3.5.41 \(\int \frac {x}{(\sqrt {a+b x}+\sqrt {a+c x})^3} \, dx\) [441]

3.5.41.1 Optimal result

Integrand size = 23, antiderivative size = 157 \[ \int \frac {x}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^3} \, dx=\frac {2 (b+3 c) \sqrt {a+b x}}{(b-c)^3}-\frac {4 a \sqrt {a+b x}}{(b-c)^3 x}-\frac {2 (3 b+c) \sqrt {a+c x}}{(b-c)^3}+\frac {4 a \sqrt {a+c x}}{(b-c)^3 x}-\frac {6 \sqrt {a} (b+c) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{(b-c)^3}+\frac {6 \sqrt {a} (b+c) \text {arctanh}\left (\frac {\sqrt {a+c x}}{\sqrt {a}}\right )}{(b-c)^3} \]

output

-6*(b+c)*arctanh((b*x+a)^(1/2)/a^(1/2))*a^(1/2)/(b-c)^3+6*(b+c)*arctanh((c 
*x+a)^(1/2)/a^(1/2))*a^(1/2)/(b-c)^3+2*(b+3*c)*(b*x+a)^(1/2)/(b-c)^3-4*a*( 
b*x+a)^(1/2)/(b-c)^3/x-2*(3*b+c)*(c*x+a)^(1/2)/(b-c)^3+4*a*(c*x+a)^(1/2)/( 
b-c)^3/x
 

3.5.41.2 Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(690\) vs. \(2(157)=314\).

Time = 10.68 (sec) , antiderivative size = 690, normalized size of antiderivative = 4.39 \[ \int \frac {x}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^3} \, dx=\frac {2 \sqrt {b-c} \left (b \sqrt {a-\frac {a b}{c}} c^2 x \left (b x+c x-2 \sqrt {a+b x} \sqrt {a+c x}\right )+a \left (4 b \sqrt {a-\frac {a b}{c}} c \sqrt {a+b x} \sqrt {a+c x}+6 \sqrt {a-\frac {a b}{c}} c^2 \sqrt {a+b x} \sqrt {a+c x}+2 c^3 x \left (\sqrt {a-\frac {a b}{c}}-\sqrt {a+b x}\right )+b^2 c x \left (\sqrt {a-\frac {a b}{c}}+\sqrt {a+b x}-5 \sqrt {a+c x}\right )+b c^2 x \left (7 \sqrt {a-\frac {a b}{c}}-5 \sqrt {a+b x}-\sqrt {a+c x}\right )\right )+a^2 \left (2 c^2 \left (\sqrt {a-\frac {a b}{c}}-\sqrt {a+b x}-3 \sqrt {a+c x}\right )+b c \left (6 \sqrt {a-\frac {a b}{c}}-5 \sqrt {a+b x}-\sqrt {a+c x}\right )+b^2 \left (\sqrt {a+b x}+\sqrt {a+c x}\right )\right )\right )-12 a \sqrt {c} (b+c) \left (2 \sqrt {a-\frac {a b}{c}} c \sqrt {a+b x} \sqrt {a+c x}+2 a c \left (\sqrt {a-\frac {a b}{c}}-\sqrt {a+b x}-\sqrt {a+c x}\right )+b c x \left (2 \sqrt {a-\frac {a b}{c}}-\sqrt {a+b x}-\sqrt {a+c x}\right )+a b \left (\sqrt {a+b x}+\sqrt {a+c x}\right )\right ) \arctan \left (\frac {\sqrt {b-c} \sqrt {a+c x}}{\sqrt {c} \left (-\sqrt {a-\frac {a b}{c}}+\sqrt {a+b x}+\sqrt {a+c x}\right )}\right )}{(b-c)^{5/2} c \left (a (b-c)+\sqrt {a-\frac {a b}{c}} c \sqrt {a+b x}\right ) \left (a+b x-\sqrt {a-\frac {a b}{c}} \sqrt {a+b x}-\sqrt {a-\frac {a b}{c}} \sqrt {a+c x}+\sqrt {a+b x} \sqrt {a+c x}\right )} \]

input

Integrate[x/(Sqrt[a + b*x] + Sqrt[a + c*x])^3,x]
 

