\(\int \frac {e^{-\text {arctanh}(a x)}}{(c-\frac {c}{a x})^{3/2}} \, dx\) [573]

Optimal result

Integrand size = 24, antiderivative size = 159 \[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=-\frac {(1-a x)^{3/2} \sqrt {1+a x}}{a^2 \left (c-\frac {c}{a x}\right )^{3/2} x}-\frac {(1-a x)^{3/2} \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{a^{5/2} \left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}}+\frac {\sqrt {2} (1-a x)^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )}{a^{5/2} \left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}} \] Output:

-(-a*x+1)^(3/2)*(a*x+1)^(1/2)/a^2/(c-c/a/x)^(3/2)/x-(-a*x+1)^(3/2)*arcsinh 
(a^(1/2)*x^(1/2))/a^(5/2)/(c-c/a/x)^(3/2)/x^(3/2)+(-a*x+1)^(3/2)*arctanh(2 
^(1/2)*a^(1/2)*x^(1/2)/(a*x+1)^(1/2))*2^(1/2)/a^(5/2)/(c-c/a/x)^(3/2)/x^(3 
/2)
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.67 \[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=\frac {(1-a x)^{3/2} \left (-\frac {\sqrt {x} \sqrt {1+a x}}{a^2}-\frac {\text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{a^{5/2}}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )}{a^{5/2}}\right )}{\left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}} \] Input:

Integrate[1/(E^ArcTanh[a*x]*(c - c/(a*x))^(3/2)),x]
 

Output:

((1 - a*x)^(3/2)*(-((Sqrt[x]*Sqrt[1 + a*x])/a^2) - ArcSinh[Sqrt[a]*Sqrt[x] 
]/a^(5/2) + (Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]])/a^( 
5/2)))/((c - c/(a*x))^(3/2)*x^(3/2))
 

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.73, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6684, 6679, 113, 27, 140, 27, 63, 104, 219, 222}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 6684

\(\displaystyle \frac {(1-a x)^{3/2} \int \frac {e^{-\text {arctanh}(a x)} x^{3/2}}{(1-a x)^{3/2}}dx}{x^{3/2} \left (c-\frac {c}{a x}\right )^{3/2}}\)

\(\Big \downarrow \) 6679

\(\displaystyle \frac {(1-a x)^{3/2} \int \frac {x^{3/2}}{(1-a x) \sqrt {a x+1}}dx}{x^{3/2} \left (c-\frac {c}{a x}\right )^{3/2}}\)

\(\Big \downarrow \) 113

\(\displaystyle \frac {(1-a x)^{3/2} \left (-\frac {\int -\frac {\sqrt {a x+1}}{2 \sqrt {x} (1-a x)}dx}{a^2}-\frac {\sqrt {x} \sqrt {a x+1}}{a^2}\right )}{x^{3/2} \left (c-\frac {c}{a x}\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(1-a x)^{3/2} \left (\frac {\int \frac {\sqrt {a x+1}}{\sqrt {x} (1-a x)}dx}{2 a^2}-\frac {\sqrt {x} \sqrt {a x+1}}{a^2}\right )}{x^{3/2} \left (c-\frac {c}{a x}\right )^{3/2}}\)

\(\Big \downarrow \) 140

\(\displaystyle \frac {(1-a x)^{3/2} \left (\frac {\int \frac {2}{\sqrt {x} (1-a x) \sqrt {a x+1}}dx-\int \frac {1}{\sqrt {x} \sqrt {a x+1}}dx}{2 a^2}-\frac {\sqrt {x} \sqrt {a x+1}}{a^2}\right )}{x^{3/2} \left (c-\frac {c}{a x}\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(1-a x)^{3/2} \left (\frac {2 \int \frac {1}{\sqrt {x} (1-a x) \sqrt {a x+1}}dx-\int \frac {1}{\sqrt {x} \sqrt {a x+1}}dx}{2 a^2}-\frac {\sqrt {x} \sqrt {a x+1}}{a^2}\right )}{x^{3/2} \left (c-\frac {c}{a x}\right )^{3/2}}\)

