Integrand size = 26, antiderivative size = 396 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{(d+e x)^2} \, dx=-\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) x}{e^7}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{e^8 (d+e x)}+\frac {\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^2}{2 e^8}-\frac {5 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^3}{3 e^8}+\frac {3 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^4}{4 e^8}-\frac {7 c^3 (2 c d-b e) (d+e x)^5}{5 e^8}+\frac {c^4 (d+e x)^6}{3 e^8}+\frac {\left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) \log (d+e x)}{e^8} \] Output:
-3*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)*(7*c^2*d^2+b^2*e^2-c*e*(-3*a*e+7*b*d)) *x/e^7+(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^3/e^8/(e*x+d)+1/2*(70*c^4*d^4+b^4* e^4-4*b^2*c*e^3*(-3*a*e+5*b*d)-20*c^3*d^2*e*(-3*a*e+7*b*d)+6*c^2*e^2*(a^2* e^2-10*a*b*d*e+15*b^2*d^2))*(e*x+d)^2/e^8-5/3*c*(-b*e+2*c*d)*(7*c^2*d^2+b^ 2*e^2-c*e*(-3*a*e+7*b*d))*(e*x+d)^3/e^8+3/4*c^2*(14*c^2*d^2+3*b^2*e^2-2*c* e*(-a*e+7*b*d))*(e*x+d)^4/e^8-7/5*c^3*(-b*e+2*c*d)*(e*x+d)^5/e^8+1/3*c^4*( e*x+d)^6/e^8+(a*e^2-b*d*e+c*d^2)^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(-a*e+7*b*d ))*ln(e*x+d)/e^8
Time = 0.26 (sec) , antiderivative size = 637, normalized size of antiderivative = 1.61 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{(d+e x)^2} \, dx=\frac {2 c^4 \left (60 d^7-360 d^6 e x-210 d^5 e^2 x^2+70 d^4 e^3 x^3-35 d^3 e^4 x^4+21 d^2 e^5 x^5-14 d e^6 x^6+10 e^7 x^7\right )+30 b e^4 \left (6 a^2 b d e^2-2 a^3 e^3+6 a b^2 e \left (-d^2+d e x+e^2 x^2\right )+b^3 \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )\right )+20 c e^3 \left (6 a^3 d e^3+27 a^2 b e^2 \left (-d^2+d e x+e^2 x^2\right )+18 a b^2 e \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )-5 b^3 \left (3 d^4-9 d^3 e x-6 d^2 e^2 x^2+2 d e^3 x^3-e^4 x^4\right )\right )+15 c^2 e^2 \left (12 a^2 e^2 \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )+20 a b e \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )+3 b^2 \left (12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right )\right )+6 c^3 e \left (5 a e \left (12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right )-7 b \left (10 d^6-50 d^5 e x-30 d^4 e^2 x^2+10 d^3 e^3 x^3-5 d^2 e^4 x^4+3 d e^5 x^5-2 e^6 x^6\right )\right )+60 \left (14 c^2 d^2+3 b^2 e^2+2 c e (-7 b d+a e)\right ) \left (c d^2+e (-b d+a e)\right )^2 (d+e x) \log (d+e x)}{60 e^8 (d+e x)} \] Input:
Integrate[((b + 2*c*x)*(a + b*x + c*x^2)^3)/(d + e*x)^2,x]
Output:
(2*c^4*(60*d^7 - 360*d^6*e*x - 210*d^5*e^2*x^2 + 70*d^4*e^3*x^3 - 35*d^3*e ^4*x^4 + 21*d^2*e^5*x^5 - 14*d*e^6*x^6 + 10*e^7*x^7) + 30*b*e^4*(6*a^2*b*d *e^2 - 2*a^3*e^3 + 6*a*b^2*e*(-d^2 + d*e*x + e^2*x^2) + b^3*(2*d^3 - 4*d^2 *e*x - 3*d*e^2*x^2 + e^3*x^3)) + 20*c*e^3*(6*a^3*d*e^3 + 27*a^2*b*e^2*(-d^ 2 + d*e*x + e^2*x^2) + 18*a*b^2*e*(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x ^3) - 5*b^3*(3*d^4 - 9*d^3*e*x - 6*d^2*e^2*x^2 + 2*d*e^3*x^3 - e^4*x^4)) + 15*c^2*e^2*(12*a^2*e^2*(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3) + 20*a *b*e*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4) + 3*b^2* (12*d^5 - 48*d^4*e*x - 30*d^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e ^5*x^5)) + 6*c^3*e*(5*a*e*(12*d^5 - 48*d^4*e*x - 30*d^3*e^2*x^2 + 10*d^2*e ^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^5) - 7*b*(10*d^6 - 50*d^5*e*x - 30*d^4*e^2* x^2 + 10*d^3*e^3*x^3 - 5*d^2*e^4*x^4 + 3*d*e^5*x^5 - 2*e^6*x^6)) + 60*(14* c^2*d^2 + 3*b^2*e^2 + 2*c*e*(-7*b*d + a*e))*(c*d^2 + e*(-(b*d) + a*e))^2*( d + e*x)*Log[d + e*x])/(60*e^8*(d + e*x))
Time = 1.44 (sec) , antiderivative size = 396, 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.
