\(\int \frac {\sqrt {c+d x} (e+f x)}{x (a+b x)^2} \, dx\) [24]

Optimal result

Integrand size = 25, antiderivative size = 127 \[ \int \frac {\sqrt {c+d x} (e+f x)}{x (a+b x)^2} \, dx=\frac {(b e-a f) \sqrt {c+d x}}{a b (a+b x)}-\frac {2 \sqrt {c} e \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^2}+\frac {\left (2 b^2 c e-a d (b e+a f)\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^2 b^{3/2} \sqrt {b c-a d}} \] Output:

(-a*f+b*e)*(d*x+c)^(1/2)/a/b/(b*x+a)-2*c^(1/2)*e*arctanh((d*x+c)^(1/2)/c^( 
1/2))/a^2+(2*b^2*c*e-a*d*(a*f+b*e))*arctanh(b^(1/2)*(d*x+c)^(1/2)/(-a*d+b* 
c)^(1/2))/a^2/b^(3/2)/(-a*d+b*c)^(1/2)
 

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {c+d x} (e+f x)}{x (a+b x)^2} \, dx=\frac {\frac {a (b e-a f) \sqrt {c+d x}}{b (a+b x)}+\frac {\left (-2 b^2 c e+a b d e+a^2 d f\right ) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{b^{3/2} \sqrt {-b c+a d}}-2 \sqrt {c} e \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^2} \] Input:

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

Output:

((a*(b*e - a*f)*Sqrt[c + d*x])/(b*(a + b*x)) + ((-2*b^2*c*e + a*b*d*e + a^ 
2*d*f)*ArcTan[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[-(b*c) + a*d]])/(b^(3/2)*Sqrt[- 
(b*c) + a*d]) - 2*Sqrt[c]*e*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/a^2
 

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {166, 27, 174, 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.

\(\displaystyle \int \frac {\sqrt {c+d x} (e+f x)}{x (a+b x)^2} \, dx\)

\(\Big \downarrow \) 166

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

((b*e - a*f)*Sqrt[c + d*x])/(a*b*(a + b*x)) + ((-4*b*Sqrt[c]*e*ArcTanh[Sqr 
t[c + d*x]/Sqrt[c]])/a + (2*(2*b^2*c*e - a*d*(b*e + a*f))*ArcTanh[(Sqrt[b] 
*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(a*Sqrt[b]*Sqrt[b*c - a*d]))/(2*a*b)
 

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 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 166

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

rule 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d)   Int[(e + f*x)^p/(c + d 
*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, 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]
 

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.04

method	result	size
pseudoelliptic	\(-\frac {-\left (b x +a \right ) \left (a^{2} d f +a b d e -2 c e \,b^{2}\right ) \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )+\left (2 b e \sqrt {c}\, \left (b x +a \right ) \operatorname {arctanh}\left (\frac {\sqrt {x d +c}}{\sqrt {c}}\right )+a \sqrt {x d +c}\, \left (a f -b e \right )\right ) \sqrt {\left (a d -b c \right ) b}}{\sqrt {\left (a d -b c \right ) b}\, \left (b x +a \right ) b \,a^{2}}\)	\(132\)
derivativedivides	\(2 d \left (-\frac {e \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {x d +c}}{\sqrt {c}}\right )}{d \,a^{2}}+\frac {-\frac {a d \left (a f -b e \right ) \sqrt {x d +c}}{2 b \left (\left (x d +c \right ) b +a d -b c \right )}+\frac {\left (a^{2} d f +a b d e -2 c e \,b^{2}\right ) \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 b \sqrt {\left (a d -b c \right ) b}}}{a^{2} d}\right )\)	\(134\)
default	\(2 d \left (-\frac {e \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {x d +c}}{\sqrt {c}}\right )}{d \,a^{2}}+\frac {-\frac {a d \left (a f -b e \right ) \sqrt {x d +c}}{2 b \left (\left (x d +c \right ) b +a d -b c \right )}+\frac {\left (a^{2} d f +a b d e -2 c e \,b^{2}\right ) \arctan \left (\frac {b \sqrt {x d +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 b \sqrt {\left (a d -b c \right ) b}}}{a^{2} d}\right )\)	\(134\)

