\(\int \frac {d+e x}{x^2 (a+b x+c x^2)^2} \, dx\) [54]

Optimal result

Integrand size = 21, antiderivative size = 201 \[ \int \frac {d+e x}{x^2 \left (a+b x+c x^2\right )^2} \, dx=-\frac {d}{a^2 x}-\frac {b^3 d-3 a b c d-a b^2 e+2 a^2 c e+c \left (b^2 d-2 a c d-a b e\right ) x}{a^2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\left (2 b^4 d-12 a b^2 c d+12 a^2 c^2 d-a b^3 e+6 a^2 b c e\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^3 \left (b^2-4 a c\right )^{3/2}}-\frac {(2 b d-a e) \log (x)}{a^3}+\frac {(2 b d-a e) \log \left (a+b x+c x^2\right )}{2 a^3} \] Output:

-d/a^2/x-(b^3*d-3*a*b*c*d-a*b^2*e+2*a^2*c*e+c*(-a*b*e-2*a*c*d+b^2*d)*x)/a^ 
2/(-4*a*c+b^2)/(c*x^2+b*x+a)-(6*a^2*b*c*e+12*a^2*c^2*d-a*b^3*e-12*a*b^2*c* 
d+2*b^4*d)*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/a^3/(-4*a*c+b^2)^(3/2)-(- 
a*e+2*b*d)*ln(x)/a^3+1/2*(-a*e+2*b*d)*ln(c*x^2+b*x+a)/a^3
 

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.96 \[ \int \frac {d+e x}{x^2 \left (a+b x+c x^2\right )^2} \, dx=\frac {-\frac {2 a d}{x}-\frac {2 a \left (b^3 d+2 a c (a e-c d x)+b^2 (-a e+c d x)-a b c (3 d+e x)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}-\frac {2 \left (2 b^4 d-12 a b^2 c d+12 a^2 c^2 d-a b^3 e+6 a^2 b c e\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}}+2 (-2 b d+a e) \log (x)+(2 b d-a e) \log (a+x (b+c x))}{2 a^3} \] Input:

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

Output:

((-2*a*d)/x - (2*a*(b^3*d + 2*a*c*(a*e - c*d*x) + b^2*(-(a*e) + c*d*x) - a 
*b*c*(3*d + e*x)))/((b^2 - 4*a*c)*(a + x*(b + c*x))) - (2*(2*b^4*d - 12*a* 
b^2*c*d + 12*a^2*c^2*d - a*b^3*e + 6*a^2*b*c*e)*ArcTan[(b + 2*c*x)/Sqrt[-b 
^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + 2*(-2*b*d + a*e)*Log[x] + (2*b*d - a* 
e)*Log[a + x*(b + c*x)])/(2*a^3)
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.15, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {1235, 25, 1200, 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.

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

\(\Big \downarrow \) 1235

\(\displaystyle \frac {c x (b d-2 a e)-a b e-2 a c d+b^2 d}{a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\int -\frac {2 d b^2-a e b-6 a c d+2 c (b d-2 a e) x}{x^2 \left (c x^2+b x+a\right )}dx}{a \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {2 d b^2-a e b-6 a c d+2 c (b d-2 a e) x}{x^2 \left (c x^2+b x+a\right )}dx}{a \left (b^2-4 a c\right )}+\frac {c x (b d-2 a e)-a b e-2 a c d+b^2 d}{a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\)

\(\Big \downarrow \) 1200

\(\displaystyle \frac {\int \left (-\frac {\left (4 a c-b^2\right ) (a e-2 b d)}{a^2 x}+\frac {2 d b^4-a e b^3-10 a c d b^2+5 a^2 c e b+6 a^2 c^2 d+c \left (b^2-4 a c\right ) (2 b d-a e) x}{a^2 \left (c x^2+b x+a\right )}+\frac {2 d b^2-a e b-6 a c d}{a x^2}\right )dx}{a \left (b^2-4 a c\right )}+\frac {c x (b d-2 a e)-a b e-2 a c d+b^2 d}{a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {\text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (6 a^2 b c e+12 a^2 c^2 d-a b^3 e-12 a b^2 c d+2 b^4 d\right )}{a^2 \sqrt {b^2-4 a c}}+\frac {\left (b^2-4 a c\right ) (2 b d-a e) \log \left (a+b x+c x^2\right )}{2 a^2}-\frac {\log (x) \left (b^2-4 a c\right ) (2 b d-a e)}{a^2}-\frac {-a b e-6 a c d+2 b^2 d}{a x}}{a \left (b^2-4 a c\right )}+\frac {c x (b d-2 a e)-a b e-2 a c d+b^2 d}{a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\)

