3.874 \(\int \frac{e^{\tanh ^{-1}(a+b x)}}{x^2 (1-a^2-2 a b x-b^2 x^2)} \, dx\)

Optimal. Leaf size=150 \[ -\frac{\sqrt{a+b x+1}}{\left (1-a^2\right ) x \sqrt{-a-b x+1}}-\frac{2 (2 a+1) b \tanh ^{-1}\left (\frac{\sqrt{1-a} \sqrt{a+b x+1}}{\sqrt{a+1} \sqrt{-a-b x+1}}\right )}{(1-a)^2 (a+1) \sqrt{1-a^2}}+\frac{(a+2) b \sqrt{a+b x+1}}{(1-a)^2 (a+1) \sqrt{-a-b x+1}} \]

[Out]

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

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Rubi [A]  time = 0.159194, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {6164, 103, 152, 12, 93, 208} \[ -\frac{\sqrt{a+b x+1}}{\left (1-a^2\right ) x \sqrt{-a-b x+1}}-\frac{2 (2 a+1) b \tanh ^{-1}\left (\frac{\sqrt{1-a} \sqrt{a+b x+1}}{\sqrt{a+1} \sqrt{-a-b x+1}}\right )}{(1-a)^2 (a+1) \sqrt{1-a^2}}+\frac{(a+2) b \sqrt{a+b x+1}}{(1-a)^2 (a+1) \sqrt{-a-b x+1}} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a + b*x]/(x^2*(1 - a^2 - 2*a*b*x - b^2*x^2)),x]

[Out]

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

Rule 6164

Int[E^(ArcTanh[(a_) + (b_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(c/
(1 - a^2))^p, Int[u*(1 - a - b*x)^(p - n/2)*(1 + a + b*x)^(p + n/2), x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]
&& EqQ[b*d - 2*a*e, 0] && EqQ[b^2*c + e*(1 - a^2), 0] && (IntegerQ[p] || GtQ[c/(1 - a^2), 0])

Rule 103

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

Rule 152

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rule 12

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

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{e^{\tanh ^{-1}(a+b x)}}{x^2 \left (1-a^2-2 a b x-b^2 x^2\right )} \, dx &=\int \frac{1}{x^2 (1-a-b x)^{3/2} \sqrt{1+a+b x}} \, dx\\ &=-\frac{\sqrt{1+a+b x}}{\left (1-a^2\right ) x \sqrt{1-a-b x}}-\frac{\int \frac{-(1+2 a) b-b^2 x}{x (1-a-b x)^{3/2} \sqrt{1+a+b x}} \, dx}{1-a^2}\\ &=\frac{(2+a) b \sqrt{1+a+b x}}{(1-a)^2 (1+a) \sqrt{1-a-b x}}-\frac{\sqrt{1+a+b x}}{\left (1-a^2\right ) x \sqrt{1-a-b x}}+\frac{\int \frac{(1+2 a) b^2}{x \sqrt{1-a-b x} \sqrt{1+a+b x}} \, dx}{(1-a)^2 (1+a) b}\\ &=\frac{(2+a) b \sqrt{1+a+b x}}{(1-a)^2 (1+a) \sqrt{1-a-b x}}-\frac{\sqrt{1+a+b x}}{\left (1-a^2\right ) x \sqrt{1-a-b x}}+\frac{((1+2 a) b) \int \frac{1}{x \sqrt{1-a-b x} \sqrt{1+a+b x}} \, dx}{(1-a)^2 (1+a)}\\ &=\frac{(2+a) b \sqrt{1+a+b x}}{(1-a)^2 (1+a) \sqrt{1-a-b x}}-\frac{\sqrt{1+a+b x}}{\left (1-a^2\right ) x \sqrt{1-a-b x}}+\frac{(2 (1+2 a) b) \operatorname{Subst}\left (\int \frac{1}{-1-a-(-1+a) x^2} \, dx,x,\frac{\sqrt{1+a+b x}}{\sqrt{1-a-b x}}\right )}{(1-a)^2 (1+a)}\\ &=\frac{(2+a) b \sqrt{1+a+b x}}{(1-a)^2 (1+a) \sqrt{1-a-b x}}-\frac{\sqrt{1+a+b x}}{\left (1-a^2\right ) x \sqrt{1-a-b x}}-\frac{2 (1+2 a) b \tanh ^{-1}\left (\frac{\sqrt{1-a} \sqrt{1+a+b x}}{\sqrt{1+a} \sqrt{1-a-b x}}\right )}{(1-a)^2 (1+a) \sqrt{1-a^2}}\\ \end{align*}