\(\Big \downarrow \) 63

\(\displaystyle \frac {(1-a x)^{3/2} \left (\frac {2 \int \frac {1}{\sqrt {x} (1-a x) \sqrt {a x+1}}dx-2 \int \frac {1}{\sqrt {a x+1}}d\sqrt {x}}{2 a^2}-\frac {\sqrt {x} \sqrt {a x+1}}{a^2}\right )}{x^{3/2} \left (c-\frac {c}{a x}\right )^{3/2}}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {(1-a x)^{3/2} \left (\frac {4 \int \frac {1}{1-\frac {2 a x}{a x+1}}d\frac {\sqrt {x}}{\sqrt {a x+1}}-2 \int \frac {1}{\sqrt {a x+1}}d\sqrt {x}}{2 a^2}-\frac {\sqrt {x} \sqrt {a x+1}}{a^2}\right )}{x^{3/2} \left (c-\frac {c}{a x}\right )^{3/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {(1-a x)^{3/2} \left (\frac {\frac {2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )}{\sqrt {a}}-2 \int \frac {1}{\sqrt {a x+1}}d\sqrt {x}}{2 a^2}-\frac {\sqrt {x} \sqrt {a x+1}}{a^2}\right )}{x^{3/2} \left (c-\frac {c}{a x}\right )^{3/2}}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {(1-a x)^{3/2} \left (\frac {\frac {2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )}{\sqrt {a}}-\frac {2 \text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a}}}{2 a^2}-\frac {\sqrt {x} \sqrt {a x+1}}{a^2}\right )}{x^{3/2} \left (c-\frac {c}{a x}\right )^{3/2}}\)

Input:

Int[1/(E^ArcTanh[a*x]*(c - c/(a*x))^(3/2)),x]
 

Output:

((1 - a*x)^(3/2)*(-((Sqrt[x]*Sqrt[1 + a*x])/a^2) + ((-2*ArcSinh[Sqrt[a]*Sq 
rt[x]])/Sqrt[a] + (2*Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a* 
x]])/Sqrt[a])/(2*a^2)))/((c - c/(a*x))^(3/2)*x^(3/2))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 63

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2/b   S 
ubst[Int[1/Sqrt[c + d*(x^2/b)], x], x, Sqrt[b*x]], x] /; FreeQ[{b, c, d}, x 
] && GtQ[c, 0]
 

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[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] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 113

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1))   Int[(a + b*x) 
^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m 
 - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m 
 + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & 
& GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]
 

rule 140

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*d^(m + n)*f^p   Int[(a + b*x)^(m - 1)/(c + d*x)^m, x] 
, x] + Int[(a + b*x)^(m - 1)*((e + f*x)^p/(c + d*x)^m)*ExpandToSum[(a + b*x 
)*(c + d*x)^(-p - 1) - (b*d^(-p - 1)*f^p)/(e + f*x)^p, x], x] /; FreeQ[{a, 
b, c, d, e, f, m, n}, x] && EqQ[m + n + p + 1, 0] && ILtQ[p, 0] && (GtQ[m, 
0] || SumSimplerQ[m, -1] ||  !(GtQ[n, 0] || SumSimplerQ[n, -1]))
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt 
[a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
 

rule 6679

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

rule 6684

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_), x_Symbol] 
 :> Simp[x^p*((c + d/x)^p/(1 + c*(x/d))^p)   Int[u*(1 + c*(x/d))^p*(E^(n*Ar 
cTanh[a*x])/x^p), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 - a^2*d^2, 
 0] &&  !IntegerQ[p]
 

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.06

method	result	size
default	\(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \sqrt {-a^{2} x^{2}+1}\, \left (2 \sqrt {-x \left (a x +1\right )}\, a^{\frac {3}{2}} \sqrt {2}\, \sqrt {-\frac {1}{a}}-\arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-x \left (a x +1\right )}}\right ) a \sqrt {2}\, \sqrt {-\frac {1}{a}}+2 \ln \left (\frac {2 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-x \left (a x +1\right )}\, a -3 a x -1}{a x -1}\right ) \sqrt {a}\right ) \sqrt {2}}{4 a^{\frac {3}{2}} c^{2} \left (a x -1\right ) \sqrt {-x \left (a x +1\right )}\, \sqrt {-\frac {1}{a}}}\)	\(168\)
risch	\(-\frac {\left (a x +1\right ) \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}\, \left (a x -1\right )}{a \sqrt {-\left (a x +1\right ) a c x}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {-a^{2} x^{2}+1}\, c}-\frac {\left (\frac {\arctan \left (\frac {\sqrt {a^{2} c}\, \left (x +\frac {1}{2 a}\right )}{\sqrt {-a^{2} c \,x^{2}-a c x}}\right )}{2 a^{2} \sqrt {a^{2} c}}-\frac {\ln \left (\frac {-4 c -3 \left (x -\frac {1}{a}\right ) a c +2 \sqrt {-2 c}\, \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -3 \left (x -\frac {1}{a}\right ) a c -2 c}}{x -\frac {1}{a}}\right )}{a^{3} \sqrt {-2 c}}\right ) a \sqrt {\frac {c a x \left (-a^{2} x^{2}+1\right )}{a x -1}}\, \left (a x -1\right )}{x \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {-a^{2} x^{2}+1}\, c}\)	\(267\)