Input:

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

Output:

(b^2*d - 2*a*c*d - a*b*e + c*(b*d - 2*a*e)*x)/(a*(b^2 - 4*a*c)*x*(a + b*x 
+ c*x^2)) + (-((2*b^2*d - 6*a*c*d - a*b*e)/(a*x)) - ((2*b^4*d - 12*a*b^2*c 
*d + 12*a^2*c^2*d - a*b^3*e + 6*a^2*b*c*e)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 
4*a*c]])/(a^2*Sqrt[b^2 - 4*a*c]) - ((b^2 - 4*a*c)*(2*b*d - a*e)*Log[x])/a^ 
2 + ((b^2 - 4*a*c)*(2*b*d - a*e)*Log[a + b*x + c*x^2])/(2*a^2))/(a*(b^2 - 
4*a*c))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 1200

Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* 
(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* 
x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In 
tegersQ[n]
 

rule 1235

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 
*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a 
+ b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] 
 + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^m 
*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 
 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* 
m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - 
f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] 
)
 

rule 2009

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

Maple [A] (verified)

Time = 1.44 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.46

method	result	size
default	\(-\frac {d}{a^{2} x}+\frac {\left (a e -2 b d \right ) \ln \left (x \right )}{a^{3}}-\frac {\frac {\frac {a c \left (a b e +2 a c d -b^{2} d \right ) x}{4 a c -b^{2}}-\frac {a \left (2 a^{2} c e -e a \,b^{2}-3 a b c d +b^{3} d \right )}{4 a c -b^{2}}}{c \,x^{2}+b x +a}+\frac {\frac {\left (4 a^{2} c^{2} e -a \,b^{2} c e -8 a b \,c^{2} d +2 b^{3} c d \right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (5 a^{2} b c e +6 a^{2} c^{2} d -a \,b^{3} e -10 c d a \,b^{2}+2 b^{4} d -\frac {\left (4 a^{2} c^{2} e -a \,b^{2} c e -8 a b \,c^{2} d +2 b^{3} c d \right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{4 a c -b^{2}}}{a^{3}}\)	\(293\)
risch	\(\text {Expression too large to display}\)	\(15188\)

Input:

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

Output:

-d/a^2/x+(a*e-2*b*d)/a^3*ln(x)-1/a^3*((a*c*(a*b*e+2*a*c*d-b^2*d)/(4*a*c-b^ 
2)*x-a*(2*a^2*c*e-a*b^2*e-3*a*b*c*d+b^3*d)/(4*a*c-b^2))/(c*x^2+b*x+a)+1/(4 
*a*c-b^2)*(1/2*(4*a^2*c^2*e-a*b^2*c*e-8*a*b*c^2*d+2*b^3*c*d)/c*ln(c*x^2+b* 
x+a)+2*(5*a^2*b*c*e+6*a^2*c^2*d-a*b^3*e-10*c*d*a*b^2+2*b^4*d-1/2*(4*a^2*c^ 
2*e-a*b^2*c*e-8*a*b*c^2*d+2*b^3*c*d)*b/c)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+ 
b)/(4*a*c-b^2)^(1/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 798 vs. \(2 (195) = 390\).

