Integrand size = 24, antiderivative size = 194 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^2} \, dx=-\frac {(c d-b e) (2 B d (2 c d-b e)-A e (3 c d-b e)) x}{e^5}+\frac {(c d-b e) (3 B c d-b B e-2 A c e) x^2}{2 e^4}-\frac {c (2 B c d-2 b B e-A c e) x^3}{3 e^3}+\frac {B c^2 x^4}{4 e^2}+\frac {d^2 (B d-A e) (c d-b e)^2}{e^6 (d+e x)}+\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) \log (d+e x)}{e^6} \]
-(-b*e+c*d)*(2*B*d*(-b*e+2*c*d)-A*e*(-b*e+3*c*d))*x/e^5+1/2*(-b*e+c*d)*(-2 *A*c*e-B*b*e+3*B*c*d)*x^2/e^4-1/3*c*(-A*c*e-2*B*b*e+2*B*c*d)*x^3/e^3+1/4*B *c^2*x^4/e^2+d^2*(-A*e+B*d)*(-b*e+c*d)^2/e^6/(e*x+d)+d*(-b*e+c*d)*(B*d*(-3 *b*e+5*c*d)-2*A*e*(-b*e+2*c*d))*ln(e*x+d)/e^6
Time = 0.06 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.95 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^2} \, dx=\frac {12 e (-c d+b e) (2 B d (2 c d-b e)+A e (-3 c d+b e)) x+6 e^2 (-c d+b e) (-3 B c d+b B e+2 A c e) x^2+4 c e^3 (-2 B c d+2 b B e+A c e) x^3+3 B c^2 e^4 x^4+\frac {12 d^2 (B d-A e) (c d-b e)^2}{d+e x}+12 d (c d-b e) (B d (5 c d-3 b e)+2 A e (-2 c d+b e)) \log (d+e x)}{12 e^6} \]
(12*e*(-(c*d) + b*e)*(2*B*d*(2*c*d - b*e) + A*e*(-3*c*d + b*e))*x + 6*e^2* (-(c*d) + b*e)*(-3*B*c*d + b*B*e + 2*A*c*e)*x^2 + 4*c*e^3*(-2*B*c*d + 2*b* B*e + A*c*e)*x^3 + 3*B*c^2*e^4*x^4 + (12*d^2*(B*d - A*e)*(c*d - b*e)^2)/(d + e*x) + 12*d*(c*d - b*e)*(B*d*(5*c*d - 3*b*e) + 2*A*e*(-2*c*d + b*e))*Lo g[d + e*x])/(12*e^6)
Time = 0.47 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, 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.
| \(\displaystyle \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (-\frac {d^2 (B d-A e) (c d-b e)^2}{e^5 (d+e x)^2}+\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^5 (d+e x)}+\frac {(c d-b e) (A e (3 c d-b e)-2 B d (2 c d-b e))}{e^5}+\frac {x (b e-c d) (2 A c e+b B e-3 B c d)}{e^4}+\frac {c x^2 (A c e+2 b B e-2 B c d)}{e^3}+\frac {B c^2 x^3}{e^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d^2 (B d-A e) (c d-b e)^2}{e^6 (d+e x)}+\frac {d (c d-b e) \log (d+e x) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^6}-\frac {x (c d-b e) (2 B d (2 c d-b e)-A e (3 c d-b e))}{e^5}+\frac {x^2 (c d-b e) (-2 A c e-b B e+3 B c d)}{2 e^4}-\frac {c x^3 (-A c e-2 b B e+2 B c d)}{3 e^3}+\frac {B c^2 x^4}{4 e^2}\) |
