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

Optimal. Leaf size=105 \[ -\frac{\left (d^2-e^2 x^2\right )^{3/2}}{x}-\frac{1}{2} e (4 d+e x) \sqrt{d^2-e^2 x^2}-\frac{1}{2} d^2 e \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+2 d^2 e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]

[Out]

-(e*(4*d + e*x)*Sqrt[d^2 - e^2*x^2])/2 - (d^2 - e^2*x^2)^(3/2)/x - (d^2*e*ArcTan
[(e*x)/Sqrt[d^2 - e^2*x^2]])/2 + 2*d^2*e*ArcTanh[Sqrt[d^2 - e^2*x^2]/d]

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Rubi [A]  time = 0.382593, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{\left (d^2-e^2 x^2\right )^{3/2}}{x}-\frac{1}{2} e (4 d+e x) \sqrt{d^2-e^2 x^2}-\frac{1}{2} d^2 e \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+2 d^2 e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(e*(4*d + e*x)*Sqrt[d^2 - e^2*x^2])/2 - (d^2 - e^2*x^2)^(3/2)/x - (d^2*e*ArcTan
[(e*x)/Sqrt[d^2 - e^2*x^2]])/2 + 2*d^2*e*ArcTanh[Sqrt[d^2 - e^2*x^2]/d]

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Rubi in Sympy [A]  time = 36.7024, size = 105, normalized size = 1. \[ - \frac{d^{2} e \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{2} + 2 d^{2} e \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )} - \frac{d^{2} \sqrt{d^{2} - e^{2} x^{2}}}{x} - 2 d e \sqrt{d^{2} - e^{2} x^{2}} + \frac{e^{2} x \sqrt{d^{2} - e^{2} x^{2}}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d**2*e*atan(e*x/sqrt(d**2 - e**2*x**2))/2 + 2*d**2*e*atanh(sqrt(d**2 - e**2*x**
2)/d) - d**2*sqrt(d**2 - e**2*x**2)/x - 2*d*e*sqrt(d**2 - e**2*x**2) + e**2*x*sq
rt(d**2 - e**2*x**2)/2

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Mathematica [A]  time = 0.114991, size = 100, normalized size = 0.95 \[ \left (-\frac{d^2}{x}-2 d e+\frac{e^2 x}{2}\right ) \sqrt{d^2-e^2 x^2}+2 d^2 e \log \left (\sqrt{d^2-e^2 x^2}+d\right )-\frac{1}{2} d^2 e \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-2 d^2 e \log (x) \]

Antiderivative was successfully verified.

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

[Out]

(-2*d*e - d^2/x + (e^2*x)/2)*Sqrt[d^2 - e^2*x^2] - (d^2*e*ArcTan[(e*x)/Sqrt[d^2
- e^2*x^2]])/2 - 2*d^2*e*Log[x] + 2*d^2*e*Log[d + Sqrt[d^2 - e^2*x^2]]

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Maple [B]  time = 0.019, size = 425, normalized size = 4.1 \[ -{\frac{1}{{d}^{4}x} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{{e}^{2}x}{{d}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{e}^{2}x}{4\,{d}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{15\,{e}^{2}x}{8}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{15\,{d}^{2}{e}^{2}}{8}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{1}{3\,{d}^{3}e} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{d}{e}} \right ) ^{-2}}+{\frac{11\,e}{15\,{d}^{3}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{11\,{e}^{2}x}{12\,{d}^{2}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{11\,{e}^{2}x}{8}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}+{\frac{11\,{d}^{2}{e}^{2}}{8}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{2\,e}{5\,{d}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{2\,e}{3\,d} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-2\,de\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}+2\,{\frac{{d}^{3}e}{\sqrt{{d}^{2}}}\ln \left ({\frac{2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}{x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/d^4/x*(-e^2*x^2+d^2)^(7/2)-1/d^4*e^2*x*(-e^2*x^2+d^2)^(5/2)-5/4/d^2*e^2*x*(-e
^2*x^2+d^2)^(3/2)-15/8*e^2*x*(-e^2*x^2+d^2)^(1/2)-15/8*d^2*e^2/(e^2)^(1/2)*arcta
n((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))+1/3/d^3/e/(x+d/e)^2*(-(x+d/e)^2*e^2+2*d*e*
(x+d/e))^(7/2)+11/15/d^3*e*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(5/2)+11/12/d^2*e^2*(-
(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)*x+11/8*e^2*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/
2)*x+11/8*d^2*e^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-(x+d/e)^2*e^2+2*d*e*(x+d/e)
)^(1/2))-2/5/d^3*e*(-e^2*x^2+d^2)^(5/2)-2/3/d*e*(-e^2*x^2+d^2)^(3/2)-2*d*e*(-e^2
*x^2+d^2)^(1/2)+2*d^3*e/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2)
)/x)

