Integrand size = 26, antiderivative size = 223 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^2}{(d+e x)^2} \, dx=-\frac {\left (8 c^3 d^3-b^3 e^3-c^2 d e (15 b d-8 a e)+2 b c e^2 (4 b d-3 a e)\right ) x}{e^5}+\frac {c \left (3 c^2 d^2+2 b^2 e^2-c e (5 b d-2 a e)\right ) x^2}{e^4}-\frac {c^2 (4 c d-5 b e) x^3}{3 e^3}+\frac {c^3 x^4}{2 e^2}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{e^6 (d+e x)}+\frac {2 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) \log (d+e x)}{e^6} \]
-(8*c^3*d^3-b^3*e^3-c^2*d*e*(-8*a*e+15*b*d)+2*b*c*e^2*(-3*a*e+4*b*d))*x/e^ 5+c*(3*c^2*d^2+2*b^2*e^2-c*e*(-2*a*e+5*b*d))*x^2/e^4-1/3*c^2*(-5*b*e+4*c*d )*x^3/e^3+1/2*c^3*x^4/e^2+(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^2/e^6/(e*x+d)+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))*ln(e*x+d)/e^6
Time = 0.19 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.08 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^2}{(d+e x)^2} \, dx=\frac {6 e \left (-8 c^3 d^3+b^3 e^3+c^2 d e (15 b d-8 a e)+2 b c e^2 (-4 b d+3 a e)\right ) x+6 c e^2 \left (3 c^2 d^2+2 b^2 e^2+c e (-5 b d+2 a e)\right ) x^2-2 c^2 e^3 (4 c d-5 b e) x^3+3 c^3 e^4 x^4+\frac {6 (2 c d-b e) \left (c d^2+e (-b d+a e)\right )^2}{d+e x}+12 \left (5 c^3 d^4+b^2 e^3 (-b d+a e)+2 c^2 d^2 e (-5 b d+3 a e)+c e^2 \left (6 b^2 d^2-6 a b d e+a^2 e^2\right )\right ) \log (d+e x)}{6 e^6} \]
(6*e*(-8*c^3*d^3 + b^3*e^3 + c^2*d*e*(15*b*d - 8*a*e) + 2*b*c*e^2*(-4*b*d + 3*a*e))*x + 6*c*e^2*(3*c^2*d^2 + 2*b^2*e^2 + c*e*(-5*b*d + 2*a*e))*x^2 - 2*c^2*e^3*(4*c*d - 5*b*e)*x^3 + 3*c^3*e^4*x^4 + (6*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^2)/(d + e*x) + 12*(5*c^3*d^4 + b^2*e^3*(-(b*d) + a*e) + 2*c^2*d^2*e*(-5*b*d + 3*a*e) + c*e^2*(6*b^2*d^2 - 6*a*b*d*e + a^2*e^2))*L og[d + e*x])/(6*e^6)
Time = 0.51 (sec) , antiderivative size = 223, 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.
