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Rubi in Sympy [A]  time = 41.996, size = 117, normalized size = 0.85 \[ - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{3 x^{3}} + \frac{2 e \sqrt{d^{2} - e^{2} x^{2}}}{d x^{2}} + \frac{10 e^{3} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{d^{2}} - \frac{8 e^{3} \sqrt{d^{2} - e^{2} x^{2}}}{d^{2} \left (d + e x\right )} - \frac{23 e^{2} \sqrt{d^{2} - e^{2} x^{2}}}{3 d^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(d**2 - e**2*x**2)/(3*x**3) + 2*e*sqrt(d**2 - e**2*x**2)/(d*x**2) + 10*e**3
*atanh(sqrt(d**2 - e**2*x**2)/d)/d**2 - 8*e**3*sqrt(d**2 - e**2*x**2)/(d**2*(d +
e*x)) - 23*e**2*sqrt(d**2 - e**2*x**2)/(3*d**2*x)

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Mathematica [A]  time = 0.217721, size = 94, normalized size = 0.69 \[ -\frac{-30 e^3 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+\frac{\sqrt{d^2-e^2 x^2} \left (d^3-5 d^2 e x+17 d e^2 x^2+47 e^3 x^3\right )}{x^3 (d+e x)}+30 e^3 \log (x)}{3 d^2} \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[d^2 - e^2*x^2]*(d^3 - 5*d^2*e*x + 17*d*e^2*x^2 + 47*e^3*x^3))/(x^3*(d +
e*x)) + 30*e^3*Log[x] - 30*e^3*Log[d + Sqrt[d^2 - e^2*x^2]])/(3*d^2)

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Maple [B]  time = 0.024, size = 575, normalized size = 4.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3/d^6/x^3*(-e^2*x^2+d^2)^(7/2)-26/3/d^8*e^2/x*(-e^2*x^2+d^2)^(7/2)-26/3/d^8*e
^4*x*(-e^2*x^2+d^2)^(5/2)-65/6/d^6*e^4*x*(-e^2*x^2+d^2)^(3/2)-65/4/d^4*e^4*x*(-e
^2*x^2+d^2)^(1/2)-65/4/d^2*e^4/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(
1/2))-1/d^5/e/(x+d/e)^4*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(7/2)+1/d^6/(x+d/e)^3*(-(
x+d/e)^2*e^2+2*d*e*(x+d/e))^(7/2)+14/3/d^7*e/(x+d/e)^2*(-(x+d/e)^2*e^2+2*d*e*(x+
d/e))^(7/2)-2/d^7*e^3*(-e^2*x^2+d^2)^(5/2)-10/3/d^5*e^3*(-e^2*x^2+d^2)^(3/2)-10/
d^3*e^3*(-e^2*x^2+d^2)^(1/2)+10/d*e^3/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*
x^2+d^2)^(1/2))/x)+26/3/d^7*e^3*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(5/2)+65/6/d^6*e^
4*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)*x+65/4/d^4*e^4*(-(x+d/e)^2*e^2+2*d*e*(x+d
/e))^(1/2)*x+65/4/d^2*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))+2/d^7*e/x^2*(-e^2*x^2+d^2)^(7/2)

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

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^4),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.276153, size = 528, normalized size = 3.85 \[ \frac{23 \, e^{7} x^{7} + 230 \, d e^{6} x^{6} - 138 \, d^{2} e^{5} x^{5} - 434 \, d^{3} e^{4} x^{4} + 159 \, d^{4} e^{3} x^{3} + 148 \, d^{5} e^{2} x^{2} - 44 \, d^{6} e x + 8 \, d^{7} - 30 \,{\left (e^{7} x^{7} - 3 \, d e^{6} x^{6} - 8 \, d^{2} e^{5} x^{5} + 4 \, d^{3} e^{4} x^{4} + 8 \, d^{4} e^{3} x^{3} +{\left (e^{6} x^{6} + 4 \, d e^{5} x^{5} - 4 \, d^{2} e^{4} x^{4} - 8 \, d^{3} e^{3} x^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (71 \, e^{6} x^{6} - 75 \, d e^{5} x^{5} - 357 \, d^{2} e^{4} x^{4} + 137 \, d^{3} e^{3} x^{3} + 152 \, d^{4} e^{2} x^{2} - 44 \, d^{5} e x + 8 \, d^{6}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{3 \,{\left (d^{2} e^{4} x^{7} - 3 \, d^{3} e^{3} x^{6} - 8 \, d^{4} e^{2} x^{5} + 4 \, d^{5} e x^{4} + 8 \, d^{6} x^{3} +{\left (d^{2} e^{3} x^{6} + 4 \, d^{3} e^{2} x^{5} - 4 \, d^{4} e x^{4} - 8 \, d^{5} x^{3}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

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^4),x, algorithm="fricas")

[Out]

1/3*(23*e^7*x^7 + 230*d*e^6*x^6 - 138*d^2*e^5*x^5 - 434*d^3*e^4*x^4 + 159*d^4*e^
3*x^3 + 148*d^5*e^2*x^2 - 44*d^6*e*x + 8*d^7 - 30*(e^7*x^7 - 3*d*e^6*x^6 - 8*d^2
*e^5*x^5 + 4*d^3*e^4*x^4 + 8*d^4*e^3*x^3 + (e^6*x^6 + 4*d*e^5*x^5 - 4*d^2*e^4*x^
4 - 8*d^3*e^3*x^3)*sqrt(-e^2*x^2 + d^2))*log(-(d - sqrt(-e^2*x^2 + d^2))/x) - (7
1*e^6*x^6 - 75*d*e^5*x^5 - 357*d^2*e^4*x^4 + 137*d^3*e^3*x^3 + 152*d^4*e^2*x^2 -
44*d^5*e*x + 8*d^6)*sqrt(-e^2*x^2 + d^2))/(d^2*e^4*x^7 - 3*d^3*e^3*x^6 - 8*d^4*
e^2*x^5 + 4*d^5*e*x^4 + 8*d^6*x^3 + (d^2*e^3*x^6 + 4*d^3*e^2*x^5 - 4*d^4*e*x^4 -
8*d^5*x^3)*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**4/(e*x+d)**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.392081, 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^4),x, algorithm="giac")

[Out]

Done