Input:

int(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a/x)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 453, normalized size of antiderivative = 2.85 \[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=\left [\frac {4 \, \sqrt {-a^{2} x^{2} + 1} a x \sqrt {\frac {a c x - c}{a x}} + \sqrt {2} {\left (a c x - c\right )} \sqrt {-\frac {1}{c}} \log \left (-\frac {17 \, a^{3} x^{3} - 3 \, a^{2} x^{2} + 4 \, \sqrt {2} {\left (3 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-\frac {1}{c}} \sqrt {\frac {a c x - c}{a x}} - 13 \, a x - 1}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) - {\left (a x - 1\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right )}{4 \, {\left (a^{2} c^{2} x - a c^{2}\right )}}, \frac {2 \, \sqrt {-a^{2} x^{2} + 1} a x \sqrt {\frac {a c x - c}{a x}} + {\left (a x - 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - \frac {\sqrt {2} {\left (a c x - c\right )} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} a x \sqrt {\frac {a c x - c}{a x}}}{{\left (3 \, a^{2} x^{2} - 2 \, a x - 1\right )} \sqrt {c}}\right )}{\sqrt {c}}}{2 \, {\left (a^{2} c^{2} x - a c^{2}\right )}}\right ] \] Input:

integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a/x)^(3/2),x, algorithm="frica 
s")
 

Output:

[1/4*(4*sqrt(-a^2*x^2 + 1)*a*x*sqrt((a*c*x - c)/(a*x)) + sqrt(2)*(a*c*x - 
c)*sqrt(-1/c)*log(-(17*a^3*x^3 - 3*a^2*x^2 + 4*sqrt(2)*(3*a^2*x^2 + a*x)*s 
qrt(-a^2*x^2 + 1)*sqrt(-1/c)*sqrt((a*c*x - c)/(a*x)) - 13*a*x - 1)/(a^3*x^ 
3 - 3*a^2*x^2 + 3*a*x - 1)) - (a*x - 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c 
*x + 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x 
)) - c)/(a*x - 1)))/(a^2*c^2*x - a*c^2), 1/2*(2*sqrt(-a^2*x^2 + 1)*a*x*sqr 
t((a*c*x - c)/(a*x)) + (a*x - 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqr 
t(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - sqrt(2)*(a*c*x 
 - c)*arctan(2*sqrt(2)*sqrt(-a^2*x^2 + 1)*a*x*sqrt((a*c*x - c)/(a*x))/((3* 
a^2*x^2 - 2*a*x - 1)*sqrt(c)))/sqrt(c))/(a^2*c^2*x - a*c^2)]
 

Sympy [F]

\[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=\int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{\frac {3}{2}} \left (a x + 1\right )}\, dx \] Input:

integrate(1/(a*x+1)*(-a**2*x**2+1)**(1/2)/(c-c/a/x)**(3/2),x)
 

Output:

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

Maxima [F]

\[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a x + 1\right )} {\left (c - \frac {c}{a x}\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a/x)^(3/2),x, algorithm="maxim 
a")
 

Output:

integrate(sqrt(-a^2*x^2 + 1)/((a*x + 1)*(c - c/(a*x))^(3/2)), x)
 

Giac [F]

\[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a x + 1\right )} {\left (c - \frac {c}{a x}\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a/x)^(3/2),x, algorithm="giac" 
)
 

Output:

integrate(sqrt(-a^2*x^2 + 1)/((a*x + 1)*(c - c/(a*x))^(3/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=\int \frac {\sqrt {1-a^2\,x^2}}{{\left (c-\frac {c}{a\,x}\right )}^{3/2}\,\left (a\,x+1\right )} \,d x \] Input:

int((1 - a^2*x^2)^(1/2)/((c - c/(a*x))^(3/2)*(a*x + 1)),x)
 

Output:

int((1 - a^2*x^2)^(1/2)/((c - c/(a*x))^(3/2)*(a*x + 1)), x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.43 \[ \int \frac {e^{-\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx=\frac {\sqrt {c}\, \left (-\sqrt {2}\, \mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, \sqrt {2}\, i}{a x +1}\right )+\mathit {atan} \left (\frac {\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, i}{a x +1}\right )+\sqrt {x}\, \sqrt {a}\, \sqrt {a x +1}\, i \right )}{a \,c^{2}} \] Input:

int(1/(a*x+1)*(-a^2*x^2+1)^(1/2)/(c-c/a/x)^(3/2),x)
 

Output:

(sqrt(c)*( - sqrt(2)*atan((sqrt(x)*sqrt(a)*sqrt(a*x + 1)*sqrt(2)*i)/(a*x + 
 1)) + atan((sqrt(x)*sqrt(a)*sqrt(a*x + 1)*i)/(a*x + 1)) + sqrt(x)*sqrt(a) 
*sqrt(a*x + 1)*i))/(a*c**2)