Integrand size = 20, antiderivative size = 173 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^2} \, dx=\frac {e^2 (2 c d-b e) x}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^2 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {(2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^2 \left (b^2-4 a c\right )^{3/2}}+\frac {e^3 \log \left (a+b x+c x^2\right )}{2 c^2} \] Output:
e^2*(-b*e+2*c*d)*x/c/(-4*a*c+b^2)-(e*x+d)^2*(b*d-2*a*e+(-b*e+2*c*d)*x)/(-4 *a*c+b^2)/(c*x^2+b*x+a)+(-b*e+2*c*d)*(2*c^2*d^2-b^2*e^2-2*c*e*(-3*a*e+b*d) )*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/c^2/(-4*a*c+b^2)^(3/2)+1/2*e^3*ln( c*x^2+b*x+a)/c^2
Time = 0.19 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.16 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^2} \, dx=\frac {\frac {2 \left (b^3 e^3 x+b^2 e^2 (a e-3 c d x)-2 c \left (a^2 e^3+c^2 d^3 x-3 a c d e (d+e x)\right )-b c \left (c d^2 (d-3 e x)+3 a e^2 (d+e x)\right )\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\frac {2 (-2 c d+b e) \left (-2 c^2 d^2+b^2 e^2+2 c e (b d-3 a e)\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}}+e^3 \log (a+x (b+c x))}{2 c^2} \] Input:
Integrate[(d + e*x)^3/(a + b*x + c*x^2)^2,x]
Output:
((2*(b^3*e^3*x + b^2*e^2*(a*e - 3*c*d*x) - 2*c*(a^2*e^3 + c^2*d^3*x - 3*a* c*d*e*(d + e*x)) - b*c*(c*d^2*(d - 3*e*x) + 3*a*e^2*(d + e*x))))/((b^2 - 4 *a*c)*(a + x*(b + c*x))) + (2*(-2*c*d + b*e)*(-2*c^2*d^2 + b^2*e^2 + 2*c*e *(b*d - 3*a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/ 2) + e^3*Log[a + x*(b + c*x)])/(2*c^2)
Time = 0.42 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.07, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1164, 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.
\(\displaystyle \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 1164 |
\(\displaystyle -\frac {\int \frac {(d+e x) \left (2 c d^2-e (3 b d-4 a e)-e (2 c d-b e) x\right )}{c x^2+b x+a}dx}{b^2-4 a c}-\frac {(d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
\(\Big \downarrow \) 1200 |
\(\displaystyle -\frac {\int \left (\frac {2 c^2 d^3-3 c e (b d-2 a e) d-a b e^3-\left (b^2-4 a c\right ) e^3 x}{c \left (c x^2+b x+a\right )}-e^2 \left (2 d-\frac {b e}{c}\right )\right )dx}{b^2-4 a c}-\frac {(d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {-\frac {(2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (-2 c e (b d-3 a e)-b^2 e^2+2 c^2 d^2\right )}{c^2 \sqrt {b^2-4 a c}}-\frac {e^3 \left (b^2-4 a c\right ) \log \left (a+b x+c x^2\right )}{2 c^2}-e^2 x \left (2 d-\frac {b e}{c}\right )}{b^2-4 a c}-\frac {(d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
Input:
Int[(d + e*x)^3/(a + b*x + c*x^2)^2,x]
Output:
-(((d + e*x)^2*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2))) - (-(e^2*(2*d - (b*e)/c)*x) - ((2*c*d - b*e)*(2*c^2*d^2 - b^2*e^2 - 2*c*e*(b*d - 3*a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^2*Sqrt[ b^2 - 4*a*c]) - ((b^2 - 4*a*c)*e^3*Log[a + b*x + c*x^2])/(2*c^2))/(b^2 - 4 *a*c)
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m - 1)*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a* c)) Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2* c*d^2*(2*p + 3) + e*(b*e - 2*d*c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 1] && Int QuadraticQ[a, b, c, d, e, m, p, x]
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]
Time = 1.00 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.66
method | result | size |
default | \(\frac {\frac {\left (3 a b c \,e^{3}-6 d \,e^{2} a \,c^{2}-b^{3} e^{3}+3 d \,e^{2} b^{2} c -3 d^{2} e b \,c^{2}+2 d^{3} c^{3}\right ) x}{c^{2} \left (4 a c -b^{2}\right )}+\frac {2 a^{2} c \,e^{3}-b^{2} e^{3} a +3 b d \,e^{2} a c -6 a \,c^{2} d^{2} e +d^{3} b \,c^{2}}{c^{2} \left (4 a c -b^{2}\right )}}{c \,x^{2}+b x +a}+\frac {\frac {\left (4 a c \,e^{3}-b^{2} e^{3}\right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (-a \,e^{3} b +6 a d \,e^{2} c -3 d^{2} e b c +2 c^{2} d^{3}-\frac {\left (4 a c \,e^{3}-b^{2} e^{3}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{\left (4 a c -b^{2}\right ) c}\) | \(287\) |
risch | \(\text {Expression too large to display}\) | \(2178\) |
Input:
int((e*x+d)^3/(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
((3*a*b*c*e^3-6*a*c^2*d*e^2-b^3*e^3+3*b^2*c*d*e^2-3*b*c^2*d^2*e+2*c^3*d^3) /c^2/(4*a*c-b^2)*x+(2*a^2*c*e^3-a*b^2*e^3+3*a*b*c*d*e^2-6*a*c^2*d^2*e+b*c^ 2*d^3)/c^2/(4*a*c-b^2))/(c*x^2+b*x+a)+1/(4*a*c-b^2)/c*(1/2*(4*a*c*e^3-b^2* e^3)/c*ln(c*x^2+b*x+a)+2*(-a*e^3*b+6*a*d*e^2*c-3*d^2*e*b*c+2*c^2*d^3-1/2*( 4*a*c*e^3-b^2*e^3)*b/c)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/ 2)))
Leaf count of result is larger than twice the leaf count of optimal. 572 vs. \(2 (167) = 334\).
Time = 0.10 (sec) , antiderivative size = 1164, normalized size of antiderivative = 6.73 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)^3/(c*x^2+b*x+a)^2,x, algorithm="fricas")
Output:
[-1/2*(2*(b^3*c^2 - 4*a*b*c^3)*d^3 - 12*(a*b^2*c^2 - 4*a^2*c^3)*d^2*e + 6* (a*b^3*c - 4*a^2*b*c^2)*d*e^2 - 2*(a*b^4 - 6*a^2*b^2*c + 8*a^3*c^2)*e^3 - (4*a*c^3*d^3 - 6*a*b*c^2*d^2*e + 12*a^2*c^2*d*e^2 + (a*b^3 - 6*a^2*b*c)*e^ 3 + (4*c^4*d^3 - 6*b*c^3*d^2*e + 12*a*c^3*d*e^2 + (b^3*c - 6*a*b*c^2)*e^3) *x^2 + (4*b*c^3*d^3 - 6*b^2*c^2*d^2*e + 12*a*b*c^2*d*e^2 + (b^4 - 6*a*b^2* c)*e^3)*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*(2*(b^2*c^3 - 4*a*c^4)*d ^3 - 3*(b^3*c^2 - 4*a*b*c^3)*d^2*e + 3*(b^4*c - 6*a*b^2*c^2 + 8*a^2*c^3)*d *e^2 - (b^5 - 7*a*b^3*c + 12*a^2*b*c^2)*e^3)*x - ((b^4*c - 8*a*b^2*c^2 + 1 6*a^2*c^3)*e^3*x^2 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*e^3*x + (a*b^4 - 8*a ^2*b^2*c + 16*a^3*c^2)*e^3)*log(c*x^2 + b*x + a))/(a*b^4*c^2 - 8*a^2*b^2*c ^3 + 16*a^3*c^4 + (b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*x^2 + (b^5*c^2 - 8* a*b^3*c^3 + 16*a^2*b*c^4)*x), -1/2*(2*(b^3*c^2 - 4*a*b*c^3)*d^3 - 12*(a*b^ 2*c^2 - 4*a^2*c^3)*d^2*e + 6*(a*b^3*c - 4*a^2*b*c^2)*d*e^2 - 2*(a*b^4 - 6* a^2*b^2*c + 8*a^3*c^2)*e^3 - 2*(4*a*c^3*d^3 - 6*a*b*c^2*d^2*e + 12*a^2*c^2 *d*e^2 + (a*b^3 - 6*a^2*b*c)*e^3 + (4*c^4*d^3 - 6*b*c^3*d^2*e + 12*a*c^3*d *e^2 + (b^3*c - 6*a*b*c^2)*e^3)*x^2 + (4*b*c^3*d^3 - 6*b^2*c^2*d^2*e + 12* a*b*c^2*d*e^2 + (b^4 - 6*a*b^2*c)*e^3)*x)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt( -b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + 2*(2*(b^2*c^3 - 4*a*c^4)*d^3 - 3*(b^3*c^2 - 4*a*b*c^3)*d^2*e + 3*(b^4*c - 6*a*b^2*c^2 + 8*a^2*c^3)*d*e...
Leaf count of result is larger than twice the leaf count of optimal. 1238 vs. \(2 (160) = 320\).
Time = 3.43 (sec) , antiderivative size = 1238, normalized size of antiderivative = 7.16 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^2} \, dx =\text {Too large to display} \] Input:
integrate((e*x+d)**3/(c*x**2+b*x+a)**2,x)
Output:
(e**3/(2*c**2) - sqrt(-(4*a*c - b**2)**3)*(b*e - 2*c*d)*(6*a*c*e**2 - b**2 *e**2 - 2*b*c*d*e + 2*c**2*d**2)/(2*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))*log(x + (-16*a**2*c**3*(e**3/(2*c**2) - sqrt(-(4* a*c - b**2)**3)*(b*e - 2*c*d)*(6*a*c*e**2 - b**2*e**2 - 2*b*c*d*e + 2*c**2 *d**2)/(2*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) + 8*a**2*c*e**3 + 8*a*b**2*c**2*(e**3/(2*c**2) - sqrt(-(4*a*c - b**2)**3)*( b*e - 2*c*d)*(6*a*c*e**2 - b**2*e**2 - 2*b*c*d*e + 2*c**2*d**2)/(2*c**2*(6 4*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) - a*b**2*e**3 - 6* a*b*c*d*e**2 - b**4*c*(e**3/(2*c**2) - sqrt(-(4*a*c - b**2)**3)*(b*e - 2*c *d)*(6*a*c*e**2 - b**2*e**2 - 2*b*c*d*e + 2*c**2*d**2)/(2*c**2*(64*a**3*c* *3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) + 3*b**2*c*d**2*e - 2*b*c** 2*d**3)/(6*a*b*c*e**3 - 12*a*c**2*d*e**2 - b**3*e**3 + 6*b*c**2*d**2*e - 4 *c**3*d**3)) + (e**3/(2*c**2) + sqrt(-(4*a*c - b**2)**3)*(b*e - 2*c*d)*(6* a*c*e**2 - b**2*e**2 - 2*b*c*d*e + 2*c**2*d**2)/(2*c**2*(64*a**3*c**3 - 48 *a**2*b**2*c**2 + 12*a*b**4*c - b**6)))*log(x + (-16*a**2*c**3*(e**3/(2*c* *2) + sqrt(-(4*a*c - b**2)**3)*(b*e - 2*c*d)*(6*a*c*e**2 - b**2*e**2 - 2*b *c*d*e + 2*c**2*d**2)/(2*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b** 4*c - b**6))) + 8*a**2*c*e**3 + 8*a*b**2*c**2*(e**3/(2*c**2) + sqrt(-(4*a* c - b**2)**3)*(b*e - 2*c*d)*(6*a*c*e**2 - b**2*e**2 - 2*b*c*d*e + 2*c**2*d **2)/(2*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) ...
