\(\int \frac {(f+g x) \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{d+e x} \, dx\) [261]

Optimal result

Integrand size = 42, antiderivative size = 171 \[ \int \frac {(f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx=\frac {\left (a e^2 g+c d (4 e f-3 d g)+2 c d e g x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c d e^2}+\frac {\left (c d^2-a e^2\right ) \left (a e^2 g-c d (4 e f-3 d g)\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{4 c^{3/2} d^{3/2} e^{5/2}} \] Output:

1/4*(a*e^2*g+c*d*(-3*d*g+4*e*f)+2*c*d*e*g*x)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e* 
x^2)^(1/2)/c/d/e^2+1/4*(-a*e^2+c*d^2)*(a*e^2*g-c*d*(-3*d*g+4*e*f))*arctanh 
(c^(1/2)*d^(1/2)*(e*x+d)/e^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/ 
c^(3/2)/d^(3/2)/e^(5/2)
 

Mathematica [A] (verified)

Time = 1.31 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.13 \[ \int \frac {(f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (\sqrt {c} \sqrt {d} \sqrt {e} \left (a e^2 g+c d (4 e f-3 d g+2 e g x)\right )-\frac {2 \left (c d^2-a e^2\right ) \left (a e^2 g+c d (-4 e f+3 d g)\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {d-\frac {a e^2}{c d}}-\sqrt {d+e x}\right )}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{4 c^{3/2} d^{3/2} e^{5/2}} \] Input:

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

Output:

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(Sqrt[c]*Sqrt[d]*Sqrt[e]*(a*e^2*g + c*d*(4* 
e*f - 3*d*g + 2*e*g*x)) - (2*(c*d^2 - a*e^2)*(a*e^2*g + c*d*(-4*e*f + 3*d* 
g))*ArcTanh[(Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c]*Sqrt[d]*(Sqrt[d - (a*e^2) 
/(c*d)] - Sqrt[d + e*x]))])/(Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(4*c^(3/2) 
*d^(3/2)*e^(5/2))
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1215, 1225, 1092, 219}

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.

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

\(\Big \downarrow \) 1215

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

\(\Big \downarrow \) 1225

\(\displaystyle \frac {\left (c d^2-a e^2\right ) \left (a e^2 g-c d (4 e f-3 d g)\right ) \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 c d e^2}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (a e^2 g+c d (4 e f-3 d g)+2 c d e g x\right )}{4 c d e^2}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\left (c d^2-a e^2\right ) \left (a e^2 g-c d (4 e f-3 d g)\right ) \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{4 c d e^2}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (a e^2 g+c d (4 e f-3 d g)+2 c d e g x\right )}{4 c d e^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\left (c d^2-a e^2\right ) \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right ) \left (a e^2 g-c d (4 e f-3 d g)\right )}{8 c^{3/2} d^{3/2} e^{5/2}}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (a e^2 g+c d (4 e f-3 d g)+2 c d e g x\right )}{4 c d e^2}\)

Input:

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

Output:

((a*e^2*g + c*d*(4*e*f - 3*d*g) + 2*c*d*e*g*x)*Sqrt[a*d*e + (c*d^2 + a*e^2 
)*x + c*d*e*x^2])/(4*c*d*e^2) + ((c*d^2 - a*e^2)*(a*e^2*g - c*d*(4*e*f - 3 
*d*g))*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt 
[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(8*c^(3/2)*d^(3/2)*e^(5/2))
 

Defintions of rubi rules used

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1215

Int[(((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_))/( 
(d_) + (e_.)*(x_)), x_Symbol] :> Int[(a/d + c*(x/e))*(f + g*x)^n*(a + b*x + 
 c*x^2)^(p - 1), x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[c*d^2 - 
 b*d*e + a*e^2, 0] && GtQ[p, 0]
 

rule 1225

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( 
x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 
 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), 
 x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p 
+ 3))/(2*c^2*(2*p + 3))   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c 
, d, e, f, g, p}, x] &&  !LeQ[p, -1]
 

Maple [A] (verified)

Time = 1.81 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.75

method	result	size
default	\(\frac {g \left (\frac {\left (2 c d x e +a \,e^{2}+c \,d^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{8 d e c \sqrt {d e c}}\right )}{e}-\frac {\left (d g -e f \right ) \left (\sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}+\frac {\left (a \,e^{2}-c \,d^{2}\right ) \ln \left (\frac {\frac {a \,e^{2}}{2}-\frac {c \,d^{2}}{2}+d e c \left (x +\frac {d}{e}\right )}{\sqrt {d e c}}+\sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{2 \sqrt {d e c}}\right )}{e^{2}}\)	\(299\)

Input:

int((g*x+f)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)/(e*x+d),x,method=_RETU 
RNVERBOSE)
 

