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

Optimal. Leaf size=55 \[ -\frac{\sqrt{1-a^2 x^2}}{a^3 (a x+1)}-\frac{\sqrt{1-a^2 x^2}}{a^3}-\frac{\sin ^{-1}(a x)}{a^3} \]

[Out]

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

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Rubi [A]  time = 0.0863924, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {1639, 12, 793, 216} \[ -\frac{\sqrt{1-a^2 x^2}}{a^3 (a x+1)}-\frac{\sqrt{1-a^2 x^2}}{a^3}-\frac{\sin ^{-1}(a x)}{a^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2/((1 + a*x)*Sqrt[1 - a^2*x^2]),x]

[Out]

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

Rule 1639

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff
[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + c*x^2)^(p + 1))/(c*e^(q - 1)*(m + q + 2*p + 1)), x]
+ Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*
f*(m + q + 2*p + 1)*(d + e*x)^q - 2*e*f*(m + p + q)*(d + e*x)^(q - 2)*(a*e - c*d*x), x], x], x] /; NeQ[m + q +
2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 793

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d*g - e*f)*(
d + e*x)^m*(a + c*x^2)^(p + 1))/(2*c*d*(m + p + 1)), x] + Dist[(m*(g*c*d + c*e*f) + 2*e*c*f*(p + 1))/(e*(2*c*d
)*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[c*d^2
+ a*e^2, 0] && ((LtQ[m, -1] &&  !IGtQ[m + p + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) &&
NeQ[m + p + 1, 0]

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]

Rubi steps

\begin{align*} \int \frac{x^2}{(1+a x) \sqrt{1-a^2 x^2}} \, dx &=-\frac{\sqrt{1-a^2 x^2}}{a^3}-\frac{\int \frac{a^3 x}{(1+a x) \sqrt{1-a^2 x^2}} \, dx}{a^4}\\ &=-\frac{\sqrt{1-a^2 x^2}}{a^3}-\frac{\int \frac{x}{(1+a x) \sqrt{1-a^2 x^2}} \, dx}{a}\\ &=-\frac{\sqrt{1-a^2 x^2}}{a^3}-\frac{\sqrt{1-a^2 x^2}}{a^3 (1+a x)}-\frac{\int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{a^2}\\ &=-\frac{\sqrt{1-a^2 x^2}}{a^3}-\frac{\sqrt{1-a^2 x^2}}{a^3 (1+a x)}-\frac{\sin ^{-1}(a x)}{a^3}\\ \end{align*}

Mathematica [A]  time = 0.0551428, size = 37, normalized size = 0.67 \[ -\frac{\frac{\sqrt{1-a^2 x^2} (a x+2)}{a x+1}+\sin ^{-1}(a x)}{a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/((1 + a*x)*Sqrt[1 - a^2*x^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 = 0.048, size = 84, normalized size = 1.5 \begin{align*} -{\frac{1}{{a}^{3}}\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}}}}}-{\frac{1}{{a}^{4} \left ( x+{a}^{-1} \right ) }\sqrt{- \left ( x+{a}^{-1} \right ) ^{2}{a}^{2}+2\,a \left ( x+{a}^{-1} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.47628, size = 70, normalized size = 1.27 \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1}}{a^{4} x + a^{3}} - \frac{\arcsin \left (a x\right )}{a^{3}} - \frac{\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)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.62199, size = 151, normalized size = 2.75 \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)^(1/2),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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(a*x+1)/(-a**2*x**2+1)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.2485, size = 95, normalized size = 1.73 \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)^(1/2),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))