Integrand size = 20, antiderivative size = 191 \[ \int \frac {(A+B x) (d+e x)^4}{(a+b x)^3} \, dx=\frac {2 e^2 (b d-a e) (3 b B d+2 A b e-5 a B e) x}{b^5}-\frac {(A b-a B) (b d-a e)^4}{2 b^6 (a+b x)^2}-\frac {(b d-a e)^3 (b B d+4 A b e-5 a B e)}{b^6 (a+b x)}+\frac {e^3 (4 b B d+A b e-5 a B e) (a+b x)^2}{2 b^6}+\frac {B e^4 (a+b x)^3}{3 b^6}+\frac {2 e (b d-a e)^2 (2 b B d+3 A b e-5 a B e) \log (a+b x)}{b^6} \]
2*e^2*(-a*e+b*d)*(2*A*b*e-5*B*a*e+3*B*b*d)*x/b^5-1/2*(A*b-B*a)*(-a*e+b*d)^ 4/b^6/(b*x+a)^2-(-a*e+b*d)^3*(4*A*b*e-5*B*a*e+B*b*d)/b^6/(b*x+a)+1/2*e^3*( A*b*e-5*B*a*e+4*B*b*d)*(b*x+a)^2/b^6+1/3*B*e^4*(b*x+a)^3/b^6+2*e*(-a*e+b*d )^2*(3*A*b*e-5*B*a*e+2*B*b*d)*ln(b*x+a)/b^6
Time = 0.06 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.98 \[ \int \frac {(A+B x) (d+e x)^4}{(a+b x)^3} \, dx=\frac {6 b e^2 \left (6 a^2 B e^2-3 a b e (4 B d+A e)+2 b^2 d (3 B d+2 A e)\right ) x+3 b^2 e^3 (4 b B d+A b e-3 a B e) x^2+2 b^3 B e^4 x^3-\frac {3 (A b-a B) (b d-a e)^4}{(a+b x)^2}-\frac {6 (b d-a e)^3 (b B d+4 A b e-5 a B e)}{a+b x}+12 e (b d-a e)^2 (2 b B d+3 A b e-5 a B e) \log (a+b x)}{6 b^6} \]
(6*b*e^2*(6*a^2*B*e^2 - 3*a*b*e*(4*B*d + A*e) + 2*b^2*d*(3*B*d + 2*A*e))*x + 3*b^2*e^3*(4*b*B*d + A*b*e - 3*a*B*e)*x^2 + 2*b^3*B*e^4*x^3 - (3*(A*b - a*B)*(b*d - a*e)^4)/(a + b*x)^2 - (6*(b*d - a*e)^3*(b*B*d + 4*A*b*e - 5*a *B*e))/(a + b*x) + 12*e*(b*d - a*e)^2*(2*b*B*d + 3*A*b*e - 5*a*B*e)*Log[a + b*x])/(6*b^6)
Time = 0.42 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {86, 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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^4}{(a+b x)^3} \, dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {e^3 (a+b x) (-5 a B e+A b e+4 b B d)}{b^5}+\frac {2 e^2 (b d-a e) (-5 a B e+2 A b e+3 b B d)}{b^5}+\frac {2 e (b d-a e)^2 (-5 a B e+3 A b e+2 b B d)}{b^5 (a+b x)}+\frac {(b d-a e)^3 (-5 a B e+4 A b e+b B d)}{b^5 (a+b x)^2}+\frac {(A b-a B) (b d-a e)^4}{b^5 (a+b x)^3}+\frac {B e^4 (a+b x)^2}{b^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^3 (a+b x)^2 (-5 a B e+A b e+4 b B d)}{2 b^6}-\frac {(b d-a e)^3 (-5 a B e+4 A b e+b B d)}{b^6 (a+b x)}-\frac {(A b-a B) (b d-a e)^4}{2 b^6 (a+b x)^2}+\frac {2 e (b d-a e)^2 \log (a+b x) (-5 a B e+3 A b e+2 b B d)}{b^6}+\frac {2 e^2 x (b d-a e) (-5 a B e+2 A b e+3 b B d)}{b^5}+\frac {B e^4 (a+b x)^3}{3 b^6}\) |
(2*e^2*(b*d - a*e)*(3*b*B*d + 2*A*b*e - 5*a*B*e)*x)/b^5 - ((A*b - a*B)*(b* d - a*e)^4)/(2*b^6*(a + b*x)^2) - ((b*d - a*e)^3*(b*B*d + 4*A*b*e - 5*a*B* e))/(b^6*(a + b*x)) + (e^3*(4*b*B*d + A*b*e - 5*a*B*e)*(a + b*x)^2)/(2*b^6 ) + (B*e^4*(a + b*x)^3)/(3*b^6) + (2*e*(b*d - a*e)^2*(2*b*B*d + 3*A*b*e - 5*a*B*e)*Log[a + b*x])/b^6
3.12.34.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Leaf count of result is larger than twice the leaf count of optimal. \(415\) vs. \(2(185)=370\).
