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

Optimal. Leaf size=117 \[ \frac{1}{2} d e (2 d-3 e x) \sqrt{d^2-e^2 x^2}-\frac{(3 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{3 x}-\frac{3}{2} d^3 e \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-d^3 e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]

[Out]

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

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Rubi [A]  time = 0.293066, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32 \[ \frac{1}{2} d e (2 d-3 e x) \sqrt{d^2-e^2 x^2}-\frac{(3 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{3 x}-\frac{3}{2} d^3 e \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-d^3 e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 41.3694, size = 99, normalized size = 0.85 \[ - \frac{3 d^{3} e \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{2} - d^{3} e \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )} + \frac{d e \left (4 d - 6 e x\right ) \sqrt{d^{2} - e^{2} x^{2}}}{4} - \frac{\left (3 d - e x\right ) \left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{3 x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*d**3*e*atan(e*x/sqrt(d**2 - e**2*x**2))/2 - d**3*e*atanh(sqrt(d**2 - e**2*x**
2)/d) + d*e*(4*d - 6*e*x)*sqrt(d**2 - e**2*x**2)/4 - (3*d - e*x)*(d**2 - e**2*x*
*2)**(3/2)/(3*x)

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Mathematica [A]  time = 0.161032, size = 114, normalized size = 0.97 \[ d^3 e \log (x)-d^3 e \log \left (\sqrt{d^2-e^2 x^2}+d\right )-\frac{3}{2} d^3 e \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+\sqrt{d^2-e^2 x^2} \left (-\frac{d^3}{x}+\frac{4 d^2 e}{3}-\frac{1}{2} d e^2 x-\frac{e^3 x^2}{3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.033, size = 182, normalized size = 1.6 \[ -{\frac{1}{dx} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{5}{2}}}}-{\frac{{e}^{2}x}{d} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{e}^{2}dx}{2}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-{\frac{3\,{e}^{2}{d}^{3}}{2}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}+{\frac{e}{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{{\frac{3}{2}}}}+e{d}^{2}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}-{e{d}^{4}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/d/x*(-e^2*x^2+d^2)^(5/2)-e^2/d*x*(-e^2*x^2+d^2)^(3/2)-3/2*d*e^2*x*(-e^2*x^2+d
^2)^(1/2)-3/2*e^2*d^3/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))+1/3
*e*(-e^2*x^2+d^2)^(3/2)+e*d^2*(-e^2*x^2+d^2)^(1/2)-e*d^4/(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)^(3/2)*(e*x + d)/x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.288147, size = 532, normalized size = 4.55 \[ \frac{8 \, d e^{7} x^{7} + 12 \, d^{2} e^{6} x^{6} - 48 \, d^{3} e^{5} x^{5} - 12 \, d^{4} e^{4} x^{4} + 48 \, d^{5} e^{3} x^{3} - 48 \, d^{6} e^{2} x^{2} + 48 \, d^{8} + 18 \,{\left (d^{3} e^{5} x^{5} - 8 \, d^{5} e^{3} x^{3} + 8 \, d^{7} e x + 4 \,{\left (d^{4} e^{3} x^{3} - 2 \, d^{6} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) + 6 \,{\left (d^{3} e^{5} x^{5} - 8 \, d^{5} e^{3} x^{3} + 8 \, d^{7} e x + 4 \,{\left (d^{4} e^{3} x^{3} - 2 \, d^{6} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) -{\left (2 \, e^{7} x^{7} + 3 \, d e^{6} x^{6} - 24 \, d^{2} e^{5} x^{5} - 18 \, d^{3} e^{4} x^{4} + 48 \, d^{4} e^{3} x^{3} - 24 \, d^{5} e^{2} x^{2} + 48 \, d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{6 \,{\left (e^{4} x^{5} - 8 \, d^{2} e^{2} x^{3} + 8 \, d^{4} x + 4 \,{\left (d e^{2} x^{3} - 2 \, d^{3} x\right )} \sqrt{-e^{2} x^{2} + d^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(8*d*e^7*x^7 + 12*d^2*e^6*x^6 - 48*d^3*e^5*x^5 - 12*d^4*e^4*x^4 + 48*d^5*e^3
*x^3 - 48*d^6*e^2*x^2 + 48*d^8 + 18*(d^3*e^5*x^5 - 8*d^5*e^3*x^3 + 8*d^7*e*x + 4
*(d^4*e^3*x^3 - 2*d^6*e*x)*sqrt(-e^2*x^2 + d^2))*arctan(-(d - sqrt(-e^2*x^2 + d^
2))/(e*x)) + 6*(d^3*e^5*x^5 - 8*d^5*e^3*x^3 + 8*d^7*e*x + 4*(d^4*e^3*x^3 - 2*d^6
*e*x)*sqrt(-e^2*x^2 + d^2))*log(-(d - sqrt(-e^2*x^2 + d^2))/x) - (2*e^7*x^7 + 3*
d*e^6*x^6 - 24*d^2*e^5*x^5 - 18*d^3*e^4*x^4 + 48*d^4*e^3*x^3 - 24*d^5*e^2*x^2 +
48*d^7)*sqrt(-e^2*x^2 + d^2))/(e^4*x^5 - 8*d^2*e^2*x^3 + 8*d^4*x + 4*(d*e^2*x^3
- 2*d^3*x)*sqrt(-e^2*x^2 + d^2))

