Integrand size = 21, antiderivative size = 229 \[ \int \frac {x^4 (d+e x)}{a+b x+c x^2} \, dx=\frac {\left (b^2 c d-a c^2 d-b^3 e+2 a b c e\right ) x}{c^4}-\frac {\left (b c d-b^2 e+a c e\right ) x^2}{2 c^3}+\frac {(c d-b e) x^3}{3 c^2}+\frac {e x^4}{4 c}-\frac {\left (b^4 c d-4 a b^2 c^2 d+2 a^2 c^3 d-b^5 e+5 a b^3 c e-5 a^2 b c^2 e\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^5 \sqrt {b^2-4 a c}}-\frac {\left (b^3 c d-2 a b c^2 d-b^4 e+3 a b^2 c e-a^2 c^2 e\right ) \log \left (a+b x+c x^2\right )}{2 c^5} \] Output:
(2*a*b*c*e-a*c^2*d-b^3*e+b^2*c*d)*x/c^4-1/2*(a*c*e-b^2*e+b*c*d)*x^2/c^3+1/ 3*(-b*e+c*d)*x^3/c^2+1/4*e*x^4/c-(-5*a^2*b*c^2*e+2*a^2*c^3*d+5*a*b^3*c*e-4 *a*b^2*c^2*d-b^5*e+b^4*c*d)*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/c^5/(-4* a*c+b^2)^(1/2)-1/2*(-a^2*c^2*e+3*a*b^2*c*e-2*a*b*c^2*d-b^4*e+b^3*c*d)*ln(c *x^2+b*x+a)/c^5
Time = 0.14 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.97 \[ \int \frac {x^4 (d+e x)}{a+b x+c x^2} \, dx=\frac {-12 c \left (-b^2 c d+a c^2 d+b^3 e-2 a b c e\right ) x-6 c^2 \left (b c d-b^2 e+a c e\right ) x^2+4 c^3 (c d-b e) x^3+3 c^4 e x^4+\frac {12 \left (b^4 c d-4 a b^2 c^2 d+2 a^2 c^3 d-b^5 e+5 a b^3 c e-5 a^2 b c^2 e\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}+6 \left (-b^3 c d+2 a b c^2 d+b^4 e-3 a b^2 c e+a^2 c^2 e\right ) \log (a+x (b+c x))}{12 c^5} \] Input:
Integrate[(x^4*(d + e*x))/(a + b*x + c*x^2),x]
Output:
(-12*c*(-(b^2*c*d) + a*c^2*d + b^3*e - 2*a*b*c*e)*x - 6*c^2*(b*c*d - b^2*e + a*c*e)*x^2 + 4*c^3*(c*d - b*e)*x^3 + 3*c^4*e*x^4 + (12*(b^4*c*d - 4*a*b ^2*c^2*d + 2*a^2*c^3*d - b^5*e + 5*a*b^3*c*e - 5*a^2*b*c^2*e)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + 6*(-(b^3*c*d) + 2*a*b*c^2 *d + b^4*e - 3*a*b^2*c*e + a^2*c^2*e)*Log[a + x*(b + c*x)])/(12*c^5)
Time = 0.58 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {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.