Exception generated. \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*x+d)^3/(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
Time = 0.37 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.41 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^2} \, dx=\frac {e^{3} \log \left (c x^{2} + b x + a\right )}{2 \, c^{2}} - \frac {{\left (4 \, c^{3} d^{3} - 6 \, b c^{2} d^{2} e + 12 \, a c^{2} d e^{2} + b^{3} e^{3} - 6 \, a b c e^{3}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {b c^{2} d^{3} - 6 \, a c^{2} d^{2} e + 3 \, a b c d e^{2} - a b^{2} e^{3} + 2 \, a^{2} c e^{3} + {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, b^{2} c d e^{2} - 6 \, a c^{2} d e^{2} - b^{3} e^{3} + 3 \, a b c e^{3}\right )} x}{{\left (c x^{2} + b x + a\right )} {\left (b^{2} - 4 \, a c\right )} c^{2}} \] Input:
integrate((e*x+d)^3/(c*x^2+b*x+a)^2,x, algorithm="giac")
Output:
1/2*e^3*log(c*x^2 + b*x + a)/c^2 - (4*c^3*d^3 - 6*b*c^2*d^2*e + 12*a*c^2*d *e^2 + b^3*e^3 - 6*a*b*c*e^3)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^2 *c^2 - 4*a*c^3)*sqrt(-b^2 + 4*a*c)) - (b*c^2*d^3 - 6*a*c^2*d^2*e + 3*a*b*c *d*e^2 - a*b^2*e^3 + 2*a^2*c*e^3 + (2*c^3*d^3 - 3*b*c^2*d^2*e + 3*b^2*c*d* e^2 - 6*a*c^2*d*e^2 - b^3*e^3 + 3*a*b*c*e^3)*x)/((c*x^2 + b*x + a)*(b^2 - 4*a*c)*c^2)
Time = 6.14 (sec) , antiderivative size = 483, normalized size of antiderivative = 2.79 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^2} \, dx=\frac {\frac {2\,a^2\,c\,e^3-a\,b^2\,e^3+3\,a\,b\,c\,d\,e^2-6\,a\,c^2\,d^2\,e+b\,c^2\,d^3}{c^2\,\left (4\,a\,c-b^2\right )}-\frac {x\,\left (b^3\,e^3-3\,b^2\,c\,d\,e^2+3\,b\,c^2\,d^2\,e-3\,a\,b\,c\,e^3-2\,c^3\,d^3+6\,a\,c^2\,d\,e^2\right )}{c^2\,\left (4\,a\,c-b^2\right )}}{c\,x^2+b\,x+a}-\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (-64\,a^3\,c^3\,e^3+48\,a^2\,b^2\,c^2\,e^3-12\,a\,b^4\,c\,e^3+b^6\,e^3\right )}{2\,\left (64\,a^3\,c^5-48\,a^2\,b^2\,c^4+12\,a\,b^4\,c^3-b^6\,c^2\right )}+\frac {\mathrm {atan}\left (\frac {c^2\,\left (\frac {2\,x\,\left (b\,e-2\,c\,d\right )\,\left (b^2\,e^2+2\,b\,c\,d\,e-2\,c^2\,d^2-6\,a\,c\,e^2\right )}{c\,{\left (4\,a\,c-b^2\right )}^3}-\frac {\left (b\,e-2\,c\,d\right )\,\left (b^3\,c-4\,a\,b\,c^2\right )\,\left (b^2\,e^2+2\,b\,c\,d\,e-2\,c^2\,d^2-6\,a\,c\,e^2\right )}{c^3\,{\left (4\,a\,c-b^2\right )}^4}\right )\,{\left (4\,a\,c-b^2\right )}^{5/2}}{b^3\,e^3-6\,b\,c^2\,d^2\,e-6\,a\,b\,c\,e^3+4\,c^3\,d^3+12\,a\,c^2\,d\,e^2}\right )\,\left (b\,e-2\,c\,d\right )\,\left (b^2\,e^2+2\,b\,c\,d\,e-2\,c^2\,d^2-6\,a\,c\,e^2\right )}{c^2\,{\left (4\,a\,c-b^2\right )}^{3/2}} \] Input:
int((d + e*x)^3/(a + b*x + c*x^2)^2,x)
Output:
((b*c^2*d^3 - a*b^2*e^3 + 2*a^2*c*e^3 - 6*a*c^2*d^2*e + 3*a*b*c*d*e^2)/(c^ 2*(4*a*c - b^2)) - (x*(b^3*e^3 - 2*c^3*d^3 - 3*a*b*c*e^3 + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2))/(c^2*(4*a*c - b^2)))/(a + b*x + c*x^2) - (log(a + b*x + c*x^2)*(b^6*e^3 - 64*a^3*c^3*e^3 + 48*a^2*b^2*c^2*e^3 - 12* a*b^4*c*e^3))/(2*(64*a^3*c^5 - b^6*c^2 + 12*a*b^4*c^3 - 48*a^2*b^2*c^4)) + (atan((c^2*((2*x*(b*e - 2*c*d)*(b^2*e^2 - 2*c^2*d^2 - 6*a*c*e^2 + 2*b*c*d *e))/(c*(4*a*c - b^2)^3) - ((b*e - 2*c*d)*(b^3*c - 4*a*b*c^2)*(b^2*e^2 - 2 *c^2*d^2 - 6*a*c*e^2 + 2*b*c*d*e))/(c^3*(4*a*c - b^2)^4))*(4*a*c - b^2)^(5 /2))/(b^3*e^3 + 4*c^3*d^3 - 6*a*b*c*e^3 + 12*a*c^2*d*e^2 - 6*b*c^2*d^2*e)) *(b*e - 2*c*d)*(b^2*e^2 - 2*c^2*d^2 - 6*a*c*e^2 + 2*b*c*d*e))/(c^2*(4*a*c - b^2)^(3/2))
Time = 0.20 (sec) , antiderivative size = 1212, normalized size of antiderivative = 7.01 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^2} \, dx =\text {Too large to display} \] Input:
int((e*x+d)^3/(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**2*b**2*c *e**3 + 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b* c**2*d*e**2 + 2*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a* b**4*e**3 - 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b **3*c*e**3*x - 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))* a*b**2*c**2*d**2*e + 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b **2))*a*b**2*c**2*d*e**2*x - 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4 *a*c - b**2))*a*b**2*c**2*e**3*x**2 + 8*sqrt(4*a*c - b**2)*atan((b + 2*c*x )/sqrt(4*a*c - b**2))*a*b*c**3*d**3 + 24*sqrt(4*a*c - b**2)*atan((b + 2*c* x)/sqrt(4*a*c - b**2))*a*b*c**3*d*e**2*x**2 + 2*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**5*e**3*x + 2*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**4*c*e**3*x**2 - 12*sqrt(4*a*c - b**2)*atan( (b + 2*c*x)/sqrt(4*a*c - b**2))*b**3*c**2*d**2*e*x + 8*sqrt(4*a*c - b**2)* atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**2*c**3*d**3*x - 12*sqrt(4*a*c - b* *2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**2*c**3*d**2*e*x**2 + 8*sqrt(4* a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b*c**4*d**3*x**2 + 16*log (a + b*x + c*x**2)*a**3*b*c**2*e**3 - 8*log(a + b*x + c*x**2)*a**2*b**3*c* e**3 + 16*log(a + b*x + c*x**2)*a**2*b**2*c**2*e**3*x + 16*log(a + b*x + c *x**2)*a**2*b*c**3*e**3*x**2 + log(a + b*x + c*x**2)*a*b**5*e**3 - 8*log(a + b*x + c*x**2)*a*b**4*c*e**3*x - 8*log(a + b*x + c*x**2)*a*b**3*c**2*...