Integrand size = 25, antiderivative size = 157 \[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx=\frac {2 (a+3 c) \sqrt {a+b x}}{(a-c)^3}+\frac {8 (a+b x)^{3/2}}{3 (a-c)^3}-\frac {2 (3 a+c) \sqrt {c+b x}}{(a-c)^3}-\frac {8 (c+b x)^{3/2}}{3 (a-c)^3}-\frac {2 \sqrt {a} (a+3 c) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{(a-c)^3}+\frac {2 \sqrt {c} (3 a+c) \text {arctanh}\left (\frac {\sqrt {c+b x}}{\sqrt {c}}\right )}{(a-c)^3} \] Output:
2*(a+3*c)*(b*x+a)^(1/2)/(a-c)^3+8/3*(b*x+a)^(3/2)/(a-c)^3-2*(3*a+c)*(b*x+c )^(1/2)/(a-c)^3-8/3*(b*x+c)^(3/2)/(a-c)^3-2*a^(1/2)*(a+3*c)*arctanh((b*x+a )^(1/2)/a^(1/2))/(a-c)^3+2*c^(1/2)*(3*a+c)*arctanh((b*x+c)^(1/2)/c^(1/2))/ (a-c)^3
Time = 1.39 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.55 \[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx=\frac {2}{3} \left (-\frac {\sqrt {c+b x} (9 a+7 c+4 b x)}{(a-c)^3}+\frac {\sqrt {a+b x} (7 a+9 c+4 b x)}{(a-c)^3}-\frac {3 \left (\sqrt {a}-\sqrt {c}\right ) \arctan \left (\frac {-\sqrt {a+b x}+\sqrt {c+b x}}{\sqrt {-\left (\sqrt {a}-\sqrt {c}\right )^2}}\right )}{\sqrt {-\left (\sqrt {a}-\sqrt {c}\right )^2} \left (\sqrt {a}+\sqrt {c}\right )^3}-\frac {3 \left (\sqrt {a}+\sqrt {c}\right ) \arctan \left (\frac {-\sqrt {a+b x}+\sqrt {c+b x}}{\sqrt {-\left (\sqrt {a}+\sqrt {c}\right )^2}}\right )}{\left (\sqrt {a}-\sqrt {c}\right )^3 \sqrt {-\left (\sqrt {a}+\sqrt {c}\right )^2}}\right ) \] Input:
Integrate[1/(x*(Sqrt[a + b*x] + Sqrt[c + b*x])^3),x]
Output:
(2*(-((Sqrt[c + b*x]*(9*a + 7*c + 4*b*x))/(a - c)^3) + (Sqrt[a + b*x]*(7*a + 9*c + 4*b*x))/(a - c)^3 - (3*(Sqrt[a] - Sqrt[c])*ArcTan[(-Sqrt[a + b*x] + Sqrt[c + b*x])/Sqrt[-(Sqrt[a] - Sqrt[c])^2]])/(Sqrt[-(Sqrt[a] - Sqrt[c] )^2]*(Sqrt[a] + Sqrt[c])^3) - (3*(Sqrt[a] + Sqrt[c])*ArcTan[(-Sqrt[a + b*x ] + Sqrt[c + b*x])/Sqrt[-(Sqrt[a] + Sqrt[c])^2]])/((Sqrt[a] - Sqrt[c])^3*S qrt[-(Sqrt[a] + Sqrt[c])^2])))/3
Time = 0.66 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.78, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {7240, 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.
