Integrand size = 29, antiderivative size = 137 \[ \int \frac {(A+B x) \sqrt {a x+b x^2}}{x^2 (c+d x)^2} \, dx=\frac {(B c-3 A d) \sqrt {a x+b x^2}}{c^2 d x}-\frac {(B c-A d) \sqrt {a x+b x^2}}{c d x (c+d x)}+\frac {(2 A b c+a B c-3 a A d) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a x+b x^2}}\right )}{c^{5/2} \sqrt {b c-a d}} \] Output:
(-3*A*d+B*c)*(b*x^2+a*x)^(1/2)/c^2/d/x-(-A*d+B*c)*(b*x^2+a*x)^(1/2)/c/d/x/ (d*x+c)+(-3*A*a*d+2*A*b*c+B*a*c)*arctanh((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^2 +a*x)^(1/2))/c^(5/2)/(-a*d+b*c)^(1/2)
Time = 10.10 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.12 \[ \int \frac {(A+B x) \sqrt {a x+b x^2}}{x^2 (c+d x)^2} \, dx=\frac {\sqrt {x (a+b x)} \left (\frac {2 (-B c+A d) (a+b x)}{c+d x}+\frac {2 (2 A b c+a B c-3 a A d) \left (\sqrt {c} \sqrt {a+b x}-\sqrt {b c-a d} \sqrt {x} \text {arctanh}\left (\frac {\sqrt {b c-a d} \sqrt {x}}{\sqrt {c} \sqrt {a+b x}}\right )\right )}{c^{3/2} \sqrt {a+b x}}\right )}{2 c (-b c+a d) x} \] Input:
Integrate[((A + B*x)*Sqrt[a*x + b*x^2])/(x^2*(c + d*x)^2),x]
Output:
(Sqrt[x*(a + b*x)]*((2*(-(B*c) + A*d)*(a + b*x))/(c + d*x) + (2*(2*A*b*c + a*B*c - 3*a*A*d)*(Sqrt[c]*Sqrt[a + b*x] - Sqrt[b*c - a*d]*Sqrt[x]*ArcTanh [(Sqrt[b*c - a*d]*Sqrt[x])/(Sqrt[c]*Sqrt[a + b*x])]))/(c^(3/2)*Sqrt[a + b* x])))/(2*c*(-(b*c) + a*d)*x)
Leaf count is larger than twice the leaf count of optimal. \(389\) vs. \(2(137)=274\).
Time = 1.34 (sec) , antiderivative size = 389, normalized size of antiderivative = 2.84, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2153, 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.
| \(\displaystyle \int \frac {\sqrt {a x+b x^2} (A+B x)}{x^2 (c+d x)^2} \, dx\) |
| \(\Big \downarrow \) 2153 |
| \(\displaystyle \int \left (\frac {\sqrt {a x+b x^2} (B c-2 A d)}{c^3 x}-\frac {d \sqrt {a x+b x^2} (B c-2 A d)}{c^3 (c+d x)}-\frac {d \sqrt {a x+b x^2} (B c-A d)}{c^2 (c+d x)^2}+\frac {A \sqrt {a x+b x^2}}{c^2 x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt {b c-a d} (B c-2 A d) \text {arctanh}\left (\frac {x (2 b c-a d)+a c}{2 \sqrt {c} \sqrt {a x+b x^2} \sqrt {b c-a d}}\right )}{c^{5/2} d}+\frac {(2 b c-a d) (B c-A d) \text {arctanh}\left (\frac {x (2 b c-a d)+a c}{2 \sqrt {c} \sqrt {a x+b x^2} \sqrt {b c-a d}}\right )}{2 c^{5/2} d \sqrt {b c-a d}}+\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right ) (2 b c-a d) (B c-2 A d)}{\sqrt {b} c^3 d}+\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right ) (B c-2 A d)}{\sqrt {b} c^3}-\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right ) (B c-A d)}{c^2 d}+\frac {2 A \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{c^2}+\frac {\sqrt {a x+b x^2} (B c-A d)}{c^2 (c+d x)}-\frac {2 A \sqrt {a x+b x^2}}{c^2 x}\) |