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Sympy [A]  time = 18.1014, size = 386, normalized size = 3.3 \[ d^{3} \left (\begin{cases} \frac{i d}{x \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + i e \operatorname{acosh}{\left (\frac{e x}{d} \right )} - \frac{i e^{2} x}{d \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \left |{\frac{e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac{d}{x \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} - e \operatorname{asin}{\left (\frac{e x}{d} \right )} + \frac{e^{2} x}{d \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) + d^{2} e \left (\begin{cases} \frac{d^{2}}{e x \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} - d \operatorname{acosh}{\left (\frac{d}{e x} \right )} - \frac{e x}{\sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}} & \text{for}\: \left |{\frac{d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac{i d^{2}}{e x \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} + i d \operatorname{asin}{\left (\frac{d}{e x} \right )} + \frac{i e x}{\sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}} & \text{otherwise} \end{cases}\right ) - d e^{2} \left (\begin{cases} - \frac{i d^{2} \operatorname{acosh}{\left (\frac{e x}{d} \right )}}{2 e} - \frac{i d x}{2 \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} + \frac{i e^{2} x^{3}}{2 d \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}} & \text{for}\: \left |{\frac{e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac{d^{2} \operatorname{asin}{\left (\frac{e x}{d} \right )}}{2 e} + \frac{d x \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}}{2} & \text{otherwise} \end{cases}\right ) - e^{3} \left (\begin{cases} \frac{x^{2} \sqrt{d^{2}}}{2} & \text{for}\: e^{2} = 0 \\- \frac{\left (d^{2} - e^{2} x^{2}\right )^{\frac{3}{2}}}{3 e^{2}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**3*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)) + d**2*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)) -
d*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*asin(e*x/d)/(2*e) + d*x*sqrt(1 - e**2*x**2/d**2)/2, True)) - e**3*Piecewise((x
**2*sqrt(d**2)/2, Eq(e**2, 0)), (-(d**2 - e**2*x**2)**(3/2)/(3*e**2), True))

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GIAC/XCAS [A]  time = 0.319975, size = 212, normalized size = 1.81 \[ -\frac{3}{2} \, d^{3} \arcsin \left (\frac{x e}{d}\right ) e{\rm sign}\left (d\right ) - d^{3} e{\rm ln}\left (\frac{{\left | -2 \, d e - 2 \, \sqrt{-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \,{\left | x \right |}}\right ) + \frac{d^{3} x e^{3}}{2 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}} - \frac{{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} d^{3} e^{\left (-1\right )}}{2 \, x} + \frac{1}{6} \, \sqrt{-x^{2} e^{2} + d^{2}}{\left (8 \, d^{2} e -{\left (2 \, x e^{3} + 3 \, d e^{2}\right )} x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/2*d^3*arcsin(x*e/d)*e*sign(d) - d^3*e*ln(1/2*abs(-2*d*e - 2*sqrt(-x^2*e^2 + d
^2)*e)*e^(-2)/abs(x)) + 1/2*d^3*x*e^3/(d*e + sqrt(-x^2*e^2 + d^2)*e) - 1/2*(d*e
+ sqrt(-x^2*e^2 + d^2)*e)*d^3*e^(-1)/x + 1/6*sqrt(-x^2*e^2 + d^2)*(8*d^2*e - (2*
x*e^3 + 3*d*e^2)*x)