3.577 \(\int \frac{(a+b x)^2}{\sqrt{1-x^2}} \, dx\)

Optimal. Leaf size=54 \[ \frac{1}{2} \left (2 a^2+b^2\right ) \sin ^{-1}(x)-\frac{3}{2} a b \sqrt{1-x^2}-\frac{1}{2} b \sqrt{1-x^2} (a+b x) \]

[Out]

(-3*a*b*Sqrt[1 - x^2])/2 - (b*(a + b*x)*Sqrt[1 - x^2])/2 + ((2*a^2 + b^2)*ArcSin
[x])/2

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Rubi [A]  time = 0.0796412, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{1}{2} \left (2 a^2+b^2\right ) \sin ^{-1}(x)-\frac{3}{2} a b \sqrt{1-x^2}-\frac{1}{2} b \sqrt{1-x^2} (a+b x) \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^2/Sqrt[1 - x^2],x]

[Out]

(-3*a*b*Sqrt[1 - x^2])/2 - (b*(a + b*x)*Sqrt[1 - x^2])/2 + ((2*a^2 + b^2)*ArcSin
[x])/2

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Rubi in Sympy [A]  time = 11.193, size = 42, normalized size = 0.78 \[ - \frac{3 a b \sqrt{- x^{2} + 1}}{2} - \frac{b \left (a + b x\right ) \sqrt{- x^{2} + 1}}{2} + \left (a^{2} + \frac{b^{2}}{2}\right ) \operatorname{asin}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**2/(-x**2+1)**(1/2),x)

[Out]

-3*a*b*sqrt(-x**2 + 1)/2 - b*(a + b*x)*sqrt(-x**2 + 1)/2 + (a**2 + b**2/2)*asin(
x)

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Mathematica [A]  time = 0.0537149, size = 38, normalized size = 0.7 \[ \frac{1}{2} \left (\left (2 a^2+b^2\right ) \sin ^{-1}(x)-b \sqrt{1-x^2} (4 a+b x)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^2/Sqrt[1 - x^2],x]

[Out]

(-(b*(4*a + b*x)*Sqrt[1 - x^2]) + (2*a^2 + b^2)*ArcSin[x])/2

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Maple [A]  time = 0.009, size = 42, normalized size = 0.8 \[{a}^{2}\arcsin \left ( x \right ) +{b}^{2} \left ( -{\frac{x}{2}\sqrt{-{x}^{2}+1}}+{\frac{\arcsin \left ( x \right ) }{2}} \right ) -2\,ab\sqrt{-{x}^{2}+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^2/(-x^2+1)^(1/2),x)

[Out]

a^2*arcsin(x)+b^2*(-1/2*x*(-x^2+1)^(1/2)+1/2*arcsin(x))-2*a*b*(-x^2+1)^(1/2)

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Maxima [A]  time = 0.800446, size = 57, normalized size = 1.06 \[ -\frac{1}{2} \, \sqrt{-x^{2} + 1} b^{2} x + a^{2} \arcsin \left (x\right ) + \frac{1}{2} \, b^{2} \arcsin \left (x\right ) - 2 \, \sqrt{-x^{2} + 1} a b \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^2/sqrt(-x^2 + 1),x, algorithm="maxima")

[Out]

-1/2*sqrt(-x^2 + 1)*b^2*x + a^2*arcsin(x) + 1/2*b^2*arcsin(x) - 2*sqrt(-x^2 + 1)
*a*b

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Fricas [A]  time = 0.228372, size = 184, normalized size = 3.41 \[ \frac{2 \, b^{2} x^{3} + 4 \, a b x^{2} - 2 \, b^{2} x - 2 \,{\left ({\left (2 \, a^{2} + b^{2}\right )} x^{2} - 4 \, a^{2} - 2 \, b^{2} + 2 \,{\left (2 \, a^{2} + b^{2}\right )} \sqrt{-x^{2} + 1}\right )} \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) -{\left (b^{2} x^{3} + 4 \, a b x^{2} - 2 \, b^{2} x\right )} \sqrt{-x^{2} + 1}}{2 \,{\left (x^{2} + 2 \, \sqrt{-x^{2} + 1} - 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^2/sqrt(-x^2 + 1),x, algorithm="fricas")

[Out]

1/2*(2*b^2*x^3 + 4*a*b*x^2 - 2*b^2*x - 2*((2*a^2 + b^2)*x^2 - 4*a^2 - 2*b^2 + 2*
(2*a^2 + b^2)*sqrt(-x^2 + 1))*arctan((sqrt(-x^2 + 1) - 1)/x) - (b^2*x^3 + 4*a*b*
x^2 - 2*b^2*x)*sqrt(-x^2 + 1))/(x^2 + 2*sqrt(-x^2 + 1) - 2)

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Sympy [A]  time = 0.609793, size = 42, normalized size = 0.78 \[ a^{2} \operatorname{asin}{\left (x \right )} - 2 a b \sqrt{- x^{2} + 1} - \frac{b^{2} x \sqrt{- x^{2} + 1}}{2} + \frac{b^{2} \operatorname{asin}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**2/(-x**2+1)**(1/2),x)

[Out]

a**2*asin(x) - 2*a*b*sqrt(-x**2 + 1) - b**2*x*sqrt(-x**2 + 1)/2 + b**2*asin(x)/2

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GIAC/XCAS [A]  time = 0.216583, size = 47, normalized size = 0.87 \[ \frac{1}{2} \,{\left (2 \, a^{2} + b^{2}\right )} \arcsin \left (x\right ) - \frac{1}{2} \,{\left (b^{2} x + 4 \, a b\right )} \sqrt{-x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^2/sqrt(-x^2 + 1),x, algorithm="giac")

[Out]

1/2*(2*a^2 + b^2)*arcsin(x) - 1/2*(b^2*x + 4*a*b)*sqrt(-x^2 + 1)