Time = 0.72 (sec) , antiderivative size = 416, normalized size of antiderivative = 2.18
| method | result | size |
| norman | \(\frac {\frac {\left (12 A \,a^{3} b \,e^{4}-24 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e -20 B \,a^{4} e^{4}+48 B \,a^{3} b d \,e^{3}-36 B \,a^{2} b^{2} d^{2} e^{2}+8 B a \,b^{3} d^{3} e -B \,b^{4} d^{4}\right ) x}{b^{5}}+\frac {18 A \,a^{4} b \,e^{4}-36 A \,a^{3} b^{2} d \,e^{3}+18 A \,a^{2} b^{3} d^{2} e^{2}-4 A a \,b^{4} d^{3} e -A \,b^{5} d^{4}-30 B \,a^{5} e^{4}+72 B \,a^{4} b d \,e^{3}-54 B \,a^{3} b^{2} d^{2} e^{2}+12 B \,a^{2} b^{3} d^{3} e -B a \,b^{4} d^{4}}{2 b^{6}}+\frac {B \,e^{4} x^{5}}{3 b}-\frac {2 e^{2} \left (3 A a b \,e^{2}-6 A \,b^{2} d e -5 B \,a^{2} e^{2}+12 B a b d e -9 b^{2} B \,d^{2}\right ) x^{3}}{3 b^{3}}+\frac {e^{3} \left (3 A b e -5 B a e +12 B b d \right ) x^{4}}{6 b^{2}}}{\left (b x +a \right )^{2}}+\frac {2 e \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e -5 B \,a^{3} e^{3}+12 B \,a^{2} b d \,e^{2}-9 B a \,b^{2} d^{2} e +2 b^{3} B \,d^{3}\right ) \ln \left (b x +a \right )}{b^{6}}\) | \(416\) |
| default | \(-\frac {e^{2} \left (-\frac {1}{3} b^{2} B \,x^{3} e^{2}-\frac {1}{2} A \,b^{2} e^{2} x^{2}+\frac {3}{2} B a b \,e^{2} x^{2}-2 B \,b^{2} d e \,x^{2}+3 A a b \,e^{2} x -4 A \,b^{2} d e x -6 B \,a^{2} e^{2} x +12 B a b d e x -6 b^{2} B \,d^{2} x \right )}{b^{5}}+\frac {2 e \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e -5 B \,a^{3} e^{3}+12 B \,a^{2} b d \,e^{2}-9 B a \,b^{2} d^{2} e +2 b^{3} B \,d^{3}\right ) \ln \left (b x +a \right )}{b^{6}}-\frac {A \,a^{4} b \,e^{4}-4 A \,a^{3} b^{2} d \,e^{3}+6 A \,a^{2} b^{3} d^{2} e^{2}-4 A a \,b^{4} d^{3} e +A \,b^{5} d^{4}-B \,a^{5} e^{4}+4 B \,a^{4} b d \,e^{3}-6 B \,a^{3} b^{2} d^{2} e^{2}+4 B \,a^{2} b^{3} d^{3} e -B a \,b^{4} d^{4}}{2 b^{6} \left (b x +a \right )^{2}}-\frac {-4 A \,a^{3} b \,e^{4}+12 A \,a^{2} b^{2} d \,e^{3}-12 A a \,b^{3} d^{2} e^{2}+4 A \,b^{4} d^{3} e +5 B \,a^{4} e^{4}-16 B \,a^{3} b d \,e^{3}+18 B \,a^{2} b^{2} d^{2} e^{2}-8 B a \,b^{3} d^{3} e +B \,b^{4} d^{4}}{b^{6} \left (b x +a \right )}\) | \(427\) |
| risch | \(\frac {e^{4} B \,x^{3}}{3 b^{3}}+\frac {e^{4} A \,x^{2}}{2 b^{3}}-\frac {3 e^{4} B a \,x^{2}}{2 b^{4}}+\frac {2 e^{3} B d \,x^{2}}{b^{3}}-\frac {3 e^{4} A a x}{b^{4}}+\frac {4 e^{3} A d x}{b^{3}}+\frac {6 e^{4} B \,a^{2} x}{b^{5}}-\frac {12 e^{3} B a d x}{b^{4}}+\frac {6 e^{2} B \,d^{2} x}{b^{3}}+\frac {\left (4 A \,a^{3} b \,e^{4}-12 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e -5 B \,a^{4} e^{4}+16 B \,a^{3} b d \,e^{3}-18 B \,a^{2} b^{2} d^{2} e^{2}+8 B a \,b^{3} d^{3} e -B \,b^{4} d^{4}\right ) x +\frac {7 A \,a^{4} b \,e^{4}-20 A \,a^{3} b^{2} d \,e^{3}+18 A \,a^{2} b^{3} d^{2} e^{2}-4 A a \,b^{4} d^{3} e -A \,b^{5} d^{4}-9 B \,a^{5} e^{4}+28 B \,a^{4} b d \,e^{3}-30 B \,a^{3} b^{2} d^{2} e^{2}+12 B \,a^{2} b^{3} d^{3} e -B a \,b^{4} d^{4}}{2 b}}{b^{5} \left (b x +a \right )^{2}}+\frac {6 e^{4} \ln \left (b x +a \right ) A \,a^{2}}{b^{5}}-\frac {12 e^{3} \ln \left (b x +a \right ) A a d}{b^{4}}+\frac {6 e^{2} \ln \left (b x +a \right ) A \,d^{2}}{b^{3}}-\frac {10 e^{4} \ln \left (b x +a \right ) B \,a^{3}}{b^{6}}+\frac {24 e^{3} \ln \left (b x +a \right ) B \,a^{2} d}{b^{5}}-\frac {18 e^{2} \ln \left (b x +a \right ) B a \,d^{2}}{b^{4}}+\frac {4 e \ln \left (b x +a \right ) B \,d^{3}}{b^{3}}\) | \(475\) |
| parallelrisch | \(\frac {36 A \ln \left (b x +a \right ) x^{2} a^{2} b^{3} e^{4}+36 A \ln \left (b x +a \right ) x^{2} b^{5} d^{2} e^{2}-60 B \ln \left (b x +a \right ) x^{2} a^{3} b^{2} e^{4}+72 A \ln \left (b x +a \right ) x \,a^{3} b^{2} e^{4}-120 B \ln \left (b x +a \right ) x \,a^{4} b \,e^{4}+24 B \ln \left (b x +a \right ) x^{2} b^{5} d^{3} e +36 A \ln \left (b x +a \right ) a^{4} b \,e^{4}-5 B \,x^{4} a \,b^{4} e^{4}+12 B \,x^{4} b^{5} d \,e^{3}-12 A \,x^{3} a \,b^{4} e^{4}+24 A \,x^{3} b^{5} d \,e^{3}+20 B \,x^{3} a^{2} b^{3} e^{4}+36 B \,x^{3} b^{5} d^{2} e^{2}+72 A x \,a^{3} b^{2} e^{4}-24 A x \,b^{5} d^{3} e -120 B x \,a^{4} b \,e^{4}-216 B x \,a^{2} b^{3} d^{2} e^{2}-72 A \ln \left (b x +a \right ) a^{3} b^{2} d \,e^{3}+36 A \ln \left (b x +a \right ) a^{2} b^{3} d^{2} e^{2}+144 B \ln \left (b x +a \right ) a^{4} b d \,e^{3}-108 B \ln \left (b x +a \right ) a^{3} b^{2} d^{2} e^{2}+3 A \,x^{4} b^{5} e^{4}-6 B x \,b^{5} d^{4}-60 B \ln \left (b x +a \right ) a^{5} e^{4}-12 A a \,b^{4} d^{3} e +216 B \,a^{4} b d \,e^{3}-162 B \,a^{3} b^{2} d^{2} e^{2}+36 B \,a^{2} b^{3} d^{3} e -108 A \,a^{3} b^{2} d \,e^{3}+54 A \,a^{2} b^{3} d^{2} e^{2}-3 A \,b^{5} d^{4}+24 B \ln \left (b x +a \right ) a^{2} b^{3} d^{3} e +48 B x a \,b^{4} d^{3} e -144 A \ln \left (b x +a \right ) x \,a^{2} b^{3} d \,e^{3}+72 A \ln \left (b x +a \right ) x a \,b^{4} d^{2} e^{2}+288 B \ln \left (b x +a \right ) x \,a^{3} b^{2} d \,e^{3}-216 B \ln \left (b x +a \right ) x \,a^{2} b^{3} d^{2} e^{2}+48 B \ln \left (b x +a \right ) x a \,b^{4} d^{3} e -72 A \ln \left (b x +a \right ) x^{2} a \,b^{4} d \,e^{3}+144 B \ln \left (b x +a \right ) x^{2} a^{2} b^{3} d \,e^{3}-108 B \ln \left (b x +a \right ) x^{2} a \,b^{4} d^{2} e^{2}-48 B \,x^{3} a \,b^{4} d \,e^{3}-144 A x \,a^{2} b^{3} d \,e^{3}+72 A x a \,b^{4} d^{2} e^{2}+288 B x \,a^{3} b^{2} d \,e^{3}-90 B \,a^{5} e^{4}-3 B a \,b^{4} d^{4}+2 B \,x^{5} e^{4} b^{5}+54 A \,a^{4} b \,e^{4}}{6 b^{6} \left (b x +a \right )^{2}}\) | \(775\) |
