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

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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 54, normalized size of antiderivative = 1.00, 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 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]

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 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]

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 = 0.03, 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])

________________________________________________________________________________________

fricas [A]  time = 0.77, size = 66, normalized size = 1.22 \[ -\frac {2 \, a x - 2 \, {\left (a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + \sqrt {-a^{2} x^{2} + 1} {\left (a x + 2\right )} + 2}{a^{4} x + a^{3}} \]

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)

________________________________________________________________________________________

giac [A]  time = 0.21, size = 70, normalized size = 1.30 \[ -\frac {\arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{a^{2} {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{a^{3}} + \frac {2}{a^{2} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )} {\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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))

________________________________________________________________________________________

maple [A]  time = 0.02, size = 85, normalized size = 1.57 \[ \frac {x^{2}}{\sqrt {-a^{2} x^{2}+1}\, a}+\frac {x}{\sqrt {-a^{2} x^{2}+1}\, a^{2}}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}\, a^{2}}-\frac {2}{\sqrt {-a^{2} x^{2}+1}\, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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))

________________________________________________________________________________________

maxima [A]  time = 0.96, size = 63, normalized size = 1.17 \[ \frac {x^{2}}{\sqrt {-a^{2} x^{2} + 1} a} + \frac {x}{\sqrt {-a^{2} x^{2} + 1} a^{2}} - \frac {\arcsin \left (a x\right )}{a^{3}} - \frac {2}{\sqrt {-a^{2} x^{2} + 1} a^{3}} \]

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*x)/a^3 - 2/(sqrt(-a^2*x^2 + 1)*a^3)

________________________________________________________________________________________

mupad [B]  time = 0.09, size = 84, normalized size = 1.56 \[ \frac {\sqrt {1-a^2\,x^2}}{\left (a\,\sqrt {-a^2}+a^2\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{a^2\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{a^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^2*(a*x - 1))/(1 - a^2*x^2)^(3/2),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 8.33, 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*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))

________________________________________________________________________________________