Mathematica [A]  time = 0.298647, size = 149, normalized size = 0.99 \[ -\frac{\sqrt{-a^2-2 a b x-b^2 x^2+1} \left (\frac{b}{a+b x-1}+\frac{1}{a x+x}\right )+\frac{(2 a+1) b \log \left (\sqrt{1-a^2} \sqrt{-a^2-2 a b x-b^2 x^2+1}-a^2-a b x+1\right )}{(a+1) \sqrt{1-a^2}}-\frac{(2 a+1) b \log (x)}{(a+1) \sqrt{1-a^2}}}{(a-1)^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcTanh[a + b*x]/(x^2*(1 - a^2 - 2*a*b*x - b^2*x^2)),x]

[Out]

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

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Maple [B]  time = 0.062, size = 671, normalized size = 4.5 \begin{align*} -{\frac{1}{ \left ( -{a}^{2}+1 \right ) x}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}}+3\,{\frac{ab}{ \left ( -{a}^{2}+1 \right ) ^{2}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}+3\,{\frac{{a}^{2}{b}^{2}x}{ \left ( -{a}^{2}+1 \right ) ^{2}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}+3\,{\frac{{a}^{3}b}{ \left ( -{a}^{2}+1 \right ) ^{2}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}-3\,{\frac{ab}{ \left ( -{a}^{2}+1 \right ) ^{5/2}}\ln \left ({\frac{-2\,{a}^{2}+2-2\,xab+2\,\sqrt{-{a}^{2}+1}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}{x}} \right ) }+2\,{\frac{{b}^{2}x}{ \left ( -{a}^{2}+1 \right ) \sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}+2\,{\frac{ab}{ \left ( -{a}^{2}+1 \right ) \sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}-{\frac{a}{ \left ( -{a}^{2}+1 \right ) x}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}}+3\,{\frac{{a}^{2}b}{ \left ( -{a}^{2}+1 \right ) ^{2}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}+3\,{\frac{{a}^{3}{b}^{2}x}{ \left ( -{a}^{2}+1 \right ) ^{2}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}+3\,{\frac{{a}^{4}b}{ \left ( -{a}^{2}+1 \right ) ^{2}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}-3\,{\frac{{a}^{2}b}{ \left ( -{a}^{2}+1 \right ) ^{5/2}}\ln \left ({\frac{-2\,{a}^{2}+2-2\,xab+2\,\sqrt{-{a}^{2}+1}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}{x}} \right ) }+3\,{\frac{a{b}^{2}x}{ \left ( -{a}^{2}+1 \right ) \sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}+3\,{\frac{{a}^{2}b}{ \left ( -{a}^{2}+1 \right ) \sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}+{\frac{b}{-{a}^{2}+1}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}}}}-{b\ln \left ({\frac{1}{x} \left ( -2\,{a}^{2}+2-2\,xab+2\,\sqrt{-{a}^{2}+1}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1} \right ) } \right ) \left ( -{a}^{2}+1 \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(-a^2+1)/x/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+3*a*b/(-a^2+1)^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+3*a^2*b^2/(-a^2+1
)^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*x+3*a^3*b/(-a^2+1)^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-3*a*b/(-a^2+1)^(5/2)*ln
((-2*a^2+2-2*x*a*b+2*(-a^2+1)^(1/2)*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2))/x)+2/(-a^2+1)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/
2)*x*b^2+2/(-a^2+1)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*a*b-1/(-a^2+1)/x/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*a+3*a^2*b/(
-a^2+1)^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)+3*a^3*b^2/(-a^2+1)^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*x+3*a^4*b/(-a^2+1