-(((c*d - b*e)*(2*B*d*(2*c*d - b*e) - A*e*(3*c*d - b*e))*x)/e^5) + ((c*d - b*e)*(3*B*c*d - b*B*e - 2*A*c*e)*x^2)/(2*e^4) - (c*(2*B*c*d - 2*b*B*e - A *c*e)*x^3)/(3*e^3) + (B*c^2*x^4)/(4*e^2) + (d^2*(B*d - A*e)*(c*d - b*e)^2) /(e^6*(d + e*x)) + (d*(c*d - b*e)*(B*d*(5*c*d - 3*b*e) - 2*A*e*(2*c*d - b* e))*Log[d + e*x])/e^6
3.12.20.3.1 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 = 0.22 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.57
| method | result | size |
| norman | \(\frac {\frac {\left (2 A \,b^{2} d^{2} e^{3}-6 A b c \,d^{3} e^{2}+4 A \,c^{2} d^{4} e -3 B \,b^{2} d^{3} e^{2}+8 B b c \,d^{4} e -5 B \,c^{2} d^{5}\right ) x}{e^{5} d}+\frac {\left (2 A \,b^{2} e^{3}-6 A b c d \,e^{2}+4 A \,c^{2} d^{2} e -3 B \,b^{2} d \,e^{2}+8 B b c \,d^{2} e -5 B \,c^{2} d^{3}\right ) x^{2}}{2 e^{4}}+\frac {\left (6 A b c \,e^{2}-4 A \,c^{2} d e +3 B \,b^{2} e^{2}-8 B b c d e +5 B \,c^{2} d^{2}\right ) x^{3}}{6 e^{3}}+\frac {B \,c^{2} x^{5}}{4 e}+\frac {c \left (4 A c e +8 B b e -5 B c d \right ) x^{4}}{12 e^{2}}}{e x +d}-\frac {d \left (2 A \,b^{2} e^{3}-6 A b c d \,e^{2}+4 A \,c^{2} d^{2} e -3 B \,b^{2} d \,e^{2}+8 B b c \,d^{2} e -5 B \,c^{2} d^{3}\right ) \ln \left (e x +d \right )}{e^{6}}\) | \(304\) |
| default | \(\frac {\frac {1}{4} B \,c^{2} x^{4} e^{3}+\frac {1}{3} A \,c^{2} e^{3} x^{3}+\frac {2}{3} B b c \,e^{3} x^{3}-\frac {2}{3} B \,c^{2} d \,e^{2} x^{3}+A b c \,e^{3} x^{2}-A \,c^{2} d \,e^{2} x^{2}+\frac {1}{2} B \,b^{2} e^{3} x^{2}-2 B b c d \,e^{2} x^{2}+\frac {3}{2} B \,c^{2} d^{2} e \,x^{2}+A \,b^{2} e^{3} x -4 A b c d \,e^{2} x +3 A \,c^{2} d^{2} e x -2 B \,b^{2} d \,e^{2} x +6 B b c \,d^{2} e x -4 B \,c^{2} d^{3} x}{e^{5}}-\frac {d^{2} \left (A \,b^{2} e^{3}-2 A b c d \,e^{2}+A \,c^{2} d^{2} e -B \,b^{2} d \,e^{2}+2 B b c \,d^{2} e -B \,c^{2} d^{3}\right )}{e^{6} \left (e x +d \right )}-\frac {d \left (2 A \,b^{2} e^{3}-6 A b c d \,e^{2}+4 A \,c^{2} d^{2} e -3 B \,b^{2} d \,e^{2}+8 B b c \,d^{2} e -5 B \,c^{2} d^{3}\right ) \ln \left (e x +d \right )}{e^{6}}\) | \(315\) |
| risch | \(\frac {b^{2} B \,x^{2}}{2 e^{2}}+\frac {B \,c^{2} x^{4}}{4 e^{2}}+\frac {A \,b^{2} x}{e^{2}}+\frac {A \,c^{2} x^{3}}{3 e^{2}}-\frac {2 d^{4} B b c}{e^{5} \left (e x +d \right )}+\frac {6 d^{2} \ln \left (e x +d \right ) A b c}{e^{4}}-\frac {8 d^{3} \ln \left (e x +d \right ) B b c}{e^{5}}-\frac {2 B b c d \,x^{2}}{e^{3}}-\frac {4 A b c d x}{e^{3}}+\frac {6 B b c \,d^{2} x}{e^{4}}+\frac {2 d^{3} A b c}{e^{4} \left (e x +d \right )}-\frac {2 d \ln \left (e x +d \right ) A \,b^{2}}{e^{3}}-\frac {4 