| \(\displaystyle \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^2}{(d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {c^2 d e (15 b d-8 a e)-2 b c e^2 (4 b d-3 a e)+b^3 e^3-8 c^3 d^3}{e^5}+\frac {2 \left (a e^2-b d e+c d^2\right ) \left (a c e^2+b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^5 (d+e x)}+\frac {2 c x \left (-c e (5 b d-2 a e)+2 b^2 e^2+3 c^2 d^2\right )}{e^4}+\frac {(b e-2 c d) \left (a e^2-b d e+c d^2\right )^2}{e^5 (d+e x)^2}-\frac {c^2 x^2 (4 c d-5 b e)}{e^3}+\frac {2 c^3 x^3}{e^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {x \left (-c^2 d e (15 b d-8 a e)+2 b c e^2 (4 b d-3 a e)-b^3 e^3+8 c^3 d^3\right )}{e^5}+\frac {2 \log (d+e x) \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6}+\frac {c x^2 \left (-c e (5 b d-2 a e)+2 b^2 e^2+3 c^2 d^2\right )}{e^4}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^6 (d+e x)}-\frac {c^2 x^3 (4 c d-5 b e)}{3 e^3}+\frac {c^3 x^4}{2 e^2}\) |
-(((8*c^3*d^3 - b^3*e^3 - c^2*d*e*(15*b*d - 8*a*e) + 2*b*c*e^2*(4*b*d - 3* a*e))*x)/e^5) + (c*(3*c^2*d^2 + 2*b^2*e^2 - c*e*(5*b*d - 2*a*e))*x^2)/e^4 - (c^2*(4*c*d - 5*b*e)*x^3)/(3*e^3) + (c^3*x^4)/(2*e^2) + ((2*c*d - b*e)*( c*d^2 - b*d*e + a*e^2)^2)/(e^6*(d + e*x)) + (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))*Log[d + e*x])/e^6
3.16.10.3.1 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 = 0.40 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.48
| method | result | size |
| norman | \(\frac {\frac {\left (6 c \,e^{3} b a -6 a \,c^{2} d \,e^{2}+b^{3} e^{3}-6 b^{2} c d \,e^{2}+10 b \,c^{2} d^{2} e -5 c^{3} d^{3}\right ) x^{2}}{e^{4}}+\frac {\left (a^{2} b \,e^{5}-2 a^{2} c d \,e^{4}-2 a \,b^{2} d \,e^{4}+12 a b c \,d^{2} e^{3}-12 a \,c^{2} d^{3} e^{2}+2 b^{3} d^{2} e^{3}-12 b^{2} c \,d^{3} e^{2}+20 b \,c^{2} d^{4} e -10 c^{3} d^{5}\right ) x}{e^{5} d}+\frac {c^{3} x^{5}}{2 e}+\frac {c \left (6 a c \,e^{2}+6 b^{2} e^{2}-10 b c d e +5 c^{2} d^{2}\right ) x^{3}}{3 e^{3}}+\frac {5 c^{2} \left (2 b e -c d \right ) x^{4}}{6 e^{2}}}{e x +d}+\frac {2 \left (c \,e^{4} a^{2}+a \,b^{2} e^{4}-6 a b c d \,e^{3}+6 a \,c^{2} d^{2} e^{2}-b^{3} d \,e^{3}+6 b^{2} c \,d^{2} e^{2}-10 b \,c^{2} d^{3} e +5 c^{3} d^{4}\right ) \ln \left (e x +d \right )}{e^{6}}\) | \(331\) |