| \(\displaystyle \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {b x+c}\right )^3} \, dx\) |
| \(\Big \downarrow \) 7240 |
| \(\displaystyle \frac {\int \left (4 \sqrt {a+b x} b-4 \sqrt {c+b x} b+\frac {(a+3 c) \sqrt {a+b x}}{x}-\frac {(3 a+c) \sqrt {c+b x}}{x}\right )dx}{(a-c)^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-2 \sqrt {a} (a+3 c) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )+2 \sqrt {c} (3 a+c) \text {arctanh}\left (\frac {\sqrt {b x+c}}{\sqrt {c}}\right )+2 (a+3 c) \sqrt {a+b x}-2 (3 a+c) \sqrt {b x+c}+\frac {8}{3} (a+b x)^{3/2}-\frac {8}{3} (b x+c)^{3/2}}{(a-c)^3}\) |
Input:
Int[1/(x*(Sqrt[a + b*x] + Sqrt[c + b*x])^3),x]
Output:
(2*(a + 3*c)*Sqrt[a + b*x] + (8*(a + b*x)^(3/2))/3 - 2*(3*a + c)*Sqrt[c + b*x] - (8*(c + b*x)^(3/2))/3 - 2*Sqrt[a]*(a + 3*c)*ArcTanh[Sqrt[a + b*x]/S qrt[a]] + 2*Sqrt[c]*(3*a + c)*ArcTanh[Sqrt[c + b*x]/Sqrt[c]])/(a - c)^3
Int[(u_.)*((e_.)*Sqrt[(a_.) + (b_.)*(x_)^(n_.)] + (f_.)*Sqrt[(c_.) + (d_.)* (x_)^(n_.)])^(m_), x_Symbol] :> Simp[(a*e^2 - c*f^2)^m Int[ExpandIntegran d[u/(e*Sqrt[a + b*x^n] - f*Sqrt[c + d*x^n])^m, x], x], x] /; FreeQ[{a, b, c , d, e, f, n}, x] && ILtQ[m, 0] && EqQ[b*e^2 - d*f^2, 0]
Time = 0.01 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.15
| method | result | size |
| default | \(\frac {a \left (2 \sqrt {b x +a}-2 \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )\right )}{\left (a -c \right )^{3}}+\frac {8 \left (b x +a \right )^{\frac {3}{2}}}{3 \left (a -c \right )^{3}}-\frac {8 \left (b x +c \right )^{\frac {3}{2}}}{3 \left (a -c \right )^{3}}+\frac {3 c \left (2 \sqrt {b x +a}-2 \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )\right )}{\left (a -c \right )^{3}}-\frac {3 a \left (2 \sqrt {b x +c}-2 \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {b x +c}}{\sqrt {c}}\right )\right )}{\left (a -c \right )^{3}}-\frac {c \left (2 \sqrt {b x +c}-2 \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {b x +c}}{\sqrt {c}}\right )\right )}{\left (a -c \right )^{3}}\) | \(181\) |
Input:
int(1/x/((b*x+a)^(1/2)+(b*x+c)^(1/2))^3,x,method=_RETURNVERBOSE)
Output:
1/(a-c)^3*a*(2*(b*x+a)^(1/2)-2*a^(1/2)*arctanh((b*x+a)^(1/2)/a^(1/2)))+8/3 *(b*x+a)^(3/2)/(a-c)^3-8/3*(b*x+c)^(3/2)/(a-c)^3+3/(a-c)^3*c*(2*(b*x+a)^(1 /2)-2*a^(1/2)*arctanh((b*x+a)^(1/2)/a^(1/2)))-3/(a-c)^3*a*(2*(b*x+c)^(1/2) -2*c^(1/2)*arctanh((b*x+c)^(1/2)/c^(1/2)))-1/(a-c)^3*c*(2*(b*x+c)^(1/2)-2* c^(1/2)*arctanh((b*x+c)^(1/2)/c^(1/2)))
Time = 0.10 (sec) , antiderivative size = 504, normalized size of antiderivative = 3.21 \[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx=\left [-\frac {3 \, {\left (a + 3 \, c\right )} \sqrt {a} \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 3 \, {\left (3 \, a + c\right )} \sqrt {c} \log \left (\frac {b x - 2 \, \sqrt {b x + c} \sqrt {c} + 2 \, c}{x}\right ) - 2 \, {\left (4 \, b x + 7 \, a + 9 \, c\right )} \sqrt {b x + a} + 2 \, {\left (4 \, b x + 9 \, a + 7 \, c\right )} \sqrt {b x + c}}{3 \, {\left (a^{3} - 3 \, a^{2} c + 3 \, a