output

(2*Sqrt[b - c]*(b*Sqrt[a - (a*b)/c]*c^2*x*(b*x + c*x - 2*Sqrt[a + b*x]*Sqr 
t[a + c*x]) + a*(4*b*Sqrt[a - (a*b)/c]*c*Sqrt[a + b*x]*Sqrt[a + c*x] + 6*S 
qrt[a - (a*b)/c]*c^2*Sqrt[a + b*x]*Sqrt[a + c*x] + 2*c^3*x*(Sqrt[a - (a*b) 
/c] - Sqrt[a + b*x]) + b^2*c*x*(Sqrt[a - (a*b)/c] + Sqrt[a + b*x] - 5*Sqrt 
[a + c*x]) + b*c^2*x*(7*Sqrt[a - (a*b)/c] - 5*Sqrt[a + b*x] - Sqrt[a + c*x 
])) + a^2*(2*c^2*(Sqrt[a - (a*b)/c] - Sqrt[a + b*x] - 3*Sqrt[a + c*x]) + b 
*c*(6*Sqrt[a - (a*b)/c] - 5*Sqrt[a + b*x] - Sqrt[a + c*x]) + b^2*(Sqrt[a + 
 b*x] + Sqrt[a + c*x]))) - 12*a*Sqrt[c]*(b + c)*(2*Sqrt[a - (a*b)/c]*c*Sqr 
t[a + b*x]*Sqrt[a + c*x] + 2*a*c*(Sqrt[a - (a*b)/c] - Sqrt[a + b*x] - Sqrt 
[a + c*x]) + b*c*x*(2*Sqrt[a - (a*b)/c] - Sqrt[a + b*x] - Sqrt[a + c*x]) + 
 a*b*(Sqrt[a + b*x] + Sqrt[a + c*x]))*ArcTan[(Sqrt[b - c]*Sqrt[a + c*x])/( 
Sqrt[c]*(-Sqrt[a - (a*b)/c] + Sqrt[a + b*x] + Sqrt[a + c*x]))])/((b - c)^( 
5/2)*c*(a*(b - c) + Sqrt[a - (a*b)/c]*c*Sqrt[a + b*x])*(a + b*x - Sqrt[a - 
 (a*b)/c]*Sqrt[a + b*x] - Sqrt[a - (a*b)/c]*Sqrt[a + c*x] + Sqrt[a + b*x]* 
Sqrt[a + c*x]))
 

3.5.41.3 Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.11, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {7241, 2009}

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.

\(\displaystyle \int \frac {x}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^3} \, dx\)

\(\Big \downarrow \) 7241

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-2 \sqrt {a} (b+3 c) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )+2 \sqrt {a} (3 b+c) \text {arctanh}\left (\frac {\sqrt {a+c x}}{\sqrt {a}}\right )-4 \sqrt {a} b \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )+4 \sqrt {a} c \text {arctanh}\left (\frac {\sqrt {a+c x}}{\sqrt {a}}\right )+2 (b+3 c) \sqrt {a+b x}-2 (3 b+c) \sqrt {a+c x}-\frac {4 a \sqrt {a+b x}}{x}+\frac {4 a \sqrt {a+c x}}{x}}{(b-c)^3}\)

input

Int[x/(Sqrt[a + b*x] + Sqrt[a + c*x])^3,x]
 

output

(2*(b + 3*c)*Sqrt[a + b*x] - (4*a*Sqrt[a + b*x])/x - 2*(3*b + c)*Sqrt[a + 
c*x] + (4*a*Sqrt[a + c*x])/x - 4*Sqrt[a]*b*ArcTanh[Sqrt[a + b*x]/Sqrt[a]] 
- 2*Sqrt[a]*(b + 3*c)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]] + 4*Sqrt[a]*c*ArcTanh 
[Sqrt[a + c*x]/Sqrt[a]] + 2*Sqrt[a]*(3*b + c)*ArcTanh[Sqrt[a + c*x]/Sqrt[a 
]])/(b - c)^3
 

