Integrand size = 31, antiderivative size = 186 \[ \int \frac {(A+B x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {e^3 (4 b B d+A b e-4 a B e) x}{b^5}+\frac {B e^4 x^2}{2 b^4}-\frac {(A b-a B) (b d-a e)^4}{3 b^6 (a+b x)^3}-\frac {(b d-a e)^3 (b B d+4 A b e-5 a B e)}{2 b^6 (a+b x)^2}-\frac {2 e (b d-a e)^2 (2 b B d+3 A b e-5 a B e)}{b^6 (a+b x)}+\frac {2 e^2 (b d-a e) (3 b B d+2 A b e-5 a B e) \log (a+b x)}{b^6} \]
e^3*(A*b*e-4*B*a*e+4*B*b*d)*x/b^5+1/2*B*e^4*x^2/b^4-1/3*(A*b-B*a)*(-a*e+b* d)^4/b^6/(b*x+a)^3-1/2*(-a*e+b*d)^3*(4*A*b*e-5*B*a*e+B*b*d)/b^6/(b*x+a)^2- 2*e*(-a*e+b*d)^2*(3*A*b*e-5*B*a*e+2*B*b*d)/b^6/(b*x+a)+2*e^2*(-a*e+b*d)*(2 *A*b*e-5*B*a*e+3*B*b*d)*ln(b*x+a)/b^6
Time = 0.12 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.95 \[ \int \frac {(A+B x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {-2 A b \left (13 a^4 e^4+a^3 b e^3 (-22 d+27 e x)+3 a^2 b^2 e^2 \left (2 d^2-18 d e x+3 e^2 x^2\right )+a b^3 e \left (2 d^3+18 d^2 e x-36 d e^2 x^2-9 e^3 x^3\right )+b^4 \left (d^4+6 d^3 e x+18 d^2 e^2 x^2-3 e^4 x^4\right )\right )+B \left (47 a^5 e^4+a^4 b e^3 (-104 d+81 e x)-3 a^3 b^2 e^2 \left (-22 d^2+72 d e x+3 e^2 x^2\right )-a^2 b^3 e \left (8 d^3-162 d^2 e x+72 d e^2 x^2+63 e^3 x^3\right )+3 b^5 x \left (-d^4-8 d^3 e x+8 d e^3 x^3+e^4 x^4\right )-a b^4 \left (d^4+24 d^3 e x-108 d^2 e^2 x^2-72 d e^3 x^3+15 e^4 x^4\right )\right )+12 e^2 (b d-a e) (3 b B d+2 A b e-5 a B e) (a+b x)^3 \log (a+b x)}{6 b^6 (a+b x)^3} \]
(-2*A*b*(13*a^4*e^4 + a^3*b*e^3*(-22*d + 27*e*x) + 3*a^2*b^2*e^2*(2*d^2 - 18*d*e*x + 3*e^2*x^2) + a*b^3*e*(2*d^3 + 18*d^2*e*x - 36*d*e^2*x^2 - 9*e^3 *x^3) + b^4*(d^4 + 6*d^3*e*x + 18*d^2*e^2*x^2 - 3*e^4*x^4)) + B*(47*a^5*e^ 4 + a^4*b*e^3*(-104*d + 81*e*x) - 3*a^3*b^2*e^2*(-22*d^2 + 72*d*e*x + 3*e^ 2*x^2) - a^2*b^3*e*(8*d^3 - 162*d^2*e*x + 72*d*e^2*x^2 + 63*e^3*x^3) + 3*b ^5*x*(-d^4 - 8*d^3*e*x + 8*d*e^3*x^3 + e^4*x^4) - a*b^4*(d^4 + 24*d^3*e*x - 108*d^2*e^2*x^2 - 72*d*e^3*x^3 + 15*e^4*x^4)) + 12*e^2*(b*d - a*e)*(3*b* B*d + 2*A*b*e - 5*a*B*e)*(a + b*x)^3*Log[a + b*x])/(6*b^6*(a + b*x)^3)
Time = 0.46 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1184, 27, 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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle b^4 \int \frac {(A+B x) (d+e x)^4}{b^4 (a+b x)^4}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(A+B x) (d+e x)^4}{(a+b x)^4}dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {e^3 (-4 a B e+A b e+4 b B d)}{b^5}+\frac {2 e^2 (b d-a e) (-5 a B e+2 A b e+3 b B d)}{b^5 (a+b x)}+\frac {2 e (b d-a e)^2 (-5 a B e+3 A b e+2 b B d)}{b^5 (a+b x)^2}+\frac {(b d-a e)^3 (-5 a B e+4 A b e+b B d)}{b^5 (a+b x)^3}+\frac {(A b-a B) (b d-a e)^4}{b^5 (a+b x)^4}+\frac {B e^4 x}{b^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 e^2 (b d-a e) \log (a+b x) (-5 a B e+2 A b e+3 b B d)}{b^6}-\frac {2 e (b d-a e)^2 (-5 a B e+3 A b e+2 b B d)}{b^6 (a+b x)}-\frac {(b d-a e)^3 (-5 a B e+4 A b e+b B d)}{2 b^6 (a+b x)^2}-\frac {(A b-a B) (b d-a e)^4}{3 b^6 (a+b x)^3}+\frac {e^3 x (-4 a B e+A b e+4 b B d)}{b^5}+\frac {B e^4 x^2}{2 b^4}\) |
(e^3*(4*b*B*d + A*b*e - 4*a*B*e)*x)/b^5 + (B*e^4*x^2)/(2*b^4) - ((A*b - a* B)*(b*d - a*e)^4)/(3*b^6*(a + b*x)^3) - ((b*d - a*e)^3*(b*B*d + 4*A*b*e - 5*a*B*e))/(2*b^6*(a + b*x)^2) - (2*e*(b*d - a*e)^2*(2*b*B*d + 3*A*b*e - 5* a*B*e))/(b^6*(a + b*x)) + (2*e^2*(b*d - a*e)*(3*b*B*d + 2*A*b*e - 5*a*B*e) *Log[a + b*x])/b^6
3.18.1.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]))))
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(f + g*x )^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(415\) vs. \(2(180)=360\).
