3.1105 \(\int \frac{(1+a x) \sqrt{1-a^2 x^2}}{1-a x} \, dx\)

Optimal. Leaf size=62 \[ -\frac{\left (1-a^2 x^2\right )^{3/2}}{2 a (1-a x)}-\frac{3 \sqrt{1-a^2 x^2}}{2 a}+\frac{3 \sin ^{-1}(a x)}{2 a} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.028291, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {795, 665, 216} \[ -\frac{\left (1-a^2 x^2\right )^{3/2}}{2 a (1-a x)}-\frac{3 \sqrt{1-a^2 x^2}}{2 a}+\frac{3 \sin ^{-1}(a x)}{2 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 795

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(g*(d + e*x)^m
*(a + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[(m*(d*g + e*f) + 2*e*f*(p + 1))/(e*(m + 2*p + 2)), Int[(d +
e*x)^m*(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] && NeQ[m + 2*p +
2, 0] && NeQ[m, 2]

Rule 665

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

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

Mathematica [A]  time = 0.103726, size = 91, normalized size = 1.47 \[ \frac{\sqrt{1-a^2 x^2} \left (6 \sqrt{a x+1} \sin ^{-1}\left (\frac{\sqrt{a x+1}}{\sqrt{2}}\right )-\sqrt{1-a x} \left (a^2 x^2+5 a x+4\right )\right )}{2 a \sqrt{1-a x} (a x+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - a^2*x^2]*(-(Sqrt[1 - a*x]*(4 + 5*a*x + a^2*x^2)) + 6*Sqrt[1 + a*x]*ArcSin[Sqrt[1 + a*x]/Sqrt[2]]))/(
2*a*Sqrt[1 - a*x]*(1 + a*x))

________________________________________________________________________________________

Maple [B]  time = 0.011, size = 118, normalized size = 1.9 \begin{align*} -{\frac{x}{2}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{1}{2}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}-2\,{\frac{1}{a}\sqrt{- \left ( x-{a}^{-1} \right ) ^{2}{a}^{2}-2\,a \left ( x-{a}^{-1} \right ) }}+2\,{\frac{1}{\sqrt{{a}^{2}}}\arctan \left ({\sqrt{{a}^{2}}x{\frac{1}{\sqrt{- \left ( x-{a}^{-1} \right ) ^{2}{a}^{2}-2\,a \left ( x-{a}^{-1} \right ) }}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.48671, size = 57, normalized size = 0.92 \begin{align*} -\frac{1}{2} \, \sqrt{-a^{2} x^{2} + 1} x + \frac{3 \, \arcsin \left (a x\right )}{2 \, a} - \frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.5794, size = 111, normalized size = 1.79 \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1}{\left (a x + 4\right )} + 6 \, \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right )}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(sqrt(-a^2*x^2 + 1)*(a*x + 4) + 6*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)))/a

________________________________________________________________________________________

Sympy [A]  time = 4.49177, size = 76, normalized size = 1.23 \begin{align*} - \begin{cases} - \frac{- \sqrt{- a^{2} x^{2} + 1} + \operatorname{asin}{\left (a x \right )}}{a} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases} - \begin{cases} - \frac{- \frac{a x \sqrt{- a^{2} x^{2} + 1}}{2} - \sqrt{- a^{2} x^{2} + 1} + \frac{\operatorname{asin}{\left (a x \right )}}{2}}{a} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Piecewise((-(-sqrt(-a**2*x**2 + 1) + asin(a*x))/a, (a*x > -1) & (a*x < 1))) - Piecewise((-(-a*x*sqrt(-a**2*x*
*2 + 1)/2 - sqrt(-a**2*x**2 + 1) + asin(a*x)/2)/a, (a*x > -1) & (a*x < 1)))

________________________________________________________________________________________

Giac [A]  time = 1.07473, size = 46, normalized size = 0.74 \begin{align*} -\frac{1}{2} \, \sqrt{-a^{2} x^{2} + 1}{\left (x + \frac{4}{a}\right )} + \frac{3 \, \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{2 \,{\left | a \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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