3.5.41.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7241

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

3.5.41.4 Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.51

method	result	size
default	\(\frac {b \left (2 \sqrt {b x +a}-2 \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )\right )}{\left (b -c \right )^{3}}+\frac {8 a b \left (-\frac {\sqrt {b x +a}}{2 x b}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2 \sqrt {a}}\right )}{\left (b -c \right )^{3}}-\frac {8 a c \left (-\frac {\sqrt {c x +a}}{2 c x}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {c x +a}}{\sqrt {a}}\right )}{2 \sqrt {a}}\right )}{\left (b -c \right )^{3}}+\frac {3 c \left (2 \sqrt {b x +a}-2 \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )\right )}{\left (b -c \right )^{3}}-\frac {3 b \left (2 \sqrt {c x +a}-2 \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {c x +a}}{\sqrt {a}}\right )\right )}{\left (b -c \right )^{3}}-\frac {c \left (2 \sqrt {c x +a}-2 \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {c x +a}}{\sqrt {a}}\right )\right )}{\left (b -c \right )^{3}}\)	\(237\)

input

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

output

1/(b-c)^3*b*(2*(b*x+a)^(1/2)-2*a^(1/2)*arctanh((b*x+a)^(1/2)/a^(1/2)))+8*a 
/(b-c)^3*b*(-1/2*(b*x+a)^(1/2)/x/b-1/2*arctanh((b*x+a)^(1/2)/a^(1/2))/a^(1 
/2))-8*a/(b-c)^3*c*(-1/2*(c*x+a)^(1/2)/c/x-1/2/a^(1/2)*arctanh((c*x+a)^(1/ 
2)/a^(1/2)))+3/(b-c)^3*c*(2*(b*x+a)^(1/2)-2*a^(1/2)*arctanh((b*x+a)^(1/2)/ 
a^(1/2)))-3/(b-c)^3*b*(2*(c*x+a)^(1/2)-2*a^(1/2)*arctanh((c*x+a)^(1/2)/a^( 
1/2)))-1/(b-c)^3*c*(2*(c*x+a)^(1/2)-2*a^(1/2)*arctanh((c*x+a)^(1/2)/a^(1/2 
)))
 

3.5.41.5 Fricas [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.66 \[ \int \frac {x}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^3} \, dx=\left [-\frac {3 \, \sqrt {a} {\left (b + c\right )} x \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 3 \, \sqrt {a} {\left (b + c\right )} x \log \left (\frac {c x - 2 \, \sqrt {c x + a} \sqrt {a} + 2 \, a}{x}\right ) - 2 \, {\left ({\left (b + 3 \, c\right )} x - 2 \, a\right )} \sqrt {b x + a} + 2 \, {\left ({\left (3 \, b + c\right )} x - 2 \, a\right )} \sqrt {c x + a}}{{\left (b^{3} - 3 \, b^{2} c + 3 \, b c^{2} - c^{3}\right )} x}, \frac {2 \, {\left (3 \, \sqrt {-a} {\left (b + c\right )} x \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - 3 \, \sqrt {-a} {\left (b + c\right )} x \arctan \left (\frac {\sqrt {c x + a} \sqrt {-a}}{a}\right ) + {\left ({\left (b + 3 \, c\right )} x - 2 \, a\right )} \sqrt {b x + a} - {\left ({\left (3 \, b + c\right )} x - 2 \, a\right )} \sqrt {c x + a}\right )}}{{\left (b^{3} - 3 \, b^{2} c + 3 \, b c^{2} - c^{3}\right )} x}\right ] \]

input

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

output

[-(3*sqrt(a)*(b + c)*x*log((b*x + 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 3*sq 
rt(a)*(b + c)*x*log((c*x - 2*sqrt(c*x + a)*sqrt(a) + 2*a)/x) - 2*((b + 3*c 
)*x - 2*a)*sqrt(b*x + a) + 2*((3*b + c)*x - 2*a)*sqrt(c*x + a))/((b^3 - 3* 
b^2*c + 3*b*c^2 - c^3)*x), 2*(3*sqrt(-a)*(b + c)*x*arctan(sqrt(b*x + a)*sq 
rt(-a)/a) - 3*sqrt(-a)*(b + c)*x*arctan(sqrt(c*x + a)*sqrt(-a)/a) + ((b + 
3*c)*x - 2*a)*sqrt(b*x + a) - ((3*b + c)*x - 2*a)*sqrt(c*x + a))/((b^3 - 3 
*b^2*c + 3*b*c^2 - c^3)*x)]
 

