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

Optimal. Leaf size=121 \[ \frac{3 d e (d-e x) \sqrt{d^2-e^2 x^2}}{2 x}-\frac{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^2}+\frac{3}{2} d^2 e^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\frac{3}{2} d^2 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]

[Out]

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

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Rubi [A]  time = 0.369046, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296 \[ \frac{3 d e (d-e x) \sqrt{d^2-e^2 x^2}}{2 x}-\frac{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^2}+\frac{3}{2} d^2 e^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\frac{3}{2} d^2 e^2 \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 50.646, size = 112, normalized size = 0.93 \[ \frac{3 d^{2} e^{2} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{2} + \frac{3 d^{2} e^{2} \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )}}{2} + \frac{3 d e \left (4 d - 4 e x\right ) \sqrt{d^{2} - e^{2} x^{2}}}{8 x} - \frac{\left (2 d + 2 e x\right ) \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{4 x^{2}} \]

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),x)

[Out]

3*d**2*e**2*atan(e*x/sqrt(d**2 - e**2*x**2))/2 + 3*d**2*e**2*atanh(sqrt(d**2 - e
**2*x**2)/d)/2 + 3*d*e*(4*d - 4*e*x)*sqrt(d**2 - e**2*x**2)/(8*x) - (2*d + 2*e*x
)*(d**2 - e**2*x**2)**(3/2)/(4*x**2)

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Mathematica [A]  time = 0.200284, size = 119, normalized size = 0.98 \[ \frac{1}{2} \left (3 d^2 e^2 \log \left (\sqrt{d^2-e^2 x^2}+d\right )+3 d^2 e^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-3 d^2 e^2 \log (x)+\frac{\sqrt{d^2-e^2 x^2} \left (-d^3+2 d^2 e x-2 d e^2 x^2+e^3 x^3\right )}{x^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.018, size = 411, normalized size = 3.4 \[ -{\frac{1}{2\,{d}^{3}{x}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}-{\frac{3\,{e}^{2}}{10\,{d}^{3}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{{e}^{2}}{2\,d} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{3\,d{e}^{2}}{2}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{3\,{e}^{2}{d}^{3}}{2}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}-{\frac{{e}^{2}}{5\,{d}^{3}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{5}{2}}}}-{\frac{{e}^{3}x}{4\,{d}^{2}} \left ( - \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{e}^{3}x}{8}\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}-{\frac{3\,{d}^{2}{e}^{3}}{8}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{- \left ( x+{\frac{d}{e}} \right ) ^{2}{e}^{2}+2\,de \left ( x+{\frac{d}{e}} \right ) }}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{e}{{d}^{4}x} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{7}{2}}}}+{\frac{{e}^{3}x}{{d}^{4}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}+{\frac{5\,{e}^{3}x}{4\,{d}^{2}} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+{\frac{15\,{e}^{3}x}{8}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}+{\frac{15\,{d}^{2}{e}^{3}}{8}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}} \]

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),x)

[Out]

-1/2/d^3/x^2*(-e^2*x^2+d^2)^(7/2)-3/10/d^3*e^2*(-e^2*x^2+d^2)^(5/2)-1/2*e^2/d*(-
e^2*x^2+d^2)^(3/2)-3/2*d*e^2*(-e^2*x^2+d^2)^(1/2)+3/2*e^2*d^3/(d^2)^(1/2)*ln((2*
d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)-1/5/d^3*e^2*(-(x+d/e)^2*e^2+2*d*e*(x+
d/e))^(5/2)-1/4/d^2*e^3*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(3/2)*x-3/8*e^3*(-(x+d/e)
^2*e^2+2*d*e*(x+d/e))^(1/2)*x-3/8*d^2*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))+e/d^4/x*(-e^2*x^2+d^2)^(7/2)+e^3/d^4*x*(-e^2*x^
2+d^2)^(5/2)+5/4*e^3/d^2*x*(-e^2*x^2+d^2)^(3/2)+15/8*e^3*x*(-e^2*x^2+d^2)^(1/2)+
15/8*e^3*d^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))