((12*A*a^3*b*e^4-24*A*a^2*b^2*d*e^3+12*A*a*b^3*d^2*e^2-4*A*b^4*d^3*e-20*B* a^4*e^4+48*B*a^3*b*d*e^3-36*B*a^2*b^2*d^2*e^2+8*B*a*b^3*d^3*e-B*b^4*d^4)/b ^5*x+1/2*(18*A*a^4*b*e^4-36*A*a^3*b^2*d*e^3+18*A*a^2*b^3*d^2*e^2-4*A*a*b^4 *d^3*e-A*b^5*d^4-30*B*a^5*e^4+72*B*a^4*b*d*e^3-54*B*a^3*b^2*d^2*e^2+12*B*a ^2*b^3*d^3*e-B*a*b^4*d^4)/b^6+1/3*B/b*e^4*x^5-2/3*e^2*(3*A*a*b*e^2-6*A*b^2 *d*e-5*B*a^2*e^2+12*B*a*b*d*e-9*B*b^2*d^2)/b^3*x^3+1/6*e^3*(3*A*b*e-5*B*a* e+12*B*b*d)/b^2*x^4)/(b*x+a)^2+2/b^6*e*(3*A*a^2*b*e^3-6*A*a*b^2*d*e^2+3*A* b^3*d^2*e-5*B*a^3*e^3+12*B*a^2*b*d*e^2-9*B*a*b^2*d^2*e+2*B*b^3*d^3)*ln(b*x +a)
Leaf count of result is larger than twice the leaf count of optimal. 668 vs. \(2 (185) = 370\).
Time = 0.23 (sec) , antiderivative size = 668, normalized size of antiderivative = 3.50 \[ \int \frac {(A+B x) (d+e x)^4}{(a+b x)^3} \, dx=\frac {2 \, B b^{5} e^{4} x^{5} - 3 \, {\left (B a b^{4} + A b^{5}\right )} d^{4} + 12 \, {\left (3 \, B a^{2} b^{3} - A a b^{4}\right )} d^{3} e - 18 \, {\left (5 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} d^{2} e^{2} + 12 \, {\left (7 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} d e^{3} - 3 \, {\left (9 \, B a^{5} - 7 \, A a^{4} b\right )} e^{4} + {\left (12 \, B b^{5} d e^{3} - {\left (5 \, B a b^{4} - 3 \, A b^{5}\right )} e^{4}\right )} x^{4} + 4 \, {\left (9 \, B b^{5} d^{2} e^{2} - 6 \, {\left (2 \, B a b^{4} - A b^{5}\right )} d e^{3} + {\left (5 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} e^{4}\right )} x^{3} + 3 \, {\left (24 \, B a b^{4} d^{2} e^{2} - 4 \, {\left (11 \, B a^{2} b^{3} - 4 \, A a b^{4}\right )} d e^{3} + {\left (21 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} e^{4}\right )} x^{2} - 6 \, {\left (B b^{5} d^{4} - 4 \, {\left (2 \, B a b^{4} - A b^{5}\right )} d^{3} e + 12 \, {\left (B a^{2} b^{3} - A a b^{4}\right )} d^{2} e^{2} - 4 \, {\left (B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} d e^{3} - {\left (B a^{4} b + A a^{3} b^{2}\right )} e^{4}\right )} x + 12 \, {\left (2 \, B a^{2} b^{3} d^{3} e - 3 \, {\left (3 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} d^{2} e^{2} + 6 \, {\left (2 \, B a^{4} b - A a^{3} b^{2}\right )} d e^{3} - {\left (5 \, B a^{5} - 3 \, A a^{4} b\right )} e^{4} + {\left (2 \, B b^{5} d^{3} e - 3 \, {\left (3 \, B a b^{4} - A b^{5}\right )} d^{2} e^{2} + 6 \, {\left (2 \, B a^{2} b^{3} - A a b^{4}\right )} d e^{3} - {\left (5 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} e^{4}\right )} x^{2} + 2 \, {\left (2 \, B a b^{4} d^{3} e - 3 \, {\left (3 \, B a^{2} b^{3} - A a b^{4}\right )} d^{2} e^{2} + 6 \, {\left (2 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} d e^{3} - {\left (5 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} e^{4}\right )} x\right )} \log \left (b x + a\right )}{6 \, {\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} \]
