3.1.32 \(\int \frac {x^2 (1-a x)}{(1-a^2 x^2)^{3/2}} \, dx\) [32]

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]

-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

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Rubi [A]
time = 0.02, 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 = {811, 655, 222, 651} \begin {gather*} -\frac {\text {ArcSin}(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} \end {gather*}

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 222

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 651

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 655

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 811

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*}

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Mathematica [A]
time = 0.20, size = 71, normalized size = 1.31 \begin {gather*} \frac {(-2-a x) \sqrt {1-a^2 x^2}}{a^3 (1+a x)}-\frac {\log \left (-\sqrt {-a^2} x+\sqrt {1-a^2 x^2}\right )}{\left (-a^2\right )^{3/2}} \end {gather*}

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

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Maple [A]
time = 0.11, size = 90, normalized size = 1.67

method	result	size
default	\(-a \left (-\frac {x^{2}}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {2}{a^{4} \sqrt {-a^{2} x^{2}+1}}\right )+\frac {x}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}\)	\(90\)
risch	\(\frac {a^{2} x^{2}-1}{a^{3} \sqrt {-a^{2} x^{2}+1}}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}-\frac {\sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2}+2 \left (x +\frac {1}{a}\right ) a}}{a^{4} \left (x +\frac {1}{a}\right )}\)	\(92\)
meijerg	\(-\frac {-2 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right )}{4 \sqrt {-a^{2} x^{2}+1}}}{a^{3} \sqrt {\pi }}-\frac {\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}}{a^{2} \sqrt {\pi }\, \sqrt {-a^{2}}}\)	\(105\)

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,method=_RETURNVERBOSE)

[Out]

-a*(-x^2/a^2/(-a^2*x^2+1)^(1/2)+2/a^4/(-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 = 0.51, size = 63, normalized size = 1.17 \begin {gather*} \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}} \end {gather*}

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)

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Fricas [A]
time = 2.04, size = 66, normalized size = 1.22 \begin {gather*} -\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 {gather*}

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 = 3.53, size = 119, normalized size = 2.20 \begin {gather*} - a \left (\begin {cases} \frac {a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1}}{a^{6} x^{2} - a^{4}} - \frac {2 \sqrt {- a^{2} x^{2} + 1}}{a^{6} x^{2} - a^{4}} & \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 {gather*}

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((a**2*x**2*sqrt(-a**2*x**2 + 1)/(a**6*x**2 - a**4) - 2*sqrt(-a**2*x**2 + 1)/(a**6*x**2 - a**4), N
e(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 = 0.76, size = 70, normalized size = 1.30 \begin {gather*} -\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 {gather*}

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))

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Mupad [B]
time = 0.09, size = 84, normalized size = 1.56 \begin {gather*} \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} \end {gather*}

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

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