\(\int \frac {1}{x^2 (\sqrt {a+b x}+\sqrt {c+b x})} \, dx\) [5]

Optimal result

Integrand size = 25, antiderivative size = 103 \[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {c+b x}\right )} \, dx=-\frac {\sqrt {a+b x}}{(a-c) x}+\frac {\sqrt {c+b x}}{(a-c) x}-\frac {b \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a} (a-c)}+\frac {b \text {arctanh}\left (\frac {\sqrt {c+b x}}{\sqrt {c}}\right )}{(a-c) \sqrt {c}} \] Output:

-(b*x+a)^(1/2)/(a-c)/x+(b*x+c)^(1/2)/(a-c)/x-b*arctanh((b*x+a)^(1/2)/a^(1/ 
2))/a^(1/2)/(a-c)+b*arctanh((b*x+c)^(1/2)/c^(1/2))/(a-c)/c^(1/2)
 

Mathematica [A] (verified)

Time = 1.70 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.82 \[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {c+b x}\right )} \, dx=\frac {\sqrt {a} \sqrt {c} \left (-\sqrt {a+b x}+\sqrt {c+b x}\right )+b \sqrt {-\left (\sqrt {a}-\sqrt {c}\right )^2} x \arctan \left (\frac {\sqrt {a+b x}-\sqrt {c+b x}}{\sqrt {-\left (\sqrt {a}-\sqrt {c}\right )^2}}\right )-b \sqrt {-\left (\sqrt {a}+\sqrt {c}\right )^2} x \arctan \left (\frac {\sqrt {a+b x}-\sqrt {c+b x}}{\sqrt {-\left (\sqrt {a}+\sqrt {c}\right )^2}}\right )}{\sqrt {a} (a-c) \sqrt {c} x} \] Input:

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

Output:

(Sqrt[a]*Sqrt[c]*(-Sqrt[a + b*x] + Sqrt[c + b*x]) + b*Sqrt[-(Sqrt[a] - Sqr 
t[c])^2]*x*ArcTan[(Sqrt[a + b*x] - Sqrt[c + b*x])/Sqrt[-(Sqrt[a] - Sqrt[c] 
)^2]] - b*Sqrt[-(Sqrt[a] + Sqrt[c])^2]*x*ArcTan[(Sqrt[a + b*x] - Sqrt[c + 
b*x])/Sqrt[-(Sqrt[a] + Sqrt[c])^2]])/(Sqrt[a]*(a - c)*Sqrt[c]*x)
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.93, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2529, 51, 73, 221}

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.

\(\displaystyle \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {b x+c}\right )} \, dx\)

\(\Big \downarrow \) 2529

\(\displaystyle \frac {\int \frac {\sqrt {a+b x}}{x^2}dx}{a-c}-\frac {\int \frac {\sqrt {c+b x}}{x^2}dx}{a-c}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {\frac {1}{2} b \int \frac {1}{x \sqrt {a+b x}}dx-\frac {\sqrt {a+b x}}{x}}{a-c}-\frac {\frac {1}{2} b \int \frac {1}{x \sqrt {c+b x}}dx-\frac {\sqrt {b x+c}}{x}}{a-c}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\int \frac {1}{\frac {a+b x}{b}-\frac {a}{b}}d\sqrt {a+b x}-\frac {\sqrt {a+b x}}{x}}{a-c}-\frac {\int \frac {1}{\frac {c+b x}{b}-\frac {c}{b}}d\sqrt {c+b x}-\frac {\sqrt {b x+c}}{x}}{a-c}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {-\frac {b \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\sqrt {a+b x}}{x}}{a-c}-\frac {-\frac {b \text {arctanh}\left (\frac {\sqrt {b x+c}}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {\sqrt {b x+c}}{x}}{a-c}\)

Input:

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

Output:

(-(Sqrt[a + b*x]/x) - (b*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/Sqrt[a])/(a - c) 
- (-(Sqrt[c + b*x]/x) - (b*ArcTanh[Sqrt[c + b*x]/Sqrt[c]])/Sqrt[c])/(a - c 
)
 

Defintions of rubi rules used

rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x 
] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

