3.165 \(\int \frac{\left (d^2-e^2 x^2\right )^{5/2}}{x^3 (d+e x)^2} \, dx\)

Optimal. Leaf size=110 \[ \frac{e (4 d+e x) \sqrt{d^2-e^2 x^2}}{2 x}-\frac{\left (d^2-e^2 x^2\right )^{3/2}}{2 x^2}+2 d e^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\frac{1}{2} d e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]

[Out]

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

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Rubi [A]  time = 0.397948, antiderivative size = 110, 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 \[ \frac{e (4 d+e x) \sqrt{d^2-e^2 x^2}}{2 x}-\frac{\left (d^2-e^2 x^2\right )^{3/2}}{2 x^2}+2 d e^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-\frac{1}{2} d e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(d^2 - e^2*x^2)^(5/2)/(x^3*(d + e*x)^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 39.8365, size = 107, normalized size = 0.97 \[ - \frac{d^{2} \sqrt{d^{2} - e^{2} x^{2}}}{2 x^{2}} + 2 d e^{2} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )} - \frac{d e^{2} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{2} + \frac{2 d e \sqrt{d^{2} - e^{2} x^{2}}}{x} + e^{2} \sqrt{d^{2} - e^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-e**2*x**2+d**2)**(5/2)/x**3/(e*x+d)**2,x)

[Out]

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

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Mathematica [A]  time = 0.127396, size = 102, normalized size = 0.93 \[ \left (-\frac{d^2}{2 x^2}+\frac{2 d e}{x}+e^2\right ) \sqrt{d^2-e^2 x^2}-\frac{1}{2} d e^2 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+2 d e^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\frac{1}{2} d e^2 \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[(d^2 - e^2*x^2)^(5/2)/(x^3*(d + e*x)^2),x]

[Out]

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

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Maple [B]  time = 0.018, size = 456, normalized size = 4.2 \[ -{\frac{1}{2\,{d}^{4}{x}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{{e}^{2}}{10\,{d}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{{e}^{2}}{6\,{d}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{{e}^{2}}{2}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{{d}^{2}{e}^{2}}{2}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}-{\frac{14\,{e}^{2}}{15\,{d}^{4}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{5}{2}}}}-{\frac{7\,{e}^{3}x}{6\,{d}^{3}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{7\,{e}^{3}x}{4\,d}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}-{\frac{7\,d{e}^{3}}{4}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+2\,{\frac{e \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{7/2}}{{d}^{5}x}}+2\,{\frac{{e}^{3}x \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{5/2}}{{d}^{5}}}+{\frac{5\,{e}^{3}x}{2\,{d}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{15\,{e}^{3}x}{4\,d}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{15\,d{e}^{3}}{4}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{1}{3\,{d}^{4}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{7}{2}}} \left ( x+{\frac{d}{e}} \right ) ^{-2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-e^2*x^2+d^2)^(5/2)/x^3/(e*x+d)^2,x)

[Out]

-1/2/d^4/x^2*(-e^2*x^2+d^2)^(7/2)+1/10/d^4*e^2*(-e^2*x^2+d^2)^(5/2)+1/6/d^2*e^2*
(-e^2*x^2+d^2)^(3/2)+1/2*e^2*(-e^2*x^2+d^2)^(1/2)-1/2*d^2*e^2/(d^2)^(1/2)*ln((2*
d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)-14/15/d^4*e^2*(-(x+d/e)^2*e^2+2*d*e*(
x+d/e))^(5/2)-7/6/d^3*e^3*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)*x-7/4/d*e^3*(-(x+
d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)*x-7/4*d*e^3/(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^5*e/x*(-e^2*x^2+d^2)^(7/2)+2/d^5*e^3*x*(-
e^2*x^2+d^2)^(5/2)+5/2/d^3*e^3*x*(-e^2*x^2+d^2)^(3/2)+15/4/d*e^3*x*(-e^2*x^2+d^2
)^(1/2)+15/4*d*e^3/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-1/3/d^
4/(x+d/e)^2*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(7/2)