| \(\displaystyle \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{(d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {(d+e x) \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{e^7}+\frac {3 c^2 (d+e x)^3 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^7}+\frac {5 c (d+e x)^2 (2 c d-b e) \left (c e (7 b d-3 a e)-b^2 e^2-7 c^2 d^2\right )}{e^7}+\frac {\left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^7 (d+e x)}+\frac {3 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-3 a c e^2-b^2 e^2+7 b c d e-7 c^2 d^2\right )}{e^7}+\frac {(b e-2 c d) \left (a e^2-b d e+c d^2\right )^3}{e^7 (d+e x)^2}-\frac {7 c^3 (d+e x)^4 (2 c d-b e)}{e^7}+\frac {2 c^4 (d+e x)^5}{e^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(d+e x)^2 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{2 e^8}+\frac {3 c^2 (d+e x)^4 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{4 e^8}-\frac {5 c (d+e x)^3 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^8}+\frac {\log (d+e x) \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8}-\frac {3 x (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^7}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^8 (d+e x)}-\frac {7 c^3 (d+e x)^5 (2 c d-b e)}{5 e^8}+\frac {c^4 (d+e x)^6}{3 e^8}\) |
Input:
Int[((b + 2*c*x)*(a + b*x + c*x^2)^3)/(d + e*x)^2,x]
Output:
(-3*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b* d - 3*a*e))*x)/e^7 + ((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(e^8*(d + e *x)) + ((70*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(5*b*d - 3*a*e) - 20*c^3*d^2*e *(7*b*d - 3*a*e) + 6*c^2*e^2*(15*b^2*d^2 - 10*a*b*d*e + a^2*e^2))*(d + e*x )^2)/(2*e^8) - (5*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a* e))*(d + e*x)^3)/(3*e^8) + (3*c^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^4)/(4*e^8) - (7*c^3*(2*c*d - b*e)*(d + e*x)^5)/(5*e^8) + (c^4*(d + e*x)^6)/(3*e^8) + ((c*d^2 - b*d*e + a*e^2)^2*(14*c^2*d^2 + 3*b^2 *e^2 - 2*c*e*(7*b*d - a*e))*Log[d + e*x])/e^8
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 = 1.32 (sec) , antiderivative size = 709, normalized size of antiderivative = 1.79
| method | result | size |
| norman | \(\frac {\frac {\left (a^{3} b \,e^{7}-2 d \,e^{6} c \,a^{3}-3 a^{2} b^{2} d \,e^{6}+18 a^{2} b c \,d^{2} e^{5}-18 d^{3} e^{4} a^{2} c^{2}+6 a \,b^{3} d^{2} e^{5}-36 a \,b^{2} c \,d^{3} e^{4}+60 a b \,c^{2} d^{4} e^{3}-30 d^{5} e^{2} a \,c^{3}-3 b^{4} d^{3} e^{4}+20 b^{3} c \,d^{4} e^{3}-45 b^{2} c^{2} d^{5} e^{2}+42 b \,c^{3} d^{6} e -14 d^{7} c^{4}\right ) x}{d \,e^{7}}+\frac {c^{4} x^{7}}{3 e}+\frac {\left (18 e^{4} a^{2} c^{2}+36 a \,b^{2} c \,e^{4}-60 a b \,c^{2} d \,e^{3}+30 d^{2} e^{2} a \,c^{3}+3 b^{4} e^{4}-20 d \,e^{3} b^{3} c +45 d^{2} e^{2} b^{2} c^{2}-42 d^{3} e b \,c^{3}+14 d^{4} c^{4}\right ) x^{3}}{6 e^{5}}+\frac {\left (18 a^{2} b c \,e^{5}-18 a^{2} c^{2} d \,e^{4}+6 a \,b^{3} e^{5}-36 a \,b^{2} c d \,e^{4}+60 a b \,c^{2} d^{2} e^{3}-30 a \,c^{3} d^{3} e^{2}-3 b^{4} d \,e^{4}+20 b^{3} c \,d^{2} e^{3}-45 b^{2} c^{2} d^{3} e^{2}+42 b \,c^{3} d^{4} e -14 c^{4} d^{5}\right ) x^{2}}{2 e^{6}}+\frac {c \left (60 a b c \,e^{3}-30 d \,e^{2} a \,c^{2}+20 b^{3} e^{3}-45 d \,e^{2} b^{2} c +42 d^{2} e b \,c^{2}-14 d^{3} c^{3}\right ) x^{4}}{12 e^{4}}+\frac {c^{2} \left (30 a c \,e^{2}+45 b^{2} e^{2}-42 b c d e +14 c^{2} d^{2}\right ) x^{5}}{20 e^{3}}+\frac {7 c^{3} \left (3 b e -c d \right ) x^{6}}{15 e^{2}}}{e x +d}+\frac {\left (2 e^{6} c \,a^{3}+3 a^{2} b^{2} e^{6}-18 a^{2} b c d \,e^{5}+18 d^{2} e^{4} a^{2} c^{2}-6 a \,b^{3} d \,e^{5}+36 a \,b^{2} c \,d^{2} e^{4}-60 a b \,c^{2} d^{3} e^{3}+30 d^{4} e^{2} a \,c^{3}+3 b^{4} d^{2} e^{4}-20 b^{3} c \,d^{3} e^{3}+45 b^{2} c^{2} d^{4} e^{2}-42 b \,c^{3} d^{5} e +14 d^{6} c^{4}\right ) \ln \left (e x +d \right )}{e^{8}}\) | \(709\) |
| default | \(\frac {-12 a^{2} c^{2} d \,e^{4} x +\frac {3}{2} a \,c^{3} e^{5} x^{4}-36 b^{2} c^{2} d^{3} e^{2} x -\frac {7}{2} b \,c^{3} d \,e^{4} x^{4}+5 a b \,c^{2} e^{5} x^{3}-15 a b \,c^{2} d \,e^{4} x^{2}+5 c^{4} d^{4} e \,x^{2}+3 a \,b^{3} e^{5} x -2 b^{4} d \,e^{4} x +35 b \,c^{3} d^{4} e x +15 b^{3} c \,d^{2} e^{3} x -24 a \,c^{3} d^{3} e^{2} x -24 a \,b^{2} c d \,e^{4} x +45 a b \,c^{2} d^{2} e^{3} x -6 b^{2} c^{2} d \,e^{4} x^{3}-5 b^{3} c d \,e^{4} x^{2}+\frac {27}{2} b^{2} c^{2} d^{2} e^{3} x^{2}-14 b \,c^{3} d^{3} e^{2} x^{2}+7 b \,c^{3} d^{2} e^{3} x^{3}+6 a \,b^{2} c \,e^{5} x^{2}+9 a \,c^{3} d^{2} e^{3} x^{2}-4 a \,c^{3} d \,e^{4} x^{3}+\frac {7}{5} b \,c^{3} e^{5} x^{5}-\frac {4}{5} c^{4} d \,e^{4} x^{5}+\frac {9}{4} b^{2} c^{2} e^{5} x^{4}+\frac {3}{2} c^{4} d^{2} e^{3} x^{4}+9 a^{2} b c \,e^{5} x +\frac {5}{3} b^{3} c \,e^{5} x^{3}-\frac {8}{3} c^{4} d^{3} e^{2} x^{3}+3 a^{2} c^{2} e^{5} x^{2}-12 c^{4} d^{5} x +\frac {1}{2} b^{4} e^{5} x^{2}+\frac {1}{3} c^{4} x^{6} e^{5}}{e^{7}}+\frac {\left (2 e^{6} c \,a^{3}+3 a^{2} b^{2} e^{6}-18 a^{2} b c d \,e^{5}+18 d^{2} e^{4} a^{2} c^{2}-6 a \,b^{3} d \,e^{5}+36 a \,b^{2} c \,d^{2} e^{4}-60 a b \,c^{2} d^{3} e^{3}+30 d^{4} e^{2} a \,c^{3}+3 b^{4} d^{2} e^{4}-20 b^{3} c \,d^{3} e^{3}+45 b^{2} c^{2} d^{4} e^{2}-42 b \,c^{3} d^{5} e +14 d^{6} c^{4}\right ) \ln \left (e x +d \right )}{e^{8}}-\frac {a^{3} b \,e^{7}-2 d \,e^{6} c \,a^{3}-3 a^{2} b^{2} d \,e^{6}+9 a^{2} b c \,d^{2} e^{5}-6 d^{3} e^{4} a^{2} c^{2}+3 a \,b^{3} d^{2} e^{5}-12 a \,b^{2} c \,d^{3} e^{4}+15 a b \,c^{2} d^{4} e^{3}-6 d^{5} e^{2} a \,c^{3}-b^{4} d^{3} e^{4}+5 b^{3} c \,d^{4} e^{3}-9 b^{2} c^{2} d^{5} e^{2}+7 b \,c^{3} d^{6} e -2 d^{7} c^{4}}{e^{8} \left (e x +d \right )}\) | \(766\) |
| risch | \(\frac {6 d^{5} a \,c^{3}}{e^{6} \left (e x +d \right )}-\frac {5 b^{3} c \,d^{4}}{e^{5} \left (e x +d \right )}+\frac {9 b^{2} c^{2} d^{5}}{e^{6} \left (e x +d \right )}-\frac {7 b \,c^{3} d^{6}}{e^{7} \left (e x +d \right )}+\frac {2 \ln \left (e x +d \right ) c \,a^{3}}{e^{2}}+\frac {3 \ln \left (e x +d \right ) a^{2} b^{2}}{e^{2}}+\frac {3 \ln \left (e x +d \right ) b^{4} d^{2}}{e^{4}}+\frac {6 d^{3} a^{2} c^{2}}{e^{4} \left (e x +d \right )}-\frac {3 a \,b^{3} d^{2}}{e^{3} \left (e x +d \right )}+\frac {3 a \,c^{3} x^{4}}{2 e^{2}}+\frac {5 c^{4} d^{4} x^{2}}{e^{6}}+\frac {3 a \,b^{3} x}{e^{2}}-\frac {2 b^{4} d x}{e^{3}}+\frac {7 b \,c^{3} x^{5}}{5 e^{2}}-\frac {4 c^{4} d \,x^{5}}{5 e^{3}}+\frac {9 b^{2} c^{2} x^{4}}{4 e^{2}}+\frac {3 c^{4} d^{2} x^{4}}{2 e^{4}}+\frac {5 b^{3} c \,x^{3}}{3 e^{2}}-\frac {8 c^{4} d^{3} x^{3}}{3 e^{5}}+\frac {3 a^{2} c^{2} x^{2}}{e^{2}}-\frac {12 c^{4} d^{5} x}{e^{7}}+\frac {9 a \,c^{3} d^{2} x^{2}}{e^{4}}-\frac {4 a \,c^{3} d \,x^{3}}{e^{3}}+\frac {9 a^{2} b c x}{e^{2}}-\frac {7 b \,c^{3} d \,x^{4}}{2 e^{3}}+\frac {5 a b \,c^{2} x^{3}}{e^{2}}+\frac {35 b \,c^{3} d^{4} x}{e^{6}}+\frac {15 b^{3} c \,d^{2} x}{e^{4}}-\frac {24 a \,c^{3} d^{3} x}{e^{5}}-\frac {6 b^{2} c^{2} d \,x^{3}}{e^{3}}-\frac {5 b^{3} c d \,x^{2}}{e^{3}}+\frac {27 b^{2} c^{2} d^{2} x^{2}}{2 e^{4}}-\frac {14 b \,c^{3} d^{3} x^{2}}{e^{5}}+\frac {7 b \,c^{3} d^{2} x^{3}}{e^{4}}+\frac {6 a \,b^{2} c \,x^{2}}{e^{2}}+\frac {18 \ln \left (e x +d \right ) d^{2} a^{2} c^{2}}{e^{4}}-\frac {6 \ln \left (e x +d \right ) a \,b^{3} d}{e^{3}}+\frac {30 \ln \left (e x +d \right ) d^{4} a \,c^{3}}{e^{6}}-\frac {20 \ln \left (e x +d \right ) b^{3} c \,d^{3}}{e^{5}}+\frac {45 \ln \left (e x +d \right ) b^{2} c^{2} d^{4}}{e^{6}}-\frac {12 a^{2} c^{2} d x}{e^{3}}-\frac {36 b^{2} c^{2} d^{3} x}{e^{5}}+\frac {14 \ln \left (e x +d \right ) d^{6} c^{4}}{e^{8}}-\frac {a^{3} b}{e \left (e x +d \right )}+\frac {b^{4} d^{3}}{e^{4} \left (e x +d \right )}+\frac {2 d^{7} c^{4}}{e^{8} \left (e x +d \right )}-\frac {42 \ln \left (e x +d \right ) b \,c^{3} d^{5}}{e^{7}}+\frac {2 d c \,a^{3}}{e^{2} \left (e x +d \right )}+\frac {3 a^{2} b^{2} d}{e^{2} \left (e x +d \right )}-\frac {15 a b \,c^{2} d \,x^{2}}{e^{3}}-\frac {24 a \,b^{2} c d x}{e^{3}}+\frac {45 a b \,c^{2} d^{2} x}{e^{4}}-\frac {18 \ln \left (e x +d \right ) a^{2} b c d}{e^{3}}+\frac {36 \ln \left (e x +d \right ) a \,b^{2} c \,d^{2}}{e^{4}}-\frac {60 \ln \left (e x +d \right ) a b \,c^{2} d^{3}}{e^{5}}-\frac {9 a^{2} b c \,d^{2}}{e^{3} \left (e x +d \right )}+\frac {12 a \,b^{2} c \,d^{3}}{e^{4} \left (e x +d \right )}-\frac {15 a b \,c^{2} d^{4}}{e^{5} \left (e x +d \right )}+\frac {b^{4} x^{2}}{2 e^{2}}+\frac {c^{4} x^{6}}{3 e^{2}}\) | \(930\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1105\) |
Input:
int((2*c*x+b)*(c*x^2+b*x+a)^3/(e*x+d)^2,x,method=_RETURNVERBOSE)
Output:
((a^3*b*e^7-2*a^3*c*d*e^6-3*a^2*b^2*d*e^6+18*a^2*b*c*d^2*e^5-18*a^2*c^2*d^ 3*e^4+6*a*b^3*d^2*e^5-36*a*b^2*c*d^3*e^4+60*a*b*c^2*d^4*e^3-30*a*c^3*d^5*e ^2-3*b^4*d^3*e^4+20*b^3*c*d^4*e^3-45*b^2*c^2*d^5*e^2+42*b*c^3*d^6*e-14*c^4 *d^7)/d/e^7*x+1/3*c^4/e*x^7+1/6*(18*a^2*c^2*e^4+36*a*b^2*c*e^4-60*a*b*c^2* d*e^3+30*a*c^3*d^2*e^2+3*b^4*e^4-20*b^3*c*d*e^3+45*b^2*c^2*d^2*e^2-42*b*c^ 3*d^3*e+14*c^4*d^4)/e^5*x^3+1/2*(18*a^2*b*c*e^5-18*a^2*c^2*d*e^4+6*a*b^3*e ^5-36*a*b^2*c*d*e^4+60*a*b*c^2*d^2*e^3-30*a*c^3*d^3*e^2-3*b^4*d*e^4+20*b^3 *c*d^2*e^3-45*b^2*c^2*d^3*e^2+42*b*c^3*d^4*e-14*c^4*d^5)/e^6*x^2+1/12*c*(6 0*a*b*c*e^3-30*a*c^2*d*e^2+20*b^3*e^3-45*b^2*c*d*e^2+42*b*c^2*d^2*e-14*c^3 *d^3)/e^4*x^4+1/20*c^2*(30*a*c*e^2+45*b^2*e^2-42*b*c*d*e+14*c^2*d^2)/e^3*x ^5+7/15*c^3*(3*b*e-c*d)/e^2*x^6)/(e*x+d)+1/e^8*(2*a^3*c*e^6+3*a^2*b^2*e^6- 18*a^2*b*c*d*e^5+18*a^2*c^2*d^2*e^4-6*a*b^3*d*e^5+36*a*b^2*c*d^2*e^4-60*a* b*c^2*d^3*e^3+30*a*c^3*d^4*e^2+3*b^4*d^2*e^4-20*b^3*c*d^3*e^3+45*b^2*c^2*d ^4*e^2-42*b*c^3*d^5*e+14*c^4*d^6)*ln(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 903 vs. \(2 (386) = 772\).