| default | \(\frac {\frac {1}{2} c^{3} x^{4} e^{3}+\frac {5}{3} b \,c^{2} e^{3} x^{3}-\frac {4}{3} c^{3} d \,e^{2} x^{3}+2 a \,c^{2} e^{3} x^{2}+2 b^{2} c \,e^{3} x^{2}-5 b \,c^{2} d \,e^{2} x^{2}+3 c^{3} d^{2} e \,x^{2}+6 c \,e^{3} b a x -8 a \,c^{2} d \,e^{2} x +b^{3} e^{3} x -8 b^{2} c d \,e^{2} x +15 b \,c^{2} d^{2} e x -8 c^{3} d^{3} x}{e^{5}}-\frac {a^{2} b \,e^{5}-2 a^{2} c d \,e^{4}-2 a \,b^{2} d \,e^{4}+6 a b c \,d^{2} e^{3}-4 a \,c^{2} d^{3} e^{2}+b^{3} d^{2} e^{3}-4 b^{2} c \,d^{3} e^{2}+5 b \,c^{2} d^{4} e -2 c^{3} d^{5}}{e^{6} \left (e x +d \right )}+\frac {\left (2 c \,e^{4} a^{2}+2 a \,b^{2} e^{4}-12 a b c d \,e^{3}+12 a \,c^{2} d^{2} e^{2}-2 b^{3} d \,e^{3}+12 b^{2} c \,d^{2} e^{2}-20 b \,c^{2} d^{3} e +10 c^{3} d^{4}\right ) \ln \left (e x +d \right )}{e^{6}}\) | \(343\) |
| risch | \(\frac {12 \ln \left (e x +d \right ) a \,c^{2} d^{2}}{e^{4}}+\frac {12 \ln \left (e x +d \right ) b^{2} c \,d^{2}}{e^{4}}-\frac {20 \ln \left (e x +d \right ) b \,c^{2} d^{3}}{e^{5}}+\frac {c^{3} x^{4}}{2 e^{2}}+\frac {5 b \,c^{2} x^{3}}{3 e^{2}}-\frac {4 c^{3} d \,x^{3}}{3 e^{3}}+\frac {2 a \,c^{2} x^{2}}{e^{2}}+\frac {2 b^{2} c \,x^{2}}{e^{2}}+\frac {3 c^{3} d^{2} x^{2}}{e^{4}}-\frac {8 c^{3} d^{3} x}{e^{5}}-\frac {a^{2} b}{e \left (e x +d \right )}-\frac {b^{3} d^{2}}{e^{3} \left (e x +d \right )}+\frac {2 c^{3} d^{5}}{e^{6} \left (e x +d \right )}+\frac {2 \ln \left (e x +d \right ) c \,a^{2}}{e^{2}}+\frac {2 \ln \left (e x +d \right ) a \,b^{2}}{e^{2}}-\frac {2 \ln \left (e x +d \right ) b^{3} d}{e^{3}}+\frac {10 \ln \left (e x +d \right ) c^{3} d^{4}}{e^{6}}-\frac {6 a b c \,d^{2}}{e^{3} \left (e x +d \right )}-\frac {12 \ln \left (e x +d \right ) a b c d}{e^{3}}-\frac {5 b \,c^{2} d \,x^{2}}{e^{3}}+\frac {6 c b a x}{e^{2}}-\frac {8 a \,c^{2} d x}{e^{3}}-\frac {8 b^{2} c d x}{e^{3}}+\frac {15 b \,c^{2} d^{2} x}{e^{4}}+\frac {2 a^{2} c d}{e^{2} \left (e x +d \right )}+\frac {2 a \,b^{2} d}{e^{2} \left (e x +d \right )}+\frac {4 a \,c^{2} d^{3}}{e^{4} \left (e x +d \right )}+\frac {4 b^{2} c \,d^{3}}{e^{4} \left (e x +d \right )}-\frac {5 b \,c^{2} d^{4}}{e^{5} \left (e x +d \right )}+\frac {b^{3} x}{e^{2}}\) | \(444\) |
| parallelrisch | \(\frac {-20 x^{3} b \,c^{2} d \,e^{4}+36 x^{2} a b c \,e^{5}-36 x^{2} a \,c^{2} d \,e^{4}-12 \ln \left (e x +d \right ) b^{3} d^{2} e^{3}+10 x^{4} b \,c^{2} e^{5}-5 x^{4} c^{3} d \,e^{4}+12 x^{3} a \,c^{2} e^{5}+12 x^{3} b^{2} c \,e^{5}+10 x^{3} c^{3} d^{2} e^{3}-30 x^{2} c^{3} d^{3} e^{2}+72 \ln \left (e x +d \right ) x \,b^{2} c \,d^{2} e^{3}+72 \ln \left (e x +d \right ) x a \,c^{2} d^{2} e^{3}-120 \ln \left (e x +d \right ) x b \,c^{2} d^{3} e^{2}+60 c^{3} d^{5}+72 b^{2} c \,d^{3} e^{2}-120 