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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]