Time = 1.06 (sec) , antiderivative size = 1615, normalized size of antiderivative = 8.03 \[ \int \frac {d+e x}{x^2 \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/2*(2*(2*(a*b^4*c - 7*a^2*b^2*c^2 + 12*a^3*c^3)*d - (a^2*b^3*c - 4*a^3* 
b*c^2)*e)*x^2 + ((2*(b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*d - (a*b^3*c - 6*a^2 
*b*c^2)*e)*x^3 + (2*(b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*d - (a*b^4 - 6*a^2*b^2 
*c)*e)*x^2 + (2*(a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2)*d - (a^2*b^3 - 6*a^3*b*c 
)*e)*x)*sqrt(b^2 - 4*a*c)*log((2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c + sqrt(b^ 
2 - 4*a*c)*(2*c*x + b))/(c*x^2 + b*x + a)) + 2*(a^2*b^4 - 8*a^3*b^2*c + 16 
*a^4*c^2)*d + 2*((2*a*b^5 - 15*a^2*b^3*c + 28*a^3*b*c^2)*d - (a^2*b^4 - 6* 
a^3*b^2*c + 8*a^4*c^2)*e)*x - ((2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d - 
 (a*b^4*c - 8*a^2*b^2*c^2 + 16*a^3*c^3)*e)*x^3 + (2*(b^6 - 8*a*b^4*c + 16* 
a^2*b^2*c^2)*d - (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*e)*x^2 + (2*(a*b^5 - 
 8*a^2*b^3*c + 16*a^3*b*c^2)*d - (a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)*e)*x 
)*log(c*x^2 + b*x + a) + 2*((2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d - (a 
*b^4*c - 8*a^2*b^2*c^2 + 16*a^3*c^3)*e)*x^3 + (2*(b^6 - 8*a*b^4*c + 16*a^2 
*b^2*c^2)*d - (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*e)*x^2 + (2*(a*b^5 - 8* 
a^2*b^3*c + 16*a^3*b*c^2)*d - (a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)*e)*x)*l 
og(x))/((a^3*b^4*c - 8*a^4*b^2*c^2 + 16*a^5*c^3)*x^3 + (a^3*b^5 - 8*a^4*b^ 
3*c + 16*a^5*b*c^2)*x^2 + (a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2)*x), -1/2*(2 
*(2*(a*b^4*c - 7*a^2*b^2*c^2 + 12*a^3*c^3)*d - (a^2*b^3*c - 4*a^3*b*c^2)*e 
)*x^2 + 2*((2*(b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*d - (a*b^3*c - 6*a^2*b*c^2 
)*e)*x^3 + (2*(b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*d - (a*b^4 - 6*a^2*b^2*c)...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((e*x+d)/x^2/(c*x^2+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(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.18 \[ \int \frac {d+e x}{x^2 \left (a+b x+c x^2\right )^2} \, dx=\frac {{\left (2 \, b^{4} d - 12 \, a b^{2} c d + 12 \, a^{2} c^{2} d - a b^{3} e + 6 \, a^{2} b c e\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (a^{3} b^{2} - 4 \, a^{4} c\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {2 \, b^{2} c d x^{2} - 6 \, a c^{2} d x^{2} - a b c e x^{2} + 2 \, b^{3} d x - 7 \, a b c d x - a b^{2} e x + 2 \, a^{2} c e x + a b^{2} d - 4 \, a^{2} c d}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} {\left (c x^{3} + b x^{2} + a x\right )}} + \frac {{\left (2 \, b d - a e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, a^{3}} - \frac {{\left (2 \, b d - a e\right )} \log \left ({\left | x \right |}\right )}{a^{3}} \] Input:

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

Output:

(2*b^4*d - 12*a*b^2*c*d + 12*a^2*c^2*d - a*b^3*e + 6*a^2*b*c*e)*arctan((2* 
c*x + b)/sqrt(-b^2 + 4*a*c))/((a^3*b^2 - 4*a^4*c)*sqrt(-b^2 + 4*a*c)) - (2 
*b^2*c*d*x^2 - 6*a*c^2*d*x^2 - a*b*c*e*x^2 + 2*b^3*d*x - 7*a*b*c*d*x - a*b 
^2*e*x + 2*a^2*c*e*x + a*b^2*d - 4*a^2*c*d)/((a^2*b^2 - 4*a^3*c)*(c*x^3 + 
b*x^2 + a*x)) + 1/2*(2*b*d - a*e)*log(c*x^2 + b*x + a)/a^3 - (2*b*d - a*e) 
*log(abs(x))/a^3
 

Mupad [B] (verification not implemented)

Time = 12.99 (sec) , antiderivative size = 1366, normalized size of antiderivative = 6.80 \[ \int \frac {d+e x}{x^2 \left (a+b x+c x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