d^{3} \ln \left (e x +d \right ) A \,c^{2}}{e^{5}}+\frac {3 d^{2} \ln \left (e x +d \right ) B \,b^{2}}{e^{4}}+\frac {5 d^{4} \ln \left (e x +d \right ) B \,c^{2}}{e^{6}}+\frac {3 A \,c^{2} d^{2} x}{e^{4}}-\frac {2 B \,b^{2} d x}{e^{3}}-\frac {4 B \,c^{2} d^{3} x}{e^{5}}-\frac {d^{2} A \,b^{2}}{e^{3} \left (e x +d \right )}-\frac {d^{4} A \,c^{2}}{e^{5} \left (e x +d \right )}+\frac {d^{3} B \,b^{2}}{e^{4} \left (e x +d \right )}+\frac {d^{5} B \,c^{2}}{e^{6} \left (e x +d \right )}+\frac {A b c \,x^{2}}{e^{2}}-\frac {A \,c^{2} d \,x^{2}}{e^{3}}+\frac {3 B \,c^{2} d^{2} x^{2}}{2 e^{4}}+\frac {2 B b c \,x^{3}}{3 e^{2}}-\frac {2 B \,c^{2} d \,x^{3}}{3 e^{3}}\) | \(394\) |
| parallelrisch | \(-\frac {-60 B \,c^{2} d^{5}-60 B \ln \left (e x +d \right ) x \,c^{2} d^{4} e +24 A \ln \left (e x +d \right ) x \,b^{2} d \,e^{4}+48 A \ln \left (e x +d \right ) x \,c^{2} d^{3} e^{2}-36 B \ln \left (e x +d \right ) x \,b^{2} d^{2} e^{3}-60 B \ln \left (e x +d \right ) c^{2} d^{5}-4 A \,x^{4} c^{2} e^{5}-6 B \,x^{3} b^{2} e^{5}-12 A \,x^{2} b^{2} e^{5}-3 B \,x^{5} c^{2} e^{5}-72 A \ln \left (e x +d \right ) b c \,d^{3} e^{2}+96 B \ln \left (e x +d \right ) b c \,d^{4} e -48 B \,x^{2} b c \,d^{2} e^{3}+36 A \,x^{2} b c d \,e^{4}+16 B \,x^{3} b c d \,e^{4}+24 A \,b^{2} d^{2} e^{3}+48 A \,c^{2} d^{4} e -36 B \,b^{2} d^{3} e^{2}-72 A b c \,d^{3} e^{2}+96 B b c \,d^{4} e -72 A \ln \left (e x +d \right ) x b c \,d^{2} e^{3}+96 B \ln \left (e x +d \right ) x b c \,d^{3} e^{2}-12 A \,x^{3} b c \,e^{5}+8 A \,x^{3} c^{2} d \,e^{4}-10 B \,x^{3} c^{2} d^{2} e^{3}-24 A \,x^{2} c^{2} d^{2} e^{3}+18 B \,x^{2} b^{2} d \,e^{4}+30 B \,x^{2} c^{2} d^{3} e^{2}+24 A \ln \left (e x +d \right ) b^{2} d^{2} e^{3}+48 A \ln \left (e x +d \right ) c^{2} d^{4} e -36 B \ln \left (e x +d \right ) b^{2} d^{3} e^{2}-8 B \,x^{4} b c \,e^{5}+5 B \,x^{4} c^{2} d \,e^{4}}{12 e^{6} \left (e x +d \right )}\) | \(476\) |
((2*A*b^2*d^2*e^3-6*A*b*c*d^3*e^2+4*A*c^2*d^4*e-3*B*b^2*d^3*e^2+8*B*b*c*d^ 4*e-5*B*c^2*d^5)/e^5/d*x+1/2/e^4*(2*A*b^2*e^3-6*A*b*c*d*e^2+4*A*c^2*d^2*e- 3*B*b^2*d*e^2+8*B*b*c*d^2*e-5*B*c^2*d^3)*x^2+1/6*(6*A*b*c*e^2-4*A*c^2*d*e+ 3*B*b^2*e^2-8*B*b*c*d*e+5*B*c^2*d^2)/e^3*x^3+1/4*B*c^2*x^5/e+1/12*c*(4*A*c *e+8*B*b*e-5*B*c*d)/e^2*x^4)/(e*x+d)-d/e^6*(2*A*b^2*e^3-6*A*b*c*d*e^2+4*A* c^2*d^2*e-3*B*b^2*d*e^2+8*B*b*c*d^2*e-5*B*c^2*d^3)*ln(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 420 vs. \(2 (188) = 376\).
