3.32 \(\int \frac{x^2 (1-a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.127679, antiderivative size = 54, 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 \[ -\frac{\sin ^{-1}(a x)}{a^3}-\frac{1-a x}{a^3 \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{a^3} \]

Antiderivative was successfully verified.

[In]  Int[(x^2*(1 - a*x))/(1 - a^2*x^2)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 14.3374, size = 44, normalized size = 0.81 \[ - \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{3}} - \frac{\operatorname{asin}{\left (a x \right )}}{a^{3}} - \frac{\sqrt{- a^{2} x^{2} + 1}}{a^{3} \left (a x + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(-a*x+1)/(-a**2*x**2+1)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0923865, size = 37, normalized size = 0.69 \[ -\frac{\frac{\sqrt{1-a^2 x^2} (a x+2)}{a x+1}+\sin ^{-1}(a x)}{a^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^2*(1 - a*x))/(1 - a^2*x^2)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.022, size = 85, normalized size = 1.6 \[{\frac{{x}^{2}}{a}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-2\,{\frac{1}{{a}^{3}\sqrt{-{a}^{2}{x}^{2}+1}}}+{\frac{x}{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{1}{{a}^{2}}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(-a*x+1)/(-a^2*x^2+1)^(3/2),x)

[Out]

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

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Maxima [A]  time = 0.789037, size = 101, normalized size = 1.87 \[ \frac{x^{2}}{\sqrt{-a^{2} x^{2} + 1} a} + \frac{x}{\sqrt{-a^{2} x^{2} + 1} a^{2}} - \frac{\arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{\sqrt{a^{2}} a^{2}} - \frac{2}{\sqrt{-a^{2} x^{2} + 1} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(a*x - 1)*x^2/(-a^2*x^2 + 1)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.279162, size = 197, normalized size = 3.65 \[ \frac{a^{3} x^{3} + a^{2} x^{2} + 2 \, a x + 2 \,{\left (a^{2} x^{2} - a x + \sqrt{-a^{2} x^{2} + 1}{\left (a x + 2\right )} - 2\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) -{\left (a^{2} x^{2} + 2 \, a x\right )} \sqrt{-a^{2} x^{2} + 1}}{a^{5} x^{2} - a^{4} x - 2 \, a^{3} +{\left (a^{4} x + 2 \, a^{3}\right )} \sqrt{-a^{2} x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(a*x - 1)*x^2/(-a^2*x^2 + 1)^(3/2),x, algorithm="fricas")

[Out]

(a^3*x^3 + a^2*x^2 + 2*a*x + 2*(a^2*x^2 - a*x + sqrt(-a^2*x^2 + 1)*(a*x + 2) - 2
)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (a^2*x^2 + 2*a*x)*sqrt(-a^2*x^2 + 1))
/(a^5*x^2 - a^4*x - 2*a^3 + (a^4*x + 2*a^3)*sqrt(-a^2*x^2 + 1))

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Sympy [A]  time = 10.1649, size = 102, normalized size = 1.89 \[ - a \left (\begin{cases} - \frac{x^{2}}{a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{2}{a^{4} \sqrt{- a^{2} x^{2} + 1}} & \text{for}\: a \neq 0 \\\frac{x^{4}}{4} & \text{otherwise} \end{cases}\right ) + \begin{cases} - \frac{i x}{a^{2} \sqrt{a^{2} x^{2} - 1}} + \frac{i \operatorname{acosh}{\left (a x \right )}}{a^{3}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac{x}{a^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{\operatorname{asin}{\left (a x \right )}}{a^{3}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(-a*x+1)/(-a**2*x**2+1)**(3/2),x)

[Out]

-a*Piecewise((-x**2/(a**2*sqrt(-a**2*x**2 + 1)) + 2/(a**4*sqrt(-a**2*x**2 + 1)),
Ne(a, 0)), (x**4/4, True)) + Piecewise((-I*x/(a**2*sqrt(a**2*x**2 - 1)) + I*aco
sh(a*x)/a**3, Abs(a**2*x**2) > 1), (x/(a**2*sqrt(-a**2*x**2 + 1)) - asin(a*x)/a*
*3, True))

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GIAC/XCAS [A]  time = 0.289859, size = 95, normalized size = 1.76 \[ -\frac{\arcsin \left (a x\right ){\rm sign}\left (a\right )}{a^{2}{\left | a \right |}} - \frac{\sqrt{-a^{2} x^{2} + 1}}{a^{3}} + \frac{2}{a^{2}{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} + 1\right )}{\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(a*x - 1)*x^2/(-a^2*x^2 + 1)^(3/2),x, algorithm="giac")

[Out]

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