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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^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.293698, size = 470, normalized size = 4.27 \[ \frac{2 \, e^{6} x^{6} + 4 \, d e^{5} x^{5} - 5 \, d^{2} e^{4} x^{4} - 20 \, d^{3} e^{3} x^{3} + 5 \, d^{4} e^{2} x^{2} + 16 \, d^{5} e x - 4 \, d^{6} - 8 \,{\left (3 \, d^{2} e^{4} x^{4} - 4 \, d^{4} e^{2} x^{2} -{\left (d e^{4} x^{4} - 4 \, d^{3} e^{2} x^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) +{\left (3 \, d^{2} e^{4} x^{4} - 4 \, d^{4} e^{2} x^{2} -{\left (d e^{4} x^{4} - 4 \, d^{3} e^{2} x^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) +{\left (4 \, d e^{4} x^{4} + 12 \, d^{2} e^{3} x^{3} - 3 \, d^{3} e^{2} x^{2} - 16 \, d^{4} e x + 4 \, d^{5}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{2 \,{\left (3 \, d e^{2} x^{4} - 4 \, d^{3} x^{2} -{\left (e^{2} x^{4} - 4 \, d^{2} x^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

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^3),x, algorithm="fricas")

[Out]

1/2*(2*e^6*x^6 + 4*d*e^5*x^5 - 5*d^2*e^4*x^4 - 20*d^3*e^3*x^3 + 5*d^4*e^2*x^2 +
16*d^5*e*x - 4*d^6 - 8*(3*d^2*e^4*x^4 - 4*d^4*e^2*x^2 - (d*e^4*x^4 - 4*d^3*e^2*x
^2)*sqrt(-e^2*x^2 + d^2))*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (3*d^2*e^4
*x^4 - 4*d^4*e^2*x^2 - (d*e^4*x^4 - 4*d^3*e^2*x^2)*sqrt(-e^2*x^2 + d^2))*log(-(d
- sqrt(-e^2*x^2 + d^2))/x) + (4*d*e^4*x^4 + 12*d^2*e^3*x^3 - 3*d^3*e^2*x^2 - 16
*d^4*e*x + 4*d^5)*sqrt(-e^2*x^2 + d^2))/(3*d*e^2*x^4 - 4*d^3*x^2 - (e^2*x^4 - 4*
d^2*x^2)*sqrt(-e^2*x^2 + d^2))

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Sympy [A]  time = 23.3053, size = 347, normalized size = 3.15 \[ d^{2} \left (\begin{cases} - \frac{d^{2}}{2 e x^{3} \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} + \frac{e}{2 x \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} + \frac{e^{2} \operatorname{acosh}{\left (\frac{d}{e x} \right )}}{2 d} & \text{for}\: \left |{\frac{d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac{i e \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}}{2 x} - \frac{i e^{2} \operatorname{asin}{\left (\frac{d}{e x} \right )}}{2 d} & \text{otherwise} \end{cases}\right ) - 2 d e \left (\begin{cases} \frac{i d}{x \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + i e \operatorname{acosh}{\left (\frac{e x}{d} \right )} - \frac{i e^{2} x}{d \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \left |{\frac{e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac{d}{x \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} - e \operatorname{asin}{\left (\frac{e x}{d} \right )} + \frac{e^{2} x}{d \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) + e^{2} \left (\begin{cases} \frac{d^{2}}{e x \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} - d \operatorname{acosh}{\left (\frac{d}{e x} \right )} - \frac{e x}{\sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} & \text{for}\: \left |{\frac{d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac{i d^{2}}{e x \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} + i d \operatorname{asin}{\left (\frac{d}{e x} \right )} + \frac{i e x}{\sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e**2*x**2+d**2)**(5/2)/x**3/(e*x+d)**2,x)

[Out]

d**2*Piecewise((-d**2/(2*e*x**3*sqrt(d**2/(e**2*x**2) - 1)) + e/(2*x*sqrt(d**2/(
e**2*x**2) - 1)) + e**2*acosh(d/(e*x))/(2*d), Abs(d**2/(e**2*x**2)) > 1), (-I*e*
sqrt(-d**2/(e**2*x**2) + 1)/(2*x) - I*e**2*asin(d/(e*x))/(2*d), True)) - 2*d*e*P
iecewise((I*d/(x*sqrt(-1 + e**2*x**2/d**2)) + I*e*acosh(e*x/d) - I*e**2*x/(d*sqr
t(-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)) + e**2*Piecewi
se((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))

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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^3),x, algorithm="giac")

[Out]

Timed out