3.5.41.6 Sympy [F]

\[ \int \frac {x}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^3} \, dx=\int \frac {x}{\left (\sqrt {a + b x} + \sqrt {a + c x}\right )^{3}}\, dx \]

input

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

output

Integral(x/(sqrt(a + b*x) + sqrt(a + c*x))**3, x)
 

3.5.41.7 Maxima [F]

\[ \int \frac {x}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^3} \, dx=\int { \frac {x}{{\left (\sqrt {b x + a} + \sqrt {c x + a}\right )}^{3}} \,d x } \]

input

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

output

integrate(x/(sqrt(b*x + a) + sqrt(c*x + a))^3, x)
 

3.5.41.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2318 vs. \(2 (137) = 274\).

Time = 9.20 (sec) , antiderivative size = 2318, normalized size of antiderivative = 14.76 \[ \int \frac {x}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^3} \, dx=\text {Too large to display} \]

input

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

output

-2*(sqrt(a*b^2 + (b*x + a)*b*c - a*b*c)*(3*b*abs(b) + c*abs(b))/(b^4 - 3*b 
^3*c + 3*b^2*c^2 - b*c^3) + 2*sqrt(b*x + a)*a*b/((b^3 - 3*b^2*c + 3*b*c^2 
- c^3)*x) - 3*(a*b^2 + a*b*c)*arctan(sqrt(b*x + a)/sqrt(-a))/((b^3 - 3*b^2 
*c + 3*b*c^2 - c^3)*sqrt(-a)) - (sqrt(b*x + a)*b^2 + 3*sqrt(b*x + a)*b*c)/ 
(b^3 - 3*b^2*c + 3*b*c^2 - c^3) + 4*((sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 
 + (b*x + a)*b*c - a*b*c))*a^2*b^3*c*abs(b) - (sqrt(b*c)*sqrt(b*x + a) - s 
qrt(a*b^2 + (b*x + a)*b*c - a*b*c))*a^2*b^2*c^2*abs(b) + (sqrt(b*c)*sqrt(b 
*x + a) - sqrt(a*b^2 + (b*x + a)*b*c - a*b*c))^3*a*b*c*abs(b))/((a^2*b^4 - 
 2*a^2*b^3*c + a^2*b^2*c^2 - 2*(sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 + (b* 
x + a)*b*c - a*b*c))^2*a*b^2 - 2*(sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 + ( 
b*x + a)*b*c - a*b*c))^2*a*b*c + (sqrt(b*c)*sqrt(b*x + a) - sqrt(a*b^2 + ( 
b*x + a)*b*c - a*b*c))^4)*(b^3 - 3*b^2*c + 3*b*c^2 - c^3)) + 3*(2*(a*b^4*c 
 - a*b^2*c^3)*(a*b^4 - 3*a*b^3*c + 3*a*b^2*c^2 - a*b*c^3)^2*sqrt(-a)*abs(b 
)*sgn(b^3 - 3*b^2*c + 3*b*c^2 - c^3) + 2*(a*b^4 - 3*a*b^3*c + 3*a*b^2*c^2 
- a*b*c^3)^2*(a*b^4 - a*b^2*c^2)*sqrt(-a*b*c)*abs(b) + (a^2*b^8 - 4*a^2*b^ 
7*c + 5*a^2*b^6*c^2 - 5*a^2*b^4*c^4 + 4*a^2*b^3*c^5 - a^2*b^2*c^6)*sqrt(-a 
*b*c)*abs(-a*b^4 + 3*a*b^3*c - 3*a*b^2*c^2 + a*b*c^3)*abs(b)*sgn(b^3 - 3*b 
^2*c + 3*b*c^2 - c^3) + (a^2*b^9 - 4*a^2*b^8*c + 5*a^2*b^7*c^2 - 5*a^2*b^5 
*c^4 + 4*a^2*b^4*c^5 - a^2*b^3*c^6)*sqrt(-a)*abs(-a*b^4 + 3*a*b^3*c - 3*a* 
b^2*c^2 + a*b*c^3)*abs(b) + (a^3*b^12*c - 5*a^3*b^11*c^2 + 8*a^3*b^10*c...
 