Input:
Int[((A + B*x)*Sqrt[a*x + b*x^2])/(x^2*(c + d*x)^2),x]
Output:
(-2*A*Sqrt[a*x + b*x^2])/(c^2*x) + ((B*c - A*d)*Sqrt[a*x + b*x^2])/(c^2*(c + d*x)) + (2*A*Sqrt[b]*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/c^2 + (a*( B*c - 2*A*d)*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/(Sqrt[b]*c^3) + ((2*b *c - a*d)*(B*c - 2*A*d)*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/(Sqrt[b]*c ^3*d) - (2*Sqrt[b]*(B*c - A*d)*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/(c^ 2*d) - (Sqrt[b*c - a*d]*(B*c - 2*A*d)*ArcTanh[(a*c + (2*b*c - a*d)*x)/(2*S qrt[c]*Sqrt[b*c - a*d]*Sqrt[a*x + b*x^2])])/(c^(5/2)*d) + ((2*b*c - a*d)*( B*c - A*d)*ArcTanh[(a*c + (2*b*c - a*d)*x)/(2*Sqrt[c]*Sqrt[b*c - a*d]*Sqrt [a*x + b*x^2])])/(2*c^(5/2)*d*Sqrt[b*c - a*d])
Int[(Px_)*((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b _.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Px*(d + e* x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && PolyQ[Px, x] && (IntegerQ[p] || (IntegerQ[2*p] && IntegerQ [m] && ILtQ[n, 0])) && !(IGtQ[m, 0] && IGtQ[n, 0])
Time = 0.71 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.85
| method | result | size |
| pseudoelliptic | \(\frac {-2 \sqrt {x \left (b x +a \right )}\, \left (\left (-\frac {B x}{2}+A \right ) c +\frac {3 A d x}{2}\right ) \sqrt {c \left (a d -b c \right )}+3 \left (d x +c \right ) x \left (\frac {\left (-2 A b -B a \right ) c}{3}+A a d \right ) \arctan \left (\frac {\sqrt {x \left (b x +a \right )}\, c}{x \sqrt {c \left (a d -b c \right )}}\right )}{\sqrt {c \left (a d -b c \right )}\, c^{2} x \left (d x +c \right )}\) | \(117\) |
| risch | \(-\frac {2 A \left (b x +a \right )}{c^{2} \sqrt {x \left (b x +a \right )}}-\frac {-\frac {\left (A a \,d^{2}-B b \,c^{2}\right ) \ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}+\frac {c \left (A a \,d^{2}-A b c d -B a c d +B b \,c^{2}\right ) \left (\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{c \left (a d -b c \right ) \left (x +\frac {c}{d}\right )}-\frac {\left (a d -2 b c \right ) d \ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{2 c \left (a d -b c \right ) \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}\right )}{d^{3}}}{c^{2}}\) | \(427\) |
| default | \(\text {Expression too large to display}\) | \(1022\) |
Input:
int((B*x+A)*(b*x^2+a*x)^(1/2)/x^2/(d*x+c)^2,x,method=_RETURNVERBOSE)
Output:
(-2*(x*(b*x+a))^(1/2)*((-1/2*B*x+A)*c+3/2*A*d*x)*(c*(a*d-b*c))^(1/2)+3*(d* x+c)*x*(1/3*(-2*A*b-B*a)*c+A*a*d)*arctan((x*(b*x+a))^(1/2)/x*c/(c*(a*d-b*c ))^(1/2)))/(c*(a*d-b*c))^(1/2)/c^2/x/(d*x+c)