Input:

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

Output:

-1/((a*d-b*c)*b)^(1/2)*(-(b*x+a)*(a^2*d*f+a*b*d*e-2*b^2*c*e)*arctan(b*(d*x 
+c)^(1/2)/((a*d-b*c)*b)^(1/2))+(2*b*e*c^(1/2)*(b*x+a)*arctanh((d*x+c)^(1/2 
)/c^(1/2))+a*(d*x+c)^(1/2)*(a*f-b*e))*((a*d-b*c)*b)^(1/2))/(b*x+a)/b/a^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 242 vs. \(2 (109) = 218\).

Time = 0.17 (sec) , antiderivative size = 1012, normalized size of antiderivative = 7.97 \[ \int \frac {\sqrt {c+d x} (e+f x)}{x (a+b x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/2*((a^3*d*f - (2*a*b^2*c - a^2*b*d)*e + (a^2*b*d*f - (2*b^3*c - a*b^2*d 
)*e)*x)*sqrt(b^2*c - a*b*d)*log((b*d*x + 2*b*c - a*d - 2*sqrt(b^2*c - a*b* 
d)*sqrt(d*x + c))/(b*x + a)) + 2*((b^4*c - a*b^3*d)*e*x + (a*b^3*c - a^2*b 
^2*d)*e)*sqrt(c)*log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) + 2*((a*b^3* 
c - a^2*b^2*d)*e - (a^2*b^2*c - a^3*b*d)*f)*sqrt(d*x + c))/(a^3*b^3*c - a^ 
4*b^2*d + (a^2*b^4*c - a^3*b^3*d)*x), ((a^3*d*f - (2*a*b^2*c - a^2*b*d)*e 
+ (a^2*b*d*f - (2*b^3*c - a*b^2*d)*e)*x)*sqrt(-b^2*c + a*b*d)*arctan(sqrt( 
-b^2*c + a*b*d)*sqrt(d*x + c)/(b*d*x + b*c)) + ((b^4*c - a*b^3*d)*e*x + (a 
*b^3*c - a^2*b^2*d)*e)*sqrt(c)*log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x 
) + ((a*b^3*c - a^2*b^2*d)*e - (a^2*b^2*c - a^3*b*d)*f)*sqrt(d*x + c))/(a^ 
3*b^3*c - a^4*b^2*d + (a^2*b^4*c - a^3*b^3*d)*x), 1/2*(4*((b^4*c - a*b^3*d 
)*e*x + (a*b^3*c - a^2*b^2*d)*e)*sqrt(-c)*arctan(sqrt(-c)/sqrt(d*x + c)) + 
 (a^3*d*f - (2*a*b^2*c - a^2*b*d)*e + (a^2*b*d*f - (2*b^3*c - a*b^2*d)*e)* 
x)*sqrt(b^2*c - a*b*d)*log((b*d*x + 2*b*c - a*d - 2*sqrt(b^2*c - a*b*d)*sq 
rt(d*x + c))/(b*x + a)) + 2*((a*b^3*c - a^2*b^2*d)*e - (a^2*b^2*c - a^3*b* 
d)*f)*sqrt(d*x + c))/(a^3*b^3*c - a^4*b^2*d + (a^2*b^4*c - a^3*b^3*d)*x), 
((a^3*d*f - (2*a*b^2*c - a^2*b*d)*e + (a^2*b*d*f - (2*b^3*c - a*b^2*d)*e)* 
x)*sqrt(-b^2*c + a*b*d)*arctan(sqrt(-b^2*c + a*b*d)*sqrt(d*x + c)/(b*d*x + 
 b*c)) + 2*((b^4*c - a*b^3*d)*e*x + (a*b^3*c - a^2*b^2*d)*e)*sqrt(-c)*arct 
an(sqrt(-c)/sqrt(d*x + c)) + ((a*b^3*c - a^2*b^2*d)*e - (a^2*b^2*c - a^...
 