3.5.41.9 Mupad [B] (verification not implemented)

Time = 21.49 (sec) , antiderivative size = 559, normalized size of antiderivative = 3.56 \[ \int \frac {x}{\left (\sqrt {a+b x}+\sqrt {a+c x}\right )^3} \, dx=\frac {2\,\sqrt {a}\,b^2\,\left (\sqrt {a+c\,x}-\sqrt {a}\right )\,\left (\frac {8\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {a+c\,x}-\sqrt {a}}-\frac {2\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^2}+\frac {3\,\ln \left (\frac {\sqrt {a+b\,x}-\sqrt {a}}{\sqrt {a+c\,x}-\sqrt {a}}\right )\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {a+c\,x}-\sqrt {a}}+1\right )-2\,\sqrt {a}\,c^2\,\left (\sqrt {a+c\,x}-\sqrt {a}\right )\,\left (\frac {2\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^2}-\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^4}+\frac {3\,\ln \left (\frac {\sqrt {a+b\,x}-\sqrt {a}}{\sqrt {a+c\,x}-\sqrt {a}}\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^3}\right )+2\,\sqrt {a}\,b\,c\,\left (\sqrt {a+c\,x}-\sqrt {a}\right )\,\left (\frac {8\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {a+c\,x}-\sqrt {a}}-\frac {14\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^2}+\frac {3\,\ln \left (\frac {\sqrt {a+b\,x}-\sqrt {a}}{\sqrt {a+c\,x}-\sqrt {a}}\right )\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {a+c\,x}-\sqrt {a}}-\frac {3\,\ln \left (\frac {\sqrt {a+b\,x}-\sqrt {a}}{\sqrt {a+c\,x}-\sqrt {a}}\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^3}\right )}{{\left (b-c\right )}^3\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )\,\left (b-\frac {c\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{{\left (\sqrt {a+c\,x}-\sqrt {a}\right )}^2}\right )} \]

input

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

output

(2*a^(1/2)*b^2*((a + c*x)^(1/2) - a^(1/2))*((8*((a + b*x)^(1/2) - a^(1/2)) 
)/((a + c*x)^(1/2) - a^(1/2)) - (2*((a + b*x)^(1/2) - a^(1/2))^2)/((a + c* 
x)^(1/2) - a^(1/2))^2 + (3*log(((a + b*x)^(1/2) - a^(1/2))/((a + c*x)^(1/2 
) - a^(1/2)))*((a + b*x)^(1/2) - a^(1/2)))/((a + c*x)^(1/2) - a^(1/2)) + 1 
) - 2*a^(1/2)*c^2*((a + c*x)^(1/2) - a^(1/2))*((2*((a + b*x)^(1/2) - a^(1/ 
2))^2)/((a + c*x)^(1/2) - a^(1/2))^2 - ((a + b*x)^(1/2) - a^(1/2))^4/((a + 
 c*x)^(1/2) - a^(1/2))^4 + (3*log(((a + b*x)^(1/2) - a^(1/2))/((a + c*x)^( 
1/2) - a^(1/2)))*((a + b*x)^(1/2) - a^(1/2))^3)/((a + c*x)^(1/2) - a^(1/2) 
)^3) + 2*a^(1/2)*b*c*((a + c*x)^(1/2) - a^(1/2))*((8*((a + b*x)^(1/2) - a^ 
(1/2)))/((a + c*x)^(1/2) - a^(1/2)) - (14*((a + b*x)^(1/2) - a^(1/2))^2)/( 
(a + c*x)^(1/2) - a^(1/2))^2 + (3*log(((a + b*x)^(1/2) - a^(1/2))/((a + c* 
x)^(1/2) - a^(1/2)))*((a + b*x)^(1/2) - a^(1/2)))/((a + c*x)^(1/2) - a^(1/ 
2)) - (3*log(((a + b*x)^(1/2) - a^(1/2))/((a + c*x)^(1/2) - a^(1/2)))*((a 
+ b*x)^(1/2) - a^(1/2))^3)/((a + c*x)^(1/2) - a^(1/2))^3))/((b - c)^3*((a 
+ b*x)^(1/2) - a^(1/2))*(b - (c*((a + b*x)^(1/2) - a^(1/2))^2)/((a + c*x)^ 
(1/2) - a^(1/2))^2))