c^{2} - c^{3}\right )}}, -\frac {6 \, {\left (3 \, a + c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {b x + c}}\right ) + 3 \, {\left (a + 3 \, c\right )} \sqrt {a} \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) - 2 \, {\left (4 \, b x + 7 \, a + 9 \, c\right )} \sqrt {b x + a} + 2 \, {\left (4 \, b x + 9 \, a + 7 \, c\right )} \sqrt {b x + c}}{3 \, {\left (a^{3} - 3 \, a^{2} c + 3 \, a c^{2} - c^{3}\right )}}, \frac {6 \, \sqrt {-a} {\left (a + 3 \, c\right )} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x + a}}\right ) - 3 \, {\left (3 \, a + c\right )} \sqrt {c} \log \left (\frac {b x - 2 \, \sqrt {b x + c} \sqrt {c} + 2 \, c}{x}\right ) + 2 \, {\left (4 \, b x + 7 \, a + 9 \, c\right )} \sqrt {b x + a} - 2 \, {\left (4 \, b x + 9 \, a + 7 \, c\right )} \sqrt {b x + c}}{3 \, {\left (a^{3} - 3 \, a^{2} c + 3 \, a c^{2} - c^{3}\right )}}, \frac {2 \, {\left (3 \, \sqrt {-a} {\left (a + 3 \, c\right )} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x + a}}\right ) - 3 \, {\left (3 \, a + c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {b x + c}}\right ) + {\left (4 \, b x + 7 \, a + 9 \, c\right )} \sqrt {b x + a} - {\left (4 \, b x + 9 \, a + 7 \, c\right )} \sqrt {b x + c}\right )}}{3 \, {\left (a^{3} - 3 \, a^{2} c + 3 \, a c^{2} - c^{3}\right )}}\right ] \] Input:
integrate(1/x/((b*x+a)^(1/2)+(b*x+c)^(1/2))^3,x, algorithm="fricas")
Output:
[-1/3*(3*(a + 3*c)*sqrt(a)*log((b*x + 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 3*(3*a + c)*sqrt(c)*log((b*x - 2*sqrt(b*x + c)*sqrt(c) + 2*c)/x) - 2*(4*b* x + 7*a + 9*c)*sqrt(b*x + a) + 2*(4*b*x + 9*a + 7*c)*sqrt(b*x + c))/(a^3 - 3*a^2*c + 3*a*c^2 - c^3), -1/3*(6*(3*a + c)*sqrt(-c)*arctan(sqrt(-c)/sqrt (b*x + c)) + 3*(a + 3*c)*sqrt(a)*log((b*x + 2*sqrt(b*x + a)*sqrt(a) + 2*a) /x) - 2*(4*b*x + 7*a + 9*c)*sqrt(b*x + a) + 2*(4*b*x + 9*a + 7*c)*sqrt(b*x + c))/(a^3 - 3*a^2*c + 3*a*c^2 - c^3), 1/3*(6*sqrt(-a)*(a + 3*c)*arctan(s qrt(-a)/sqrt(b*x + a)) - 3*(3*a + c)*sqrt(c)*log((b*x - 2*sqrt(b*x + c)*sq rt(c) + 2*c)/x) + 2*(4*b*x + 7*a + 9*c)*sqrt(b*x + a) - 2*(4*b*x + 9*a + 7 *c)*sqrt(b*x + c))/(a^3 - 3*a^2*c + 3*a*c^2 - c^3), 2/3*(3*sqrt(-a)*(a + 3 *c)*arctan(sqrt(-a)/sqrt(b*x + a)) - 3*(3*a + c)*sqrt(-c)*arctan(sqrt(-c)/ sqrt(b*x + c)) + (4*b*x + 7*a + 9*c)*sqrt(b*x + a) - (4*b*x + 9*a + 7*c)*s qrt(b*x + c))/(a^3 - 3*a^2*c + 3*a*c^2 - c^3)]
\[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx=\int \frac {1}{x \left (\sqrt {a + b x} + \sqrt {b x + c}\right )^{3}}\, dx \] Input:
integrate(1/x/((b*x+a)**(1/2)+(b*x+c)**(1/2))**3,x)
Output:
Integral(1/(x*(sqrt(a + b*x) + sqrt(b*x + c))**3), x)
\[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx=\int { \frac {1}{x {\left (\sqrt {b x + a} + \sqrt {b x + c}\right )}^{3}} \,d x } \] Input:
integrate(1/x/((b*x+a)^(1/2)+(b*x+c)^(1/2))^3,x, algorithm="maxima")
Output:
integrate(1/(x*(sqrt(b*x + a) + sqrt(b*x + c))^3), x)
Leaf count of result is larger than twice the leaf count of optimal. 2652 vs. \(2 (133) = 266\).