Time = 0.09 (sec) , antiderivative size = 412, normalized size of antiderivative = 3.01 \[ \int \frac {(A+B x) \sqrt {a x+b x^2}}{x^2 (c+d x)^2} \, dx=\left [\frac {\sqrt {b c^{2} - a c d} {\left ({\left (3 \, A a d^{2} - {\left (B a + 2 \, A b\right )} c d\right )} x^{2} + {\left (3 \, A a c d - {\left (B a + 2 \, A b\right )} c^{2}\right )} x\right )} \log \left (\frac {a c + {\left (2 \, b c - a d\right )} x - 2 \, \sqrt {b c^{2} - a c d} \sqrt {b x^{2} + a x}}{d x + c}\right ) - 2 \, {\left (2 \, A b c^{3} - 2 \, A a c^{2} d - {\left (B b c^{3} + 3 \, A a c d^{2} - {\left (B a + 3 \, A b\right )} c^{2} d\right )} x\right )} \sqrt {b x^{2} + a x}}{2 \, {\left ({\left (b c^{4} d - a c^{3} d^{2}\right )} x^{2} + {\left (b c^{5} - a c^{4} d\right )} x\right )}}, \frac {\sqrt {-b c^{2} + a c d} {\left ({\left (3 \, A a d^{2} - {\left (B a + 2 \, A b\right )} c d\right )} x^{2} + {\left (3 \, A a c d - {\left (B a + 2 \, A b\right )} c^{2}\right )} x\right )} \arctan \left (\frac {\sqrt {-b c^{2} + a c d} \sqrt {b x^{2} + a x}}{b c x + a c}\right ) - {\left (2 \, A b c^{3} - 2 \, A a c^{2} d - {\left (B b c^{3} + 3 \, A a c d^{2} - {\left (B a + 3 \, A b\right )} c^{2} d\right )} x\right )} \sqrt {b x^{2} + a x}}{{\left (b c^{4} d - a c^{3} d^{2}\right )} x^{2} + {\left (b c^{5} - a c^{4} d\right )} x}\right ] \] Input:
integrate((B*x+A)*(b*x^2+a*x)^(1/2)/x^2/(d*x+c)^2,x, algorithm="fricas")
Output:
[1/2*(sqrt(b*c^2 - a*c*d)*((3*A*a*d^2 - (B*a + 2*A*b)*c*d)*x^2 + (3*A*a*c* d - (B*a + 2*A*b)*c^2)*x)*log((a*c + (2*b*c - a*d)*x - 2*sqrt(b*c^2 - a*c* d)*sqrt(b*x^2 + a*x))/(d*x + c)) - 2*(2*A*b*c^3 - 2*A*a*c^2*d - (B*b*c^3 + 3*A*a*c*d^2 - (B*a + 3*A*b)*c^2*d)*x)*sqrt(b*x^2 + a*x))/((b*c^4*d - a*c^ 3*d^2)*x^2 + (b*c^5 - a*c^4*d)*x), (sqrt(-b*c^2 + a*c*d)*((3*A*a*d^2 - (B* a + 2*A*b)*c*d)*x^2 + (3*A*a*c*d - (B*a + 2*A*b)*c^2)*x)*arctan(sqrt(-b*c^ 2 + a*c*d)*sqrt(b*x^2 + a*x)/(b*c*x + a*c)) - (2*A*b*c^3 - 2*A*a*c^2*d - ( B*b*c^3 + 3*A*a*c*d^2 - (B*a + 3*A*b)*c^2*d)*x)*sqrt(b*x^2 + a*x))/((b*c^4 *d - a*c^3*d^2)*x^2 + (b*c^5 - a*c^4*d)*x)]
\[ \int \frac {(A+B x) \sqrt {a x+b x^2}}{x^2 (c+d x)^2} \, dx=\int \frac {\sqrt {x \left (a + b x\right )} \left (A + B x\right )}{x^{2} \left (c + d x\right )^{2}}\, dx \] Input:
integrate((B*x+A)*(b*x**2+a*x)**(1/2)/x**2/(d*x+c)**2,x)
Output:
Integral(sqrt(x*(a + b*x))*(A + B*x)/(x**2*(c + d*x)**2), x)
\[ \int \frac {(A+B x) \sqrt {a x+b x^2}}{x^2 (c+d x)^2} \, dx=\int { \frac {\sqrt {b x^{2} + a x} {\left (B x + A\right )}}{{\left (d x + c\right )}^{2} x^{2}} \,d x } \] Input:
integrate((B*x+A)*(b*x^2+a*x)^(1/2)/x^2/(d*x+c)^2,x, algorithm="maxima")
Output:
integrate(sqrt(b*x^2 + a*x)*(B*x + A)/((d*x + c)^2*x^2), x)
Leaf count of result is larger than twice the leaf count of optimal. 801 vs. \(2 (123) = 246\).