b \,c^{2} d^{4} e +72 a \,c^{2} d^{3} e^{2}+12 a \,b^{2} d \,e^{4}-6 a^{2} b \,e^{5}-12 b^{3} d^{2} e^{3}-36 x^{2} b^{2} c d \,e^{4}+60 x^{2} b \,c^{2} d^{2} e^{3}+12 \ln \left (e x +d \right ) a^{2} c d \,e^{4}+12 \ln \left (e x +d \right ) a \,b^{2} d \,e^{4}+72 \ln \left (e x +d \right ) a \,c^{2} d^{3} e^{2}+72 \ln \left (e x +d \right ) b^{2} c \,d^{3} e^{2}-120 \ln \left (e x +d \right ) b \,c^{2} d^{4} e +12 \ln \left (e x +d \right ) x a \,b^{2} e^{5}-12 \ln \left (e x +d \right ) x \,b^{3} d \,e^{4}+60 \ln \left (e x +d \right ) x \,c^{3} d^{4} e -72 a b c \,d^{2} e^{3}+6 x^{2} b^{3} e^{5}+60 \ln \left (e x +d \right ) c^{3} d^{5}-72 \ln \left (e x +d \right ) a b c \,d^{2} e^{3}+3 x^{5} c^{3} e^{5}+12 \ln \left (e x +d \right ) x \,a^{2} c \,e^{5}+12 a^{2} c d \,e^{4}-72 \ln \left (e x +d \right ) x a b c d \,e^{4}}{6 e^{6} \left (e x +d \right )}\) | \(539\) |
((6*a*b*c*e^3-6*a*c^2*d*e^2+b^3*e^3-6*b^2*c*d*e^2+10*b*c^2*d^2*e-5*c^3*d^3 )/e^4*x^2+(a^2*b*e^5-2*a^2*c*d*e^4-2*a*b^2*d*e^4+12*a*b*c*d^2*e^3-12*a*c^2 *d^3*e^2+2*b^3*d^2*e^3-12*b^2*c*d^3*e^2+20*b*c^2*d^4*e-10*c^3*d^5)/e^5/d*x +1/2*c^3/e*x^5+1/3*c*(6*a*c*e^2+6*b^2*e^2-10*b*c*d*e+5*c^2*d^2)/e^3*x^3+5/ 6*c^2*(2*b*e-c*d)/e^2*x^4)/(e*x+d)+2*(a^2*c*e^4+a*b^2*e^4-6*a*b*c*d*e^3+6* a*c^2*d^2*e^2-b^3*d*e^3+6*b^2*c*d^2*e^2-10*b*c^2*d^3*e+5*c^3*d^4)/e^6*ln(e *x+d)
Leaf count of result is larger than twice the leaf count of optimal. 444 vs. \(2 (219) = 438\).
Time = 0.40 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.99 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^2}{(d+e x)^2} \, dx=\frac {3 \, c^{3} e^{5} x^{5} + 12 \, c^{3} d^{5} - 30 \, b c^{2} d^{4} e - 6 \, a^{2} b e^{5} + 24 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} - 6 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 12 \, {\left (a b^{2} + a^{2} c\right )} d e^{4} - 5 \, {\left (c^{3} d e^{4} - 2 \, b c^{2} e^{5}\right )} x^{4} + 2 \, {\left (5 \, c^{3} d^{2} e^{3} - 10 \, b c^{2} d e^{4} + 6 \, {\left (b^{2} c + a c^{2}\right )} e^{5}\right )} x^{3} - 6 \, {\left (5 \, c^{3} d^{3} e^{2} - 10 \, b c^{2} d^{2} e^{3} + 6 \, {\left (b^{2} c + a c^{2}\right )} d e^{4} - {\left (b^{3} + 6 \, a b c\right )} e^{5}\right )} x^{2} - 6 \, {\left (8 \, c^{3} d^{4} e - 15 \, b c^{2} d^{3} e^{2} + 8 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{3} - {\left (b^{3} + 6 \, a b c\right )} d e^{4}\right )} x + 12 \, {\left (5 \, c^{3} d^{5} - 10 \, b c^{2} d^{4} e + 6 