Integrand size = 24, antiderivative size = 100 \[ \int \frac {(b d+2 c d x)^6}{\left (a+b x+c x^2\right )^2} \, dx=20 c \left (b^2-4 a c\right ) d^6 (b+2 c x)+\frac {20}{3} c d^6 (b+2 c x)^3-\frac {d^6 (b+2 c x)^5}{a+b x+c x^2}-20 c \left (b^2-4 a c\right )^{3/2} d^6 \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \] Output:
20*c*(-4*a*c+b^2)*d^6*(2*c*x+b)+20/3*c*d^6*(2*c*x+b)^3-d^6*(2*c*x+b)^5/(c* x^2+b*x+a)-20*c*(-4*a*c+b^2)^(3/2)*d^6*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2 ))
Time = 0.05 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.08 \[ \int \frac {(b d+2 c d x)^6}{\left (a+b x+c x^2\right )^2} \, dx=d^6 \left (-16 c^2 \left (-3 b^2+8 a c\right ) x+32 b c^3 x^2+\frac {64 c^4 x^3}{3}-\frac {\left (b^2-4 a c\right )^2 (b+2 c x)}{a+x (b+c x)}+20 c \left (-b^2+4 a c\right )^{3/2} \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )\right ) \] Input:
Integrate[(b*d + 2*c*d*x)^6/(a + b*x + c*x^2)^2,x]
Output:
d^6*(-16*c^2*(-3*b^2 + 8*a*c)*x + 32*b*c^3*x^2 + (64*c^4*x^3)/3 - ((b^2 - 4*a*c)^2*(b + 2*c*x))/(a + x*(b + c*x)) + 20*c*(-b^2 + 4*a*c)^(3/2)*ArcTan [(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])
Time = 0.27 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.97, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1110, 27, 1116, 1116, 1083, 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.
\(\displaystyle \int \frac {(b d+2 c d x)^6}{\left (a+b x+c x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 1110 |
\(\displaystyle 10 c d^2 \int \frac {d^4 (b+2 c x)^4}{c x^2+b x+a}dx-\frac {d^6 (b+2 c x)^5}{a+b x+c x^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 10 c d^6 \int \frac {(b+2 c x)^4}{c x^2+b x+a}dx-\frac {d^6 (b+2 c x)^5}{a+b x+c x^2}\) |
\(\Big \downarrow \) 1116 |
\(\displaystyle 10 c d^6 \left (\left (b^2-4 a c\right ) \int \frac {(b+2 c x)^2}{c x^2+b x+a}dx+\frac {2}{3} (b+2 c x)^3\right )-\frac {d^6 (b+2 c x)^5}{a+b x+c x^2}\) |
\(\Big \downarrow \) 1116 |
\(\displaystyle 10 c d^6 \left (\left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \int \frac {1}{c x^2+b x+a}dx+2 (b+2 c x)\right )+\frac {2}{3} (b+2 c x)^3\right )-\frac {d^6 (b+2 c x)^5}{a+b x+c x^2}\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle 10 c d^6 \left (\left (b^2-4 a c\right ) \left (2 (b+2 c x)-2 \left (b^2-4 a c\right ) \int \frac {1}{b^2-(b+2 c x)^2-4 a c}d(b+2 c x)\right )+\frac {2}{3} (b+2 c x)^3\right )-\frac {d^6 (b+2 c x)^5}{a+b x+c x^2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle 10 c d^6 \left (\left (b^2-4 a c\right ) \left (2 (b+2 c x)-2 \sqrt {b^2-4 a c} \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )\right )+\frac {2}{3} (b+2 c x)^3\right )-\frac {d^6 (b+2 c x)^5}{a+b x+c x^2}\) |
Input:
Int[(b*d + 2*c*d*x)^6/(a + b*x + c*x^2)^2,x]
Output:
-((d^6*(b + 2*c*x)^5)/(a + b*x + c*x^2)) + 10*c*d^6*((2*(b + 2*c*x)^3)/3 + (b^2 - 4*a*c)*(2*(b + 2*c*x) - 2*Sqrt[b^2 - 4*a*c]*ArcTanh[(b + 2*c*x)/Sq rt[b^2 - 4*a*c]]))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Sy mbol] :> Simp[d*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(b*(p + 1))), x] - Simp[d*e*((m - 1)/(b*(p + 1))) Int[(d + e*x)^(m - 2)*(a + b*x + c*x^ 2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0] && N eQ[m + 2*p + 3, 0] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[2*p]
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S ymbol] :> Simp[2*d*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] + Simp[d^2*(m - 1)*((b^2 - 4*a*c)/(b^2*(m + 2*p + 1))) Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && (IntegerQ[2*p] || (IntegerQ[m] && RationalQ[p]) || OddQ[m])
Time = 0.89 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.51
method | result | size |
default | \(d^{6} \left (\frac {64 c^{4} x^{3}}{3}+32 b \,c^{3} x^{2}-128 a \,c^{3} x +48 x \,b^{2} c^{2}+\frac {-2 c \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right ) x -16 a^{2} b \,c^{2}+8 a \,b^{3} c -b^{5}}{c \,x^{2}+b x +a}+\frac {20 c \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}\right )\) | \(151\) |
risch | \(\frac {64 d^{6} c^{4} x^{3}}{3}+32 d^{6} c^{3} b \,x^{2}-128 d^{6} c^{3} a x +48 d^{6} c^{2} b^{2} x +\frac {-2 c \,d^{6} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right ) x -16 b \,c^{2} d^{6} a^{2}+8 a \,b^{3} c \,d^{6}-b^{5} d^{6}}{c \,x^{2}+b x +a}-10 \left (-4 a c +b^{2}\right )^{\frac {3}{2}} d^{6} c \ln \left (2 \left (-4 a c +b^{2}\right )^{\frac {3}{2}} c x +\left (-4 a c +b^{2}\right )^{\frac {3}{2}} b +16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )+10 \left (-4 a c +b^{2}\right )^{\frac {3}{2}} d^{6} c \ln \left (-2 \left (-4 a c +b^{2}\right )^{\frac {3}{2}} c x -\left (-4 a c +b^{2}\right )^{\frac {3}{2}} b +16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )\) | \(242\) |
Input:
int((2*c*d*x+b*d)^6/(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
d^6*(64/3*c^4*x^3+32*b*c^3*x^2-128*a*c^3*x+48*x*b^2*c^2+(-2*c*(16*a^2*c^2- 8*a*b^2*c+b^4)*x-16*a^2*b*c^2+8*a*b^3*c-b^5)/(c*x^2+b*x+a)+20*c*(16*a^2*c^ 2-8*a*b^2*c+b^4)/(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. 242 vs. \(2 (94) = 188\).
Time = 0.09 (sec) , antiderivative size = 504, normalized size of antiderivative = 5.04 \[ \int \frac {(b d+2 c d x)^6}{\left (a+b x+c x^2\right )^2} \, dx=\left [\frac {64 \, c^{5} d^{6} x^{5} + 160 \, b c^{4} d^{6} x^{4} + 80 \, {\left (3 \, b^{2} c^{3} - 4 \, a c^{4}\right )} d^{6} x^{3} + 144 \, {\left (b^{3} c^{2} - 2 \, a b c^{3}\right )} d^{6} x^{2} - 6 \, {\left (b^{4} c - 32 \, a b^{2} c^{2} + 80 \, a^{2} c^{3}\right )} d^{6} x - 3 \, {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} d^{6} - 30 \, {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{6} x^{2} + {\left (b^{3} c - 4 \, a b c^{2}\right )} d^{6} x + {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} d^{6}\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right )}{3 \, {\left (c x^{2} + b x + a\right )}}, \frac {64 \, c^{5} d^{6} x^{5} + 160 \, b c^{4} d^{6} x^{4} + 80 \, {\left (3 \, b^{2} c^{3} - 4 \, a c^{4}\right )} d^{6} x^{3} + 144 \, {\left (b^{3} c^{2} - 2 \, a b c^{3}\right )} d^{6} x^{2} - 6 \, {\left (b^{4} c - 32 \, a b^{2} c^{2} + 80 \, a^{2} c^{3}\right )} d^{6} x - 3 \, {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} d^{6} - 60 \, {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{6} x^{2} + {\left (b^{3} c - 4 \, a b c^{2}\right )} d^{6} x + {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} d^{6}\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right )}{3 \, {\left (c x^{2} + b x + a\right )}}\right ] \] Input:
integrate((2*c*d*x+b*d)^6/(c*x^2+b*x+a)^2,x, algorithm="fricas")
Output:
[1/3*(64*c^5*d^6*x^5 + 160*b*c^4*d^6*x^4 + 80*(3*b^2*c^3 - 4*a*c^4)*d^6*x^ 3 + 144*(b^3*c^2 - 2*a*b*c^3)*d^6*x^2 - 6*(b^4*c - 32*a*b^2*c^2 + 80*a^2*c ^3)*d^6*x - 3*(b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d^6 - 30*((b^2*c^2 - 4*a*c^ 3)*d^6*x^2 + (b^3*c - 4*a*b*c^2)*d^6*x + (a*b^2*c - 4*a^2*c^2)*d^6)*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)))/(c*x^2 + b*x + a), 1/3*(64*c^5*d^6*x^5 + 160 *b*c^4*d^6*x^4 + 80*(3*b^2*c^3 - 4*a*c^4)*d^6*x^3 + 144*(b^3*c^2 - 2*a*b*c ^3)*d^6*x^2 - 6*(b^4*c - 32*a*b^2*c^2 + 80*a^2*c^3)*d^6*x - 3*(b^5 - 8*a*b ^3*c + 16*a^2*b*c^2)*d^6 - 60*((b^2*c^2 - 4*a*c^3)*d^6*x^2 + (b^3*c - 4*a* b*c^2)*d^6*x + (a*b^2*c - 4*a^2*c^2)*d^6)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt( -b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)))/(c*x^2 + b*x + a)]
Leaf count of result is larger than twice the leaf count of optimal. 313 vs. \(2 (99) = 198\).
Time = 0.81 (sec) , antiderivative size = 313, normalized size of antiderivative = 3.13 \[ \int \frac {(b d+2 c d x)^6}{\left (a+b x+c x^2\right )^2} \, dx=32 b c^{3} d^{6} x^{2} + \frac {64 c^{4} d^{6} x^{3}}{3} - 10 c d^{6} \sqrt {- \left (4 a c - b^{2}\right )^{3}} \log {\left (x + \frac {40 a b c^{2} d^{6} - 10 b^{3} c d^{6} - 10 c d^{6} \sqrt {- \left (4 a c - b^{2}\right )^{3}}}{80 a c^{3} d^{6} - 20 b^{2} c^{2} d^{6}} \right )} + 10 c d^{6} \sqrt {- \left (4 a c - b^{2}\right )^{3}} \log {\left (x + \frac {40 a b c^{2} d^{6} - 10 b^{3} c d^{6} + 10 c d^{6} \sqrt {- \left (4 a c - b^{2}\right )^{3}}}{80 a c^{3} d^{6} - 20 b^{2} c^{2} d^{6}} \right )} + x \left (- 128 a c^{3} d^{6} + 48 b^{2} c^{2} d^{6}\right ) + \frac {- 16 a^{2} b c^{2} d^{6} + 8 a b^{3} c d^{6} - b^{5} d^{6} + x \left (- 32 a^{2} c^{3} d^{6} + 16 a b^{2} c^{2} d^{6} - 2 b^{4} c d^{6}\right )}{a + b x + c x^{2}} \] Input:
integrate((2*c*d*x+b*d)**6/(c*x**2+b*x+a)**2,x)
Output:
32*b*c**3*d**6*x**2 + 64*c**4*d**6*x**3/3 - 10*c*d**6*sqrt(-(4*a*c - b**2) **3)*log(x + (40*a*b*c**2*d**6 - 10*b**3*c*d**6 - 10*c*d**6*sqrt(-(4*a*c - b**2)**3))/(80*a*c**3*d**6 - 20*b**2*c**2*d**6)) + 10*c*d**6*sqrt(-(4*a*c - b**2)**3)*log(x + (40*a*b*c**2*d**6 - 10*b**3*c*d**6 + 10*c*d**6*sqrt(- (4*a*c - b**2)**3))/(80*a*c**3*d**6 - 20*b**2*c**2*d**6)) + x*(-128*a*c**3 *d**6 + 48*b**2*c**2*d**6) + (-16*a**2*b*c**2*d**6 + 8*a*b**3*c*d**6 - b** 5*d**6 + x*(-32*a**2*c**3*d**6 + 16*a*b**2*c**2*d**6 - 2*b**4*c*d**6))/(a + b*x + c*x**2)
Exception generated. \[ \int \frac {(b d+2 c d x)^6}{\left (a+b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((2*c*d*x+b*d)^6/(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
Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (94) = 188\).
Time = 0.14 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.97 \[ \int \frac {(b d+2 c d x)^6}{\left (a+b x+c x^2\right )^2} \, dx=\frac {20 \, {\left (b^{4} c d^{6} - 8 \, a b^{2} c^{2} d^{6} + 16 \, a^{2} c^{3} d^{6}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c}} - \frac {2 \, b^{4} c d^{6} x - 16 \, a b^{2} c^{2} d^{6} x + 32 \, a^{2} c^{3} d^{6} x + b^{5} d^{6} - 8 \, a b^{3} c d^{6} + 16 \, a^{2} b c^{2} d^{6}}{c x^{2} + b x + a} + \frac {16 \, {\left (4 \, c^{10} d^{6} x^{3} + 6 \, b c^{9} d^{6} x^{2} + 9 \, b^{2} c^{8} d^{6} x - 24 \, a c^{9} d^{6} x\right )}}{3 \, c^{6}} \] Input:
integrate((2*c*d*x+b*d)^6/(c*x^2+b*x+a)^2,x, algorithm="giac")
Output:
20*(b^4*c*d^6 - 8*a*b^2*c^2*d^6 + 16*a^2*c^3*d^6)*arctan((2*c*x + b)/sqrt( -b^2 + 4*a*c))/sqrt(-b^2 + 4*a*c) - (2*b^4*c*d^6*x - 16*a*b^2*c^2*d^6*x + 32*a^2*c^3*d^6*x + b^5*d^6 - 8*a*b^3*c*d^6 + 16*a^2*b*c^2*d^6)/(c*x^2 + b* x + a) + 16/3*(4*c^10*d^6*x^3 + 6*b*c^9*d^6*x^2 + 9*b^2*c^8*d^6*x - 24*a*c ^9*d^6*x)/c^6
Time = 0.05 (sec) , antiderivative size = 230, normalized size of antiderivative = 2.30 \[ \int \frac {(b d+2 c d x)^6}{\left (a+b x+c x^2\right )^2} \, dx=\frac {64\,c^4\,d^6\,x^3}{3}-\frac {x\,\left (32\,a^2\,c^3\,d^6-16\,a\,b^2\,c^2\,d^6+2\,b^4\,c\,d^6\right )+b^5\,d^6+16\,a^2\,b\,c^2\,d^6-8\,a\,b^3\,c\,d^6}{c\,x^2+b\,x+a}-x\,\left (64\,c^2\,d^6\,\left (b^2+2\,a\,c\right )-112\,b^2\,c^2\,d^6\right )+32\,b\,c^3\,d^6\,x^2+20\,c\,d^6\,\mathrm {atan}\left (\frac {20\,c^2\,d^6\,x\,{\left (4\,a\,c-b^2\right )}^{3/2}+10\,b\,c\,d^6\,{\left (4\,a\,c-b^2\right )}^{3/2}}{160\,a^2\,c^3\,d^6-80\,a\,b^2\,c^2\,d^6+10\,b^4\,c\,d^6}\right )\,{\left (4\,a\,c-b^2\right )}^{3/2} \] Input:
int((b*d + 2*c*d*x)^6/(a + b*x + c*x^2)^2,x)
Output:
(64*c^4*d^6*x^3)/3 - (x*(2*b^4*c*d^6 + 32*a^2*c^3*d^6 - 16*a*b^2*c^2*d^6) + b^5*d^6 + 16*a^2*b*c^2*d^6 - 8*a*b^3*c*d^6)/(a + b*x + c*x^2) - x*(64*c^ 2*d^6*(2*a*c + b^2) - 112*b^2*c^2*d^6) + 32*b*c^3*d^6*x^2 + 20*c*d^6*atan( (20*c^2*d^6*x*(4*a*c - b^2)^(3/2) + 10*b*c*d^6*(4*a*c - b^2)^(3/2))/(10*b^ 4*c*d^6 + 160*a^2*c^3*d^6 - 80*a*b^2*c^2*d^6))*(4*a*c - b^2)^(3/2)
Time = 0.21 (sec) , antiderivative size = 373, normalized size of antiderivative = 3.73 \[ \int \frac {(b d+2 c d x)^6}{\left (a+b x+c x^2\right )^2} \, dx=\frac {d^{6} \left (240 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a^{2} b \,c^{2}-60 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{3} c +240 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{2} c^{2} x +240 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a b \,c^{3} x^{2}-60 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{4} c x -60 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{3} c^{2} x^{2}+480 a^{3} c^{3}-240 a^{2} b^{2} c^{2}+480 a^{2} c^{4} x^{2}+30 a \,b^{4} c -480 a \,b^{2} c^{3} x^{2}-320 a b \,c^{4} x^{3}-3 b^{6}+150 b^{4} c^{2} x^{2}+240 b^{3} c^{3} x^{3}+160 b^{2} c^{4} x^{4}+64 b \,c^{5} x^{5}\right )}{3 b \left (c \,x^{2}+b x +a \right )} \] Input:
int((2*c*d*x+b*d)^6/(c*x^2+b*x+a)^2,x)
Output:
(d**6*(240*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b* c**2 - 60*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**3*c + 240*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**2*c**2 *x + 240*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b*c**3* x**2 - 60*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**4*c*x - 60*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**3*c**2*x* *2 + 480*a**3*c**3 - 240*a**2*b**2*c**2 + 480*a**2*c**4*x**2 + 30*a*b**4*c - 480*a*b**2*c**3*x**2 - 320*a*b*c**4*x**3 - 3*b**6 + 150*b**4*c**2*x**2 + 240*b**3*c**3*x**3 + 160*b**2*c**4*x**4 + 64*b*c**5*x**5))/(3*b*(a + b*x + c*x**2))