Time = 0.70 (sec) , antiderivative size = 2652, normalized size of antiderivative = 16.89 \[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(1/x/((b*x+a)^(1/2)+(b*x+c)^(1/2))^3,x, algorithm="giac")
Output:
-2/3*sqrt(b*x + c)*(4*(a^3 - 3*a^2*c + 3*a*c^2 - c^3)*(b*x + a)/(a^6 - 6*a ^5*c + 15*a^4*c^2 - 20*a^3*c^3 + 15*a^2*c^4 - 6*a*c^5 + c^6) + (5*a^4 - 8* a^3*c - 6*a^2*c^2 + 16*a*c^3 - 7*c^4)/(a^6 - 6*a^5*c + 15*a^4*c^2 - 20*a^3 *c^3 + 15*a^2*c^4 - 6*a*c^5 + c^6)) + 2*(a^2 + 3*a*c)*arctan(sqrt(b*x + a) /sqrt(-a))/((a^3 - 3*a^2*c + 3*a*c^2 - c^3)*sqrt(-a)) + 2/3*(4*(b*x + a)^( 3/2)*a^6 + 3*sqrt(b*x + a)*a^7 - 24*(b*x + a)^(3/2)*a^5*c - 9*sqrt(b*x + a )*a^6*c + 60*(b*x + a)^(3/2)*a^4*c^2 - 9*sqrt(b*x + a)*a^5*c^2 - 80*(b*x + a)^(3/2)*a^3*c^3 + 75*sqrt(b*x + a)*a^4*c^3 + 60*(b*x + a)^(3/2)*a^2*c^4 - 135*sqrt(b*x + a)*a^3*c^4 - 24*(b*x + a)^(3/2)*a*c^5 + 117*sqrt(b*x + a) *a^2*c^5 + 4*(b*x + a)^(3/2)*c^6 - 51*sqrt(b*x + a)*a*c^6 + 9*sqrt(b*x + a )*c^7)/(a^9 - 9*a^8*c + 36*a^7*c^2 - 84*a^6*c^3 + 126*a^5*c^4 - 126*a^4*c^ 5 + 84*a^3*c^6 - 36*a^2*c^7 + 9*a*c^8 - c^9) - 2*(3*a^9*c - 14*a^8*c^2 + 2 2*a^7*c^3 - 6*a^6*c^4 - 20*a^5*c^5 + 22*a^4*c^6 - 6*a^3*c^7 - 2*a^2*c^8 + a*c^9 - 2*(3*a^2*c^2 + a*c^3 + (3*a*c^2 + c^3)*sqrt(a*c))*(a^3 - 3*a^2*c + 3*a*c^2 - c^3)^2*sgn(a^3 - 3*a^2*c + 3*a*c^2 - c^3) - 2*(3*a^2*c^2 + a*c^ 3 - (3*a^2*c + a*c^2)*sqrt(a*c))*(a^3 - 3*a^2*c + 3*a*c^2 - c^3)^2 - (3*a^ 5*c^2 - 11*a^4*c^3 + 14*a^3*c^4 - 6*a^2*c^5 - a*c^6 + c^7 - (3*a^5*c - 11* a^4*c^2 + 14*a^3*c^3 - 6*a^2*c^4 - a*c^5 + c^6)*sqrt(a*c))*abs(-a^3 + 3*a^ 2*c - 3*a*c^2 + c^3)*sgn(a^3 - 3*a^2*c + 3*a*c^2 - c^3) - (3*a^6*c - 11*a^ 5*c^2 + 14*a^4*c^3 - 6*a^3*c^4 - a^2*c^5 + a*c^6 + (3*a^5*c - 11*a^4*c^...