Time = 0.08 (sec) , antiderivative size = 903, normalized size of antiderivative = 2.28 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{(d+e x)^2} \, dx =\text {Too large to display} \] Input:
integrate((2*c*x+b)*(c*x^2+b*x+a)^3/(e*x+d)^2,x, algorithm="fricas")
Output:
1/60*(20*c^4*e^7*x^7 + 120*c^4*d^7 - 420*b*c^3*d^6*e - 60*a^3*b*e^7 + 180* (3*b^2*c^2 + 2*a*c^3)*d^5*e^2 - 300*(b^3*c + 3*a*b*c^2)*d^4*e^3 + 60*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^4 - 180*(a*b^3 + 3*a^2*b*c)*d^2*e^5 + 60*( 3*a^2*b^2 + 2*a^3*c)*d*e^6 - 28*(c^4*d*e^6 - 3*b*c^3*e^7)*x^6 + 3*(14*c^4* d^2*e^5 - 42*b*c^3*d*e^6 + 15*(3*b^2*c^2 + 2*a*c^3)*e^7)*x^5 - 5*(14*c^4*d ^3*e^4 - 42*b*c^3*d^2*e^5 + 15*(3*b^2*c^2 + 2*a*c^3)*d*e^6 - 20*(b^3*c + 3 *a*b*c^2)*e^7)*x^4 + 10*(14*c^4*d^4*e^3 - 42*b*c^3*d^3*e^4 + 15*(3*b^2*c^2 + 2*a*c^3)*d^2*e^5 - 20*(b^3*c + 3*a*b*c^2)*d*e^6 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^7)*x^3 - 30*(14*c^4*d^5*e^2 - 42*b*c^3*d^4*e^3 + 15*(3*b^2*c ^2 + 2*a*c^3)*d^3*e^4 - 20*(b^3*c + 3*a*b*c^2)*d^2*e^5 + 3*(b^4 + 12*a*b^2 *c + 6*a^2*c^2)*d*e^6 - 6*(a*b^3 + 3*a^2*b*c)*e^7)*x^2 - 60*(12*c^4*d^6*e - 35*b*c^3*d^5*e^2 + 12*(3*b^2*c^2 + 2*a*c^3)*d^4*e^3 - 15*(b^3*c + 3*a*b* c^2)*d^3*e^4 + 2*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^5 - 3*(a*b^3 + 3*a^2 *b*c)*d*e^6)*x + 60*(14*c^4*d^7 - 42*b*c^3*d^6*e + 15*(3*b^2*c^2 + 2*a*c^3 )*d^5*e^2 - 20*(b^3*c + 3*a*b*c^2)*d^4*e^3 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c ^2)*d^3*e^4 - 6*(a*b^3 + 3*a^2*b*c)*d^2*e^5 + (3*a^2*b^2 + 2*a^3*c)*d*e^6 + (14*c^4*d^6*e - 42*b*c^3*d^5*e^2 + 15*(3*b^2*c^2 + 2*a*c^3)*d^4*e^3 - 20 *(b^3*c + 3*a*b*c^2)*d^3*e^4 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^5 - 6*(a*b^3 + 3*a^2*b*c)*d*e^6 + (3*a^2*b^2 + 2*a^3*c)*e^7)*x)*log(e*x + d))/ (e^9*x + d*e^8)
Time = 2.00 (sec) , antiderivative size = 688, normalized size of antiderivative = 1.74 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{(d+e x)^2} \, dx=\frac {c^{4} x^{6}}{3 e^{2}} + x^{5} \cdot \left (\frac {7 b c^{3}}{5 e^{2}} - \frac {4 c^{4} d}{5 e^{3}}\right ) + x^{4} \cdot \left (\frac {3 a c^{3}}{2 e^{2}} + \frac {9 b^{2} c^{2}}{4 e^{2}} - \frac {7 b c^{3} d}{2 e^{3}} + \frac {3 c^{4} d^{2}}{2 e^{4}}\right ) + x^{3} \cdot \left (\frac {5 a b c^{2}}{e^{2}} - \frac {4 a c^{3} d}{e^{3}} + \frac {5 b^{3} c}{3 e^{2}} - \frac {6 b^{2} c^{2} d}{e^{3}} + \frac {7 b c^{3} d^{2}}{e^{4}} - \frac {8 c^{4} d^{3}}{3 e^{5}}\right ) + x^{2} \cdot \left (\frac {3 a^{2} c^{2}}{e^{2}} + \frac {6 a b^{2} c}{e^{2}} - \frac {15 a b c^{2} d}{e^{3}} + \frac {9 a c^{3} d^{2}}{e^{4}} + \frac {b^{4}}{2 e^{2}} - \frac {5 b^{3} c d}{e^{3}} + \frac {27 b^{2} c^{2} d^{2}}{2 e^{4}} - \frac {14 b c^{3} d^{3}}{e^{5}} + \frac {5 c^{4} d^{4}}{e^{6}}\right ) + x \left (\frac {9 a^{2} b c}{e^{2}} - \frac {12 a^{2} c^{2} d}{e^{3}} + \frac {3 a b^{3}}{e^{2}} - \frac {24 a b^{2} c d}{e^{3}} + \frac {45 a b c^{2} d^{2}}{e^{4}} - \frac {24 a c^{3} d^{3}}{e^{5}} - \frac {2 b^{4} d}{e^{3}} + \frac {15 b^{3} c d^{2}}{e^{4}} - \frac {36 b^{2} c^{2} d^{3}}{e^{5}} + \frac {35 b c^{3} d^{4}}{e^{6}} - \frac {12 c^{4} d^{5}}{e^{7}}\right ) + \frac {- a^{3} b e^{7} + 2 a^{3} c d e^{6} + 3 a^{2} b^{2} d e^{6} - 9 a^{2} b c d^{2} e^{5} + 6 a^{2} c^{2} d^{3} e^{4} - 3 a b^{3} d^{2} e^{5} + 12 a b^{2} c d^{3} e^{4} - 15 a b c^{2} d^{4} e^{3} + 6 a c^{3} d^{5} e^{2} + b^{4} d^{3} e^{4} - 5 b^{3} c d^{4} e^{3} + 9 b^{2} c^{2} d^{5} e^{2} - 7 b c^{3} d^{6} e + 2 c^{4} d^{7}}{d e^{8} + e^{9} x} + \frac {\left (a e^{2} - b d e + c d^{2}\right )^{2} \cdot \left (2 a c e^{2} + 3 b^{2} e^{2} - 14 b c d e + 14 c^{2} d^{2}\right ) \log {\left (d + e x \right )}}{e^{8}} \] Input:
integrate((2*c*x+b)*(c*x**2+b*x+a)**3/(e*x+d)**2,x)
Output:
c**4*x**6/(3*e**2) + x**5*(7*b*c**3/(5*e**2) - 4*c**4*d/(5*e**3)) + x**4*( 3*a*c**3/(2*e**2) + 9*b**2*c**2/(4*e**2) - 7*b*c**3*d/(2*e**3) + 3*c**4*d* *2/(2*e**4)) + x**3*(5*a*b*c**2/e**2 - 4*a*c**3*d/e**3 + 5*b**3*c/(3*e**2) - 6*b**2*c**2*d/e**3 + 7*b*c**3*d**2/e**4 - 8*c**4*d**3/(3*e**5)) + x**2* (3*a**2*c**2/e**2 + 6*a*b**2*c/e**2 - 15*a*b*c**2*d/e**3 + 9*a*c**3*d**2/e **4 + b**4/(2*e**2) - 5*b**3*c*d/e**3 + 27*b**2*c**2*d**2/(2*e**4) - 14*b* c**3*d**3/e**5 + 5*c**4*d**4/e**6) + x*(9*a**2*b*c/e**2 - 12*a**2*c**2*d/e **3 + 3*a*b**3/e**2 - 24*a*b**2*c*d/e**3 + 45*a*b*c**2*d**2/e**4 - 24*a*c* *3*d**3/e**5 - 2*b**4*d/e**3 + 15*b**3*c*d**2/e**4 - 36*b**2*c**2*d**3/e** 5 + 35*b*c**3*d**4/e**6 - 12*c**4*d**5/e**7) + (-a**3*b*e**7 + 2*a**3*c*d* e**6 + 3*a**2*b**2*d*e**6 - 9*a**2*b*c*d**2*e**5 + 6*a**2*c**2*d**3*e**4 - 3*a*b**3*d**2*e**5 + 12*a*b**2*c*d**3*e**4 - 15*a*b*c**2*d**4*e**3 + 6*a* c**3*d**5*e**2 + b**4*d**3*e**4 - 5*b**3*c*d**4*e**3 + 9*b**2*c**2*d**5*e* *2 - 7*b*c**3*d**6*e + 2*c**4*d**7)/(d*e**8 + e**9*x) + (a*e**2 - b*d*e + c*d**2)**2*(2*a*c*e**2 + 3*b**2*e**2 - 14*b*c*d*e + 14*c**2*d**2)*log(d + e*x)/e**8
Time = 0.04 (sec) , antiderivative size = 649, normalized size of antiderivative = 1.64 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{(d+e x)^2} \, dx=\frac {2 \, c^{4} d^{7} - 7 \, b c^{3} d^{6} e - a^{3} b e^{7} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{2} - 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{4} - 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{5} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{6}}{e^{9} x + d e^{8}} + \frac {20 \, c^{4} e^{5} x^{6} - 12 \, {\left (4 \, c^{4} d e^{4} - 7 \, b c^{3} e^{5}\right )} x^{5} + 15 \, {\left (6 \, c^{4} d^{2} e^{3} - 14 \, b c^{3} d e^{4} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{5}\right )} x^{4} - 20 \, {\left (8 \, c^{4} d^{3} e^{2} - 21 \, b c^{3} d^{2} e^{3} + 6 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{4} - 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{5}\right )} x^{3} + 30 \, {\left (10 \, c^{4} d^{4} e - 28 \, b c^{3} d^{3} e^{2} + 9 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{3} - 10 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{4} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{5}\right )} x^{2} - 60 \, {\left (12 \, c^{4} d^{5} - 35 \, b c^{3} d^{4} e + 12 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{2} - 15 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{3} + 2 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{4} - 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{5}\right )} x}{60 \, e^{7}} + \frac {{\left (14 \, c^{4} d^{6} - 42 \, b c^{3} d^{5} e + 15 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{2} - 20 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{3} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{4} - 6 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{5} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{6}\right )} \log \left (e x + d\right )}{e^{8}} \] Input:
integrate((2*c*x+b)*(c*x^2+b*x+a)^3/(e*x+d)^2,x, algorithm="maxima")
Output:
(2*c^4*d^7 - 7*b*c^3*d^6*e - a^3*b*e^7 + 3*(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 - 5*(b^3*c + 3*a*b*c^2)*d^4*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^4 - 3*(a*b^3 + 3*a^2*b*c)*d^2*e^5 + (3*a^2*b^2 + 2*a^3*c)*d*e^6)/(e^9*x + d*e^ 8) + 1/60*(20*c^4*e^5*x^6 - 12*(4*c^4*d*e^4 - 7*b*c^3*e^5)*x^5 + 15*(6*c^4 *d^2*e^3 - 14*b*c^3*d*e^4 + 3*(3*b^2*c^2 + 2*a*c^3)*e^5)*x^4 - 20*(8*c^4*d ^3*e^2 - 21*b*c^3*d^2*e^3 + 6*(3*b^2*c^2 + 2*a*c^3)*d*e^4 - 5*(b^3*c + 3*a *b*c^2)*e^5)*x^3 + 30*(10*c^4*d^4*e - 28*b*c^3*d^3*e^2 + 9*(3*b^2*c^2 + 2* a*c^3)*d^2*e^3 - 10*(b^3*c + 3*a*b*c^2)*d*e^4 + (b^4 + 12*a*b^2*c + 6*a^2* c^2)*e^5)*x^2 - 60*(12*c^4*d^5 - 35*b*c^3*d^4*e + 12*(3*b^2*c^2 + 2*a*c^3) *d^3*e^2 - 15*(b^3*c + 3*a*b*c^2)*d^2*e^3 + 2*(b^4 + 12*a*b^2*c + 6*a^2*c^ 2)*d*e^4 - 3*(a*b^3 + 3*a^2*b*c)*e^5)*x)/e^7 + (14*c^4*d^6 - 42*b*c^3*d^5* e + 15*(3*b^2*c^2 + 2*a*c^3)*d^4*e^2 - 20*(b^3*c + 3*a*b*c^2)*d^3*e^3 + 3* (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^4 - 6*(a*b^3 + 3*a^2*b*c)*d*e^5 + (3* a^2*b^2 + 2*a^3*c)*e^6)*log(e*x + d)/e^8
Leaf count of result is larger than twice the leaf count of optimal. 870 vs. \(2 (386) = 772\).
