Integrand size = 22, antiderivative size = 139 \[ \int \frac {x^8}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=\frac {x}{c^2}-\frac {a b \left (b^2-3 a c\right )+\left (b^4-4 a b^2 c+2 a^2 c^2\right ) x}{c^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {2 \left (b^4-6 a b^2 c+6 a^2 c^2\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{3/2}}-\frac {b \log \left (a+b x+c x^2\right )}{c^3} \] Output:
x/c^2-(a*b*(-3*a*c+b^2)+(2*a^2*c^2-4*a*b^2*c+b^4)*x)/c^3/(-4*a*c+b^2)/(c*x ^2+b*x+a)-2*(6*a^2*c^2-6*a*b^2*c+b^4)*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2) )/c^3/(-4*a*c+b^2)^(3/2)-b*ln(c*x^2+b*x+a)/c^3
Time = 0.21 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.95 \[ \int \frac {x^8}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=\frac {c x+\frac {-b^4 x-a b^2 (b-4 c x)+a^2 c (3 b-2 c x)}{\left (b^2-4 a c\right ) (a+x (b+c x))}-\frac {2 \left (b^4-6 a b^2 c+6 a^2 c^2\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}}-b \log (a+x (b+c x))}{c^3} \] Input:
Integrate[x^8/(a*x^2 + b*x^3 + c*x^4)^2,x]
Output:
(c*x + (-(b^4*x) - a*b^2*(b - 4*c*x) + a^2*c*(3*b - 2*c*x))/((b^2 - 4*a*c) *(a + x*(b + c*x))) - (2*(b^4 - 6*a*b^2*c + 6*a^2*c^2)*ArcTan[(b + 2*c*x)/ Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) - b*Log[a + x*(b + c*x)])/c^3
Time = 0.36 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.11, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {9, 1164, 27, 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.
\(\displaystyle \int \frac {x^8}{\left (a x^2+b x^3+c x^4\right )^2} \, dx\) |
\(\Big \downarrow \) 9 |
\(\displaystyle \int \frac {x^4}{\left (a+b x+c x^2\right )^2}dx\) |
\(\Big \downarrow \) 1164 |
\(\displaystyle \frac {x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\int \frac {2 x^2 (3 a+b x)}{c x^2+b x+a}dx}{b^2-4 a c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {2 \int \frac {x^2 (3 a+b x)}{c x^2+b x+a}dx}{b^2-4 a c}\) |
\(\Big \downarrow \) 1200 |
\(\displaystyle \frac {x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {2 \int \left (-\frac {b^2-3 a c}{c^2}+\frac {b x}{c}+\frac {a \left (b^2-3 a c\right )+b \left (b^2-4 a c\right ) x}{c^2 \left (c x^2+b x+a\right )}\right )dx}{b^2-4 a c}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {2 \left (\frac {\left (6 a^2 c^2-6 a b^2 c+b^4\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3 \sqrt {b^2-4 a c}}+\frac {b \left (b^2-4 a c\right ) \log \left (a+b x+c x^2\right )}{2 c^3}-\frac {x \left (b^2-3 a c\right )}{c^2}+\frac {b x^2}{2 c}\right )}{b^2-4 a c}\) |
Input:
Int[x^8/(a*x^2 + b*x^3 + c*x^4)^2,x]
Output:
(x^3*(2*a + b*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2)) - (2*(-(((b^2 - 3*a*c) *x)/c^2) + (b*x^2)/(2*c) + ((b^4 - 6*a*b^2*c + 6*a^2*c^2)*ArcTanh[(b + 2*c *x)/Sqrt[b^2 - 4*a*c]])/(c^3*Sqrt[b^2 - 4*a*c]) + (b*(b^2 - 4*a*c)*Log[a + b*x + c*x^2])/(2*c^3)))/(b^2 - 4*a*c)
Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && !MonomialQ[Px, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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 = 0.15 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.42
method | result | size |
default | \(\frac {x}{c^{2}}-\frac {\frac {-\frac {\left (2 a^{2} c^{2}-4 a \,b^{2} c +b^{4}\right ) x}{c \left (4 a c -b^{2}\right )}+\frac {b a \left (3 a c -b^{2}\right )}{c \left (4 a c -b^{2}\right )}}{c \,x^{2}+b x +a}+\frac {\frac {\left (4 a b c -b^{3}\right ) \ln \left (c \,x^{2}+b x +a \right )}{c}+\frac {4 \left (3 a^{2} c -b^{2} a -\frac {\left (4 a b c -b^{3}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{4 a c -b^{2}}}{c^{2}}\) | \(198\) |
risch | \(\text {Expression too large to display}\) | \(1176\) |
Input:
int(x^8/(c*x^4+b*x^3+a*x^2)^2,x,method=_RETURNVERBOSE)
Output:
x/c^2-1/c^2*((-(2*a^2*c^2-4*a*b^2*c+b^4)/c/(4*a*c-b^2)*x+b*a/c*(3*a*c-b^2) /(4*a*c-b^2))/(c*x^2+b*x+a)+2/(4*a*c-b^2)*(1/2*(4*a*b*c-b^3)/c*ln(c*x^2+b* x+a)+2*(3*a^2*c-b^2*a-1/2*(4*a*b*c-b^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. 409 vs. \(2 (135) = 270\).
Time = 0.08 (sec) , antiderivative size = 837, normalized size of antiderivative = 6.02 \[ \int \frac {x^8}{\left (a x^2+b x^3+c x^4\right )^2} \, dx =\text {Too large to display} \] Input:
integrate(x^8/(c*x^4+b*x^3+a*x^2)^2,x, algorithm="fricas")
Output:
[-(a*b^5 - 7*a^2*b^3*c + 12*a^3*b*c^2 - (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^ 4)*x^3 - (b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^2 + (a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2 + (b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*x^2 + (b^5 - 6*a*b^3*c + 6* a^2*b*c^2)*x)*sqrt(b^2 - 4*a*c)*log((2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c + s qrt(b^2 - 4*a*c)*(2*c*x + b))/(c*x^2 + b*x + a)) + (b^6 - 9*a*b^4*c + 26*a ^2*b^2*c^2 - 24*a^3*c^3)*x + (a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2 + (b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^2 + (b^6 - 8*a*b^4*c + 16*a^2*b^2*c^2)*x)* log(c*x^2 + b*x + a))/(a*b^4*c^3 - 8*a^2*b^2*c^4 + 16*a^3*c^5 + (b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*x^2 + (b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*x) , -(a*b^5 - 7*a^2*b^3*c + 12*a^3*b*c^2 - (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c ^4)*x^3 - (b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^2 + 2*(a*b^4 - 6*a^2*b^2* c + 6*a^3*c^2 + (b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*x^2 + (b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*x)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b) /(b^2 - 4*a*c)) + (b^6 - 9*a*b^4*c + 26*a^2*b^2*c^2 - 24*a^3*c^3)*x + (a*b ^5 - 8*a^2*b^3*c + 16*a^3*b*c^2 + (b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^2 + (b^6 - 8*a*b^4*c + 16*a^2*b^2*c^2)*x)*log(c*x^2 + b*x + a))/(a*b^4*c^3 - 8*a^2*b^2*c^4 + 16*a^3*c^5 + (b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*x^2 + (b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*x)]
Leaf count of result is larger than twice the leaf count of optimal. 842 vs. \(2 (134) = 268\).
Time = 1.09 (sec) , antiderivative size = 842, normalized size of antiderivative = 6.06 \[ \int \frac {x^8}{\left (a x^2+b x^3+c x^4\right )^2} \, dx =\text {Too large to display} \] Input:
integrate(x**8/(c*x**4+b*x**3+a*x**2)**2,x)
Output:
(-b/c**3 - sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(c** 3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))*log(x + (-10*a **2*b*c - 16*a**2*c**4*(-b/c**3 - sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) + 2*a*b**3 + 8*a*b**2*c**3*(-b/c**3 - sqrt(-(4*a*c - b**2)**3)*(6 *a**2*c**2 - 6*a*b**2*c + b**4)/(c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) - b**4*c**2*(-b/c**3 - sqrt(-(4*a*c - b**2)**3)*(6*a **2*c**2 - 6*a*b**2*c + b**4)/(c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12 *a*b**4*c - b**6))))/(12*a**2*c**2 - 12*a*b**2*c + 2*b**4)) + (-b/c**3 + s qrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(c**3*(64*a**3*c **3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))*log(x + (-10*a**2*b*c - 16 *a**2*c**4*(-b/c**3 + sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) + 2 *a*b**3 + 8*a*b**2*c**3*(-b/c**3 + sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) - b**4*c**2*(-b/c**3 + sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6 *a*b**2*c + b**4)/(c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))))/(12*a**2*c**2 - 12*a*b**2*c + 2*b**4)) + (-3*a**2*b*c + a*b**3 + x*(2*a**2*c**2 - 4*a*b**2*c + b**4))/(4*a**2*c**4 - a*b**2*c**3 + x**2*(4* a*c**5 - b**2*c**4) + x*(4*a*b*c**4 - b**3*c**3)) + x/c**2
Exception generated. \[ \int \frac {x^8}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^8/(c*x^4+b*x^3+a*x^2)^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 = 0.13 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.16 \[ \int \frac {x^8}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=\frac {2 \, {\left (b^{4} - 6 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {x}{c^{2}} - \frac {b \log \left (c x^{2} + b x + a\right )}{c^{3}} - \frac {\frac {{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} x}{c} + \frac {a b^{3} - 3 \, a^{2} b c}{c}}{{\left (c x^{2} + b x + a\right )} {\left (b^{2} - 4 \, a c\right )} c^{2}} \] Input:
integrate(x^8/(c*x^4+b*x^3+a*x^2)^2,x, algorithm="giac")
Output:
2*(b^4 - 6*a*b^2*c + 6*a^2*c^2)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b ^2*c^3 - 4*a*c^4)*sqrt(-b^2 + 4*a*c)) + x/c^2 - b*log(c*x^2 + b*x + a)/c^3 - ((b^4 - 4*a*b^2*c + 2*a^2*c^2)*x/c + (a*b^3 - 3*a^2*b*c)/c)/((c*x^2 + b *x + a)*(b^2 - 4*a*c)*c^2)
Time = 21.39 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.88 \[ \int \frac {x^8}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=\frac {x}{c^2}+\frac {\frac {a\,\left (b^3-3\,a\,b\,c\right )}{c\,\left (4\,a\,c-b^2\right )}+\frac {x\,\left (2\,a^2\,c^2-4\,a\,b^2\,c+b^4\right )}{c\,\left (4\,a\,c-b^2\right )}}{c^3\,x^2+b\,c^2\,x+a\,c^2}+\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (-128\,a^3\,b\,c^3+96\,a^2\,b^3\,c^2-24\,a\,b^5\,c+2\,b^7\right )}{2\,\left (64\,a^3\,c^6-48\,a^2\,b^2\,c^5+12\,a\,b^4\,c^4-b^6\,c^3\right )}-\frac {2\,\mathrm {atan}\left (\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}-\frac {b^3\,c^2-4\,a\,b\,c^3}{c^2\,{\left (4\,a\,c-b^2\right )}^{3/2}}\right )\,\left (6\,a^2\,c^2-6\,a\,b^2\,c+b^4\right )}{c^3\,{\left (4\,a\,c-b^2\right )}^{3/2}} \] Input:
int(x^8/(a*x^2 + b*x^3 + c*x^4)^2,x)
Output:
x/c^2 + ((a*(b^3 - 3*a*b*c))/(c*(4*a*c - b^2)) + (x*(b^4 + 2*a^2*c^2 - 4*a *b^2*c))/(c*(4*a*c - b^2)))/(a*c^2 + c^3*x^2 + b*c^2*x) + (log(a + b*x + c *x^2)*(2*b^7 - 128*a^3*b*c^3 + 96*a^2*b^3*c^2 - 24*a*b^5*c))/(2*(64*a^3*c^ 6 - b^6*c^3 + 12*a*b^4*c^4 - 48*a^2*b^2*c^5)) - (2*atan((2*c*x)/(4*a*c - b ^2)^(1/2) - (b^3*c^2 - 4*a*b*c^3)/(c^2*(4*a*c - b^2)^(3/2)))*(b^4 + 6*a^2* c^2 - 6*a*b^2*c))/(c^3*(4*a*c - b^2)^(3/2))
Time = 0.16 (sec) , antiderivative size = 752, normalized size of antiderivative = 5.41 \[ \int \frac {x^8}{\left (a x^2+b x^3+c x^4\right )^2} \, dx=\frac {-\mathrm {log}\left (c \,x^{2}+b x +a \right ) a \,b^{6}-\mathrm {log}\left (c \,x^{2}+b x +a \right ) b^{7} x -\mathrm {log}\left (c \,x^{2}+b x +a \right ) b^{6} c \,x^{2}-24 a^{4} c^{3}+14 a^{3} b^{2} c^{2}-24 a^{3} c^{4} x^{2}-2 a^{2} b^{4} c +2 b^{6} c \,x^{2}+b^{5} c^{2} x^{3}-2 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{5}-2 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{6} x -16 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a^{3} b^{2} c^{2}+8 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a^{2} b^{4} c +42 a^{2} b^{2} c^{3} x^{2}+16 a^{2} b \,c^{4} x^{3}-17 a \,b^{4} c^{2} x^{2}-8 a \,b^{3} c^{3} x^{3}-12 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a^{2} b^{2} c^{2} x -12 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a^{2} b \,c^{3} x^{2}+12 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{4} c x +12 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{3} c^{2} x^{2}-12 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a^{3} b \,c^{2}+12 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a^{2} b^{3} c -2 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{5} c \,x^{2}-16 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a^{2} b^{3} c^{2} x -16 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a^{2} b^{2} c^{3} x^{2}+8 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a \,b^{5} c x +8 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a \,b^{4} c^{2} x^{2}}{b \,c^{3} \left (16 a^{2} c^{3} x^{2}-8 a \,b^{2} c^{2} x^{2}+b^{4} c \,x^{2}+16 a^{2} b \,c^{2} x -8 a \,b^{3} c x +b^{5} x +16 a^{3} c^{2}-8 a^{2} b^{2} c +a \,b^{4}\right )} \] Input:
int(x^8/(c*x^4+b*x^3+a*x^2)^2,x)
Output:
( - 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b*c**2 + 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**3*c - 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**2*c** 2*x - 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b*c* *3*x**2 - 2*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**5 + 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**4*c*x + 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**3*c**2*x* *2 - 2*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**6*x - 2* sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**5*c*x**2 - 16*l og(a + b*x + c*x**2)*a**3*b**2*c**2 + 8*log(a + b*x + c*x**2)*a**2*b**4*c - 16*log(a + b*x + c*x**2)*a**2*b**3*c**2*x - 16*log(a + b*x + c*x**2)*a** 2*b**2*c**3*x**2 - log(a + b*x + c*x**2)*a*b**6 + 8*log(a + b*x + c*x**2)* a*b**5*c*x + 8*log(a + b*x + c*x**2)*a*b**4*c**2*x**2 - log(a + b*x + c*x* *2)*b**7*x - log(a + b*x + c*x**2)*b**6*c*x**2 - 24*a**4*c**3 + 14*a**3*b* *2*c**2 - 24*a**3*c**4*x**2 - 2*a**2*b**4*c + 42*a**2*b**2*c**3*x**2 + 16* a**2*b*c**4*x**3 - 17*a*b**4*c**2*x**2 - 8*a*b**3*c**3*x**3 + 2*b**6*c*x** 2 + b**5*c**2*x**3)/(b*c**3*(16*a**3*c**2 - 8*a**2*b**2*c + 16*a**2*b*c**2 *x + 16*a**2*c**3*x**2 + a*b**4 - 8*a*b**3*c*x - 8*a*b**2*c**2*x**2 + b**5 *x + b**4*c*x**2))