Time = 0.67 (sec) , antiderivative size = 801, normalized size of antiderivative = 5.85 \[ \int \frac {(A+B x) \sqrt {a x+b x^2}}{x^2 (c+d x)^2} \, dx =\text {Too large to display} \] Input:
integrate((B*x+A)*(b*x^2+a*x)^(1/2)/x^2/(d*x+c)^2,x, algorithm="giac")
Output:
-1/6*d^2*((B*a*c*sgn(1/(d*x + c))*sgn(d) + (2*b*c - 3*a*d)*A*sgn(1/(d*x + c))*sgn(d))*log(abs(-2*sqrt(b*c^2 - a*c*d)*c*(sqrt(b - 2*b*c/(d*x + c) + b *c^2/(d*x + c)^2 + a*d/(d*x + c) - a*c*d/(d*x + c)^2) + sqrt(b*c^2*d^2 - a *c*d^3)/((d*x + c)*d))^3*abs(d) + 2*b^2*c^2*d - 3*a*b*c*d^2 + a^2*d^3 - 2* sqrt(b*c^2 - a*c*d)*(3*b*c - 2*a*d)*(sqrt(b - 2*b*c/(d*x + c) + b*c^2/(d*x + c)^2 + a*d/(d*x + c) - a*c*d/(d*x + c)^2) + sqrt(b*c^2*d^2 - a*c*d^3)/( (d*x + c)*d))*abs(d) + (6*b*c^2*d - 5*a*c*d^2)*(sqrt(b - 2*b*c/(d*x + c) + b*c^2/(d*x + c)^2 + a*d/(d*x + c) - a*c*d/(d*x + c)^2) + sqrt(b*c^2*d^2 - a*c*d^3)/((d*x + c)*d))^2))/(sqrt(b*c^2 - a*c*d)*c^2*d*abs(d)) - (B*a*c*d ^2*log(abs(8*b^2*c^2*d - 8*a*b*c*d^2 + a^2*d^3 - 8*sqrt(b*c^2 - a*c*d)*b^( 3/2)*c*abs(d) + 4*sqrt(b*c^2 - a*c*d)*a*sqrt(b)*d*abs(d))) + 2*A*b*c*d^2*l og(abs(8*b^2*c^2*d - 8*a*b*c*d^2 + a^2*d^3 - 8*sqrt(b*c^2 - a*c*d)*b^(3/2) *c*abs(d) + 4*sqrt(b*c^2 - a*c*d)*a*sqrt(b)*d*abs(d))) - 3*A*a*d^3*log(abs (8*b^2*c^2*d - 8*a*b*c*d^2 + a^2*d^3 - 8*sqrt(b*c^2 - a*c*d)*b^(3/2)*c*abs (d) + 4*sqrt(b*c^2 - a*c*d)*a*sqrt(b)*d*abs(d))) - 6*sqrt(b*c^2 - a*c*d)*B *sqrt(b)*c*abs(d) + 6*sqrt(b*c^2 - a*c*d)*A*sqrt(b)*d*abs(d))*sgn(1/(d*x + c))*sgn(d)/(sqrt(b*c^2 - a*c*d)*c^2*d^3*abs(d)) - 6*(B*c^3*d^2*sgn(1/(d*x + c))*sgn(d) - A*c^2*d^3*sgn(1/(d*x + c))*sgn(d))*sqrt(b - 2*b*c/(d*x + c ) + b*c^2/(d*x + c)^2 + a*d/(d*x + c) - a*c*d/(d*x + c)^2)/(c^4*d^5))
Timed out. \[ \int \frac {(A+B x) \sqrt {a x+b x^2}}{x^2 (c+d x)^2} \, dx=\int \frac {\sqrt {b\,x^2+a\,x}\,\left (A+B\,x\right )}{x^2\,{\left (c+d\,x\right )}^2} \,d x \] Input:
int(((a*x + b*x^2)^(1/2)*(A + B*x))/(x^2*(c + d*x)^2),x)
Output:
int(((a*x + b*x^2)^(1/2)*(A + B*x))/(x^2*(c + d*x)^2), x)
Time = 0.25 (sec) , antiderivative size = 624, normalized size of antiderivative = 4.55 \[ \int \frac {(A+B x) \sqrt {a x+b x^2}}{x^2 (c+d x)^2} \, dx=\frac {9 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b x +a}-\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a^{2} c d x +9 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b x +a}-\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a^{2} d^{2} x^{2}-12 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b x +a}-\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a b \,c^{2} x -12 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b x +a}-\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a b c d \,x^{2}+9 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a^{2} c d x +9 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a^{2} d^{2} x^{2}-12 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a b \,c^{2} x -12 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a b c d \,x^{2}-6 \sqrt {x}\, \sqrt {b x +a}\, a^{2} c^{2} d -9 \sqrt {x}\, \sqrt {b x +a}\, a^{2} c \,d^{2} x +8 \sqrt {x}\, \sqrt {b x +a}\, a b \,c^{3}+15 \sqrt {x}\, \sqrt {b x +a}\, a b \,c^{2} d x -4 \sqrt {x}\, \sqrt {b x +a}\, b^{2} c^{3} x +3 \sqrt {b}\, a^{2} c^{2} d x +3 \sqrt {b}\, a^{2} c \,d^{2} x^{2}-9 \sqrt {b}\, a b \,c^{3} x -9 \sqrt {b}\, a b \,c^{2} d \,x^{2}}{c^{3} x \left (3 a \,d^{2} x -4 b c d x +3 a c d -4 b \,c^{2}\right )} \] Input:
int((B*x+A)*(b*x^2+a*x)^(1/2)/x^2/(d*x+c)^2,x)
Output:
(9*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x) - sqrt(x)*sqrt(d)*sqrt(b))/(sqrt(c)*sqrt(b)))*a**2*c*d*x + 9*sqrt(c)*sqrt(a *d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x) - sqrt(x)*sqrt(d)* sqrt(b))/(sqrt(c)*sqrt(b)))*a**2*d**2*x**2 - 12*sqrt(c)*sqrt(a*d - b*c)*at an((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x) - sqrt(x)*sqrt(d)*sqrt(b))/(sq rt(c)*sqrt(b)))*a*b*c**2*x - 12*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b *c) - sqrt(d)*sqrt(a + b*x) - sqrt(x)*sqrt(d)*sqrt(b))/(sqrt(c)*sqrt(b)))* a*b*c*d*x**2 + 9*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) + sqrt(d)*s qrt(a + b*x) + sqrt(x)*sqrt(d)*sqrt(b))/(sqrt(c)*sqrt(b)))*a**2*c*d*x + 9* sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) + sqrt(d)*sqrt(a + b*x) + sq rt(x)*sqrt(d)*sqrt(b))/(sqrt(c)*sqrt(b)))*a**2*d**2*x**2 - 12*sqrt(c)*sqrt (a*d - b*c)*atan((sqrt(a*d - b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(x)*sqrt(d )*sqrt(b))/(sqrt(c)*sqrt(b)))*a*b*c**2*x - 12*sqrt(c)*sqrt(a*d - b*c)*atan ((sqrt(a*d - b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(x)*sqrt(d)*sqrt(b))/(sqrt (c)*sqrt(b)))*a*b*c*d*x**2 - 6*sqrt(x)*sqrt(a + b*x)*a**2*c**2*d - 9*sqrt( x)*sqrt(a + b*x)*a**2*c*d**2*x + 8*sqrt(x)*sqrt(a + b*x)*a*b*c**3 + 15*sqr t(x)*sqrt(a + b*x)*a*b*c**2*d*x - 4*sqrt(x)*sqrt(a + b*x)*b**2*c**3*x + 3* sqrt(b)*a**2*c**2*d*x + 3*sqrt(b)*a**2*c*d**2*x**2 - 9*sqrt(b)*a*b*c**3*x - 9*sqrt(b)*a*b*c**2*d*x**2)/(c**3*x*(3*a*c*d + 3*a*d**2*x - 4*b*c**2 - 4* b*c*d*x))