Output:

g/e*(1/4*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)/c 
/d/e+1/8*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/d/e/c*ln((1/2*a*e^2+1/2*c*d^2+c*d 
*x*e)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2))/(d*e*c)^(1/2) 
)-(d*g-e*f)/e^2*((d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)+1/2*(a*e^2- 
c*d^2)*ln((1/2*a*e^2-1/2*c*d^2+d*e*c*(x+d/e))/(d*e*c)^(1/2)+(d*e*c*(x+d/e) 
^2+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(d*e*c)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 502, normalized size of antiderivative = 2.94 \[ \int \frac {(f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx=\left [\frac {\sqrt {c d e} {\left (4 \, {\left (c^{2} d^{3} e - a c d e^{3}\right )} f - {\left (3 \, c^{2} d^{4} - 2 \, a c d^{2} e^{2} - a^{2} e^{4}\right )} g\right )} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {c d e} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) + 4 \, {\left (2 \, c^{2} d^{2} e^{2} g x + 4 \, c^{2} d^{2} e^{2} f - {\left (3 \, c^{2} d^{3} e - a c d e^{3}\right )} g\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{16 \, c^{2} d^{2} e^{3}}, \frac {\sqrt {-c d e} {\left (4 \, {\left (c^{2} d^{3} e - a c d e^{3}\right )} f - {\left (3 \, c^{2} d^{4} - 2 \, a c d^{2} e^{2} - a^{2} e^{4}\right )} g\right )} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) + 2 \, {\left (2 \, c^{2} d^{2} e^{2} g x + 4 \, c^{2} d^{2} e^{2} f - {\left (3 \, c^{2} d^{3} e - a c d e^{3}\right )} g\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{8 \, c^{2} d^{2} e^{3}}\right ] \] Input:

integrate((g*x+f)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d),x, algor 
ithm="fricas")
 

Output:

[1/16*(sqrt(c*d*e)*(4*(c^2*d^3*e - a*c*d*e^3)*f - (3*c^2*d^4 - 2*a*c*d^2*e 
^2 - a^2*e^4)*g)*log(8*c^2*d^2*e^2*x^2 + c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4 
 - 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*e*x + c*d^2 + a*e^ 
2)*sqrt(c*d*e) + 8*(c^2*d^3*e + a*c*d*e^3)*x) + 4*(2*c^2*d^2*e^2*g*x + 4*c 
^2*d^2*e^2*f - (3*c^2*d^3*e - a*c*d*e^3)*g)*sqrt(c*d*e*x^2 + a*d*e + (c*d^ 
2 + a*e^2)*x))/(c^2*d^2*e^3), 1/8*(sqrt(-c*d*e)*(4*(c^2*d^3*e - a*c*d*e^3) 
*f - (3*c^2*d^4 - 2*a*c*d^2*e^2 - a^2*e^4)*g)*arctan(1/2*sqrt(c*d*e*x^2 + 
a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(-c*d*e)/(c^2*d 
^2*e^2*x^2 + a*c*d^2*e^2 + (c^2*d^3*e + a*c*d*e^3)*x)) + 2*(2*c^2*d^2*e^2* 
g*x + 4*c^2*d^2*e^2*f - (3*c^2*d^3*e - a*c*d*e^3)*g)*sqrt(c*d*e*x^2 + a*d* 
e + (c*d^2 + a*e^2)*x))/(c^2*d^2*e^3)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((g*x+f)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d),x, algor 
ithm="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(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.12 \[ \int \frac {(f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx=\frac {1}{4} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (\frac {2 \, g x}{e} + \frac {4 \, c d e f - 3 \, c d^{2} g + a e^{2} g}{c d e^{2}}\right )} + \frac {{\left (4 \, c^{2} d^{3} e f - 4 \, a c d e^{3} f - 3 \, c^{2} d^{4} g + 2 \, a c d^{2} e^{2} g + a^{2} e^{4} g\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{8 \, \sqrt {c d e} c d e^{2}} \] Input:

integrate((g*x+f)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d),x, algor 
ithm="giac")
 

Output:

1/4*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*g*x/e + (4*c*d*e*f - 3* 
c*d^2*g + a*e^2*g)/(c*d*e^2)) + 1/8*(4*c^2*d^3*e*f - 4*a*c*d*e^3*f - 3*c^2 
*d^4*g + 2*a*c*d^2*e^2*g + a^2*e^4*g)*log(abs(-c*d^2 - a*e^2 - 2*sqrt(c*d* 
e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))))/(sqrt(c 
*d*e)*c*d*e^2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 398, normalized size of antiderivative = 2.33 \[ \int \frac {(f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x} \, dx=\frac {\sqrt {e x +d}\, \sqrt {c d x +a e}\, a c d \,e^{3} g -3 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{2} d^{3} e g +4 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{2} d^{2} e^{2} f +2 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{2} d^{2} e^{2} g x -\sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{2} e^{4} g -2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a c \,d^{2} e^{2} g +4 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a c d \,e^{3} f +3 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) c^{2} d^{4} g -4 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) c^{2} d^{3} e f}{4 c^{2} d^{2} e^{3}} \] Input:

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

Output:

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