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

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

[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 = 0.0343437, 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, Rules used = {797, 641, 216, 637} \[ -\frac{1-a x}{a^3 \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{a^3}-\frac{\sin ^{-1}(a x)}{a^3} \]

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

Rule 797

Int[(x_)^2*((f_) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/c, Int[(f + g*x)*(a + c*x^2)^(p
+ 1), x], x] - Dist[a/c, Int[(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, f, g, p}, x] && EqQ[a*g^2 + f^2*
c, 0]

Rule 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 637

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(-(a*e) + c*d*x)/(a*c*Sqrt[a + c*x^2]),
x] /; FreeQ[{a, c, d, e}, x]

Rubi steps

\begin{align*} \int \frac{x^2 (1-a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx &=\frac{\int \frac{1-a x}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{a^2}-\frac{\int \frac{1-a x}{\sqrt{1-a^2 x^2}} \, dx}{a^2}\\ &=-\frac{1-a x}{a^3 \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{a^3}-\frac{\int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{a^2}\\ &=-\frac{1-a x}{a^3 \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{a^3}-\frac{\sin ^{-1}(a x)}{a^3}\\ \end{align*}

Mathematica [A]  time = 0.0313935, size = 50, normalized size = 0.93 \[ \frac{a^2 x^2-\sqrt{1-a^2 x^2} \sin ^{-1}(a x)+a x-2}{a^3 \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.053, size = 85, normalized size = 1.6 \begin{align*}{\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}}}}} \end{align*}

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 = 1.47832, size = 101, normalized size = 1.87 \begin{align*} \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}} \end{align*}

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, 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 = 1.84537, size = 151, normalized size = 2.8 \begin{align*} -\frac{2 \, a x - 2 \,{\left (a x + 1\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) + \sqrt{-a^{2} x^{2} + 1}{\left (a x + 2\right )} + 2}{a^{4} x + a^{3}} \end{align*}

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, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 5.72089, size = 102, normalized size = 1.89 \begin{align*} - 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} \end{align*}

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*acosh(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 [A]  time = 1.15445, size = 95, normalized size = 1.76 \begin{align*} -\frac{\arcsin \left (a x\right ) \mathrm{sgn}\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 |}} \end{align*}

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, algorithm="giac")

[Out]

-arcsin(a*x)*sgn(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))