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

Optimal. Leaf size=66 \[ -\sqrt{d^2-e^2 x^2}+2 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-d \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]

[Out]

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

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Rubi [A]  time = 0.235696, antiderivative size = 66, 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 \[ -\sqrt{d^2-e^2 x^2}+2 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )-d \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{d}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 23.2742, size = 53, normalized size = 0.8 \[ 2 d \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )} - d \operatorname{atanh}{\left (\frac{\sqrt{d^{2} - e^{2} x^{2}}}{d} \right )} - \sqrt{d^{2} - e^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.049789, size = 68, normalized size = 1.03 \[ -\sqrt{d^2-e^2 x^2}-d \log \left (\sqrt{d^2-e^2 x^2}+d\right )+2 d \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )+d \log (x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 91, normalized size = 1.4 \[ -{{d}^{2}\ln \left ({\frac{1}{x} \left ( 2\,{d}^{2}+2\,\sqrt{{d}^{2}}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}} \right ) } \right ){\frac{1}{\sqrt{{d}^{2}}}}}-\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}+2\,{\frac{ed}{\sqrt{{e}^{2}}}\arctan \left ({\frac{\sqrt{{e}^{2}}x}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.28202, size = 171, normalized size = 2.59 \[ -\frac{e^{2} x^{2} + 4 \,{\left (d^{2} - \sqrt{-e^{2} x^{2} + d^{2}} d\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (d^{2} - \sqrt{-e^{2} x^{2} + d^{2}} d\right )} \log \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{x}\right )}{d - \sqrt{-e^{2} x^{2} + d^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(e^2*x^2 + 4*(d^2 - sqrt(-e^2*x^2 + d^2)*d)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/
(e*x)) - (d^2 - sqrt(-e^2*x^2 + d^2)*d)*log(-(d - sqrt(-e^2*x^2 + d^2))/x))/(d -
sqrt(-e^2*x^2 + d^2))

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Sympy [A]  time = 6.54943, size = 189, normalized size = 2.86 \[ d^{2} \left (\begin{cases} - \frac{\operatorname{acosh}{\left (\frac{d}{e x} \right )}}{d} & \text{for}\: \left |{\frac{d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac{i \operatorname{asin}{\left (\frac{d}{e x} \right )}}{d} & \text{otherwise} \end{cases}\right ) + 2 d e \left (\begin{cases} \frac{\sqrt{\frac{d^{2}}{e^{2}}} \operatorname{asin}{\left (x \sqrt{\frac{e^{2}}{d^{2}}} \right )}}{\sqrt{d^{2}}} & \text{for}\: d^{2} > 0 \wedge - e^{2} < 0 \\\frac{\sqrt{- \frac{d^{2}}{e^{2}}} \operatorname{asinh}{\left (x \sqrt{- \frac{e^{2}}{d^{2}}} \right )}}{\sqrt{d^{2}}} & \text{for}\: d^{2} > 0 \wedge - e^{2} > 0 \\\frac{\sqrt{\frac{d^{2}}{e^{2}}} \operatorname{acosh}{\left (x \sqrt{\frac{e^{2}}{d^{2}}} \right )}}{\sqrt{- d^{2}}} & \text{for}\: - e^{2} > 0 \wedge d^{2} < 0 \end{cases}\right ) + e^{2} \left (\begin{cases} \frac{x^{2}}{2 \sqrt{d^{2}}} & \text{for}\: e^{2} = 0 \\- \frac{\sqrt{d^{2} - e^{2} x^{2}}}{e^{2}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.291511, size = 88, normalized size = 1.33 \[ 2 \, d \arcsin \left (\frac{x e}{d}\right ){\rm sign}\left (d\right ) - d{\rm ln}\left (\frac{{\left | -2 \, d e - 2 \, \sqrt{-x^{2} e^{2} + d^{2}} e \right |} e^{\left (-2\right )}}{2 \,{\left | x \right |}}\right ) - \sqrt{-x^{2} e^{2} + d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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