1/6*(2*B*b^5*e^4*x^5 - 3*(B*a*b^4 + A*b^5)*d^4 + 12*(3*B*a^2*b^3 - A*a*b^4 )*d^3*e - 18*(5*B*a^3*b^2 - 3*A*a^2*b^3)*d^2*e^2 + 12*(7*B*a^4*b - 5*A*a^3 *b^2)*d*e^3 - 3*(9*B*a^5 - 7*A*a^4*b)*e^4 + (12*B*b^5*d*e^3 - (5*B*a*b^4 - 3*A*b^5)*e^4)*x^4 + 4*(9*B*b^5*d^2*e^2 - 6*(2*B*a*b^4 - A*b^5)*d*e^3 + (5 *B*a^2*b^3 - 3*A*a*b^4)*e^4)*x^3 + 3*(24*B*a*b^4*d^2*e^2 - 4*(11*B*a^2*b^3 - 4*A*a*b^4)*d*e^3 + (21*B*a^3*b^2 - 11*A*a^2*b^3)*e^4)*x^2 - 6*(B*b^5*d^ 4 - 4*(2*B*a*b^4 - A*b^5)*d^3*e + 12*(B*a^2*b^3 - A*a*b^4)*d^2*e^2 - 4*(B* a^3*b^2 - 2*A*a^2*b^3)*d*e^3 - (B*a^4*b + A*a^3*b^2)*e^4)*x + 12*(2*B*a^2* b^3*d^3*e - 3*(3*B*a^3*b^2 - A*a^2*b^3)*d^2*e^2 + 6*(2*B*a^4*b - A*a^3*b^2 )*d*e^3 - (5*B*a^5 - 3*A*a^4*b)*e^4 + (2*B*b^5*d^3*e - 3*(3*B*a*b^4 - A*b^ 5)*d^2*e^2 + 6*(2*B*a^2*b^3 - A*a*b^4)*d*e^3 - (5*B*a^3*b^2 - 3*A*a^2*b^3) *e^4)*x^2 + 2*(2*B*a*b^4*d^3*e - 3*(3*B*a^2*b^3 - A*a*b^4)*d^2*e^2 + 6*(2* B*a^3*b^2 - A*a^2*b^3)*d*e^3 - (5*B*a^4*b - 3*A*a^3*b^2)*e^4)*x)*log(b*x + a))/(b^8*x^2 + 2*a*b^7*x + a^2*b^6)
Leaf count of result is larger than twice the leaf count of optimal. 444 vs. \(2 (194) = 388\).
Time = 3.31 (sec) , antiderivative size = 444, normalized size of antiderivative = 2.32 \[ \int \frac {(A+B x) (d+e x)^4}{(a+b x)^3} \, dx=\frac {B e^{4} x^{3}}{3 b^{3}} + x^{2} \left (\frac {A e^{4}}{2 b^{3}} - \frac {3 B a e^{4}}{2 b^{4}} + \frac {2 B d e^{3}}{b^{3}}\right ) + x \left (- \frac {3 A a e^{4}}{b^{4}} + \frac {4 A d e^{3}}{b^{3}} + \frac {6 B a^{2} e^{4}}{b^{5}} - \frac {12 B a d e^{3}}{b^{4}} + \frac {6 B d^{2} e^{2}}{b^{3}}\right ) + \frac {7 A a^{4} b e^{4} - 20 A a^{3} b^{2} d e^{3} + 18 A a^{2} b^{3} d^{2} e^{2} - 4 A a b^{4} d^{3} e - A b^{5} d^{4} - 9 B a^{5} e^{4} + 28 B a^{4} b d e^{3} - 30 B a^{3} b^{2} d^{2} e^{2} + 12 B a^{2} b^{3} d^{3} e - B a b^{4} d^{4} + x \left (8 A a^{3} b^{2} e^{4} - 24 A a^{2} b^{3} d e^{3} + 24 A a b^{4} d^{2} e^{2} - 8 A b^{5} d^{3} e - 10 B a^{4} b e^{4} + 32 B a^{3} b^{2} d e^{3} - 36 B a^{2} b^{3} d^{2} e^{2} + 16 B a b^{4} d^{3} e - 2 B b^{5} d^{4}\right )}{2 a^{2} b^{6} + 4 a b^{7} x + 2 b^{8} x^{2}} - \frac {2 e \left (a e - b d\right )^{2} \left (- 3 A b e + 5 B a e - 2 B b d\right ) \log {\left (a + b x \right )}}{b^{6}} \]