log(96*a^5*c^3*e - 2*a^2*b^6*e + 4*a*b^7*d + 4*b^8*d*x + 174*a^3*b^3*c^2*d 
 - 2*a^2*b^3*e*(-(4*a*c - b^2)^3)^(1/2) + 6*a^3*c^2*d*(-(4*a*c - b^2)^3)^( 
1/2) - 84*a^4*b^2*c^2*e - 2*a*b^7*e*x + 4*a*b^4*d*(-(4*a*c - b^2)^3)^(1/2) 
 - 46*a^2*b^5*c*d - 216*a^4*b*c^3*d + 23*a^3*b^4*c*e + 48*a^4*c^4*d*x + 4* 
b^5*d*x*(-(4*a*c - b^2)^3)^(1/2) + 9*a^3*b*c*e*(-(4*a*c - b^2)^3)^(1/2) - 
2*a*b^4*e*x*(-(4*a*c - b^2)^3)^(1/2) + 24*a^2*b^5*c*e*x + 120*a^4*b*c^3*e* 
x - 18*a^2*b^2*c*d*(-(4*a*c - b^2)^3)^(1/2) + 194*a^2*b^4*c^2*d*x - 276*a^ 
3*b^2*c^3*d*x - 94*a^3*b^3*c^2*e*x - 12*a^3*c^2*e*x*(-(4*a*c - b^2)^3)^(1/ 
2) - 48*a*b^6*c*d*x - 24*a*b^3*c*d*x*(-(4*a*c - b^2)^3)^(1/2) + 30*a^2*b*c 
^2*d*x*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b^2*c*e*x*(-(4*a*c - b^2)^3)^(1/2 
))*((b^4*d*(-(4*a*c - b^2)^3)^(1/2) + 6*a^2*c^2*d*(-(4*a*c - b^2)^3)^(1/2) 
 - (a*b^3*e*(-(4*a*c - b^2)^3)^(1/2))/2 - 6*a*b^2*c*d*(-(4*a*c - b^2)^3)^( 
1/2) + 3*a^2*b*c*e*(-(4*a*c - b^2)^3)^(1/2))/(a^3*b^6 - 64*a^6*c^3 - 12*a^ 
4*b^4*c + 48*a^5*b^2*c^2) - e/(2*a^2) + (b*d)/a^3) - log(2*a^2*b^6*e - 96* 
a^5*c^3*e - 4*a*b^7*d - 4*b^8*d*x - 174*a^3*b^3*c^2*d - 2*a^2*b^3*e*(-(4*a 
*c - b^2)^3)^(1/2) + 6*a^3*c^2*d*(-(4*a*c - b^2)^3)^(1/2) + 84*a^4*b^2*c^2 
*e + 2*a*b^7*e*x + 4*a*b^4*d*(-(4*a*c - b^2)^3)^(1/2) + 46*a^2*b^5*c*d + 2 
16*a^4*b*c^3*d - 23*a^3*b^4*c*e - 48*a^4*c^4*d*x + 4*b^5*d*x*(-(4*a*c - b^ 
2)^3)^(1/2) + 9*a^3*b*c*e*(-(4*a*c - b^2)^3)^(1/2) - 2*a*b^4*e*x*(-(4*a*c 
- b^2)^3)^(1/2) - 24*a^2*b^5*c*e*x - 120*a^4*b*c^3*e*x - 18*a^2*b^2*c*d...
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 1575, normalized size of antiderivative = 7.84 \[ \int \frac {d+e x}{x^2 \left (a+b x+c x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b**2*c 
*e*x - 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b*c 
**2*d*x + 2*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b 
**4*e*x + 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2* 
b**3*c*d*x - 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a* 
*2*b**3*c*e*x**2 - 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b** 
2))*a**2*b**2*c**2*d*x**2 - 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4* 
a*c - b**2))*a**2*b**2*c**2*e*x**3 - 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x 
)/sqrt(4*a*c - b**2))*a**2*b*c**3*d*x**3 - 4*sqrt(4*a*c - b**2)*atan((b + 
2*c*x)/sqrt(4*a*c - b**2))*a*b**5*d*x + 2*sqrt(4*a*c - b**2)*atan((b + 2*c 
*x)/sqrt(4*a*c - b**2))*a*b**5*e*x**2 + 24*sqrt(4*a*c - b**2)*atan((b + 2* 
c*x)/sqrt(4*a*c - b**2))*a*b**4*c*d*x**2 + 2*sqrt(4*a*c - b**2)*atan((b + 
2*c*x)/sqrt(4*a*c - b**2))*a*b**4*c*e*x**3 + 24*sqrt(4*a*c - b**2)*atan((b 
 + 2*c*x)/sqrt(4*a*c - b**2))*a*b**3*c**2*d*x**3 - 4*sqrt(4*a*c - b**2)*at 
an((b + 2*c*x)/sqrt(4*a*c - b**2))*b**6*d*x**2 - 4*sqrt(4*a*c - b**2)*atan 
((b + 2*c*x)/sqrt(4*a*c - b**2))*b**5*c*d*x**3 - 16*log(a + b*x + c*x**2)* 
a**4*b*c**2*e*x + 8*log(a + b*x + c*x**2)*a**3*b**3*c*e*x + 32*log(a + b*x 
 + c*x**2)*a**3*b**2*c**2*d*x - 16*log(a + b*x + c*x**2)*a**3*b**2*c**2*e* 
x**2 - 16*log(a + b*x + c*x**2)*a**3*b*c**3*e*x**3 - log(a + b*x + c*x**2) 
*a**2*b**5*e*x - 16*log(a + b*x + c*x**2)*a**2*b**4*c*d*x + 8*log(a + b...