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + {\left (a b^{2} + a^{2} c\right )} d e^{4} + {\left (5 \, c^{3} d^{4} e - 10 \, b c^{2} d^{3} e^{2} + 6 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{3} - {\left (b^{3} + 6 \, a b c\right )} d e^{4} + {\left (a b^{2} + a^{2} c\right )} e^{5}\right )} x\right )} \log \left (e x + d\right )}{6 \, {\left (e^{7} x + d e^{6}\right )}} \]
1/6*(3*c^3*e^5*x^5 + 12*c^3*d^5 - 30*b*c^2*d^4*e - 6*a^2*b*e^5 + 24*(b^2*c + a*c^2)*d^3*e^2 - 6*(b^3 + 6*a*b*c)*d^2*e^3 + 12*(a*b^2 + a^2*c)*d*e^4 - 5*(c^3*d*e^4 - 2*b*c^2*e^5)*x^4 + 2*(5*c^3*d^2*e^3 - 10*b*c^2*d*e^4 + 6*( b^2*c + a*c^2)*e^5)*x^3 - 6*(5*c^3*d^3*e^2 - 10*b*c^2*d^2*e^3 + 6*(b^2*c + a*c^2)*d*e^4 - (b^3 + 6*a*b*c)*e^5)*x^2 - 6*(8*c^3*d^4*e - 15*b*c^2*d^3*e ^2 + 8*(b^2*c + a*c^2)*d^2*e^3 - (b^3 + 6*a*b*c)*d*e^4)*x + 12*(5*c^3*d^5 - 10*b*c^2*d^4*e + 6*(b^2*c + a*c^2)*d^3*e^2 - (b^3 + 6*a*b*c)*d^2*e^3 + ( a*b^2 + a^2*c)*d*e^4 + (5*c^3*d^4*e - 10*b*c^2*d^3*e^2 + 6*(b^2*c + a*c^2) *d^2*e^3 - (b^3 + 6*a*b*c)*d*e^4 + (a*b^2 + a^2*c)*e^5)*x)*log(e*x + d))/( e^7*x + d*e^6)
Time = 0.79 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.46 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^2}{(d+e x)^2} \, dx=\frac {c^{3} x^{4}}{2 e^{2}} + x^{3} \cdot \left (\frac {5 b c^{2}}{3 e^{2}} - \frac {4 c^{3} d}{3 e^{3}}\right ) + x^{2} \cdot \left (\frac {2 a c^{2}}{e^{2}} + \frac {2 b^{2} c}{e^{2}} - \frac {5 b c^{2} d}{e^{3}} + \frac {3 c^{3} d^{2}}{e^{4}}\right ) + x \left (\frac {6 a b c}{e^{2}} - \frac {8 a c^{2} d}{e^{3}} + \frac {b^{3}}{e^{2}} - \frac {8 b^{2} c d}{e^{3}} + \frac {15 b c^{2} d^{2}}{e^{4}} - \frac {8 c^{3} d^{3}}{e^{5}}\right ) + \frac {- a^{2} b e^{5} + 2 a^{2} c d e^{4} + 2 a b^{2} d e^{4} - 6 a b c d^{2} e^{3} + 4 a c^{2} d^{3} e^{2} - b^{3} d^{2} e^{3} + 4 b^{2} c d^{3} e^{2} - 5 b c^{2} d^{4} e + 2 c^{3} d^{5}}{d e^{6} + e^{7} x} + \frac {2 \left (a e^{2} - b d e + c d^{2}\right ) \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) \log {\left (d + e x \right )}}{e^{6}} \]
c**3*x**4/(2*e**2) + x**3*(5*b*c**2/(3*e**2) - 4*c**3*d/(3*e**3)) + x**2*( 2*a*c**2/e**2 + 2*b**2*c/e**2 - 5*b*c**2*d/e**3 + 3*c**3*d**2/e**4) + x*(6 *a*b*c/e**2 - 8*a*c**2*d/e**3 + b**3/e**2 - 8*b**2*c*d/e**3 + 15*b*c**2*d* *2/e**4 - 8*c**3*d**3/e**5) + (-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)/(d*e**6 + e**7*x) + 2*(a*e**2 - b*d*e + c*d**2)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2)*log(d + e*x)/e**6