B*e**4*x**3/(3*b**3) + x**2*(A*e**4/(2*b**3) - 3*B*a*e**4/(2*b**4) + 2*B*d *e**3/b**3) + x*(-3*A*a*e**4/b**4 + 4*A*d*e**3/b**3 + 6*B*a**2*e**4/b**5 - 12*B*a*d*e**3/b**4 + 6*B*d**2*e**2/b**3) + (7*A*a**4*b*e**4 - 20*A*a**3*b **2*d*e**3 + 18*A*a**2*b**3*d**2*e**2 - 4*A*a*b**4*d**3*e - A*b**5*d**4 - 9*B*a**5*e**4 + 28*B*a**4*b*d*e**3 - 30*B*a**3*b**2*d**2*e**2 + 12*B*a**2* b**3*d**3*e - B*a*b**4*d**4 + x*(8*A*a**3*b**2*e**4 - 24*A*a**2*b**3*d*e** 3 + 24*A*a*b**4*d**2*e**2 - 8*A*b**5*d**3*e - 10*B*a**4*b*e**4 + 32*B*a**3 *b**2*d*e**3 - 36*B*a**2*b**3*d**2*e**2 + 16*B*a*b**4*d**3*e - 2*B*b**5*d* *4))/(2*a**2*b**6 + 4*a*b**7*x + 2*b**8*x**2) - 2*e*(a*e - b*d)**2*(-3*A*b *e + 5*B*a*e - 2*B*b*d)*log(a + b*x)/b**6
Leaf count of result is larger than twice the leaf count of optimal. 424 vs. \(2 (185) = 370\).
Time = 0.20 (sec) , antiderivative size = 424, normalized size of antiderivative = 2.22 \[ \int \frac {(A+B x) (d+e x)^4}{(a+b x)^3} \, dx=-\frac {{\left (B a b^{4} + A b^{5}\right )} d^{4} - 4 \, {\left (3 \, B a^{2} b^{3} - A a b^{4}\right )} d^{3} e + 6 \, {\left (5 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} d^{2} e^{2} - 4 \, {\left (7 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} d e^{3} + {\left (9 \, B a^{5} - 7 \, A a^{4} b\right )} e^{4} + 2 \, {\left (B b^{5} d^{4} - 4 \, {\left (2 \, B a b^{4} - A b^{5}\right )} d^{3} e + 6 \, {\left (3 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} d^{2} e^{2} - 4 \, {\left (4 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} d e^{3} + {\left (5 \, B a^{4} b - 4 \, A a^{3} b^{2}\right )} e^{4}\right )} x}{2 \, {\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} + \frac {2 \, B b^{2} e^{4} x^{3} + 3 \, {\left (4 \, B b^{2} d e^{3} - {\left (3 \, B a b - A b^{2}\right )} e^{4}\right )} x^{2} + 6 \, {\left (6 \, B b^{2} d^{2} e^{2} - 4 \, {\left (3 \, B a b - A b^{2}\right )} d e^{3} + 3 \, {\left (2 \, B a^{2} - A a b\right )} e^{4}\right )} x}{6 \, b^{5}} + \frac {2 \, {\left (2 \, B b^{3} d^{3} e - 3 \, {\left (3 \, B a b^{2} - A b^{3}\right )} d^{2} e^{2} + 6 \, {\left (2 \, B a^{2} b - A a b^{2}\right )} d e^{3} - {\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} e^{4}\right )} \log \left (b x + a\right )}{b^{6}} \]
-1/2*((B*a*b^4 + A*b^5)*d^4 - 4*(3*B*a^2*b^3 - A*a*b^4)*d^3*e + 6*(5*B*a^3 *b^2 - 3*A*a^2*b^3)*d^2*e^2 - 4*(7*B*a^4*b - 5*A*a^3*b^2)*d*e^3 + (9*B*a^5 - 7*A*a^4*b)*e^4 + 2*(B*b^5*d^4 - 4*(2*B*a*b^4 - A*b^5)*d^3*e + 6*(3*B*a^ 2*b^3 - 2*A*a*b^4)*d^2*e^2 - 4*(4*B*a^3*b^2 - 3*A*a^2*b^3)*d*e^3 + (5*B*a^ 4*b - 4*A*a^3*b^2)*e^4)*x)/(b^8*x^2 + 2*a*b^7*x + a^2*b^6) + 1/6*(2*B*b^2* e^4*x^3 + 3*(4*B*b^2*d*e^3 - (3*B*a*b - A*b^2)*e^4)*x^2 + 6*(6*B*b^2*d^2*e ^2 - 4*(3*B*a*b - A*b^2)*d*e^3 + 3*(2*B*a^2 - A*a*b)*e^4)*x)/b^5 + 2*(2*B* b^3*d^3*e - 3*(3*B*a*b^2 - A*b^3)*d^2*e^2 + 6*(2*B*a^2*b - A*a*b^2)*d*e^3 - (5*B*a^3 - 3*A*a^2*b)*e^4)*log(b*x + a)/b^6
Leaf count of result is larger than twice the leaf count of optimal. 442 vs. \(2 (185) = 370\).