Time = 0.34 (sec) , antiderivative size = 420, normalized size of antiderivative = 2.16 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^2} \, dx=\frac {3 \, B c^{2} e^{5} x^{5} + 12 \, B c^{2} d^{5} - 12 \, A b^{2} d^{2} e^{3} - 12 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + 12 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} - {\left (5 \, B c^{2} d e^{4} - 4 \, {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 2 \, {\left (5 \, B c^{2} d^{2} e^{3} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} - 6 \, {\left (5 \, B c^{2} d^{3} e^{2} - 2 \, A b^{2} e^{5} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} - 12 \, {\left (4 \, B c^{2} d^{4} e - A b^{2} d e^{4} - 3 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 2 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x + 12 \, {\left (5 \, B c^{2} d^{5} - 2 \, A b^{2} d^{2} e^{3} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} + {\left (5 \, B c^{2} d^{4} e - 2 \, A b^{2} d e^{4} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x\right )} \log \left (e x + d\right )}{12 \, {\left (e^{7} x + d e^{6}\right )}} \]
1/12*(3*B*c^2*e^5*x^5 + 12*B*c^2*d^5 - 12*A*b^2*d^2*e^3 - 12*(2*B*b*c + A* c^2)*d^4*e + 12*(B*b^2 + 2*A*b*c)*d^3*e^2 - (5*B*c^2*d*e^4 - 4*(2*B*b*c + A*c^2)*e^5)*x^4 + 2*(5*B*c^2*d^2*e^3 - 4*(2*B*b*c + A*c^2)*d*e^4 + 3*(B*b^ 2 + 2*A*b*c)*e^5)*x^3 - 6*(5*B*c^2*d^3*e^2 - 2*A*b^2*e^5 - 4*(2*B*b*c + A* c^2)*d^2*e^3 + 3*(B*b^2 + 2*A*b*c)*d*e^4)*x^2 - 12*(4*B*c^2*d^4*e - A*b^2* d*e^4 - 3*(2*B*b*c + A*c^2)*d^3*e^2 + 2*(B*b^2 + 2*A*b*c)*d^2*e^3)*x + 12* (5*B*c^2*d^5 - 2*A*b^2*d^2*e^3 - 4*(2*B*b*c + A*c^2)*d^4*e + 3*(B*b^2 + 2* A*b*c)*d^3*e^2 + (5*B*c^2*d^4*e - 2*A*b^2*d*e^4 - 4*(2*B*b*c + A*c^2)*d^3* e^2 + 3*(B*b^2 + 2*A*b*c)*d^2*e^3)*x)*log(e*x + d))/(e^7*x + d*e^6)
Time = 0.65 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.63 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^2} \, dx=\frac {B c^{2} x^{4}}{4 e^{2}} + \frac {d \left (b e - c d\right ) \left (- 2 A b e^{2} + 4 A c d e + 3 B b d e - 5 B c d^{2}\right ) \log {\left (d + e x \right )}}{e^{6}} + x^{3} \left (\frac {A c^{2}}{3 e^{2}} + \frac {2 B b c}{3 e^{2}} - \frac {2 B c^{2} d}{3 e^{3}}\right ) + x^{2} \left (\frac {A b c}{e^{2}} - \frac {A c^{2} d}{e^{3}} + \frac {B b^{2}}{2 e^{2}} - \frac {2 B b c d}{e^{3}} + \frac {3 B c^{2} d^{2}}{2 e^{4}}\right ) + x \left (\frac {A b^{2}}{e^{2}} - \frac {4 A b c d}{e^{3}} + \frac {3 A c^{2} d^{2}}{e^{4}} - \frac {2 B b^{2} d}{e^{3}} + \frac {6 B b c d^{2}}{e^{4}} - \frac {4 B c^{2} d^{3}}{e^{5}}\right ) + \frac {- A b^{2} d^{2} e^{3} + 2 A b c d^{3} e^{2} - A c^{2} d^{4} e + B b^{2} d^{3} e^{2} - 2 B b c d^{4} e + B c^{2} d^{5}}{d e^{6} + e^{7} x} \]
