\(\int \frac {1}{x^3 (d+e x) \sqrt {a+b x+c x^2}} \, dx\) [37]

Optimal result

Integrand size = 25, antiderivative size = 223 \[ \int \frac {1}{x^3 (d+e x) \sqrt {a+b x+c x^2}} \, dx=-\frac {\sqrt {a+b x+c x^2}}{2 a d x^2}+\frac {(3 b d+4 a e) \sqrt {a+b x+c x^2}}{4 a^2 d^2 x}-\frac {\left (3 b^2 d^2-4 a c d^2+4 a b d e+8 a^2 e^2\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 a^{5/2} d^3}-\frac {e^3 \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{d^3 \sqrt {c d^2-b d e+a e^2}} \] Output:

-1/2*(c*x^2+b*x+a)^(1/2)/a/d/x^2+1/4*(4*a*e+3*b*d)*(c*x^2+b*x+a)^(1/2)/a^2 
/d^2/x-1/8*(8*a^2*e^2+4*a*b*d*e-4*a*c*d^2+3*b^2*d^2)*arctanh(1/2*(b*x+2*a) 
/a^(1/2)/(c*x^2+b*x+a)^(1/2))/a^(5/2)/d^3-e^3*arctanh(1/2*(b*d-2*a*e+(-b*e 
+2*c*d)*x)/(a*e^2-b*d*e+c*d^2)^(1/2)/(c*x^2+b*x+a)^(1/2))/d^3/(a*e^2-b*d*e 
+c*d^2)^(1/2)
 

Mathematica [A] (verified)

Time = 2.37 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x^3 (d+e x) \sqrt {a+b x+c x^2}} \, dx=\frac {\frac {d (-2 a d+3 b d x+4 a e x) \sqrt {a+x (b+c x)}}{a^2 x^2}-\frac {8 e^3 \sqrt {-c d^2+b d e-a e^2} \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+x (b+c x)}}{\sqrt {-c d^2+e (b d-a e)}}\right )}{c d^2+e (-b d+a e)}+\frac {8 e^2 \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {d \left (-3 b^2 d+4 a c d-4 a b e\right ) \text {arctanh}\left (\frac {-\sqrt {c} x+\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{a^{5/2}}}{4 d^3} \] Input:

Integrate[1/(x^3*(d + e*x)*Sqrt[a + b*x + c*x^2]),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.39, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1289, 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.

\(\displaystyle \int \frac {1}{x^3 (d+e x) \sqrt {a+b x+c x^2}} \, dx\)

\(\Big \downarrow \) 1289

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

\(\Big \downarrow \) 2009

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

Input:

Int[1/(x^3*(d + e*x)*Sqrt[a + b*x + c*x^2]),x]
 

Output:

-1/2*Sqrt[a + b*x + c*x^2]/(a*d*x^2) + (3*b*Sqrt[a + b*x + c*x^2])/(4*a^2* 
d*x) + (e*Sqrt[a + b*x + c*x^2])/(a*d^2*x) - ((3*b^2 - 4*a*c)*ArcTanh[(2*a 
 + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(8*a^(5/2)*d) - (b*e*ArcTanh[( 
2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(2*a^(3/2)*d^2) - (e^2*ArcT 
anh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(Sqrt[a]*d^3) - (e^3*A 
rcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt 
[a + b*x + c*x^2])])/(d^3*Sqrt[c*d^2 - b*d*e + a*e^2])
 

Defintions of rubi rules used

rule 1289

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(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}, x] && ( 
IntegerQ[p] || (ILtQ[m, 0] && ILtQ[n, 0]))
 

rule 2009

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

Maple [A] (verified)

Time = 1.56 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.25

method	result	size
risch	\(-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (-4 a e x -3 b d x +2 a d \right )}{4 a^{2} d^{2} x^{2}}+\frac {-\frac {\left (8 e^{2} a^{2}+4 a b d e -4 a \,d^{2} c +3 b^{2} d^{2}\right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{d \sqrt {a}}+\frac {8 e^{2} a^{2} \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{d \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}}{8 a^{2} d^{2}}\)	\(278\)
default	\(\frac {-\frac {\sqrt {c \,x^{2}+b x +a}}{2 a \,x^{2}}-\frac {3 b \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{a x}+\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}+\frac {c \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}}{d}-\frac {e^{2} \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{d^{3} \sqrt {a}}-\frac {e \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{a x}+\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{d^{2}}+\frac {e^{2} \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{d^{3} \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}\)	\(384\)

Input:

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

Output:

-1/4*(c*x^2+b*x+a)^(1/2)*(-4*a*e*x-3*b*d*x+2*a*d)/a^2/d^2/x^2+1/8/a^2/d^2* 
(-(8*a^2*e^2+4*a*b*d*e-4*a*c*d^2+3*b^2*d^2)/d/a^(1/2)*ln((2*a+b*x+2*a^(1/2 
)*(c*x^2+b*x+a)^(1/2))/x)+8*e^2*a^2/d/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln(( 
2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2 
)^(1/2)*(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)) 
/(x+d/e)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 409 vs. \(2 (197) = 394\).

