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

Optimal. Leaf size=34 \[ \frac{\sqrt{1-a^2 x^2}}{a^2 (a x+1)}+\frac{\sin ^{-1}(a x)}{a^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0174339, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {793, 216} \[ \frac{\sqrt{1-a^2 x^2}}{a^2 (a x+1)}+\frac{\sin ^{-1}(a x)}{a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0247696, size = 31, normalized size = 0.91 \[ \frac{\frac{\sqrt{1-a^2 x^2}}{a x+1}+\sin ^{-1}(a x)}{a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.053, size = 65, normalized size = 1.9 \begin{align*}{\frac{1}{a}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{\frac{1}{{a}^{3} \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/(a*x+1)/(-a^2*x^2+1)^(1/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.45976, size = 45, normalized size = 1.32 \begin{align*} \frac{\sqrt{-a^{2} x^{2} + 1}}{a^{3} x + a^{2}} + \frac{\arcsin \left (a x\right )}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.63347, size = 134, normalized size = 3.94 \begin{align*} \frac{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} + 1}{a^{3} x + a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\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/(a*x+1)/(-a**2*x**2+1)**(1/2),x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.25238, size = 70, normalized size = 2.06 \begin{align*} \frac{\arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{a{\left | a \right |}} - \frac{2}{a{\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/(a*x+1)/(-a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

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