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

Optimal. Leaf size=83 \[ -\frac{3 d \sqrt{d^2-e^2 x^2}}{2 e}-\frac{(d+e x) \sqrt{d^2-e^2 x^2}}{2 e}+\frac{3 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e} \]

[Out]

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

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Rubi [A]  time = 0.0767847, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{3 d \sqrt{d^2-e^2 x^2}}{2 e}-\frac{(d+e x) \sqrt{d^2-e^2 x^2}}{2 e}+\frac{3 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.6011, size = 68, normalized size = 0.82 \[ \frac{3 d^{2} \operatorname{atan}{\left (\frac{e x}{\sqrt{d^{2} - e^{2} x^{2}}} \right )}}{2 e} - \frac{3 d \sqrt{d^{2} - e^{2} x^{2}}}{2 e} - \frac{\left (d + e x\right ) \sqrt{d^{2} - e^{2} x^{2}}}{2 e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0518996, size = 60, normalized size = 0.72 \[ \left (-\frac{2 d}{e}-\frac{x}{2}\right ) \sqrt{d^2-e^2 x^2}+\frac{3 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d^2-e^2 x^2}}\right )}{2 e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 71, normalized size = 0.9 \[{\frac{3\,{d}^{2}}{2}\arctan \left ({x\sqrt{{e}^{2}}{\frac{1}{\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}}} \right ){\frac{1}{\sqrt{{e}^{2}}}}}-{\frac{x}{2}\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}-2\,{\frac{d\sqrt{-{e}^{2}{x}^{2}+{d}^{2}}}{e}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/2*d^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-1/2*x*(-e^2*x^2+d
^2)^(1/2)-2*d*(-e^2*x^2+d^2)^(1/2)/e

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Maxima [A]  time = 0.792153, size = 85, normalized size = 1.02 \[ \frac{3 \, d^{2} \arcsin \left (\frac{e^{2} x}{\sqrt{d^{2} e^{2}}}\right )}{2 \, \sqrt{e^{2}}} - \frac{1}{2} \, \sqrt{-e^{2} x^{2} + d^{2}} x - \frac{2 \, \sqrt{-e^{2} x^{2} + d^{2}} d}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/2*d^2*arcsin(e^2*x/sqrt(d^2*e^2))/sqrt(e^2) - 1/2*sqrt(-e^2*x^2 + d^2)*x - 2*s
qrt(-e^2*x^2 + d^2)*d/e

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Fricas [A]  time = 0.274417, size = 227, normalized size = 2.73 \[ \frac{2 \, d e^{3} x^{3} + 4 \, d^{2} e^{2} x^{2} - 2 \, d^{3} e x - 6 \,{\left (d^{2} e^{2} x^{2} - 2 \, d^{4} + 2 \, \sqrt{-e^{2} x^{2} + d^{2}} d^{3}\right )} \arctan \left (-\frac{d - \sqrt{-e^{2} x^{2} + d^{2}}}{e x}\right ) -{\left (e^{3} x^{3} + 4 \, d e^{2} x^{2} - 2 \, d^{2} e x\right )} \sqrt{-e^{2} x^{2} + d^{2}}}{2 \,{\left (e^{3} x^{2} - 2 \, d^{2} e + 2 \, \sqrt{-e^{2} x^{2} + d^{2}} d e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*d*e^3*x^3 + 4*d^2*e^2*x^2 - 2*d^3*e*x - 6*(d^2*e^2*x^2 - 2*d^4 + 2*sqrt(-
e^2*x^2 + d^2)*d^3)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - (e^3*x^3 + 4*d*e
^2*x^2 - 2*d^2*e*x)*sqrt(-e^2*x^2 + d^2))/(e^3*x^2 - 2*d^2*e + 2*sqrt(-e^2*x^2 +
d^2)*d*e)

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Sympy [A]  time = 9.96247, size = 274, normalized size = 3.3 \[ d^{2} \left (\begin{cases} \frac{\sqrt{\frac{d^{2}}{e^{2}}} \operatorname{asin}{\left (x \sqrt{\frac{e^{2}}{d^{2}}} \right )}}{\sqrt{d^{2}}} & \text{for}\: d^{2} > 0 \wedge - e^{2} < 0 \\\frac{\sqrt{- \frac{d^{2}}{e^{2}}} \operatorname{asinh}{\left (x \sqrt{- \frac{e^{2}}{d^{2}}} \right )}}{\sqrt{d^{2}}} & \text{for}\: d^{2} > 0 \wedge - e^{2} > 0 \\\frac{\sqrt{\frac{d^{2}}{e^{2}}} \operatorname{acosh}{\left (x \sqrt{\frac{e^{2}}{d^{2}}} \right )}}{\sqrt{- d^{2}}} & \text{for}\: - e^{2} > 0 \wedge d^{2} < 0 \end{cases}\right ) + 2 d e \left (\begin{cases} \frac{x^{2}}{2 \sqrt{d^{2}}} & \text{for}\: e^{2} = 0 \\- \frac{\sqrt{d^{2} - e^{2} x^{2}}}{e^{2}} & \text{otherwise} \end{cases}\right ) + e^{2} \left (\begin{cases} - \frac{i d^{2} \operatorname{acosh}{\left (\frac{e x}{d} \right )}}{2 e^{3}} - \frac{i d x \sqrt{-1 + \frac{e^{2} x^{2}}{d^{2}}}}{2 e^{2}} & \text{for}\: \left |{\frac{e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac{d^{2} \operatorname{asin}{\left (\frac{e x}{d} \right )}}{2 e^{3}} - \frac{d x}{2 e^{2} \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} + \frac{x^{3}}{2 d \sqrt{1 - \frac{e^{2} x^{2}}{d^{2}}}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.292262, size = 54, normalized size = 0.65 \[ \frac{3}{2} \, d^{2} \arcsin \left (\frac{x e}{d}\right ) e^{\left (-1\right )}{\rm sign}\left (d\right ) - \frac{1}{2} \, \sqrt{-x^{2} e^{2} + d^{2}}{\left (4 \, d e^{\left (-1\right )} + x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/2*d^2*arcsin(x*e/d)*e^(-1)*sign(d) - 1/2*sqrt(-x^2*e^2 + d^2)*(4*d*e^(-1) + x)