Time = 1.42 (sec) , antiderivative size = 1737, normalized size of antiderivative = 7.79 \[ \int \frac {1}{x^3 (d+e x) \sqrt {a+b x+c x^2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/16*(8*sqrt(c*d^2 - b*d*e + a*e^2)*a^3*e^3*x^2*log((8*a*b*d*e - 8*a^2*e^ 
2 - (b^2 + 4*a*c)*d^2 - (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 + 
4*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d 
- b*e)*x) - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)/(e^2*x^2 + 
2*d*e*x + d^2)) - (4*a^2*b*d*e^3 - 8*a^3*e^4 - (3*b^2*c - 4*a*c^2)*d^4 + ( 
3*b^3 - 8*a*b*c)*d^3*e + (a*b^2 - 4*a^2*c)*d^2*e^2)*sqrt(a)*x^2*log(-(8*a* 
b*x + (b^2 + 4*a*c)*x^2 - 4*sqrt(c*x^2 + b*x + a)*(b*x + 2*a)*sqrt(a) + 8* 
a^2)/x^2) - 4*(2*a^2*c*d^4 - 2*a^2*b*d^3*e + 2*a^3*d^2*e^2 - (3*a*b*c*d^4 
- a^2*b*d^2*e^2 + 4*a^3*d*e^3 - (3*a*b^2 - 4*a^2*c)*d^3*e)*x)*sqrt(c*x^2 + 
 b*x + a))/((a^3*c*d^5 - a^3*b*d^4*e + a^4*d^3*e^2)*x^2), -1/16*(16*sqrt(- 
c*d^2 + b*d*e - a*e^2)*a^3*e^3*x^2*arctan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2 
)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x)/(a*c*d^2 - a*b*d*e 
 + a^2*e^2 + (c^2*d^2 - b*c*d*e + a*c*e^2)*x^2 + (b*c*d^2 - b^2*d*e + a*b* 
e^2)*x)) + (4*a^2*b*d*e^3 - 8*a^3*e^4 - (3*b^2*c - 4*a*c^2)*d^4 + (3*b^3 - 
 8*a*b*c)*d^3*e + (a*b^2 - 4*a^2*c)*d^2*e^2)*sqrt(a)*x^2*log(-(8*a*b*x + ( 
b^2 + 4*a*c)*x^2 - 4*sqrt(c*x^2 + b*x + a)*(b*x + 2*a)*sqrt(a) + 8*a^2)/x^ 
2) + 4*(2*a^2*c*d^4 - 2*a^2*b*d^3*e + 2*a^3*d^2*e^2 - (3*a*b*c*d^4 - a^2*b 
*d^2*e^2 + 4*a^3*d*e^3 - (3*a*b^2 - 4*a^2*c)*d^3*e)*x)*sqrt(c*x^2 + b*x + 
a))/((a^3*c*d^5 - a^3*b*d^4*e + a^4*d^3*e^2)*x^2), 1/8*(4*sqrt(c*d^2 - b*d 
*e + a*e^2)*a^3*e^3*x^2*log((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a*c)*d^2 ...
 

Sympy [F]

\[ \int \frac {1}{x^3 (d+e x) \sqrt {a+b x+c x^2}} \, dx=\int \frac {1}{x^{3} \left (d + e x\right ) \sqrt {a + b x + c x^{2}}}\, dx \] Input:

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

Output:

Integral(1/(x**3*(d + e*x)*sqrt(a + b*x + c*x**2)), x)
 

Maxima [F]

\[ \int \frac {1}{x^3 (d+e x) \sqrt {a+b x+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )} x^{3}} \,d x } \] Input:

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

Output:

integrate(1/(sqrt(c*x^2 + b*x + a)*(e*x + d)*x^3), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 399 vs. \(2 (197) = 394\).