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

rule 2529

Int[(u_)/((e_.)*Sqrt[(a_.) + (b_.)*(x_)] + (f_.)*Sqrt[(c_.) + (d_.)*(x_)]), 
 x_Symbol] :> Simp[-d/(e*(b*c - a*d))   Int[u*Sqrt[a + b*x], x], x] + Simp[ 
b/(f*(b*c - a*d))   Int[u*Sqrt[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, f} 
, x] && NeQ[b*c - a*d, 0] && EqQ[b*e^2 - d*f^2, 0]
 

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.85

method	result	size
default	\(\frac {2 b \left (-\frac {\sqrt {b x +a}}{2 x b}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2 \sqrt {a}}\right )}{a -c}-\frac {2 b \left (-\frac {\sqrt {b x +c}}{2 x b}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {b x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}\right )}{a -c}\)	\(88\)

Input:

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

Output:

2/(a-c)*b*(-1/2*(b*x+a)^(1/2)/x/b-1/2/a^(1/2)*arctanh((b*x+a)^(1/2)/a^(1/2 
)))-2/(a-c)*b*(-1/2*(b*x+c)^(1/2)/x/b-1/2/c^(1/2)*arctanh((b*x+c)^(1/2)/c^ 
(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 387, normalized size of antiderivative = 3.76 \[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {c+b x}\right )} \, dx=\left [-\frac {\sqrt {a} b c x \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + a b \sqrt {c} x \log \left (\frac {b x - 2 \, \sqrt {b x + c} \sqrt {c} + 2 \, c}{x}\right ) + 2 \, \sqrt {b x + a} a c - 2 \, \sqrt {b x + c} a c}{2 \, {\left (a^{2} c - a c^{2}\right )} x}, -\frac {2 \, a b \sqrt {-c} x \arctan \left (\frac {\sqrt {-c}}{\sqrt {b x + c}}\right ) + \sqrt {a} b c x \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, \sqrt {b x + a} a c - 2 \, \sqrt {b x + c} a c}{2 \, {\left (a^{2} c - a c^{2}\right )} x}, \frac {2 \, \sqrt {-a} b c x \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x + a}}\right ) - a b \sqrt {c} x \log \left (\frac {b x - 2 \, \sqrt {b x + c} \sqrt {c} + 2 \, c}{x}\right ) - 2 \, \sqrt {b x + a} a c + 2 \, \sqrt {b x + c} a c}{2 \, {\left (a^{2} c - a c^{2}\right )} x}, \frac {\sqrt {-a} b c x \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x + a}}\right ) - a b \sqrt {-c} x \arctan \left (\frac {\sqrt {-c}}{\sqrt {b x + c}}\right ) - \sqrt {b x + a} a c + \sqrt {b x + c} a c}{{\left (a^{2} c - a c^{2}\right )} x}\right ] \] Input:

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

Output:

[-1/2*(sqrt(a)*b*c*x*log((b*x + 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + a*b*sq 
rt(c)*x*log((b*x - 2*sqrt(b*x + c)*sqrt(c) + 2*c)/x) + 2*sqrt(b*x + a)*a*c 
 - 2*sqrt(b*x + c)*a*c)/((a^2*c - a*c^2)*x), -1/2*(2*a*b*sqrt(-c)*x*arctan 
(sqrt(-c)/sqrt(b*x + c)) + sqrt(a)*b*c*x*log((b*x + 2*sqrt(b*x + a)*sqrt(a 
) + 2*a)/x) + 2*sqrt(b*x + a)*a*c - 2*sqrt(b*x + c)*a*c)/((a^2*c - a*c^2)* 
x), 1/2*(2*sqrt(-a)*b*c*x*arctan(sqrt(-a)/sqrt(b*x + a)) - a*b*sqrt(c)*x*l 
og((b*x - 2*sqrt(b*x + c)*sqrt(c) + 2*c)/x) - 2*sqrt(b*x + a)*a*c + 2*sqrt 
(b*x + c)*a*c)/((a^2*c - a*c^2)*x), (sqrt(-a)*b*c*x*arctan(sqrt(-a)/sqrt(b 
*x + a)) - a*b*sqrt(-c)*x*arctan(sqrt(-c)/sqrt(b*x + c)) - sqrt(b*x + a)*a 
*c + sqrt(b*x + c)*a*c)/((a^2*c - a*c^2)*x)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

\[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {c+b x}\right )} \, dx=\int { \frac {1}{x^{2} {\left (\sqrt {b x + a} + \sqrt {b x + c}\right )}} \,d x } \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1190 vs. \(2 (87) = 174\).