Time = 0.28 (sec) , antiderivative size = 442, normalized size of antiderivative = 2.31 \[ \int \frac {(A+B x) (d+e x)^4}{(a+b x)^3} \, dx=\frac {2 \, {\left (2 \, B b^{3} d^{3} e - 9 \, B a b^{2} d^{2} e^{2} + 3 \, A b^{3} d^{2} e^{2} + 12 \, B a^{2} b d e^{3} - 6 \, A a b^{2} d e^{3} - 5 \, B a^{3} e^{4} + 3 \, A a^{2} b e^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} - \frac {B a b^{4} d^{4} + A b^{5} d^{4} - 12 \, B a^{2} b^{3} d^{3} e + 4 \, A a b^{4} d^{3} e + 30 \, B a^{3} b^{2} d^{2} e^{2} - 18 \, A a^{2} b^{3} d^{2} e^{2} - 28 \, B a^{4} b d e^{3} + 20 \, A a^{3} b^{2} d e^{3} + 9 \, B a^{5} e^{4} - 7 \, A a^{4} b e^{4} + 2 \, {\left (B b^{5} d^{4} - 8 \, B a b^{4} d^{3} e + 4 \, A b^{5} d^{3} e + 18 \, B a^{2} b^{3} d^{2} e^{2} - 12 \, A a b^{4} d^{2} e^{2} - 16 \, B a^{3} b^{2} d e^{3} + 12 \, A a^{2} b^{3} d e^{3} + 5 \, B a^{4} b e^{4} - 4 \, A a^{3} b^{2} e^{4}\right )} x}{2 \, {\left (b x + a\right )}^{2} b^{6}} + \frac {2 \, B b^{6} e^{4} x^{3} + 12 \, B b^{6} d e^{3} x^{2} - 9 \, B a b^{5} e^{4} x^{2} + 3 \, A b^{6} e^{4} x^{2} + 36 \, B b^{6} d^{2} e^{2} x - 72 \, B a b^{5} d e^{3} x + 24 \, A b^{6} d e^{3} x + 36 \, B a^{2} b^{4} e^{4} x - 18 \, A a b^{5} e^{4} x}{6 \, b^{9}} \]
2*(2*B*b^3*d^3*e - 9*B*a*b^2*d^2*e^2 + 3*A*b^3*d^2*e^2 + 12*B*a^2*b*d*e^3 - 6*A*a*b^2*d*e^3 - 5*B*a^3*e^4 + 3*A*a^2*b*e^4)*log(abs(b*x + a))/b^6 - 1 /2*(B*a*b^4*d^4 + A*b^5*d^4 - 12*B*a^2*b^3*d^3*e + 4*A*a*b^4*d^3*e + 30*B* a^3*b^2*d^2*e^2 - 18*A*a^2*b^3*d^2*e^2 - 28*B*a^4*b*d*e^3 + 20*A*a^3*b^2*d *e^3 + 9*B*a^5*e^4 - 7*A*a^4*b*e^4 + 2*(B*b^5*d^4 - 8*B*a*b^4*d^3*e + 4*A* b^5*d^3*e + 18*B*a^2*b^3*d^2*e^2 - 12*A*a*b^4*d^2*e^2 - 16*B*a^3*b^2*d*e^3 + 12*A*a^2*b^3*d*e^3 + 5*B*a^4*b*e^4 - 4*A*a^3*b^2*e^4)*x)/((b*x + a)^2*b ^6) + 1/6*(2*B*b^6*e^4*x^3 + 12*B*b^6*d*e^3*x^2 - 9*B*a*b^5*e^4*x^2 + 3*A* b^6*e^4*x^2 + 36*B*b^6*d^2*e^2*x - 72*B*a*b^5*d*e^3*x + 24*A*b^6*d*e^3*x + 36*B*a^2*b^4*e^4*x - 18*A*a*b^5*e^4*x)/b^9