Time = 49.47 (sec) , antiderivative size = 4060, normalized size of antiderivative = 25.86 \[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx=\text {Too large to display} \] Input:
int(1/(x*((a + b*x)^(1/2) + (c + b*x)^(1/2))^3),x)
Output:
((((a^(1/2)*(16*a + 16*c))/(3*a*c^2 - 3*a^2*c + a^3 - c^3) + (c^(1/2)*(16* a + 16*c))/(3*a*c^2 - 3*a^2*c + a^3 - c^3))*((a + b*x)^(1/2) - a^(1/2)))/( (c + b*x)^(1/2) - c^(1/2)) + (((a^(1/2)*(12*a + 20*c))/(3*a*c^2 - 3*a^2*c + a^3 - c^3) + (c^(1/2)*(20*a + 12*c))/(3*a*c^2 - 3*a^2*c + a^3 - c^3))*(( a + b*x)^(1/2) - a^(1/2))^2)/((c + b*x)^(1/2) - c^(1/2))^2 + (a^(1/2)*((28 *a)/3 + 12*c))/(3*a*c^2 - 3*a^2*c + a^3 - c^3) + (c^(1/2)*(12*a + (28*c)/3 ))/(3*a*c^2 - 3*a^2*c + a^3 - c^3))/((3*((a + b*x)^(1/2) - a^(1/2)))/((c + b*x)^(1/2) - c^(1/2)) + (3*((a + b*x)^(1/2) - a^(1/2))^2)/((c + b*x)^(1/2 ) - c^(1/2))^2 + ((a + b*x)^(1/2) - a^(1/2))^3/((c + b*x)^(1/2) - c^(1/2)) ^3 + 1) + (log(((a + b*x)^(1/2) - a^(1/2))/((c + b*x)^(1/2) - c^(1/2)))*(a *(a^(1/2) + 3*c^(1/2)) + c*(3*a^(1/2) + c^(1/2))))/(3*a*c^2 - 3*a^2*c + a^ 3 - c^3) + (atan(((((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)*((6*a*c^(11/2) - 6*a^(11/2)*c + 2 *a^(3/2)*c^5 - 2*a^5*c^(3/2) + 12*a^3*c^(7/2) - 12*a^(7/2)*c^3 - 16*a^2*c^ (9/2) + 16*a^(9/2)*c^2)/(a*c^7 + a^7*c - 6*a^2*c^6 + 15*a^3*c^5 - 20*a^4*c ^4 + 15*a^5*c^3 - 6*a^6*c^2) + (((a^(1/2)*c^(15/2) - 5*a^(3/2)*c^(13/2) + 9*a^(5/2)*c^(11/2) - 5*a^(7/2)*c^(9/2) - 5*a^(9/2)*c^(7/2) + 9*a^(11/2)*c^ (5/2) - 5*a^(13/2)*c^(3/2) + a^(15/2)*c^(1/2))/(a*c^7 + a^7*c - 6*a^2*c^6 + 15*a^3*c^5 - 20*a^4*c^4 + 15*a^5*c^3 - 6*a^6*c^2) - (2*((a + b*x)^(1/2) - a^(1/2))*(a*c^9 + a^9*c - 7*a^2*c^8 + 22*a^3*c^7 - 41*a^4*c^6 + 50*a^...
Time = 0.17 (sec) , antiderivative size = 802, normalized size of antiderivative = 5.11 \[ \int \frac {1}{x \left (\sqrt {a+b x}+\sqrt {c+b x}\right )^3} \, dx =\text {Too large to display} \] Input:
int(1/x/((b*x+a)^(1/2)+(b*x+c)^(1/2))^3,x)
Output:
( - 24*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 - 24*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))*c - 6*sqrt(2*sqrt(c)*sqrt(a) - a - c)*atan((sq rt(b*x + c) + sqrt(a + b*x))/sqrt(2*sqrt(c)*sqrt(a) - a - c))*a**2 - 36*sq rt(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*c - 6*sqrt(2*sqrt(c)*sqrt(a) - a - c)*atan((sq rt(b*x + c) + sqrt(a + b*x))/sqrt(2*sqrt(c)*sqrt(a) - a - c))*c**2 - 18*sq rt(b*x + c)*a**2 - 8*sqrt(b*x + c)*a*b*x + 4*sqrt(b*x + c)*a*c + 8*sqrt(b* x + c)*b*c*x + 14*sqrt(b*x + c)*c**2 - 12*sqrt(c)*sqrt(a)*sqrt(2*sqrt(c)*s qrt(a) + a + c)*log(sqrt(b*x + c) - sqrt(2*sqrt(c)*sqrt(a) + a + c) + sqrt (a + b*x))*a - 12*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))*c + 12*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 + 12*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) + sq rt(a + b*x))*c + 3*sqrt(2*sqrt(c)*sqrt(a) + a + c)*log(sqrt(b*x + c) - sqr t(2*sqrt(c)*sqrt(a) + a + c) + sqrt(a + b*x))*a**2 + 18*sqrt(2*sqrt(c)*sqr t(a) + a + c)*log(sqrt(b*x + c) - sqrt(2*sqrt(c)*sqrt(a) + a + c) + sqrt(a + b*x))*a*c + 3*sqrt(2*sqrt(c)*sqrt(a) + a + c)*log(sqrt(b*x + c) - sq...