Time = 2.01 (sec) , antiderivative size = 1190, normalized size of antiderivative = 11.55 \[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {c+b x}\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

b*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*(a - c)) + (2*(a*c^2 + sqrt(a*c 
)*c^2)*(a - c)^2*b*sgn(2*a - 2*c) + 2*(a*c^2 + sqrt(a*c)*a*c)*(a - c)^2*b 
+ (a^2*c^2 - 2*a*c^3 + c^4 + (a^2*c - 2*a*c^2 + c^3)*sqrt(a*c))*b*abs(a - 
c)*sgn(2*a - 2*c) + (a^3*c - 2*a^2*c^2 + a*c^3 + (a^2*c - 2*a*c^2 + c^3)*s 
qrt(a*c))*b*abs(a - c) - (a^4*c - a^3*c^2 - a^2*c^3 + a*c^4 + (a^3*c - a^2 
*c^2 - a*c^3 + c^4)*sqrt(a*c))*b*sgn(2*a - 2*c) - (a^4*c - a^3*c^2 - a^2*c 
^3 + a*c^4 + (a^4 - a^3*c - a^2*c^2 + a*c^3)*sqrt(a*c))*b)*arctan(-(sqrt(b 
*x + a) - sqrt(b*x + c))/sqrt(-(a^2 - c^2 + sqrt((a^2 - c^2)^2 - (a^3 - 3* 
a^2*c + 3*a*c^2 - c^3)*(a - c)))/(a - c)))/((sqrt(-a)*a^4*c - a^4*sqrt(-c) 
*c - 4*sqrt(-a)*a^3*c^2 + 4*a^3*sqrt(-c)*c^2 + 6*sqrt(-a)*a^2*c^3 - 6*a^2* 
sqrt(-c)*c^3 - 4*sqrt(-a)*a*c^4 + 4*a*sqrt(-c)*c^4 + sqrt(-a)*c^5 - sqrt(- 
c)*c^5)*abs(a - c)) - (2*(a*c^2 + sqrt(a*c)*c^2)*(a - c)^2*b*sgn(2*a - 2*c 
) - 2*(a*c^2 - sqrt(a*c)*a*c)*(a - c)^2*b + (a^2*c^2 - 2*a*c^3 + c^4 + (a^ 
2*c - 2*a*c^2 + c^3)*sqrt(a*c))*b*abs(a - c)*sgn(2*a - 2*c) - (a^3*c - 2*a 
^2*c^2 + a*c^3 - (a^2*c - 2*a*c^2 + c^3)*sqrt(a*c))*b*abs(a - c) - (a^4*c 
- a^3*c^2 - a^2*c^3 + a*c^4 - (a^3*c - a^2*c^2 - a*c^3 + c^4)*sqrt(a*c))*b 
*sgn(2*a - 2*c) + (a^4*c - a^3*c^2 - a^2*c^3 + a*c^4 + (a^4 - a^3*c - a^2* 
c^2 + a*c^3)*sqrt(a*c))*b)*arctan(-(sqrt(b*x + a) - sqrt(b*x + c))/sqrt(-( 
a^2 - c^2 - sqrt((a^2 - c^2)^2 - (a^3 - 3*a^2*c + 3*a*c^2 - c^3)*(a - c))) 
/(a - c)))/((sqrt(-a)*a^4*c - a^4*sqrt(-c)*c - 4*sqrt(-a)*a^3*c^2 + 4*a...
 