)^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-3*a^2*b/(-a^2+1)^(5/2)*ln((-2*a^2+2-2*x*a*b+2*(-a^2+1)^(1/2)*(-b^2*x^2-2*a*
b*x-a^2+1)^(1/2))/x)+3/(-a^2+1)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*x*b^2*a+3/(-a^2+1)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/
2)*a^2*b+b/(-a^2+1)/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)-b/(-a^2+1)^(3/2)*ln((-2*a^2+2-2*x*a*b+2*(-a^2+1)^(1/2)*(-b^
2*x^2-2*a*b*x-a^2+1)^(1/2))/x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{b x + a + 1}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )} \sqrt{-{\left (b x + a\right )}^{2} + 1} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.83045, size = 1021, normalized size = 6.81 \begin{align*} \left [-\frac{{\left ({\left (2 \, a + 1\right )} b^{2} x^{2} +{\left (2 \, a^{2} - a - 1\right )} b x\right )} \sqrt{-a^{2} + 1} \log \left (\frac{{\left (2 \, a^{2} - 1\right )} b^{2} x^{2} + 2 \, a^{4} + 4 \,{\left (a^{3} - a\right )} b x + 2 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left (a b x + a^{2} - 1\right )} \sqrt{-a^{2} + 1} - 4 \, a^{2} + 2}{x^{2}}\right ) + 2 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left (a^{3} +{\left (a^{3} + 2 \, a^{2} - a - 2\right )} b x - a^{2} - a + 1\right )}}{2 \,{\left ({\left (a^{5} - a^{4} - 2 \, a^{3} + 2 \, a^{2} + a - 1\right )} b x^{2} +{\left (a^{6} - 2 \, a^{5} - a^{4} + 4 \, a^{3} - a^{2} - 2 \, a + 1\right )} x\right )}}, \frac{{\left ({\left (2 \, a + 1\right )} b^{2} x^{2} +{\left (2 \, a^{2} - a - 1\right )} b x\right )} \sqrt{a^{2} - 1} \arctan \left (\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left (a b x + a^{2} - 1\right )} \sqrt{a^{2} - 1}}{{\left (a^{2} - 1\right )} b^{2} x^{2} + a^{4} + 2 \,{\left (a^{3} - a\right )} b x - 2 \, a^{2} + 1}\right ) - \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left (a^{3} +{\left (a^{3} + 2 \, a^{2} - a - 2\right )} b x - a^{2} - a + 1\right )}}{{\left (a^{5} - a^{4} - 2 \, a^{3} + 2 \, a^{2} + a - 1\right )} b x^{2} +{\left (a^{6} - 2 \, a^{5} - a^{4} + 4 \, a^{3} - a^{2} - 2 \, a + 1\right )} x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*(((2*a + 1)*b^2*x^2 + (2*a^2 - a - 1)*b*x)*sqrt(-a^2 + 1)*log(((2*a^2 - 1)*b^2*x^2 + 2*a^4 + 4*(a^3 - a)
*b*x + 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(-a^2 + 1) - 4*a^2 + 2)/x^2) + 2*sqrt(-b^2*x
^2 - 2*a*b*x - a^2 + 1)*(a^3 + (a^3 + 2*a^2 - a - 2)*b*x - a^2 - a + 1))/((a^5 - a^4 - 2*a^3 + 2*a^2 + a - 1)*
b*x^2 + (a^6 - 2*a^5 - a^4 + 4*a^3 - a^2 - 2*a + 1)*x), (((2*a + 1)*b^2*x^2 + (2*a^2 - a - 1)*b*x)*sqrt(a^2 -
1)*arctan(sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a*b*x + a^2 - 1)*sqrt(a^2 - 1)/((a^2 - 1)*b^2*x^2 + a^4 + 2*(a^3
- a)*b*x - 2*a^2 + 1)) - sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(a^3 + (a^3 + 2*a^2 - a - 2)*b*x - a^2 - a + 1))/
((a^5 - a^4 - 2*a^3 + 2*a^2 + a - 1)*b*x^2 + (a^6 - 2*a^5 - a^4 + 4*a^3 - a^2 - 2*a + 1)*x)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{1}{a x^{2} \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} + b x^{3} \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} - x^{2} \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(1/(a*x**2*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1) + b*x**3*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1) - x**
2*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)), x)