Time = 0.15 (sec) , antiderivative size = 451, normalized size of antiderivative = 2.36 \[ \int \frac {(A+B x) (d+e x)^4}{(a+b x)^3} \, dx=x^2\,\left (\frac {A\,e^4+4\,B\,d\,e^3}{2\,b^3}-\frac {3\,B\,a\,e^4}{2\,b^4}\right )-\frac {\frac {9\,B\,a^5\,e^4-28\,B\,a^4\,b\,d\,e^3-7\,A\,a^4\,b\,e^4+30\,B\,a^3\,b^2\,d^2\,e^2+20\,A\,a^3\,b^2\,d\,e^3-12\,B\,a^2\,b^3\,d^3\,e-18\,A\,a^2\,b^3\,d^2\,e^2+B\,a\,b^4\,d^4+4\,A\,a\,b^4\,d^3\,e+A\,b^5\,d^4}{2\,b}+x\,\left (5\,B\,a^4\,e^4-16\,B\,a^3\,b\,d\,e^3-4\,A\,a^3\,b\,e^4+18\,B\,a^2\,b^2\,d^2\,e^2+12\,A\,a^2\,b^2\,d\,e^3-8\,B\,a\,b^3\,d^3\,e-12\,A\,a\,b^3\,d^2\,e^2+B\,b^4\,d^4+4\,A\,b^4\,d^3\,e\right )}{a^2\,b^5+2\,a\,b^6\,x+b^7\,x^2}-x\,\left (\frac {3\,a\,\left (\frac {A\,e^4+4\,B\,d\,e^3}{b^3}-\frac {3\,B\,a\,e^4}{b^4}\right )}{b}-\frac {2\,d\,e^2\,\left (2\,A\,e+3\,B\,d\right )}{b^3}+\frac {3\,B\,a^2\,e^4}{b^5}\right )+\frac {\ln \left (a+b\,x\right )\,\left (-10\,B\,a^3\,e^4+24\,B\,a^2\,b\,d\,e^3+6\,A\,a^2\,b\,e^4-18\,B\,a\,b^2\,d^2\,e^2-12\,A\,a\,b^2\,d\,e^3+4\,B\,b^3\,d^3\,e+6\,A\,b^3\,d^2\,e^2\right )}{b^6}+\frac {B\,e^4\,x^3}{3\,b^3} \]
x^2*((A*e^4 + 4*B*d*e^3)/(2*b^3) - (3*B*a*e^4)/(2*b^4)) - ((A*b^5*d^4 + 9* B*a^5*e^4 - 7*A*a^4*b*e^4 + B*a*b^4*d^4 + 20*A*a^3*b^2*d*e^3 - 12*B*a^2*b^ 3*d^3*e - 18*A*a^2*b^3*d^2*e^2 + 30*B*a^3*b^2*d^2*e^2 + 4*A*a*b^4*d^3*e - 28*B*a^4*b*d*e^3)/(2*b) + x*(5*B*a^4*e^4 + B*b^4*d^4 - 4*A*a^3*b*e^4 + 4*A *b^4*d^3*e - 12*A*a*b^3*d^2*e^2 + 12*A*a^2*b^2*d*e^3 + 18*B*a^2*b^2*d^2*e^ 2 - 8*B*a*b^3*d^3*e - 16*B*a^3*b*d*e^3))/(a^2*b^5 + b^7*x^2 + 2*a*b^6*x) - x*((3*a*((A*e^4 + 4*B*d*e^3)/b^3 - (3*B*a*e^4)/b^4))/b - (2*d*e^2*(2*A*e + 3*B*d))/b^3 + (3*B*a^2*e^4)/b^5) + (log(a + b*x)*(6*A*a^2*b*e^4 - 10*B*a ^3*e^4 + 4*B*b^3*d^3*e + 6*A*b^3*d^2*e^2 - 18*B*a*b^2*d^2*e^2 - 12*A*a*b^2 *d*e^3 + 24*B*a^2*b*d*e^3))/b^6 + (B*e^4*x^3)/(3*b^3)