Mupad [B] (verification not implemented)

Time = 40.89 (sec) , antiderivative size = 2642, normalized size of antiderivative = 25.65 \[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {c+b x}\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

(b*atan(((b*(a*c^(1/2) + a^(1/2)*c)*((a^(1/2)*c^(3/2) - 2*a*c + a^(3/2)*c^ 
(1/2))*(2*a*c + a^(1/2)*c^(3/2) + a^(3/2)*c^(1/2)))^(1/2)*((a^3*b*c^(7/2) 
- a^(7/2)*b*c^3 - a^2*b*c^(9/2) + a^(9/2)*b*c^2)/(a^3*c^5 - 2*a^4*c^4 + a^ 
5*c^3) + (((a + b*x)^(1/2) - a^(1/2))*(2*a^(3/2)*b*c^5 - 2*a^5*b*c^(3/2) + 
 2*a^4*b*c^(5/2) - 2*a^(5/2)*b*c^4))/(2*((c + b*x)^(1/2) - c^(1/2))*(a^3*c 
^5 - 2*a^4*c^4 + a^5*c^3)) - (b*(a*c^(1/2) + a^(1/2)*c)*((a^(5/2)*c^(11/2) 
 - a^(7/2)*c^(9/2) - a^(9/2)*c^(7/2) + a^(11/2)*c^(5/2))/(a^3*c^5 - 2*a^4* 
c^4 + a^5*c^3) - (((a + b*x)^(1/2) - a^(1/2))*(4*a^2*c^6 - 12*a^3*c^5 + 16 
*a^4*c^4 - 12*a^5*c^3 + 4*a^6*c^2))/(2*((c + b*x)^(1/2) - c^(1/2))*(a^3*c^ 
5 - 2*a^4*c^4 + a^5*c^3)))*((a^(1/2)*c^(3/2) - 2*a*c + a^(3/2)*c^(1/2))*(2 
*a*c + a^(1/2)*c^(3/2) + a^(3/2)*c^(1/2)))^(1/2))/(2*(2*a^2*c^3 - 2*a^3*c^ 
2 + a^(3/2)*c^(7/2) - a^(7/2)*c^(3/2))))*1i)/(2*(2*a^2*c^3 - 2*a^3*c^2 + a 
^(3/2)*c^(7/2) - a^(7/2)*c^(3/2))) + (b*(a*c^(1/2) + a^(1/2)*c)*((a^(1/2)* 
c^(3/2) - 2*a*c + a^(3/2)*c^(1/2))*(2*a*c + a^(1/2)*c^(3/2) + a^(3/2)*c^(1 
/2)))^(1/2)*((a^3*b*c^(7/2) - a^(7/2)*b*c^3 - a^2*b*c^(9/2) + a^(9/2)*b*c^ 
2)/(a^3*c^5 - 2*a^4*c^4 + a^5*c^3) + (((a + b*x)^(1/2) - a^(1/2))*(2*a^(3/ 
2)*b*c^5 - 2*a^5*b*c^(3/2) + 2*a^4*b*c^(5/2) - 2*a^(5/2)*b*c^4))/(2*((c + 
b*x)^(1/2) - c^(1/2))*(a^3*c^5 - 2*a^4*c^4 + a^5*c^3)) + (b*(a*c^(1/2) + a 
^(1/2)*c)*((a^(5/2)*c^(11/2) - a^(7/2)*c^(9/2) - a^(9/2)*c^(7/2) + a^(11/2 
)*c^(5/2))/(a^3*c^5 - 2*a^4*c^4 + a^5*c^3) - (((a + b*x)^(1/2) - a^(1/2...
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 488, normalized size of antiderivative = 4.74 \[ \int \frac {1}{x^2 \left (\sqrt {a+b x}+\sqrt {c+b x}\right )} \, dx=\frac {-2 \sqrt {c}\, \sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-a -c}\, \mathit {atan} \left (\frac {\sqrt {b x +c}+\sqrt {b x +a}}{\sqrt {2 \sqrt {c}\, \sqrt {a}-a -c}}\right ) a b x -2 \sqrt {c}\, \sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-a -c}\, \mathit {atan} \left (\frac {\sqrt {b x +c}+\sqrt {b x +a}}{\sqrt {2 \sqrt {c}\, \sqrt {a}-a -c}}\right ) b c x -4 \sqrt {2 \sqrt {c}\, \sqrt {a}-a -c}\, \mathit {atan} \left (\frac {\sqrt {b x +c}+\sqrt {b x +a}}{\sqrt {2 \sqrt {c}\, \sqrt {a}-a -c}}\right ) a b c x +2 \sqrt {b x +c}\, a^{2} c -2 \sqrt {b x +c}\, a \,c^{2}-\sqrt {c}\, \sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+a +c}\, \mathrm {log}\left (\sqrt {b x +c}-\sqrt {2 \sqrt {c}\, \sqrt {a}+a +c}+\sqrt {b x +a}\right ) a b x -\sqrt {c}\, \sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+a +c}\, \mathrm {log}\left (\sqrt {b x +c}-\sqrt {2 \sqrt {c}\, \sqrt {a}+a +c}+\sqrt {b x +a}\right ) b c x +\sqrt {c}\, \sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+a +c}\, \mathrm {log}\left (\sqrt {b x +c}+\sqrt {2 \sqrt {c}\, \sqrt {a}+a +c}+\sqrt {b x +a}\right ) a b x +\sqrt {c}\, \sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+a +c}\, \mathrm {log}\left (\sqrt {b x +c}+\sqrt {2 \sqrt {c}\, \sqrt {a}+a +c}+\sqrt {b x +a}\right ) b c x +2 \sqrt {2 \sqrt {c}\, \sqrt {a}+a +c}\, \mathrm {log}\left (\sqrt {b x +c}-\sqrt {2 \sqrt {c}\, \sqrt {a}+a +c}+\sqrt {b x +a}\right ) a b c x -2 \sqrt {2 \sqrt {c}\, \sqrt {a}+a +c}\, \mathrm {log}\left (\sqrt {b x +c}+\sqrt {2 \sqrt {c}\, \sqrt {a}+a +c}+\sqrt {b x +a}\right ) a b c x -2 \sqrt {b x +a}\, a^{2} c +2 \sqrt {b x +a}\, a \,c^{2}}{2 a c x \left (a^{2}-2 a c +c^{2}\right )} \] Input:

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

Output:

( - 2*sqrt(c)*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) - a - c)*atan((sqrt(b*x + c) 
+ sqrt(a + b*x))/sqrt(2*sqrt(c)*sqrt(a) - a - c))*a*b*x - 2*sqrt(c)*sqrt(a 
)*sqrt(2*sqrt(c)*sqrt(a) - a - c)*atan((sqrt(b*x + c) + sqrt(a + b*x))/sqr 
t(2*sqrt(c)*sqrt(a) - a - c))*b*c*x - 4*sqrt(2*sqrt(c)*sqrt(a) - a - c)*at 
an((sqrt(b*x + c) + sqrt(a + b*x))/sqrt(2*sqrt(c)*sqrt(a) - a - c))*a*b*c* 
x + 2*sqrt(b*x + c)*a**2*c - 2*sqrt(b*x + c)*a*c**2 - sqrt(c)*sqrt(a)*sqrt 
(2*sqrt(c)*sqrt(a) + a + c)*log(sqrt(b*x + c) - sqrt(2*sqrt(c)*sqrt(a) + a 
 + c) + sqrt(a + b*x))*a*b*x - sqrt(c)*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + a 
+ c)*log(sqrt(b*x + c) - sqrt(2*sqrt(c)*sqrt(a) + a + c) + sqrt(a + b*x))* 
b*c*x + sqrt(c)*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + a + c)*log(sqrt(b*x + c) 
+ sqrt(2*sqrt(c)*sqrt(a) + a + c) + sqrt(a + b*x))*a*b*x + sqrt(c)*sqrt(a) 
*sqrt(2*sqrt(c)*sqrt(a) + a + c)*log(sqrt(b*x + c) + sqrt(2*sqrt(c)*sqrt(a 
) + a + c) + sqrt(a + b*x))*b*c*x + 2*sqrt(2*sqrt(c)*sqrt(a) + a + c)*log( 
sqrt(b*x + c) - sqrt(2*sqrt(c)*sqrt(a) + a + c) + sqrt(a + b*x))*a*b*c*x - 
 2*sqrt(2*sqrt(c)*sqrt(a) + a + c)*log(sqrt(b*x + c) + sqrt(2*sqrt(c)*sqrt 
(a) + a + c) + sqrt(a + b*x))*a*b*c*x - 2*sqrt(a + b*x)*a**2*c + 2*sqrt(a 
+ b*x)*a*c**2)/(2*a*c*x*(a**2 - 2*a*c + c**2))