Time = 0.26 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.79 \[ \int \frac {1}{x^3 (d+e x) \sqrt {a+b x+c x^2}} \, dx=-\frac {2 \, e^{3} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} + b d e - a e^{2}}}\right )}{\sqrt {-c d^{2} + b d e - a e^{2}} d^{3}} + \frac {{\left (3 \, b^{2} d^{2} - 4 \, a c d^{2} + 4 \, a b d e + 8 \, a^{2} e^{2}\right )} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + b x + a}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a} a^{2} d^{3}} - \frac {3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} b^{2} d - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a c d + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a b e + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a^{2} \sqrt {c} e - 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a b^{2} d - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{2} c d - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{2} b e - 8 \, a^{2} b \sqrt {c} d - 8 \, a^{3} \sqrt {c} e}{4 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} - a\right )}^{2} a^{2} d^{2}} \] Input:

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

Output:

-2*e^3*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*e + sqrt(c)*d)/sqrt(-c 
*d^2 + b*d*e - a*e^2))/(sqrt(-c*d^2 + b*d*e - a*e^2)*d^3) + 1/4*(3*b^2*d^2 
 - 4*a*c*d^2 + 4*a*b*d*e + 8*a^2*e^2)*arctan(-(sqrt(c)*x - sqrt(c*x^2 + b* 
x + a))/sqrt(-a))/(sqrt(-a)*a^2*d^3) - 1/4*(3*(sqrt(c)*x - sqrt(c*x^2 + b* 
x + a))^3*b^2*d - 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*c*d + 4*(sqrt( 
c)*x - sqrt(c*x^2 + b*x + a))^3*a*b*e + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + 
a))^2*a^2*sqrt(c)*e - 5*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^2*d - 4*(s 
qrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*c*d - 4*(sqrt(c)*x - sqrt(c*x^2 + b* 
x + a))*a^2*b*e - 8*a^2*b*sqrt(c)*d - 8*a^3*sqrt(c)*e)/(((sqrt(c)*x - sqrt 
(c*x^2 + b*x + a))^2 - a)^2*a^2*d^2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^3 (d+e x) \sqrt {a+b x+c x^2}} \, dx=\int \frac {1}{x^3\,\left (d+e\,x\right )\,\sqrt {c\,x^2+b\,x+a}} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.50 (sec) , antiderivative size = 756, normalized size of antiderivative = 3.39 \[ \int \frac {1}{x^3 (d+e x) \sqrt {a+b x+c x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(8*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e* 
*2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**3*e**3*x**2 - 8*s 
qrt(a*e**2 - b*d*e + c*d**2)*log(d + e*x)*a**3*e**3*x**2 - 4*sqrt(a + b*x 
+ c*x**2)*a**3*d**2*e**2 + 8*sqrt(a + b*x + c*x**2)*a**3*d*e**3*x + 4*sqrt 
(a + b*x + c*x**2)*a**2*b*d**3*e - 2*sqrt(a + b*x + c*x**2)*a**2*b*d**2*e* 
*2*x - 4*sqrt(a + b*x + c*x**2)*a**2*c*d**4 + 8*sqrt(a + b*x + c*x**2)*a** 
2*c*d**3*e*x - 6*sqrt(a + b*x + c*x**2)*a*b**2*d**3*e*x + 6*sqrt(a + b*x + 
 c*x**2)*a*b*c*d**4*x + 8*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2 
*a - b*x)*a**3*e**4*x**2 - 4*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) 
- 2*a - b*x)*a**2*b*d*e**3*x**2 + 4*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c 
*x**2) - 2*a - b*x)*a**2*c*d**2*e**2*x**2 - sqrt(a)*log(2*sqrt(a)*sqrt(a + 
 b*x + c*x**2) - 2*a - b*x)*a*b**2*d**2*e**2*x**2 + 8*sqrt(a)*log(2*sqrt(a 
)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a*b*c*d**3*e*x**2 - 4*sqrt(a)*log(2* 
sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a*c**2*d**4*x**2 - 3*sqrt(a)*l 
og(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*b**3*d**3*e*x**2 + 3*sqrt 
(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*b**2*c*d**4*x**2 - 8 
*sqrt(a)*log(x)*a**3*e**4*x**2 + 4*sqrt(a)*log(x)*a**2*b*d*e**3*x**2 - 4*s 
qrt(a)*log(x)*a**2*c*d**2*e**2*x**2 + sqrt(a)*log(x)*a*b**2*d**2*e**2*x**2 
 - 8*sqrt(a)*log(x)*a*b*c*d**3*e*x**2 + 4*sqrt(a)*log(x)*a*c**2*d**4*x**2 
+ 3*sqrt(a)*log(x)*b**3*d**3*e*x**2 - 3*sqrt(a)*log(x)*b**2*c*d**4*x**2...