Time = 0.19 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.39 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^2}{(d+e x)^2} \, dx=\frac {2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \, {\left (a b^{2} + a^{2} c\right )} d e^{4}}{e^{7} x + d e^{6}} + \frac {3 \, c^{3} e^{3} x^{4} - 2 \, {\left (4 \, c^{3} d e^{2} - 5 \, b c^{2} e^{3}\right )} x^{3} + 6 \, {\left (3 \, c^{3} d^{2} e - 5 \, b c^{2} d e^{2} + 2 \, {\left (b^{2} c + a c^{2}\right )} e^{3}\right )} x^{2} - 6 \, {\left (8 \, c^{3} d^{3} - 15 \, b c^{2} d^{2} e + 8 \, {\left (b^{2} c + a c^{2}\right )} d e^{2} - {\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} x}{6 \, e^{5}} + \frac {2 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d e^{3} + {\left (a b^{2} + a^{2} c\right )} e^{4}\right )} \log \left (e x + d\right )}{e^{6}} \]
(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)/(e^7*x + d*e^6) + 1/6*(3*c^3 *e^3*x^4 - 2*(4*c^3*d*e^2 - 5*b*c^2*e^3)*x^3 + 6*(3*c^3*d^2*e - 5*b*c^2*d* e^2 + 2*(b^2*c + a*c^2)*e^3)*x^2 - 6*(8*c^3*d^3 - 15*b*c^2*d^2*e + 8*(b^2* c + a*c^2)*d*e^2 - (b^3 + 6*a*b*c)*e^3)*x)/e^5 + 2*(5*c^3*d^4 - 10*b*c^2*d ^3*e + 6*(b^2*c + a*c^2)*d^2*e^2 - (b^3 + 6*a*b*c)*d*e^3 + (a*b^2 + a^2*c) *e^4)*log(e*x + d)/e^6
Time = 0.27 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.96 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^2}{(d+e x)^2} \, dx=\frac {{\left (3 \, c^{3} - \frac {10 \, {\left (2 \, c^{3} d e - b c^{2} e^{2}\right )}}{{\left (e x + d\right )} e} + \frac {12 \, {\left (5 \, c^{3} d^{2} e^{2} - 5 \, b c^{2} d e^{3} + b^{2} c e^{4} + a c^{2} e^{4}\right )}}{{\left (e x + d\right )}^{2} e^{2}} - \frac {6 \, {\left (20 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} + 12 \, b^{2} c d e^{5} + 12 \, a c^{2} d e^{5} - b^{3} e^{6} - 6 \, a b c e^{6}\right )}}{{\left (e x + d\right )}^{3} e^{3}}\right )} {\left (e x + d\right )}^{4}}{6 \, e^{6}} - \frac {2 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, b^{2} c d^{2} e^{2} + 6 \, a c^{2} d^{2} e^{2} - b^{3} d e^{3} - 6 \, a b c d e^{3} + a b^{2} e^{4} + a^{2} c e^{4}\right )} \log \left (\frac {{\left | e x + d \right |}}{{\left (e x + d\right )}^{2} {\left | e \right |}}\right )}{e^{6}} + \frac {\frac {2 \, c^{3} d^{5} e^{4}}{e x + d} - \frac {5 \, b c^{2} d^{4} e^{5}}{e x + d} + \frac {4 \, b^{2} c d^{3} e^{6}}{e x + d} + \frac {4 \, a c^{2} d^{3} e^{6}}{e x + d} - \frac {b^{3} d^{2} e^{7}}{e x + d} - \frac {6 \, a b c d^{2} e^{7}}{e x + d} + \frac {2 \, a b^{2} d e^{8}}{e x + d} + \frac {2 \, a^{2} c d e^{8}}{e x + d} - \frac {a^{2} b e^{9}}{e x + d}}{e^{10}} \]
