Optimal. Leaf size=67 \[ \frac{\left (2 a^2+1\right ) \sin ^{-1}(a+b x)}{2 b^3}+\frac{3 a \sqrt{1-(a+b x)^2}}{2 b^3}-\frac{x \sqrt{1-(a+b x)^2}}{2 b^2} \]
[Out]
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Rubi [A] time = 0.121378, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{\left (2 a^2+1\right ) \sin ^{-1}(a+b x)}{2 b^3}+\frac{3 a \sqrt{1-(a+b x)^2}}{2 b^3}-\frac{x \sqrt{1-(a+b x)^2}}{2 b^2} \]
Antiderivative was successfully verified.
[In] Int[x^2/Sqrt[1 - (a + b*x)^2],x]
[Out]
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Rubi in Sympy [A] time = 14.1268, size = 54, normalized size = 0.81 \[ \frac{3 a \sqrt{- \left (a + b x\right )^{2} + 1}}{2 b^{3}} - \frac{x \sqrt{- \left (a + b x\right )^{2} + 1}}{2 b^{2}} + \frac{\left (a^{2} + \frac{1}{2}\right ) \operatorname{asin}{\left (a + b x \right )}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(1-(b*x+a)**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.074997, size = 55, normalized size = 0.82 \[ \frac{\sqrt{-a^2-2 a b x-b^2 x^2+1} (3 a-b x)+\left (2 a^2+1\right ) \sin ^{-1}(a+b x)}{2 b^3} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/Sqrt[1 - (a + b*x)^2],x]
[Out]
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Maple [B] time = 0.022, size = 152, normalized size = 2.3 \[ -{\frac{x}{2\,{b}^{2}}\sqrt{-{b}^{2}{x}^{2}-2\,abx-{a}^{2}+1}}+{\frac{3\,a}{2\,{b}^{3}}\sqrt{-{b}^{2}{x}^{2}-2\,abx-{a}^{2}+1}}+{\frac{{a}^{2}}{{b}^{2}}\arctan \left ({1\sqrt{{b}^{2}} \left ({\frac{a}{b}}+x \right ){\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,abx-{a}^{2}+1}}}} \right ){\frac{1}{\sqrt{{b}^{2}}}}}+{\frac{1}{2\,{b}^{2}}\arctan \left ({1\sqrt{{b}^{2}} \left ({\frac{a}{b}}+x \right ){\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,abx-{a}^{2}+1}}}} \right ){\frac{1}{\sqrt{{b}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(1-(b*x+a)^2)^(1/2),x)
[Out]
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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(x^2/sqrt(-(b*x + a)^2 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.275648, size = 100, normalized size = 1.49 \[ \frac{{\left (2 \, a^{2} + 1\right )} \arctan \left (\frac{b x + a}{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}\right ) - \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left (b x - 3 \, a\right )}}{2 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/sqrt(-(b*x + a)^2 + 1),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{- \left (a + b x - 1\right ) \left (a + b x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(1-(b*x+a)**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.280923, size = 74, normalized size = 1.1 \[ -\frac{1}{2} \, \sqrt{-{\left (b x + a\right )}^{2} + 1}{\left (\frac{x}{b^{2}} - \frac{3 \, a}{b^{3}}\right )} - \frac{{\left (2 \, a^{2} + 1\right )} \arcsin \left (-b x - a\right ){\rm sign}\left (b\right )}{2 \, b^{2}{\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/sqrt(-(b*x + a)^2 + 1),x, algorithm="giac")
[Out]