| \(\displaystyle \int \frac {x^4 (d+e x)}{a+b x+c x^2} \, dx\) |
| \(\Big \downarrow \) 1200 |
| \(\displaystyle \int \left (-\frac {x \left (-a^2 c^2 e+3 a b^2 c e-2 a b c^2 d+b^4 (-e)+b^3 c d\right )+a \left (2 a b c e-a c^2 d+b^3 (-e)+b^2 c d\right )}{c^4 \left (a+b x+c x^2\right )}-\frac {x \left (a c e+b^2 (-e)+b c d\right )}{c^3}+\frac {2 a b c e-a c^2 d+b^3 (-e)+b^2 c d}{c^4}+\frac {x^2 (c d-b e)}{c^2}+\frac {e x^3}{c}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (-5 a^2 b c^2 e+2 a^2 c^3 d+5 a b^3 c e-4 a b^2 c^2 d+b^5 (-e)+b^4 c d\right )}{c^5 \sqrt {b^2-4 a c}}-\frac {\left (-a^2 c^2 e+3 a b^2 c e-2 a b c^2 d+b^4 (-e)+b^3 c d\right ) \log \left (a+b x+c x^2\right )}{2 c^5}-\frac {x^2 \left (a c e+b^2 (-e)+b c d\right )}{2 c^3}+\frac {x \left (2 a b c e-a c^2 d+b^3 (-e)+b^2 c d\right )}{c^4}+\frac {x^3 (c d-b e)}{3 c^2}+\frac {e x^4}{4 c}\) |
Input:
Int[(x^4*(d + e*x))/(a + b*x + c*x^2),x]
Output:
((b^2*c*d - a*c^2*d - b^3*e + 2*a*b*c*e)*x)/c^4 - ((b*c*d - b^2*e + a*c*e) *x^2)/(2*c^3) + ((c*d - b*e)*x^3)/(3*c^2) + (e*x^4)/(4*c) - ((b^4*c*d - 4* a*b^2*c^2*d + 2*a^2*c^3*d - b^5*e + 5*a*b^3*c*e - 5*a^2*b*c^2*e)*ArcTanh[( b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^5*Sqrt[b^2 - 4*a*c]) - ((b^3*c*d - 2*a*b *c^2*d - b^4*e + 3*a*b^2*c*e - a^2*c^2*e)*Log[a + b*x + c*x^2])/(2*c^5)
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 = 1.33 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.14
| method | result | size |
| default | \(\frac {\frac {1}{4} c^{3} x^{4} e -\frac {1}{3} b \,c^{2} x^{3} e +\frac {1}{3} c^{3} d \,x^{3}-\frac {1}{2} a \,c^{2} e \,x^{2}+\frac {1}{2} b^{2} c e \,x^{2}-\frac {1}{2} b \,c^{2} d \,x^{2}+2 a b c e x -a d x \,c^{2}-b^{3} e x +b^{2} c x d}{c^{4}}+\frac {\frac {\left (a^{2} c^{2} e -3 a \,b^{2} c e +2 a b \,c^{2} d +b^{4} e -b^{3} c d \right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (-2 a^{2} b c e +a^{2} c^{2} d +a \,b^{3} e -c d a \,b^{2}-\frac {\left (a^{2} c^{2} e -3 a \,b^{2} c e +2 a b \,c^{2} d +b^{4} e -b^{3} c d \right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{c^{4}}\) | \(260\) |
| risch | \(\text {Expression too large to display}\) | \(3986\) |
Input:
int(x^4*(e*x+d)/(c*x^2+b*x+a),x,method=_RETURNVERBOSE)
Output:
1/c^4*(1/4*c^3*x^4*e-1/3*b*c^2*x^3*e+1/3*c^3*d*x^3-1/2*a*c^2*e*x^2+1/2*b^2 *c*e*x^2-1/2*b*c^2*d*x^2+2*a*b*c*e*x-a*d*x*c^2-b^3*e*x+b^2*c*x*d)+1/c^4*(1 /2*(a^2*c^2*e-3*a*b^2*c*e+2*a*b*c^2*d+b^4*e-b^3*c*d)/c*ln(c*x^2+b*x+a)+2*( -2*a^2*b*c*e+a^2*c^2*d+a*b^3*e-c*d*a*b^2-1/2*(a^2*c^2*e-3*a*b^2*c*e+2*a*b* c^2*d+b^4*e-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)))
Time = 0.10 (sec) , antiderivative size = 730, normalized size of antiderivative = 3.19 \[ \int \frac {x^4 (d+e x)}{a+b x+c x^2} \, dx =\text {Too large to display} \] Input:
integrate(x^4*(e*x+d)/(c*x^2+b*x+a),x, algorithm="fricas")
Output:
[1/12*(3*(b^2*c^4 - 4*a*c^5)*e*x^4 + 4*((b^2*c^4 - 4*a*c^5)*d - (b^3*c^3 - 4*a*b*c^4)*e)*x^3 - 6*((b^3*c^3 - 4*a*b*c^4)*d - (b^4*c^2 - 5*a*b^2*c^3 + 4*a^2*c^4)*e)*x^2 - 6*sqrt(b^2 - 4*a*c)*((b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3 )*d - (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e)*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)) + 12*((b^4*c^2 - 5*a*b^2*c^3 + 4*a^2*c^4)*d - (b^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*e)*x - 6 *((b^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*d - (b^6 - 7*a*b^4*c + 13*a^2*b^2*c^ 2 - 4*a^3*c^3)*e)*log(c*x^2 + b*x + a))/(b^2*c^5 - 4*a*c^6), 1/12*(3*(b^2* c^4 - 4*a*c^5)*e*x^4 + 4*((b^2*c^4 - 4*a*c^5)*d - (b^3*c^3 - 4*a*b*c^4)*e) *x^3 - 6*((b^3*c^3 - 4*a*b*c^4)*d - (b^4*c^2 - 5*a*b^2*c^3 + 4*a^2*c^4)*e) *x^2 - 12*sqrt(-b^2 + 4*a*c)*((b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d - (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + 12*((b^4*c^2 - 5*a*b^2*c^3 + 4*a^2*c^4)*d - (b^5*c - 6*a*b^3*c^ 2 + 8*a^2*b*c^3)*e)*x - 6*((b^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*d - (b^6 - 7*a*b^4*c + 13*a^2*b^2*c^2 - 4*a^3*c^3)*e)*log(c*x^2 + b*x + a))/(b^2*c^5 - 4*a*c^6)]
Leaf count of result is larger than twice the leaf count of optimal. 1100 vs. \(2 (236) = 472\).
Time = 2.00 (sec) , antiderivative size = 1100, normalized size of antiderivative = 4.80 \[ \int \frac {x^4 (d+e x)}{a+b x+c x^2} \, dx =\text {Too large to display} \] Input:
integrate(x**4*(e*x+d)/(c*x**2+b*x+a),x)
Output:
x**3*(-b*e/(3*c**2) + d/(3*c)) + x**2*(-a*e/(2*c**2) + b**2*e/(2*c**3) - b *d/(2*c**2)) + x*(2*a*b*e/c**3 - a*d/c**2 - b**3*e/c**4 + b**2*d/c**3) + ( -sqrt(-4*a*c + b**2)*(5*a**2*b*c**2*e - 2*a**2*c**3*d - 5*a*b**3*c*e + 4*a *b**2*c**2*d + b**5*e - b**4*c*d)/(2*c**5*(4*a*c - b**2)) + (a**2*c**2*e - 3*a*b**2*c*e + 2*a*b*c**2*d + b**4*e - b**3*c*d)/(2*c**5))*log(x + (2*a** 3*c**2*e - 4*a**2*b**2*c*e + 3*a**2*b*c**2*d + a*b**4*e - a*b**3*c*d - 4*a *c**5*(-sqrt(-4*a*c + b**2)*(5*a**2*b*c**2*e - 2*a**2*c**3*d - 5*a*b**3*c* e + 4*a*b**2*c**2*d + b**5*e - b**4*c*d)/(2*c**5*(4*a*c - b**2)) + (a**2*c **2*e - 3*a*b**2*c*e + 2*a*b*c**2*d + b**4*e - b**3*c*d)/(2*c**5)) + b**2* c**4*(-sqrt(-4*a*c + b**2)*(5*a**2*b*c**2*e - 2*a**2*c**3*d - 5*a*b**3*c*e + 4*a*b**2*c**2*d + b**5*e - b**4*c*d)/(2*c**5*(4*a*c - b**2)) + (a**2*c* *2*e - 3*a*b**2*c*e + 2*a*b*c**2*d + b**4*e - b**3*c*d)/(2*c**5)))/(5*a**2 *b*c**2*e - 2*a**2*c**3*d - 5*a*b**3*c*e + 4*a*b**2*c**2*d + b**5*e - b**4 *c*d)) + (sqrt(-4*a*c + b**2)*(5*a**2*b*c**2*e - 2*a**2*c**3*d - 5*a*b**3* c*e + 4*a*b**2*c**2*d + b**5*e - b**4*c*d)/(2*c**5*(4*a*c - b**2)) + (a**2 *c**2*e - 3*a*b**2*c*e + 2*a*b*c**2*d + b**4*e - b**3*c*d)/(2*c**5))*log(x + (2*a**3*c**2*e - 4*a**2*b**2*c*e + 3*a**2*b*c**2*d + a*b**4*e - a*b**3* c*d - 4*a*c**5*(sqrt(-4*a*c + b**2)*(5*a**2*b*c**2*e - 2*a**2*c**3*d - 5*a *b**3*c*e + 4*a*b**2*c**2*d + b**5*e - b**4*c*d)/(2*c**5*(4*a*c - b**2)) + (a**2*c**2*e - 3*a*b**2*c*e + 2*a*b*c**2*d + b**4*e - b**3*c*d)/(2*c**...
Exception generated. \[ \int \frac {x^4 (d+e x)}{a+b x+c x^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^4*(e*x+d)/(c*x^2+b*x+a),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 = 0.23 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.03 \[ \int \frac {x^4 (d+e x)}{a+b x+c x^2} \, dx=\frac {3 \, c^{3} e x^{4} + 4 \, c^{3} d x^{3} - 4 \, b c^{2} e x^{3} - 6 \, b c^{2} d x^{2} + 6 \, b^{2} c e x^{2} - 6 \, a c^{2} e x^{2} + 12 \, b^{2} c d x - 12 \, a c^{2} d x - 12 \, b^{3} e x + 24 \, a b c e x}{12 \, c^{4}} - \frac {{\left (b^{3} c d - 2 \, a b c^{2} d - b^{4} e + 3 \, a b^{2} c e - a^{2} c^{2} e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{5}} + \frac {{\left (b^{4} c d - 4 \, a b^{2} c^{2} d + 2 \, a^{2} c^{3} d - b^{5} e + 5 \, a b^{3} c e - 5 \, a^{2} b c^{2} e\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} c^{5}} \] Input:
integrate(x^4*(e*x+d)/(c*x^2+b*x+a),x, algorithm="giac")
Output:
1/12*(3*c^3*e*x^4 + 4*c^3*d*x^3 - 4*b*c^2*e*x^3 - 6*b*c^2*d*x^2 + 6*b^2*c* e*x^2 - 6*a*c^2*e*x^2 + 12*b^2*c*d*x - 12*a*c^2*d*x - 12*b^3*e*x + 24*a*b* c*e*x)/c^4 - 1/2*(b^3*c*d - 2*a*b*c^2*d - b^4*e + 3*a*b^2*c*e - a^2*c^2*e) *log(c*x^2 + b*x + a)/c^5 + (b^4*c*d - 4*a*b^2*c^2*d + 2*a^2*c^3*d - b^5*e + 5*a*b^3*c*e - 5*a^2*b*c^2*e)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/(sq rt(-b^2 + 4*a*c)*c^5)
Time = 10.53 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.32 \[ \int \frac {x^4 (d+e x)}{a+b x+c x^2} \, dx=x^3\,\left (\frac {d}{3\,c}-\frac {b\,e}{3\,c^2}\right )+x\,\left (\frac {b\,\left (\frac {b\,\left (\frac {d}{c}-\frac {b\,e}{c^2}\right )}{c}+\frac {a\,e}{c^2}\right )}{c}-\frac {a\,\left (\frac {d}{c}-\frac {b\,e}{c^2}\right )}{c}\right )-x^2\,\left (\frac {b\,\left (\frac {d}{c}-\frac {b\,e}{c^2}\right )}{2\,c}+\frac {a\,e}{2\,c^2}\right )+\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (4\,e\,a^3\,c^3-13\,e\,a^2\,b^2\,c^2+8\,d\,a^2\,b\,c^3+7\,e\,a\,b^4\,c-6\,d\,a\,b^3\,c^2-e\,b^6+d\,b^5\,c\right )}{2\,\left (4\,a\,c^6-b^2\,c^5\right )}+\frac {e\,x^4}{4\,c}-\frac {\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )\,\left (5\,e\,a^2\,b\,c^2-2\,d\,a^2\,c^3-5\,e\,a\,b^3\,c+4\,d\,a\,b^2\,c^2+e\,b^5-d\,b^4\,c\right )}{c^5\,\sqrt {4\,a\,c-b^2}} \] Input:
int((x^4*(d + e*x))/(a + b*x + c*x^2),x)
Output:
x^3*(d/(3*c) - (b*e)/(3*c^2)) + x*((b*((b*(d/c - (b*e)/c^2))/c + (a*e)/c^2 ))/c - (a*(d/c - (b*e)/c^2))/c) - x^2*((b*(d/c - (b*e)/c^2))/(2*c) + (a*e) /(2*c^2)) + (log(a + b*x + c*x^2)*(4*a^3*c^3*e - b^6*e + b^5*c*d - 13*a^2* b^2*c^2*e + 7*a*b^4*c*e - 6*a*b^3*c^2*d + 8*a^2*b*c^3*d))/(2*(4*a*c^6 - b^ 2*c^5)) + (e*x^4)/(4*c) - (atan(b/(4*a*c - b^2)^(1/2) + (2*c*x)/(4*a*c - b ^2)^(1/2))*(b^5*e - 2*a^2*c^3*d - b^4*c*d - 5*a*b^3*c*e + 4*a*b^2*c^2*d + 5*a^2*b*c^2*e))/(c^5*(4*a*c - b^2)^(1/2))
Time = 0.23 (sec) , antiderivative size = 587, normalized size of antiderivative = 2.56 \[ \int \frac {x^4 (d+e x)}{a+b x+c x^2} \, dx=\frac {-60 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a^{2} b \,c^{2} e +60 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{3} c e -48 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{2} c^{2} d +24 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) a^{2} c^{3} d +12 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{4} c d -78 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a^{2} b^{2} c^{2} e +48 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a^{2} b \,c^{3} d +42 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a \,b^{4} c e -36 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a \,b^{3} c^{2} d +96 a^{2} b \,c^{3} e x -72 a \,b^{3} c^{2} e x +60 a \,b^{2} c^{3} d x +30 a \,b^{2} c^{3} e \,x^{2}-24 a b \,c^{4} d \,x^{2}-16 a b \,c^{4} e \,x^{3}-6 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) b^{6} e -12 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{5} e +24 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) a^{3} c^{3} e +6 \,\mathrm {log}\left (c \,x^{2}+b x +a \right ) b^{5} c d -48 a^{2} c^{4} d x -24 a^{2} c^{4} e \,x^{2}+16 a \,c^{5} d \,x^{3}+12 a \,c^{5} e \,x^{4}+12 b^{5} c e x -12 b^{4} c^{2} d x -6 b^{4} c^{2} e \,x^{2}+6 b^{3} c^{3} d \,x^{2}+4 b^{3} c^{3} e \,x^{3}-4 b^{2} c^{4} d \,x^{3}-3 b^{2} c^{4} e \,x^{4}}{12 c^{5} \left (4 a c -b^{2}\right )} \] Input:
int(x^4*(e*x+d)/(c*x^2+b*x+a),x)
Output:
( - 60*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b*c**2 *e + 24*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*c**3* d + 60*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**3*c*e - 48*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**2*c**2*d - 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**5*e + 12* sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**4*c*d + 24*log( a + b*x + c*x**2)*a**3*c**3*e - 78*log(a + b*x + c*x**2)*a**2*b**2*c**2*e + 48*log(a + b*x + c*x**2)*a**2*b*c**3*d + 42*log(a + b*x + c*x**2)*a*b**4 *c*e - 36*log(a + b*x + c*x**2)*a*b**3*c**2*d - 6*log(a + b*x + c*x**2)*b* *6*e + 6*log(a + b*x + c*x**2)*b**5*c*d + 96*a**2*b*c**3*e*x - 48*a**2*c** 4*d*x - 24*a**2*c**4*e*x**2 - 72*a*b**3*c**2*e*x + 60*a*b**2*c**3*d*x + 30 *a*b**2*c**3*e*x**2 - 24*a*b*c**4*d*x**2 - 16*a*b*c**4*e*x**3 + 16*a*c**5* d*x**3 + 12*a*c**5*e*x**4 + 12*b**5*c*e*x - 12*b**4*c**2*d*x - 6*b**4*c**2 *e*x**2 + 6*b**3*c**3*d*x**2 + 4*b**3*c**3*e*x**3 - 4*b**2*c**4*d*x**3 - 3 *b**2*c**4*e*x**4)/(12*c**5*(4*a*c - b**2))