B*c**2*x**4/(4*e**2) + d*(b*e - c*d)*(-2*A*b*e**2 + 4*A*c*d*e + 3*B*b*d*e - 5*B*c*d**2)*log(d + e*x)/e**6 + x**3*(A*c**2/(3*e**2) + 2*B*b*c/(3*e**2) - 2*B*c**2*d/(3*e**3)) + x**2*(A*b*c/e**2 - A*c**2*d/e**3 + B*b**2/(2*e** 2) - 2*B*b*c*d/e**3 + 3*B*c**2*d**2/(2*e**4)) + x*(A*b**2/e**2 - 4*A*b*c*d /e**3 + 3*A*c**2*d**2/e**4 - 2*B*b**2*d/e**3 + 6*B*b*c*d**2/e**4 - 4*B*c** 2*d**3/e**5) + (-A*b**2*d**2*e**3 + 2*A*b*c*d**3*e**2 - A*c**2*d**4*e + B* b**2*d**3*e**2 - 2*B*b*c*d**4*e + B*c**2*d**5)/(d*e**6 + e**7*x)
Time = 0.19 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.50 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^2} \, dx=\frac {B c^{2} d^{5} - A b^{2} d^{2} e^{3} - {\left (2 \, B b c + A c^{2}\right )} d^{4} e + {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2}}{e^{7} x + d e^{6}} + \frac {3 \, B c^{2} e^{3} x^{4} - 4 \, {\left (2 \, B c^{2} d e^{2} - {\left (2 \, B b c + A c^{2}\right )} e^{3}\right )} x^{3} + 6 \, {\left (3 \, B c^{2} d^{2} e - 2 \, {\left (2 \, B b c + A c^{2}\right )} d e^{2} + {\left (B b^{2} + 2 \, A b c\right )} e^{3}\right )} x^{2} - 12 \, {\left (4 \, B c^{2} d^{3} - A b^{2} e^{3} - 3 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e + 2 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{2}\right )} x}{12 \, e^{5}} + \frac {{\left (5 \, B c^{2} d^{4} - 2 \, A b^{2} d e^{3} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{2}\right )} \log \left (e x + d\right )}{e^{6}} \]
(B*c^2*d^5 - A*b^2*d^2*e^3 - (2*B*b*c + A*c^2)*d^4*e + (B*b^2 + 2*A*b*c)*d ^3*e^2)/(e^7*x + d*e^6) + 1/12*(3*B*c^2*e^3*x^4 - 4*(2*B*c^2*d*e^2 - (2*B* b*c + A*c^2)*e^3)*x^3 + 6*(3*B*c^2*d^2*e - 2*(2*B*b*c + A*c^2)*d*e^2 + (B* b^2 + 2*A*b*c)*e^3)*x^2 - 12*(4*B*c^2*d^3 - A*b^2*e^3 - 3*(2*B*b*c + A*c^2 )*d^2*e + 2*(B*b^2 + 2*A*b*c)*d*e^2)*x)/e^5 + (5*B*c^2*d^4 - 2*A*b^2*d*e^3 - 4*(2*B*b*c + A*c^2)*d^3*e + 3*(B*b^2 + 2*A*b*c)*d^2*e^2)*log(e*x + d)/e ^6
Leaf count of result is larger than twice the leaf count of optimal. 395 vs. \(2 (188) = 376\).
Time = 0.27 (sec) , antiderivative size = 395, normalized size of antiderivative = 2.04 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^2} \, dx=\frac {{\left (3 \, B c^{2} - \frac {4 \, {\left (5 \, B c^{2} d e - 2 \, B b c e^{2} - A c^{2} e^{2}\right )}}{{\left (e x + d\right )} e} + \frac {6 \, {\left (10 \, B c^{2} d^{2} e^{2} - 8 \, B b c d e^{3} - 4 \, A c^{2} d e^{3} + B b^{2} e^{4} + 2 \, A b c e^{4}\right )}}{{\left (e x + d\right )}^{2} e^{2}} - \frac {12 \, {\left (10 \, B c^{2} d^{3} e^{3} - 12 \, B b c d^{2} e^{4} - 6 \, A c^{2} d^{2} e^{4} + 3 \, B b^{2} d e^{5} + 6 \, A b c d e^{5} - A b^{2} e^{6}\right )}}{{\left (e x + d\right )}^{3} e^{3}}\right )} {\left (e x + d\right )}^{4}}{12 \, e^{6}} - \frac {{\left (5 \, B c^{2} d^{4} - 8 \, B b c