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

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]

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

_______________________________________________________________________________________

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]

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

_______________________________________________________________________________________

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]

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

_______________________________________________________________________________________

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]

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

_______________________________________________________________________________________

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]

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

_______________________________________________________________________________________

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]

Exception raised: ValueError

_______________________________________________________________________________________

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]

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

_______________________________________________________________________________________

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]

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

_______________________________________________________________________________________

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]

-1/2*sqrt(-(b*x + a)^2 + 1)*(x/b^2 - 3*a/b^3) - 1/2*(2*a^2 + 1)*arcsin(-b*x - a)
*sign(b)/(b^2*abs(b))