\(\int \frac {x}{(1+a x) \sqrt {1-a^2 x^2}} \, dx\) [152]

Optimal result

Integrand size = 23, antiderivative size = 34 \[ \int \frac {x}{(1+a x) \sqrt {1-a^2 x^2}} \, dx=\frac {\sqrt {1-a^2 x^2}}{a^2 (1+a x)}+\frac {\arcsin (a x)}{a^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {807, 222} \[ \int \frac {x}{(1+a x) \sqrt {1-a^2 x^2}} \, dx=\frac {\arcsin (a x)}{a^2}+\frac {\sqrt {1-a^2 x^2}}{a^2 (a x+1)} \]

[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 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 807

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]

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.12 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.56 \[ \int \frac {x}{(1+a x) \sqrt {1-a^2 x^2}} \, dx=\frac {\sqrt {1-a^2 x^2}}{a^2 (1+a x)}+\frac {2 \arctan \left (\frac {a x}{-1+\sqrt {1-a^2 x^2}}\right )}{a^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.91

method	result	size
default	\(\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {a^{2}}}+\frac {\sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2}+2 \left (x +\frac {1}{a}\right ) a}}{a^{3} \left (x +\frac {1}{a}\right )}\)	\(65\)

[In]

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

[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*(x+1/a)*a)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.71 \[ \int \frac {x}{(1+a x) \sqrt {1-a^2 x^2}} \, dx=\frac {a x - 2 \, {\left (a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + \sqrt {-a^{2} x^{2} + 1} + 1}{a^{3} x + a^{2}} \]

[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]

\[ \int \frac {x}{(1+a x) \sqrt {1-a^2 x^2}} \, dx=\int \frac {x}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )}\, dx \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.97 \[ \int \frac {x}{(1+a x) \sqrt {1-a^2 x^2}} \, dx=\frac {\sqrt {-a^{2} x^{2} + 1}}{a^{3} x + a^{2}} + \frac {\arcsin \left (a x\right )}{a^{2}} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.53 \[ \int \frac {x}{(1+a x) \sqrt {1-a^2 x^2}} \, dx=\frac {\arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{a {\left | a \right |}} - \frac {2}{a {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )} {\left | a \right |}} \]

[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))

Mupad [B] (verification not implemented)

Time = 11.45 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.68 \[ \int \frac {x}{(1+a x) \sqrt {1-a^2 x^2}} \, dx=\frac {1}{a^2\,\sqrt {1-a^2\,x^2}}-\frac {x}{a\,\sqrt {1-a^2\,x^2}}-\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}{a^3} \]

[In]

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

[Out]

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