d^{3} e - 4 \, A c^{2} d^{3} e + 3 \, B b^{2} d^{2} e^{2} + 6 \, A b c d^{2} e^{2} - 2 \, A b^{2} d e^{3}\right )} \log \left (\frac {{\left | e x + d \right |}}{{\left (e x + d\right )}^{2} {\left | e \right |}}\right )}{e^{6}} + \frac {\frac {B c^{2} d^{5} e^{4}}{e x + d} - \frac {2 \, B b c d^{4} e^{5}}{e x + d} - \frac {A c^{2} d^{4} e^{5}}{e x + d} + \frac {B b^{2} d^{3} e^{6}}{e x + d} + \frac {2 \, A b c d^{3} e^{6}}{e x + d} - \frac {A b^{2} d^{2} e^{7}}{e x + d}}{e^{10}} \]
1/12*(3*B*c^2 - 4*(5*B*c^2*d*e - 2*B*b*c*e^2 - A*c^2*e^2)/((e*x + d)*e) + 6*(10*B*c^2*d^2*e^2 - 8*B*b*c*d*e^3 - 4*A*c^2*d*e^3 + B*b^2*e^4 + 2*A*b*c* e^4)/((e*x + d)^2*e^2) - 12*(10*B*c^2*d^3*e^3 - 12*B*b*c*d^2*e^4 - 6*A*c^2 *d^2*e^4 + 3*B*b^2*d*e^5 + 6*A*b*c*d*e^5 - A*b^2*e^6)/((e*x + d)^3*e^3))*( e*x + d)^4/e^6 - (5*B*c^2*d^4 - 8*B*b*c*d^3*e - 4*A*c^2*d^3*e + 3*B*b^2*d^ 2*e^2 + 6*A*b*c*d^2*e^2 - 2*A*b^2*d*e^3)*log(abs(e*x + d)/((e*x + d)^2*abs (e)))/e^6 + (B*c^2*d^5*e^4/(e*x + d) - 2*B*b*c*d^4*e^5/(e*x + d) - A*c^2*d ^4*e^5/(e*x + d) + B*b^2*d^3*e^6/(e*x + d) + 2*A*b*c*d^3*e^6/(e*x + d) - A *b^2*d^2*e^7/(e*x + d))/e^10
Time = 10.60 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.91 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^2} \, dx=x\,\left (\frac {A\,b^2}{e^2}-\frac {d^2\,\left (\frac {A\,c^2+2\,B\,b\,c}{e^2}-\frac {2\,B\,c^2\,d}{e^3}\right )}{e^2}+\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {A\,c^2+2\,B\,b\,c}{e^2}-\frac {2\,B\,c^2\,d}{e^3}\right )}{e}-\frac {B\,b^2+2\,A\,c\,b}{e^2}+\frac {B\,c^2\,d^2}{e^4}\right )}{e}\right )+x^3\,\left (\frac {A\,c^2+2\,B\,b\,c}{3\,e^2}-\frac {2\,B\,c^2\,d}{3\,e^3}\right )-x^2\,\left (\frac {d\,\left (\frac {A\,c^2+2\,B\,b\,c}{e^2}-\frac {2\,B\,c^2\,d}{e^3}\right )}{e}-\frac {B\,b^2+2\,A\,c\,b}{2\,e^2}+\frac {B\,c^2\,d^2}{2\,e^4}\right )+\frac {B\,b^2\,d^3\,e^2-A\,b^2\,d^2\,e^3-2\,B\,b\,c\,d^4\,e+2\,A\,b\,c\,d^3\,e^2+B\,c^2\,d^5-A\,c^2\,d^4\,e}{e\,\left (x\,e^6+d\,e^5\right )}+\frac {\ln \left (d+e\,x\right )\,\left (3\,B\,b^2\,d^2\,e^2-2\,A\,b^2\,d\,e^3-8\,B\,b\,c\,d^3\,e+6\,A\,b\,c\,d^2\,e^2+5\,B\,c^2\,d^4-4\,A\,c^2\,d^3\,e\right )}{e^6}+\frac {B\,c^2\,x^4}{4\,e^2} \]
x*((A*b^2)/e^2 - (d^2*((A*c^2 + 2*B*b*c)/e^2 - (2*B*c^2*d)/e^3))/e^2 + (2* d*((2*d*((A*c^2 + 2*B*b*c)/e^2 - (2*B*c^2*d)/e^3))/e - (B*b^2 + 2*A*b*c)/e ^2 + (B*c^2*d^2)/e^4))/e) + x^3*((A*c^2 + 2*B*b*c)/(3*e^2) - (2*B*c^2*d)/( 3*e^3)) - x^2*((d*((A*c^2 + 2*B*b*c)/e^2 - (2*B*c^2*d)/e^3))/e - (B*b^2 + 2*A*b*c)/(2*e^2) + (B*c^2*d^2)/(2*e^4)) + (B*c^2*d^5 - A*c^2*d^4*e - A*b^2 *d^2*e^3 + B*b^2*d^3*e^2 - 2*B*b*c*d^4*e + 2*A*b*c*d^3*e^2)/(e*(d*e^5 + e^ 6*x)) + (log(d + e*x)*(5*B*c^2*d^4 - 2*A*b^2*d*e^3 - 4*A*c^2*d^3*e + 3*B*b ^2*d^2*e^2 - 8*B*b*c*d^3*e + 6*A*b*c*d^2*e^2))/e^6 + (B*c^2*x^4)/(4*e^2)