Time = 0.23 (sec) , antiderivative size = 870, normalized size of antiderivative = 2.20 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{(d+e x)^2} \, dx =\text {Too large to display} \] Input:
integrate((2*c*x+b)*(c*x^2+b*x+a)^3/(e*x+d)^2,x, algorithm="giac")
Output:
1/60*(20*c^4 - 84*(2*c^4*d*e - b*c^3*e^2)/((e*x + d)*e) + 45*(14*c^4*d^2*e ^2 - 14*b*c^3*d*e^3 + 3*b^2*c^2*e^4 + 2*a*c^3*e^4)/((e*x + d)^2*e^2) - 100 *(14*c^4*d^3*e^3 - 21*b*c^3*d^2*e^4 + 9*b^2*c^2*d*e^5 + 6*a*c^3*d*e^5 - b^ 3*c*e^6 - 3*a*b*c^2*e^6)/((e*x + d)^3*e^3) + 30*(70*c^4*d^4*e^4 - 140*b*c^ 3*d^3*e^5 + 90*b^2*c^2*d^2*e^6 + 60*a*c^3*d^2*e^6 - 20*b^3*c*d*e^7 - 60*a* b*c^2*d*e^7 + b^4*e^8 + 12*a*b^2*c*e^8 + 6*a^2*c^2*e^8)/((e*x + d)^4*e^4) - 180*(14*c^4*d^5*e^5 - 35*b*c^3*d^4*e^6 + 30*b^2*c^2*d^3*e^7 + 20*a*c^3*d ^3*e^7 - 10*b^3*c*d^2*e^8 - 30*a*b*c^2*d^2*e^8 + b^4*d*e^9 + 12*a*b^2*c*d* e^9 + 6*a^2*c^2*d*e^9 - a*b^3*e^10 - 3*a^2*b*c*e^10)/((e*x + d)^5*e^5))*(e *x + d)^6/e^8 - (14*c^4*d^6 - 42*b*c^3*d^5*e + 45*b^2*c^2*d^4*e^2 + 30*a*c ^3*d^4*e^2 - 20*b^3*c*d^3*e^3 - 60*a*b*c^2*d^3*e^3 + 3*b^4*d^2*e^4 + 36*a* b^2*c*d^2*e^4 + 18*a^2*c^2*d^2*e^4 - 6*a*b^3*d*e^5 - 18*a^2*b*c*d*e^5 + 3* a^2*b^2*e^6 + 2*a^3*c*e^6)*log(abs(e*x + d)/((e*x + d)^2*abs(e)))/e^8 + (2 *c^4*d^7*e^6/(e*x + d) - 7*b*c^3*d^6*e^7/(e*x + d) + 9*b^2*c^2*d^5*e^8/(e* x + d) + 6*a*c^3*d^5*e^8/(e*x + d) - 5*b^3*c*d^4*e^9/(e*x + d) - 15*a*b*c^ 2*d^4*e^9/(e*x + d) + b^4*d^3*e^10/(e*x + d) + 12*a*b^2*c*d^3*e^10/(e*x + d) + 6*a^2*c^2*d^3*e^10/(e*x + d) - 3*a*b^3*d^2*e^11/(e*x + d) - 9*a^2*b*c *d^2*e^11/(e*x + d) + 3*a^2*b^2*d*e^12/(e*x + d) + 2*a^3*c*d*e^12/(e*x + d ) - a^3*b*e^13/(e*x + d))/e^14
Time = 11.28 (sec) , antiderivative size = 1090, normalized size of antiderivative = 2.75 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{(d+e x)^2} \, dx =\text {Too large to display} \] Input:
int(((b + 2*c*x)*(a + b*x + c*x^2)^3)/(d + e*x)^2,x)
Output:
x^5*((7*b*c^3)/(5*e^2) - (4*c^4*d)/(5*e^3)) + x^3*((2*d*((2*d*((7*b*c^3)/e ^2 - (4*c^4*d)/e^3))/e - (6*a*c^3 + 9*b^2*c^2)/e^2 + (2*c^4*d^2)/e^4))/(3* e) - (d^2*((7*b*c^3)/e^2 - (4*c^4*d)/e^3))/(3*e^2) + (5*b*c*(3*a*c + b^2)) /(3*e^2)) - x^4*((d*((7*b*c^3)/e^2 - (4*c^4*d)/e^3))/(2*e) - (6*a*c^3 + 9* b^2*c^2)/(4*e^2) + (c^4*d^2)/(2*e^4)) - x*((2*d*((b^4 + 6*a^2*c^2 + 12*a*b ^2*c)/e^2 + (d^2*((2*d*((7*b*c^3)/e^2 - (4*c^4*d)/e^3))/e - (6*a*c^3 + 9*b ^2*c^2)/e^2 + (2*c^4*d^2)/e^4))/e^2 - (2*d*((2*d*((2*d*((7*b*c^3)/e^2 - (4 *c^4*d)/e^3))/e - (6*a*c^3 + 9*b^2*c^2)/e^2 + (2*c^4*d^2)/e^4))/e - (d^2*( (7*b*c^3)/e^2 - (4*c^4*d)/e^3))/e^2 + (5*b*c*(3*a*c + b^2))/e^2))/e))/e + (d^2*((2*d*((2*d*((7*b*c^3)/e^2 - (4*c^4*d)/e^3))/e - (6*a*c^3 + 9*b^2*c^2 )/e^2 + (2*c^4*d^2)/e^4))/e - (d^2*((7*b*c^3)/e^2 - (4*c^4*d)/e^3))/e^2 + (5*b*c*(3*a*c + b^2))/e^2))/e^2 - (3*a*b*(3*a*c + b^2))/e^2) + x^2*((b^4 + 6*a^2*c^2 + 12*a*b^2*c)/(2*e^2) + (d^2*((2*d*((7*b*c^3)/e^2 - (4*c^4*d)/e ^3))/e - (6*a*c^3 + 9*b^2*c^2)/e^2 + (2*c^4*d^2)/e^4))/(2*e^2) - (d*((2*d* ((2*d*((7*b*c^3)/e^2 - (4*c^4*d)/e^3))/e - (6*a*c^3 + 9*b^2*c^2)/e^2 + (2* c^4*d^2)/e^4))/e - (d^2*((7*b*c^3)/e^2 - (4*c^4*d)/e^3))/e^2 + (5*b*c*(3*a *c + b^2))/e^2))/e) + (log(d + e*x)*(14*c^4*d^6 + 2*a^3*c*e^6 + 3*a^2*b^2* e^6 + 3*b^4*d^2*e^4 + 30*a*c^3*d^4*e^2 - 20*b^3*c*d^3*e^3 + 18*a^2*c^2*d^2 *e^4 + 45*b^2*c^2*d^4*e^2 - 6*a*b^3*d*e^5 - 42*b*c^3*d^5*e - 18*a^2*b*c*d* e^5 - 60*a*b*c^2*d^3*e^3 + 36*a*b^2*c*d^2*e^4))/e^8 + (c^4*x^6)/(3*e^2)...
Time = 1.66 (sec) , antiderivative size = 1163, normalized size of antiderivative = 2.94 \[ \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^3}{(d+e x)^2} \, dx =\text {Too large to display} \] Input:
int((2*c*x+b)*(c*x^2+b*x+a)^3/(e*x+d)^2,x)
Output:
(120*log(d + e*x)*a**3*c*d**2*e**6 + 120*log(d + e*x)*a**3*c*d*e**7*x + 18 0*log(d + e*x)*a**2*b**2*d**2*e**6 + 180*log(d + e*x)*a**2*b**2*d*e**7*x - 1080*log(d + e*x)*a**2*b*c*d**3*e**5 - 1080*log(d + e*x)*a**2*b*c*d**2*e* *6*x + 1080*log(d + e*x)*a**2*c**2*d**4*e**4 + 1080*log(d + e*x)*a**2*c**2 *d**3*e**5*x - 360*log(d + e*x)*a*b**3*d**3*e**5 - 360*log(d + e*x)*a*b**3 *d**2*e**6*x + 2160*log(d + e*x)*a*b**2*c*d**4*e**4 + 2160*log(d + e*x)*a* b**2*c*d**3*e**5*x - 3600*log(d + e*x)*a*b*c**2*d**5*e**3 - 3600*log(d + e *x)*a*b*c**2*d**4*e**4*x + 1800*log(d + e*x)*a*c**3*d**6*e**2 + 1800*log(d + e*x)*a*c**3*d**5*e**3*x + 180*log(d + e*x)*b**4*d**4*e**4 + 180*log(d + e*x)*b**4*d**3*e**5*x - 1200*log(d + e*x)*b**3*c*d**5*e**3 - 1200*log(d + e*x)*b**3*c*d**4*e**4*x + 2700*log(d + e*x)*b**2*c**2*d**6*e**2 + 2700*lo g(d + e*x)*b**2*c**2*d**5*e**3*x - 2520*log(d + e*x)*b*c**3*d**7*e - 2520* log(d + e*x)*b*c**3*d**6*e**2*x + 840*log(d + e*x)*c**4*d**8 + 840*log(d + e*x)*c**4*d**7*e*x + 60*a**3*b*e**8*x - 120*a**3*c*d*e**7*x - 180*a**2*b* *2*d*e**7*x + 1080*a**2*b*c*d**2*e**6*x + 540*a**2*b*c*d*e**7*x**2 - 1080* a**2*c**2*d**3*e**5*x - 540*a**2*c**2*d**2*e**6*x**2 + 180*a**2*c**2*d*e** 7*x**3 + 360*a*b**3*d**2*e**6*x + 180*a*b**3*d*e**7*x**2 - 2160*a*b**2*c*d **3*e**5*x - 1080*a*b**2*c*d**2*e**6*x**2 + 360*a*b**2*c*d*e**7*x**3 + 360 0*a*b*c**2*d**4*e**4*x + 1800*a*b*c**2*d**3*e**5*x**2 - 600*a*b*c**2*d**2* e**6*x**3 + 300*a*b*c**2*d*e**7*x**4 - 1800*a*c**3*d**5*e**3*x - 900*a*...