1/6*(3*c^3 - 10*(2*c^3*d*e - b*c^2*e^2)/((e*x + d)*e) + 12*(5*c^3*d^2*e^2 - 5*b*c^2*d*e^3 + b^2*c*e^4 + a*c^2*e^4)/((e*x + d)^2*e^2) - 6*(20*c^3*d^3 *e^3 - 30*b*c^2*d^2*e^4 + 12*b^2*c*d*e^5 + 12*a*c^2*d*e^5 - b^3*e^6 - 6*a* b*c*e^6)/((e*x + d)^3*e^3))*(e*x + d)^4/e^6 - 2*(5*c^3*d^4 - 10*b*c^2*d^3* e + 6*b^2*c*d^2*e^2 + 6*a*c^2*d^2*e^2 - b^3*d*e^3 - 6*a*b*c*d*e^3 + a*b^2* e^4 + a^2*c*e^4)*log(abs(e*x + d)/((e*x + d)^2*abs(e)))/e^6 + (2*c^3*d^5*e ^4/(e*x + d) - 5*b*c^2*d^4*e^5/(e*x + d) + 4*b^2*c*d^3*e^6/(e*x + d) + 4*a *c^2*d^3*e^6/(e*x + d) - b^3*d^2*e^7/(e*x + d) - 6*a*b*c*d^2*e^7/(e*x + d) + 2*a*b^2*d*e^8/(e*x + d) + 2*a^2*c*d*e^8/(e*x + d) - a^2*b*e^9/(e*x + d) )/e^10
Time = 10.72 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.74 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^2}{(d+e x)^2} \, dx=x^3\,\left (\frac {5\,b\,c^2}{3\,e^2}-\frac {4\,c^3\,d}{3\,e^3}\right )-x^2\,\left (\frac {d\,\left (\frac {5\,b\,c^2}{e^2}-\frac {4\,c^3\,d}{e^3}\right )}{e}+\frac {c^3\,d^2}{e^4}-\frac {2\,c\,\left (b^2+a\,c\right )}{e^2}\right )+x\,\left (\frac {b^3+6\,a\,c\,b}{e^2}+\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {5\,b\,c^2}{e^2}-\frac {4\,c^3\,d}{e^3}\right )}{e}+\frac {2\,c^3\,d^2}{e^4}-\frac {4\,c\,\left (b^2+a\,c\right )}{e^2}\right )}{e}-\frac {d^2\,\left (\frac {5\,b\,c^2}{e^2}-\frac {4\,c^3\,d}{e^3}\right )}{e^2}\right )+\frac {\ln \left (d+e\,x\right )\,\left (2\,a^2\,c\,e^4+2\,a\,b^2\,e^4-12\,a\,b\,c\,d\,e^3+12\,a\,c^2\,d^2\,e^2-2\,b^3\,d\,e^3+12\,b^2\,c\,d^2\,e^2-20\,b\,c^2\,d^3\,e+10\,c^3\,d^4\right )}{e^6}+\frac {-a^2\,b\,e^5+2\,a^2\,c\,d\,e^4+2\,a\,b^2\,d\,e^4-6\,a\,b\,c\,d^2\,e^3+4\,a\,c^2\,d^3\,e^2-b^3\,d^2\,e^3+4\,b^2\,c\,d^3\,e^2-5\,b\,c^2\,d^4\,e+2\,c^3\,d^5}{e\,\left (x\,e^6+d\,e^5\right )}+\frac {c^3\,x^4}{2\,e^2} \]
x^3*((5*b*c^2)/(3*e^2) - (4*c^3*d)/(3*e^3)) - x^2*((d*((5*b*c^2)/e^2 - (4* c^3*d)/e^3))/e + (c^3*d^2)/e^4 - (2*c*(a*c + b^2))/e^2) + x*((b^3 + 6*a*b* c)/e^2 + (2*d*((2*d*((5*b*c^2)/e^2 - (4*c^3*d)/e^3))/e + (2*c^3*d^2)/e^4 - (4*c*(a*c + b^2))/e^2))/e - (d^2*((5*b*c^2)/e^2 - (4*c^3*d)/e^3))/e^2) + (log(d + e*x)*(10*c^3*d^4 + 2*a*b^2*e^4 + 2*a^2*c*e^4 - 2*b^3*d*e^3 + 12*a *c^2*d^2*e^2 + 12*b^2*c*d^2*e^2 - 20*b*c^2*d^3*e - 12*a*b*c*d*e^3))/e^6 + (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*(d*e^ 5 + e^6*x)) + (c^3*x^4)/(2*e^2)