3.39 \(\int \frac{(d+e x)^2}{x^2 \sqrt{d^2-e^2 x^2}} \, dx\)

Optimal. Leaf size=68 \[ -\frac{\sqrt{d^2-e^2 x^2}}{x}+e \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-2 e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]

[Out]

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

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Rubi [A]  time = 0.235571, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259 \[ -\frac{\sqrt{d^2-e^2 x^2}}{x}+e \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-2 e \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 24.2617, size = 54, normalized size = 0.79 \[ e \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )} - 2 e \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )} - \frac{\sqrt{d^{2} - e^{2} x^{2}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**2/x**2/(-e**2*x**2+d**2)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0671542, size = 71, normalized size = 1.04 \[ -\frac{\sqrt{d^2-e^2 x^2}}{x}-2 e \log \left (\sqrt{d^2-e^2 x^2}+d\right )+e \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+2 e \log (x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.015, size = 93, normalized size = 1.4 \[{{e}^{2}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{1}{x}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-2\,{\frac{de}{\sqrt{{d}^{2}}}\ln \left ({\frac{2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}{x}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^2/x^2/(-e^2*x^2+d^2)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.281517, size = 209, normalized size = 3.07 \[ -\frac{e^{2} x^{2} - d^{2} + 2 \,{\left (d e x - \sqrt{-e^{2} x^{2} + d^{2}} e x\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) - 2 \,{\left (d e x - \sqrt{-e^{2} x^{2} + d^{2}} e x\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right ) + \sqrt{-e^{2} x^{2} + d^{2}} d}{d x - \sqrt{-e^{2} x^{2} + d^{2}} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(e^2*x^2 - d^2 + 2*(d*e*x - sqrt(-e^2*x^2 + d^2)*e*x)*arctan(-(d - sqrt(-e^2*x^
2 + d^2))/(e*x)) - 2*(d*e*x - sqrt(-e^2*x^2 + d^2)*e*x)*log(-(d - sqrt(-e^2*x^2
+ d^2))/x) + sqrt(-e^2*x^2 + d^2)*d)/(d*x - sqrt(-e^2*x^2 + d^2)*x)

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Sympy [A]  time = 7.55546, size = 212, normalized size = 3.12 \[ d^{2} \left (\begin{cases} - \frac{e \sqrt{\frac{d^{2}}{e^{2} x^{2}} - 1}}{d^{2}} & \text{for}\: \left |{\frac{d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac{i e \sqrt{- \frac{d^{2}}{e^{2} x^{2}} + 1}}{d^{2}} & \text{otherwise} \end{cases}\right ) + 2 d e \left (\begin{cases} - \frac{\operatorname{acosh}{\left (\frac{d}{e x} \right )}}{d} & \text{for}\: \left |{\frac{d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac{i \operatorname{asin}{\left (\frac{d}{e x} \right )}}{d} & \text{otherwise} \end{cases}\right ) + e^{2} \left (\begin{cases} \frac{\sqrt{\frac{d^{2}}{e^{2}}} \operatorname{asin}{\left (x \sqrt{\frac{e^{2}}{d^{2}}} \right )}}{\sqrt{d^{2}}} & \text{for}\: d^{2} > 0 \wedge - e^{2} < 0 \\\frac{\sqrt{- \frac{d^{2}}{e^{2}}} \operatorname{asinh}{\left (x \sqrt{- \frac{e^{2}}{d^{2}}} \right )}}{\sqrt{d^{2}}} & \text{for}\: d^{2} > 0 \wedge - e^{2} > 0 \\\frac{\sqrt{\frac{d^{2}}{e^{2}}} \operatorname{acosh}{\left (x \sqrt{\frac{e^{2}}{d^{2}}} \right )}}{\sqrt{- d^{2}}} & \text{for}\: - e^{2} > 0 \wedge d^{2} < 0 \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**2/x**2/(-e**2*x**2+d**2)**(1/2),x)

[Out]

d**2*Piecewise((-e*sqrt(d**2/(e**2*x**2) - 1)/d**2, Abs(d**2/(e**2*x**2)) > 1),
(-I*e*sqrt(-d**2/(e**2*x**2) + 1)/d**2, True)) + 2*d*e*Piecewise((-acosh(d/(e*x)
)/d, Abs(d**2/(e**2*x**2)) > 1), (I*asin(d/(e*x))/d, True)) + e**2*Piecewise((sq
rt(d**2/e**2)*asin(x*sqrt(e**2/d**2))/sqrt(d**2), (d**2 > 0) & (-e**2 < 0)), (sq
rt(-d**2/e**2)*asinh(x*sqrt(-e**2/d**2))/sqrt(d**2), (d**2 > 0) & (-e**2 > 0)),
(sqrt(d**2/e**2)*acosh(x*sqrt(e**2/d**2))/sqrt(-d**2), (d**2 < 0) & (-e**2 > 0))
)

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GIAC/XCAS [A]  time = 0.290523, size = 144, normalized size = 2.12 \[ \arcsin \left (\frac{x e}{d}\right ) e{\rm sign}\left (d\right ) - 2 \, e{\rm ln}\left (\frac{{\left | -2 \, d e - 2 \, \sqrt{-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \,{\left | x \right |}}\right ) + \frac{x e^{3}}{2 \,{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )}} - \frac{{\left (d e + \sqrt{-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-1\right )}}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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