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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^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.295227, size = 429, normalized size = 4.09 \[ \frac{e^{6} x^{6} - 4 \, d e^{5} x^{5} - 7 \, d^{2} e^{4} x^{4} + 8 \, d^{3} e^{3} x^{3} + 14 \, d^{4} e^{2} x^{2} - 8 \, d^{6} + 2 \,{\left (3 \, d^{3} e^{3} x^{3} - 4 \, d^{5} e x -{\left (d^{2} e^{3} x^{3} - 4 \, d^{4} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) - 4 \,{\left (3 \, d^{3} e^{3} x^{3} - 4 \, d^{5} e x -{\left (d^{2} e^{3} x^{3} - 4 \, d^{4} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) +{\left (3 \, d e^{4} x^{4} - 8 \, d^{2} e^{3} x^{3} - 10 \, d^{3} e^{2} x^{2} + 8 \, d^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{2 \,{\left (3 \, d e^{2} x^{3} - 4 \, d^{3} x -{\left (e^{2} x^{3} - 4 \, d^{2} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

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^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 22.7376, size = 347, normalized size = 3.3 \[ d^{2} \left (\begin{cases} \frac{i d}{x \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + i e \operatorname{acosh}{\left (\frac{e x}{d} \right )} - \frac{i e^{2} x}{d \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \left |{\frac{e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac{d}{x \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} - e \operatorname{asin}{\left (\frac{e x}{d} \right )} + \frac{e^{2} x}{d \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) - 2 d e \left (\begin{cases} \frac{d^{2}}{e x \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} - d \operatorname{acosh}{\left (\frac{d}{e x} \right )} - \frac{e x}{\sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} & \text{for}\: \left |{\frac{d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac{i d^{2}}{e x \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} + i d \operatorname{asin}{\left (\frac{d}{e x} \right )} + \frac{i e x}{\sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} & \text{otherwise} \end{cases}\right ) + e^{2} \left (\begin{cases} - \frac{i d^{2} \operatorname{acosh}{\left (\frac{e x}{d} \right )}}{2 e} - \frac{i d x}{2 \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + \frac{i e^{2} x^{3}}{2 d \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \left |{\frac{e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac{d^{2} \operatorname{asin}{\left (\frac{e x}{d} \right )}}{2 e} + \frac{d x \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}}{2} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**2*Piecewise((I*d/(x*sqrt(-1 + e**2*x**2/d**2)) + I*e*acosh(e*x/d) - I*e**2*x/
(d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (-d/(x*sqrt(1 - e**2*x*
*2/d**2)) - e*asin(e*x/d) + e**2*x/(d*sqrt(1 - e**2*x**2/d**2)), True)) - 2*d*e*
Piecewise((d**2/(e*x*sqrt(d**2/(e**2*x**2) - 1)) - d*acosh(d/(e*x)) - e*x/sqrt(d
**2/(e**2*x**2) - 1), Abs(d**2/(e**2*x**2)) > 1), (-I*d**2/(e*x*sqrt(-d**2/(e**2
*x**2) + 1)) + I*d*asin(d/(e*x)) + I*e*x/sqrt(-d**2/(e**2*x**2) + 1), True)) + e
**2*Piecewise((-I*d**2*acosh(e*x/d)/(2*e) - I*d*x/(2*sqrt(-1 + e**2*x**2/d**2))
+ I*e**2*x**3/(2*d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (d**2*a
sin(e*x/d)/(2*e) + d*x*sqrt(1 - e**2*x**2/d**2)/2, True))

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GIAC/XCAS [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)/((e*x + d)^2*x^2),x, algorithm="giac")

[Out]

Timed out