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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)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.298586, size = 576, normalized size = 4.76 \[ -\frac{4 \, d e^{7} x^{7} - 6 \, d^{2} e^{6} x^{6} - 4 \, d^{3} e^{5} x^{5} + 4 \, d^{4} e^{4} x^{4} - 16 \, d^{5} e^{3} x^{3} + 12 \, d^{6} e^{2} x^{2} + 16 \, d^{7} e x - 8 \, d^{8} + 6 \,{\left (d^{2} e^{6} x^{6} - 8 \, d^{4} e^{4} x^{4} + 8 \, d^{6} e^{2} x^{2} + 4 \,{\left (d^{3} e^{4} x^{4} - 2 \, d^{5} e^{2} x^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + 3 \,{\left (d^{2} e^{6} x^{6} - 8 \, d^{4} e^{4} x^{4} + 8 \, d^{6} e^{2} x^{2} + 4 \,{\left (d^{3} e^{4} x^{4} - 2 \, d^{5} e^{2} x^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (e^{7} x^{7} - 2 \, d e^{6} x^{6} - 6 \, d^{2} e^{5} x^{5} + 7 \, d^{3} e^{4} x^{4} - 8 \, d^{4} e^{3} x^{3} + 8 \, d^{5} e^{2} x^{2} + 16 \, d^{6} e x - 8 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{2 \,{\left (e^{4} x^{6} - 8 \, d^{2} e^{2} x^{4} + 8 \, d^{4} x^{2} + 4 \,{\left (d e^{2} x^{4} - 2 \, d^{3} x^{2}\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(5/2)/((e*x + d)*x^3),x, algorithm="fricas")

[Out]

-1/2*(4*d*e^7*x^7 - 6*d^2*e^6*x^6 - 4*d^3*e^5*x^5 + 4*d^4*e^4*x^4 - 16*d^5*e^3*x
^3 + 12*d^6*e^2*x^2 + 16*d^7*e*x - 8*d^8 + 6*(d^2*e^6*x^6 - 8*d^4*e^4*x^4 + 8*d^
6*e^2*x^2 + 4*(d^3*e^4*x^4 - 2*d^5*e^2*x^2)*sqrt(-e^2*x^2 + d^2))*arctan(-(d - s
qrt(-e^2*x^2 + d^2))/(e*x)) + 3*(d^2*e^6*x^6 - 8*d^4*e^4*x^4 + 8*d^6*e^2*x^2 + 4
*(d^3*e^4*x^4 - 2*d^5*e^2*x^2)*sqrt(-e^2*x^2 + d^2))*log(-(d - sqrt(-e^2*x^2 + d
^2))/x) - (e^7*x^7 - 2*d*e^6*x^6 - 6*d^2*e^5*x^5 + 7*d^3*e^4*x^4 - 8*d^4*e^3*x^3
+ 8*d^5*e^2*x^2 + 16*d^6*e*x - 8*d^7)*sqrt(-e^2*x^2 + d^2))/(e^4*x^6 - 8*d^2*e^
2*x^4 + 8*d^4*x^2 + 4*(d*e^2*x^4 - 2*d^3*x^2)*sqrt(-e^2*x^2 + d^2))

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Sympy [A]  time = 25.719, size = 461, normalized size = 3.81 \[ d^{3} \left (\begin{cases} - \frac{d^{2}}{2 e x^{3} \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} + \frac{e}{2 x \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} + \frac{e^{2} \operatorname{acosh}{\left (\frac{d}{e x} \right )}}{2 d} & \text{for}\: \left |{\frac{d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac{i e \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}}{2 x} - \frac{i e^{2} \operatorname{asin}{\left (\frac{d}{e x} \right )}}{2 d} & \text{otherwise} \end{cases}\right ) - d^{2} e \left (\begin{cases} \frac{i d}{x \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + i e \operatorname{acosh}{\left (\frac{e x}{d} \right )} - \frac{i e^{2} x}{d \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \left |{\frac{e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac{d}{x \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} - e \operatorname{asin}{\left (\frac{e x}{d} \right )} + \frac{e^{2} x}{d \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) - d e^{2} \left (\begin{cases} \frac{d^{2}}{e x \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} - d \operatorname{acosh}{\left (\frac{d}{e x} \right )} - \frac{e x}{\sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} & \text{for}\: \left |{\frac{d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac{i d^{2}}{e x \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} + i d \operatorname{asin}{\left (\frac{d}{e x} \right )} + \frac{i e x}{\sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} & \text{otherwise} \end{cases}\right ) + e^{3} \left (\begin{cases} - \frac{i d^{2} \operatorname{acosh}{\left (\frac{e x}{d} \right )}}{2 e} - \frac{i d x}{2 \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + \frac{i e^{2} x^{3}}{2 d \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \left |{\frac{e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac{d^{2} \operatorname{asin}{\left (\frac{e x}{d} \right )}}{2 e} + \frac{d x \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}}{2} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**3*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)) - d**2*e*
Piecewise((I*d/(x*sqrt(-1 + e**2*x**2/d**2)) + I*e*acosh(e*x/d) - I*e**2*x/(d*sq
rt(-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)) - d*e**2*Piec
ewise((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**3*
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*asin(
e*x/d)/(2*e) + d*x*sqrt(1 - e**2*x**2/d**2)/2, True))

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-e^2*x^2 + d^2)^(5/2)/((e*x + d)*x^3),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError