Integrand size = 26, antiderivative size = 229 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^2}{d+e x} \, dx=\frac {2 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) x}{e^5}-\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^2}{2 e^6}+\frac {4 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^3}{3 e^6}-\frac {5 c^2 (2 c d-b e) (d+e x)^4}{4 e^6}+\frac {2 c^3 (d+e x)^5}{5 e^6}-\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 \log (d+e x)}{e^6} \]
2*(a*e^2-b*d*e+c*d^2)*(5*c^2*d^2+b^2*e^2-c*e*(-a*e+5*b*d))*x/e^5-1/2*(-b*e +2*c*d)*(10*c^2*d^2+b^2*e^2-2*c*e*(-3*a*e+5*b*d))*(e*x+d)^2/e^6+4/3*c*(5*c ^2*d^2+b^2*e^2-c*e*(-a*e+5*b*d))*(e*x+d)^3/e^6-5/4*c^2*(-b*e+2*c*d)*(e*x+d )^4/e^6+2/5*c^3*(e*x+d)^5/e^6-(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^2*ln(e*x+d) /e^6
Time = 0.13 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.00 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^2}{d+e x} \, dx=\frac {e x \left (30 b^2 e^3 (-2 b d+4 a e+b e x)+2 c^3 \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )+20 c e^2 \left (6 a^2 e^2+9 a b e (-2 d+e x)+2 b^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+5 c^2 e \left (8 a e \left (6 d^2-3 d e x+2 e^2 x^2\right )-5 b \left (12 d^3-6 d^2 e x+4 d e^2 x^2-3 e^3 x^3\right )\right )\right )-60 (2 c d-b e) \left (c d^2+e (-b d+a e)\right )^2 \log (d+e x)}{60 e^6} \]
(e*x*(30*b^2*e^3*(-2*b*d + 4*a*e + b*e*x) + 2*c^3*(60*d^4 - 30*d^3*e*x + 2 0*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4) + 20*c*e^2*(6*a^2*e^2 + 9*a*b*e *(-2*d + e*x) + 2*b^2*(6*d^2 - 3*d*e*x + 2*e^2*x^2)) + 5*c^2*e*(8*a*e*(6*d ^2 - 3*d*e*x + 2*e^2*x^2) - 5*b*(12*d^3 - 6*d^2*e*x + 4*d*e^2*x^2 - 3*e^3* x^3))) - 60*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^2*Log[d + e*x])/(60*e ^6)
Time = 0.51 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1195, 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.
| \(\displaystyle \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^2}{d+e x} \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {4 c (d+e x)^2 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^5}+\frac {(d+e x) (2 c d-b e) \left (2 c e (5 b d-3 a e)-b^2 e^2-10 c^2 d^2\right )}{e^5}+\frac {2 \left (a e^2-b d e+c d^2\right ) \left (a c e^2+b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^5}+\frac {(b e-2 c d) \left (a e^2-b d e+c d^2\right )^2}{e^5 (d+e x)}-\frac {5 c^2 (d+e x)^3 (2 c d-b e)}{e^5}+\frac {2 c^3 (d+e x)^4}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 c (d+e x)^3 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^6}-\frac {(d+e x)^2 (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{2 e^6}+\frac {2 x \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^5}-\frac {(2 c d-b e) \log (d+e x) \left (a e^2-b d e+c d^2\right )^2}{e^6}-\frac {5 c^2 (d+e x)^4 (2 c d-b e)}{4 e^6}+\frac {2 c^3 (d+e x)^5}{5 e^6}\) |
(2*(c*d^2 - b*d*e + a*e^2)*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*x)/e^ 5 - ((2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e))*(d + e*x )^2)/(2*e^6) + (4*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^3) /(3*e^6) - (5*c^2*(2*c*d - b*e)*(d + e*x)^4)/(4*e^6) + (2*c^3*(d + e*x)^5) /(5*e^6) - ((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2*Log[d + e*x])/e^6
3.16.9.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Time = 0.60 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.40
| method | result | size |
| norman | \(\frac {\left (2 c \,e^{4} a^{2}+2 a \,b^{2} e^{4}-6 a b c d \,e^{3}+4 a \,c^{2} d^{2} e^{2}-b^{3} d \,e^{3}+4 b^{2} c \,d^{2} e^{2}-5 b \,c^{2} d^{3} e +2 c^{3} d^{4}\right ) x}{e^{5}}+\frac {2 c^{3} x^{5}}{5 e}+\frac {\left (6 c \,e^{3} b a -4 a \,c^{2} d \,e^{2}+b^{3} e^{3}-4 b^{2} c d \,e^{2}+5 b \,c^{2} d^{2} e -2 c^{3} d^{3}\right ) x^{2}}{2 e^{4}}+\frac {c \left (4 a c \,e^{2}+4 b^{2} e^{2}-5 b c d e +2 c^{2} d^{2}\right ) x^{3}}{3 e^{3}}+\frac {c^{2} \left (5 b e -2 c d \right ) x^{4}}{4 e^{2}}+\frac {\left (a^{2} b \,e^{5}-2 a^{2} c d \,e^{4}-2 a \,b^{2} d \,e^{4}+6 a b c \,d^{2} e^{3}-4 a \,c^{2} d^{3} e^{2}+b^{3} d^{2} e^{3}-4 b^{2} c \,d^{3} e^{2}+5 b \,c^{2} d^{4} e -2 c^{3} d^{5}\right ) \ln \left (e x +d \right )}{e^{6}}\) | \(320\) |
| default | \(\frac {\frac {2}{5} c^{3} x^{5} e^{4}+\frac {5}{4} b \,c^{2} e^{4} x^{4}-\frac {1}{2} c^{3} d \,e^{3} x^{4}+\frac {4}{3} a \,c^{2} e^{4} x^{3}+\frac {4}{3} b^{2} c \,e^{4} x^{3}-\frac {5}{3} b \,c^{2} d \,e^{3} x^{3}+\frac {2}{3} c^{3} d^{2} e^{2} x^{3}+3 a b c \,e^{4} x^{2}-2 a \,c^{2} d \,e^{3} x^{2}+\frac {1}{2} b^{3} e^{4} x^{2}-2 b^{2} c d \,e^{3} x^{2}+\frac {5}{2} b \,c^{2} d^{2} e^{2} x^{2}-c^{3} d^{3} e \,x^{2}+2 a^{2} c \,e^{4} x +2 a \,b^{2} e^{4} x -6 a b c d \,e^{3} x +4 a \,c^{2} d^{2} e^{2} x -b^{3} d \,e^{3} x +4 b^{2} c \,d^{2} e^{2} x -5 b \,c^{2} d^{3} e x +2 c^{3} d^{4} x}{e^{5}}+\frac {\left (a^{2} b \,e^{5}-2 a^{2} c d \,e^{4}-2 a \,b^{2} d \,e^{4}+6 a b c \,d^{2} e^{3}-4 a \,c^{2} d^{3} e^{2}+b^{3} d^{2} e^{3}-4 b^{2} c \,d^{3} e^{2}+5 b \,c^{2} d^{4} e -2 c^{3} d^{5}\right ) \ln \left (e x +d \right )}{e^{6}}\) | \(356\) |
| risch | \(-\frac {2 a \,c^{2} d \,x^{2}}{e^{2}}-\frac {2 b^{2} c d \,x^{2}}{e^{2}}-\frac {4 \ln \left (e x +d \right ) a \,c^{2} d^{3}}{e^{4}}-\frac {4 \ln \left (e x +d \right ) b^{2} c \,d^{3}}{e^{4}}+\frac {\ln \left (e x +d \right ) b^{3} d^{2}}{e^{3}}-\frac {2 \ln \left (e x +d \right ) c^{3} d^{5}}{e^{6}}-\frac {b^{3} d x}{e^{2}}+\frac {2 c^{3} d^{4} x}{e^{5}}+\frac {\ln \left (e x +d \right ) a^{2} b}{e}+\frac {2 a^{2} c x}{e}+\frac {2 a \,b^{2} x}{e}+\frac {2 c^{3} d^{2} x^{3}}{3 e^{3}}-\frac {c^{3} d^{3} x^{2}}{e^{4}}+\frac {4 b^{2} c \,x^{3}}{3 e}+\frac {5 b \,c^{2} x^{4}}{4 e}-\frac {c^{3} d \,x^{4}}{2 e^{2}}+\frac {4 a \,c^{2} x^{3}}{3 e}-\frac {6 a b c d x}{e^{2}}+\frac {6 \ln \left (e x +d \right ) a b c \,d^{2}}{e^{3}}-\frac {5 b \,c^{2} d \,x^{3}}{3 e^{2}}+\frac {3 a b c \,x^{2}}{e}-\frac {5 b \,c^{2} d^{3} x}{e^{4}}+\frac {5 b \,c^{2} d^{2} x^{2}}{2 e^{3}}+\frac {4 a \,c^{2} d^{2} x}{e^{3}}+\frac {4 b^{2} c \,d^{2} x}{e^{3}}+\frac {5 \ln \left (e x +d \right ) b \,c^{2} d^{4}}{e^{5}}-\frac {2 \ln \left (e x +d \right ) a^{2} c d}{e^{2}}-\frac {2 \ln \left (e x +d \right ) a \,b^{2} d}{e^{2}}+\frac {2 c^{3} x^{5}}{5 e}+\frac {b^{3} x^{2}}{2 e}\) | \(406\) |
| parallelrisch | \(\frac {-100 x^{3} b \,c^{2} d \,e^{4}+180 x^{2} a b c \,e^{5}-120 x^{2} a \,c^{2} d \,e^{4}+120 x a \,b^{2} e^{5}-60 x \,b^{3} d \,e^{4}+120 x \,c^{3} d^{4} e +60 \ln \left (e x +d \right ) a^{2} b \,e^{5}+60 \ln \left (e x +d \right ) b^{3} d^{2} e^{3}+75 x^{4} b \,c^{2} e^{5}-30 x^{4} c^{3} d \,e^{4}+80 x^{3} a \,c^{2} e^{5}+80 x^{3} b^{2} c \,e^{5}+40 x^{3} c^{3} d^{2} e^{3}-60 x^{2} c^{3} d^{3} e^{2}+120 x \,a^{2} c \,e^{5}-120 x^{2} b^{2} c d \,e^{4}+150 x^{2} b \,c^{2} d^{2} e^{3}+240 x a \,c^{2} d^{2} e^{3}+240 x \,b^{2} c \,d^{2} e^{3}-300 x b \,c^{2} d^{3} e^{2}-120 \ln \left (e x +d \right ) a^{2} c d \,e^{4}-120 \ln \left (e x +d \right ) a \,b^{2} d \,e^{4}-240 \ln \left (e x +d \right ) a \,c^{2} d^{3} e^{2}-240 \ln \left (e x +d \right ) b^{2} c \,d^{3} e^{2}+300 \ln \left (e x +d \right ) b \,c^{2} d^{4} e +30 x^{2} b^{3} e^{5}-120 \ln \left (e x +d \right ) c^{3} d^{5}-360 x a b c d \,e^{4}+360 \ln \left (e x +d \right ) a b c \,d^{2} e^{3}+24 x^{5} c^{3} e^{5}}{60 e^{6}}\) | \(406\) |
(2*a^2*c*e^4+2*a*b^2*e^4-6*a*b*c*d*e^3+4*a*c^2*d^2*e^2-b^3*d*e^3+4*b^2*c*d ^2*e^2-5*b*c^2*d^3*e+2*c^3*d^4)/e^5*x+2/5*c^3/e*x^5+1/2/e^4*(6*a*b*c*e^3-4 *a*c^2*d*e^2+b^3*e^3-4*b^2*c*d*e^2+5*b*c^2*d^2*e-2*c^3*d^3)*x^2+1/3*c/e^3* (4*a*c*e^2+4*b^2*e^2-5*b*c*d*e+2*c^2*d^2)*x^3+1/4*c^2/e^2*(5*b*e-2*c*d)*x^ 4+(a^2*b*e^5-2*a^2*c*d*e^4-2*a*b^2*d*e^4+6*a*b*c*d^2*e^3-4*a*c^2*d^3*e^2+b ^3*d^2*e^3-4*b^2*c*d^3*e^2+5*b*c^2*d^4*e-2*c^3*d^5)/e^6*ln(e*x+d)
Time = 0.60 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.34 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^2}{d+e x} \, dx=\frac {24 \, c^{3} e^{5} x^{5} - 15 \, {\left (2 \, c^{3} d e^{4} - 5 \, b c^{2} e^{5}\right )} x^{4} + 20 \, {\left (2 \, c^{3} d^{2} e^{3} - 5 \, b c^{2} d e^{4} + 4 \, {\left (b^{2} c + a c^{2}\right )} e^{5}\right )} x^{3} - 30 \, {\left (2 \, c^{3} d^{3} e^{2} - 5 \, b c^{2} d^{2} e^{3} + 4 \, {\left (b^{2} c + a c^{2}\right )} d e^{4} - {\left (b^{3} + 6 \, a b c\right )} e^{5}\right )} x^{2} + 60 \, {\left (2 \, c^{3} d^{4} e - 5 \, b c^{2} d^{3} e^{2} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{3} - {\left (b^{3} + 6 \, a b c\right )} d e^{4} + 2 \, {\left (a b^{2} + a^{2} c\right )} e^{5}\right )} x - 60 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \, {\left (a b^{2} + a^{2} c\right )} d e^{4}\right )} \log \left (e x + d\right )}{60 \, e^{6}} \]
1/60*(24*c^3*e^5*x^5 - 15*(2*c^3*d*e^4 - 5*b*c^2*e^5)*x^4 + 20*(2*c^3*d^2* e^3 - 5*b*c^2*d*e^4 + 4*(b^2*c + a*c^2)*e^5)*x^3 - 30*(2*c^3*d^3*e^2 - 5*b *c^2*d^2*e^3 + 4*(b^2*c + a*c^2)*d*e^4 - (b^3 + 6*a*b*c)*e^5)*x^2 + 60*(2* c^3*d^4*e - 5*b*c^2*d^3*e^2 + 4*(b^2*c + a*c^2)*d^2*e^3 - (b^3 + 6*a*b*c)* d*e^4 + 2*(a*b^2 + a^2*c)*e^5)*x - 60*(2*c^3*d^5 - 5*b*c^2*d^4*e - a^2*b*e ^5 + 4*(b^2*c + a*c^2)*d^3*e^2 - (b^3 + 6*a*b*c)*d^2*e^3 + 2*(a*b^2 + a^2* c)*d*e^4)*log(e*x + d))/e^6
Time = 0.36 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.22 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^2}{d+e x} \, dx=\frac {2 c^{3} x^{5}}{5 e} + x^{4} \cdot \left (\frac {5 b c^{2}}{4 e} - \frac {c^{3} d}{2 e^{2}}\right ) + x^{3} \cdot \left (\frac {4 a c^{2}}{3 e} + \frac {4 b^{2} c}{3 e} - \frac {5 b c^{2} d}{3 e^{2}} + \frac {2 c^{3} d^{2}}{3 e^{3}}\right ) + x^{2} \cdot \left (\frac {3 a b c}{e} - \frac {2 a c^{2} d}{e^{2}} + \frac {b^{3}}{2 e} - \frac {2 b^{2} c d}{e^{2}} + \frac {5 b c^{2} d^{2}}{2 e^{3}} - \frac {c^{3} d^{3}}{e^{4}}\right ) + x \left (\frac {2 a^{2} c}{e} + \frac {2 a b^{2}}{e} - \frac {6 a b c d}{e^{2}} + \frac {4 a c^{2} d^{2}}{e^{3}} - \frac {b^{3} d}{e^{2}} + \frac {4 b^{2} c d^{2}}{e^{3}} - \frac {5 b c^{2} d^{3}}{e^{4}} + \frac {2 c^{3} d^{4}}{e^{5}}\right ) + \frac {\left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2} \log {\left (d + e x \right )}}{e^{6}} \]
2*c**3*x**5/(5*e) + x**4*(5*b*c**2/(4*e) - c**3*d/(2*e**2)) + x**3*(4*a*c* *2/(3*e) + 4*b**2*c/(3*e) - 5*b*c**2*d/(3*e**2) + 2*c**3*d**2/(3*e**3)) + x**2*(3*a*b*c/e - 2*a*c**2*d/e**2 + b**3/(2*e) - 2*b**2*c*d/e**2 + 5*b*c** 2*d**2/(2*e**3) - c**3*d**3/e**4) + x*(2*a**2*c/e + 2*a*b**2/e - 6*a*b*c*d /e**2 + 4*a*c**2*d**2/e**3 - b**3*d/e**2 + 4*b**2*c*d**2/e**3 - 5*b*c**2*d **3/e**4 + 2*c**3*d**4/e**5) + (b*e - 2*c*d)*(a*e**2 - b*d*e + c*d**2)**2* log(d + e*x)/e**6
Time = 0.19 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.34 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^2}{d+e x} \, dx=\frac {24 \, c^{3} e^{4} x^{5} - 15 \, {\left (2 \, c^{3} d e^{3} - 5 \, b c^{2} e^{4}\right )} x^{4} + 20 \, {\left (2 \, c^{3} d^{2} e^{2} - 5 \, b c^{2} d e^{3} + 4 \, {\left (b^{2} c + a c^{2}\right )} e^{4}\right )} x^{3} - 30 \, {\left (2 \, c^{3} d^{3} e - 5 \, b c^{2} d^{2} e^{2} + 4 \, {\left (b^{2} c + a c^{2}\right )} d e^{3} - {\left (b^{3} + 6 \, a b c\right )} e^{4}\right )} x^{2} + 60 \, {\left (2 \, c^{3} d^{4} - 5 \, b c^{2} d^{3} e + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d e^{3} + 2 \, {\left (a b^{2} + a^{2} c\right )} e^{4}\right )} x}{60 \, e^{5}} - \frac {{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \, {\left (a b^{2} + a^{2} c\right )} d e^{4}\right )} \log \left (e x + d\right )}{e^{6}} \]
1/60*(24*c^3*e^4*x^5 - 15*(2*c^3*d*e^3 - 5*b*c^2*e^4)*x^4 + 20*(2*c^3*d^2* e^2 - 5*b*c^2*d*e^3 + 4*(b^2*c + a*c^2)*e^4)*x^3 - 30*(2*c^3*d^3*e - 5*b*c ^2*d^2*e^2 + 4*(b^2*c + a*c^2)*d*e^3 - (b^3 + 6*a*b*c)*e^4)*x^2 + 60*(2*c^ 3*d^4 - 5*b*c^2*d^3*e + 4*(b^2*c + a*c^2)*d^2*e^2 - (b^3 + 6*a*b*c)*d*e^3 + 2*(a*b^2 + a^2*c)*e^4)*x)/e^5 - (2*c^3*d^5 - 5*b*c^2*d^4*e - a^2*b*e^5 + 4*(b^2*c + a*c^2)*d^3*e^2 - (b^3 + 6*a*b*c)*d^2*e^3 + 2*(a*b^2 + a^2*c)*d *e^4)*log(e*x + d)/e^6
Time = 0.26 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.57 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^2}{d+e x} \, dx=\frac {24 \, c^{3} e^{4} x^{5} - 30 \, c^{3} d e^{3} x^{4} + 75 \, b c^{2} e^{4} x^{4} + 40 \, c^{3} d^{2} e^{2} x^{3} - 100 \, b c^{2} d e^{3} x^{3} + 80 \, b^{2} c e^{4} x^{3} + 80 \, a c^{2} e^{4} x^{3} - 60 \, c^{3} d^{3} e x^{2} + 150 \, b c^{2} d^{2} e^{2} x^{2} - 120 \, b^{2} c d e^{3} x^{2} - 120 \, a c^{2} d e^{3} x^{2} + 30 \, b^{3} e^{4} x^{2} + 180 \, a b c e^{4} x^{2} + 120 \, c^{3} d^{4} x - 300 \, b c^{2} d^{3} e x + 240 \, b^{2} c d^{2} e^{2} x + 240 \, a c^{2} d^{2} e^{2} x - 60 \, b^{3} d e^{3} x - 360 \, a b c d e^{3} x + 120 \, a b^{2} e^{4} x + 120 \, a^{2} c e^{4} x}{60 \, e^{5}} - \frac {{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} + 4 \, a c^{2} d^{3} e^{2} - b^{3} d^{2} e^{3} - 6 \, a b c d^{2} e^{3} + 2 \, a b^{2} d e^{4} + 2 \, a^{2} c d e^{4} - a^{2} b e^{5}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{6}} \]
1/60*(24*c^3*e^4*x^5 - 30*c^3*d*e^3*x^4 + 75*b*c^2*e^4*x^4 + 40*c^3*d^2*e^ 2*x^3 - 100*b*c^2*d*e^3*x^3 + 80*b^2*c*e^4*x^3 + 80*a*c^2*e^4*x^3 - 60*c^3 *d^3*e*x^2 + 150*b*c^2*d^2*e^2*x^2 - 120*b^2*c*d*e^3*x^2 - 120*a*c^2*d*e^3 *x^2 + 30*b^3*e^4*x^2 + 180*a*b*c*e^4*x^2 + 120*c^3*d^4*x - 300*b*c^2*d^3* e*x + 240*b^2*c*d^2*e^2*x + 240*a*c^2*d^2*e^2*x - 60*b^3*d*e^3*x - 360*a*b *c*d*e^3*x + 120*a*b^2*e^4*x + 120*a^2*c*e^4*x)/e^5 - (2*c^3*d^5 - 5*b*c^2 *d^4*e + 4*b^2*c*d^3*e^2 + 4*a*c^2*d^3*e^2 - b^3*d^2*e^3 - 6*a*b*c*d^2*e^3 + 2*a*b^2*d*e^4 + 2*a^2*c*d*e^4 - a^2*b*e^5)*log(abs(e*x + d))/e^6
Time = 10.72 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.42 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^2}{d+e x} \, dx=x^4\,\left (\frac {5\,b\,c^2}{4\,e}-\frac {c^3\,d}{2\,e^2}\right )+x^2\,\left (\frac {b^3+6\,a\,c\,b}{2\,e}+\frac {d\,\left (\frac {d\,\left (\frac {5\,b\,c^2}{e}-\frac {2\,c^3\,d}{e^2}\right )}{e}-\frac {4\,c\,\left (b^2+a\,c\right )}{e}\right )}{2\,e}\right )-x^3\,\left (\frac {d\,\left (\frac {5\,b\,c^2}{e}-\frac {2\,c^3\,d}{e^2}\right )}{3\,e}-\frac {4\,c\,\left (b^2+a\,c\right )}{3\,e}\right )+x\,\left (\frac {2\,a\,\left (b^2+a\,c\right )}{e}-\frac {d\,\left (\frac {b^3+6\,a\,c\,b}{e}+\frac {d\,\left (\frac {d\,\left (\frac {5\,b\,c^2}{e}-\frac {2\,c^3\,d}{e^2}\right )}{e}-\frac {4\,c\,\left (b^2+a\,c\right )}{e}\right )}{e}\right )}{e}\right )+\frac {2\,c^3\,x^5}{5\,e}-\frac {\ln \left (d+e\,x\right )\,\left (-a^2\,b\,e^5+2\,a^2\,c\,d\,e^4+2\,a\,b^2\,d\,e^4-6\,a\,b\,c\,d^2\,e^3+4\,a\,c^2\,d^3\,e^2-b^3\,d^2\,e^3+4\,b^2\,c\,d^3\,e^2-5\,b\,c^2\,d^4\,e+2\,c^3\,d^5\right )}{e^6} \]
x^4*((5*b*c^2)/(4*e) - (c^3*d)/(2*e^2)) + x^2*((b^3 + 6*a*b*c)/(2*e) + (d* ((d*((5*b*c^2)/e - (2*c^3*d)/e^2))/e - (4*c*(a*c + b^2))/e))/(2*e)) - x^3* ((d*((5*b*c^2)/e - (2*c^3*d)/e^2))/(3*e) - (4*c*(a*c + b^2))/(3*e)) + x*(( 2*a*(a*c + b^2))/e - (d*((b^3 + 6*a*b*c)/e + (d*((d*((5*b*c^2)/e - (2*c^3* d)/e^2))/e - (4*c*(a*c + b^2))/e))/e))/e) + (2*c^3*x^5)/(5*e) - (log(d + e *x)*(2*c^3*d^5 - a^2*b*e^5 - b^3*d^2*e^3 + 4*a*c^2*d^3*e^2 + 4*b^2*c*d^3*e ^2 + 2*a*b^2*d*e^4 + 2*a^2*c*d*e^4 - 5*b*c^2*d^4*e - 6*a*b*c*d^2*e^3))/e^6