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Giac [B]  time = 1.22364, size = 701, normalized size = 4.67 \begin{align*} \frac{2 \,{\left (2 \, a b^{2} + b^{2}\right )} \arctan \left (\frac{\frac{{\left (\sqrt{-{\left (b x + a\right )}^{2} + 1}{\left | b \right |} + b\right )} a}{b^{2} x + a b} - 1}{\sqrt{a^{2} - 1}}\right )}{{\left (a^{3}{\left | b \right |} - a^{2}{\left | b \right |} - a{\left | b \right |} +{\left | b \right |}\right )} \sqrt{a^{2} - 1}} + \frac{2 \,{\left (\frac{{\left (\sqrt{-{\left (b x + a\right )}^{2} + 1}{\left | b \right |} + b\right )}^{2} a^{3} b^{2}}{{\left (b^{2} x + a b\right )}^{2}} + a^{3} b^{2} - \frac{2 \,{\left (\sqrt{-{\left (b x + a\right )}^{2} + 1}{\left | b \right |} + b\right )} a^{2} b^{2}}{b^{2} x + a b} + \frac{{\left (\sqrt{-{\left (b x + a\right )}^{2} + 1}{\left | b \right |} + b\right )}^{2} a^{2} b^{2}}{{\left (b^{2} x + a b\right )}^{2}} + a^{2} b^{2} - \frac{3 \,{\left (\sqrt{-{\left (b x + a\right )}^{2} + 1}{\left | b \right |} + b\right )} a b^{2}}{b^{2} x + a b} + a b^{2} - \frac{{\left (\sqrt{-{\left (b x + a\right )}^{2} + 1}{\left | b \right |} + b\right )} b^{2}}{b^{2} x + a b} + \frac{{\left (\sqrt{-{\left (b x + a\right )}^{2} + 1}{\left | b \right |} + b\right )}^{2} b^{2}}{{\left (b^{2} x + a b\right )}^{2}}\right )}}{{\left (a^{4}{\left | b \right |} - a^{3}{\left | b \right |} - a^{2}{\left | b \right |} + a{\left | b \right |}\right )}{\left (\frac{{\left (\sqrt{-{\left (b x + a\right )}^{2} + 1}{\left | b \right |} + b\right )} a}{b^{2} x + a b} - \frac{{\left (\sqrt{-{\left (b x + a\right )}^{2} + 1}{\left | b \right |} + b\right )}^{2} a}{{\left (b^{2} x + a b\right )}^{2}} + \frac{{\left (\sqrt{-{\left (b x + a\right )}^{2} + 1}{\left | b \right |} + b\right )}^{3} a}{{\left (b^{2} x + a b\right )}^{3}} - a + \frac{2 \,{\left (\sqrt{-{\left (b x + a\right )}^{2} + 1}{\left | b \right |} + b\right )}}{b^{2} x + a b} - \frac{2 \,{\left (\sqrt{-{\left (b x + a\right )}^{2} + 1}{\left | b \right |} + b\right )}^{2}}{{\left (b^{2} x + a b\right )}^{2}}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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