Sympy [F]

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

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

Output:

Integral(sqrt(c + d*x)*(e + f*x)/(x*(a + b*x)**2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {c+d x} (e+f x)}{x (a+b x)^2} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m 
ore detail
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt {c+d x} (e+f x)}{x (a+b x)^2} \, dx=\frac {2 \, c e \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{a^{2} \sqrt {-c}} - \frac {{\left (2 \, b^{2} c e - a b d e - a^{2} d f\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} a^{2} b} + \frac {\sqrt {d x + c} b d e - \sqrt {d x + c} a d f}{{\left ({\left (d x + c\right )} b - b c + a d\right )} a b} \] Input:

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

Output:

2*c*e*arctan(sqrt(d*x + c)/sqrt(-c))/(a^2*sqrt(-c)) - (2*b^2*c*e - a*b*d*e 
 - a^2*d*f)*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a* 
b*d)*a^2*b) + (sqrt(d*x + c)*b*d*e - sqrt(d*x + c)*a*d*f)/(((d*x + c)*b - 
b*c + a*d)*a*b)
 

Mupad [B] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 1827, normalized size of antiderivative = 14.39 \[ \int \frac {\sqrt {c+d x} (e+f x)}{x (a+b x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

(atan(((((((2*(2*a^4*b^3*c*d^3*e - 2*a^5*b^2*c*d^3*f))/(a^3*b) + ((4*a^5*b 
^3*d^3 - 8*a^4*b^4*c*d^2)*(-b^3*(a*d - b*c))^(1/2)*(c + d*x)^(1/2)*(a^2*d* 
f - 2*b^2*c*e + a*b*d*e))/(a^2*b*(a^2*b^4*c - a^3*b^3*d)))*(-b^3*(a*d - b* 
c))^(1/2)*(a^2*d*f - 2*b^2*c*e + a*b*d*e))/(2*(a^2*b^4*c - a^3*b^3*d)) + ( 
2*(c + d*x)^(1/2)*(a^4*d^4*f^2 + a^2*b^2*d^4*e^2 + 8*b^4*c^2*d^2*e^2 + 2*a 
^3*b*d^4*e*f - 4*a*b^3*c*d^3*e^2 - 4*a^2*b^2*c*d^3*e*f))/(a^2*b))*(-b^3*(a 
*d - b*c))^(1/2)*(a^2*d*f - 2*b^2*c*e + a*b*d*e)*1i)/(2*(a^2*b^4*c - a^3*b 
^3*d)) - (((((2*(2*a^4*b^3*c*d^3*e - 2*a^5*b^2*c*d^3*f))/(a^3*b) - ((4*a^5 
*b^3*d^3 - 8*a^4*b^4*c*d^2)*(-b^3*(a*d - b*c))^(1/2)*(c + d*x)^(1/2)*(a^2* 
d*f - 2*b^2*c*e + a*b*d*e))/(a^2*b*(a^2*b^4*c - a^3*b^3*d)))*(-b^3*(a*d - 
b*c))^(1/2)*(a^2*d*f - 2*b^2*c*e + a*b*d*e))/(2*(a^2*b^4*c - a^3*b^3*d)) - 
 (2*(c + d*x)^(1/2)*(a^4*d^4*f^2 + a^2*b^2*d^4*e^2 + 8*b^4*c^2*d^2*e^2 + 2 
*a^3*b*d^4*e*f - 4*a*b^3*c*d^3*e^2 - 4*a^2*b^2*c*d^3*e*f))/(a^2*b))*(-b^3* 
(a*d - b*c))^(1/2)*(a^2*d*f - 2*b^2*c*e + a*b*d*e)*1i)/(2*(a^2*b^4*c - a^3 
*b^3*d)))/((4*(a*b^2*c*d^4*e^3 - 2*b^3*c^2*d^3*e^3 + a^3*c*d^4*e*f^2 - 2*a 
*b^2*c^2*d^3*e^2*f + 2*a^2*b*c*d^4*e^2*f))/(a^3*b) + (((((2*(2*a^4*b^3*c*d 
^3*e - 2*a^5*b^2*c*d^3*f))/(a^3*b) + ((4*a^5*b^3*d^3 - 8*a^4*b^4*c*d^2)*(- 
b^3*(a*d - b*c))^(1/2)*(c + d*x)^(1/2)*(a^2*d*f - 2*b^2*c*e + a*b*d*e))/(a 
^2*b*(a^2*b^4*c - a^3*b^3*d)))*(-b^3*(a*d - b*c))^(1/2)*(a^2*d*f - 2*b^2*c 
*e + a*b*d*e))/(2*(a^2*b^4*c - a^3*b^3*d)) + (2*(c + d*x)^(1/2)*(a^4*d^...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 516, normalized size of antiderivative = 4.06 \[ \int \frac {\sqrt {c+d x} (e+f x)}{x (a+b x)^2} \, dx=\frac {\sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) a^{3} d f +\sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) a^{2} b d e +\sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) a^{2} b d f x -2 \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) a \,b^{2} c e +\sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) a \,b^{2} d e x -2 \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d x +c}\, b}{\sqrt {b}\, \sqrt {a d -b c}}\right ) b^{3} c e x -\sqrt {d x +c}\, a^{3} b d f +\sqrt {d x +c}\, a^{2} b^{2} c f +\sqrt {d x +c}\, a^{2} b^{2} d e -\sqrt {d x +c}\, a \,b^{3} c e +\sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}-\sqrt {c}\right ) a^{2} b^{2} d e -\sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}-\sqrt {c}\right ) a \,b^{3} c e +\sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}-\sqrt {c}\right ) a \,b^{3} d e x -\sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}-\sqrt {c}\right ) b^{4} c e x -\sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}+\sqrt {c}\right ) a^{2} b^{2} d e +\sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}+\sqrt {c}\right ) a \,b^{3} c e -\sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}+\sqrt {c}\right ) a \,b^{3} d e x +\sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}+\sqrt {c}\right ) b^{4} c e x}{a^{2} b^{2} \left (a b d x -b^{2} c x +a^{2} d -a b c \right )} \] Input:

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

Output:

(sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c))) 
*a**3*d*f + sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a 
*d - b*c)))*a**2*b*d*e + sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(s 
qrt(b)*sqrt(a*d - b*c)))*a**2*b*d*f*x - 2*sqrt(b)*sqrt(a*d - b*c)*atan((sq 
rt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a*b**2*c*e + sqrt(b)*sqrt(a*d - 
b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*a*b**2*d*e*x - 2*sq 
rt(b)*sqrt(a*d - b*c)*atan((sqrt(c + d*x)*b)/(sqrt(b)*sqrt(a*d - b*c)))*b* 
*3*c*e*x - sqrt(c + d*x)*a**3*b*d*f + sqrt(c + d*x)*a**2*b**2*c*f + sqrt(c 
 + d*x)*a**2*b**2*d*e - sqrt(c + d*x)*a*b**3*c*e + sqrt(c)*log(sqrt(c + d* 
x) - sqrt(c))*a**2*b**2*d*e - sqrt(c)*log(sqrt(c + d*x) - sqrt(c))*a*b**3* 
c*e + sqrt(c)*log(sqrt(c + d*x) - sqrt(c))*a*b**3*d*e*x - sqrt(c)*log(sqrt 
(c + d*x) - sqrt(c))*b**4*c*e*x - sqrt(c)*log(sqrt(c + d*x) + sqrt(c))*a** 
2*b**2*d*e + sqrt(c)*log(sqrt(c + d*x) + sqrt(c))*a*b**3*c*e - sqrt(c)*log 
(sqrt(c + d*x) + sqrt(c))*a*b**3*d*e*x + sqrt(c)*log(sqrt(c + d*x) + sqrt( 
c))*b**4*c*e*x)/(a**2*b**2*(a**2*d - a*b*c + a*b*d*x - b**2*c*x))