Time = 0.24 (sec) , antiderivative size = 416, normalized size of antiderivative = 2.24
| method | result | size |
| default | \(\frac {e^{3} \left (\frac {1}{2} B b e \,x^{2}+A b e x -4 B a e x +4 B b d x \right )}{b^{5}}-\frac {A \,a^{4} b \,e^{4}-4 A \,a^{3} b^{2} d \,e^{3}+6 A \,a^{2} b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e a +A \,b^{5} d^{4}-B \,a^{5} e^{4}+4 B \,a^{4} b d \,e^{3}-6 B \,a^{3} b^{2} d^{2} e^{2}+4 B \,a^{2} b^{3} d^{3} e -B \,b^{4} d^{4} a}{3 b^{6} \left (b x +a \right )^{3}}-\frac {2 e^{2} \left (2 A a b \,e^{2}-2 A \,b^{2} d e -5 a^{2} B \,e^{2}+8 B a b d e -3 B \,b^{2} d^{2}\right ) \ln \left (b x +a \right )}{b^{6}}-\frac {2 e \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e -5 B \,e^{3} a^{3}+12 B \,a^{2} b d \,e^{2}-9 B a \,b^{2} d^{2} e +2 B \,b^{3} d^{3}\right )}{b^{6} \left (b x +a \right )}-\frac {-4 A \,a^{3} b \,e^{4}+12 A \,a^{2} b^{2} d \,e^{3}-12 A a \,b^{3} d^{2} e^{2}+4 A \,b^{4} d^{3} e +5 B \,a^{4} e^{4}-16 B \,a^{3} b d \,e^{3}+18 B \,a^{2} b^{2} d^{2} e^{2}-8 B a \,b^{3} d^{3} e +b^{4} B \,d^{4}}{2 b^{6} \left (b x +a \right )^{2}}\) | \(416\) |
| norman | \(\frac {-\frac {44 A \,a^{4} b \,e^{4}-44 A \,a^{3} b^{2} d \,e^{3}+12 A \,a^{2} b^{3} d^{2} e^{2}+4 A \,b^{4} d^{3} e a +2 A \,b^{5} d^{4}-110 B \,a^{5} e^{4}+176 B \,a^{4} b d \,e^{3}-66 B \,a^{3} b^{2} d^{2} e^{2}+8 B \,a^{2} b^{3} d^{3} e +B \,b^{4} d^{4} a}{6 b^{6}}-\frac {\left (12 A \,a^{2} b \,e^{4}-12 A a \,b^{2} d \,e^{3}+6 A \,b^{3} d^{2} e^{2}-30 B \,e^{4} a^{3}+48 B \,a^{2} b d \,e^{3}-18 B a \,b^{2} d^{2} e^{2}+4 B \,b^{3} d^{3} e \right ) x^{2}}{b^{4}}-\frac {\left (36 A \,a^{3} b \,e^{4}-36 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}+4 A \,b^{4} d^{3} e -90 B \,a^{4} e^{4}+144 B \,a^{3} b d \,e^{3}-54 B \,a^{2} b^{2} d^{2} e^{2}+8 B a \,b^{3} d^{3} e +b^{4} B \,d^{4}\right ) x}{2 b^{5}}+\frac {B \,e^{4} x^{5}}{2 b}+\frac {e^{3} \left (2 A b e -5 B a e +8 B b d \right ) x^{4}}{2 b^{2}}}{\left (b x +a \right )^{3}}-\frac {2 e^{2} \left (2 A a b \,e^{2}-2 A \,b^{2} d e -5 a^{2} B \,e^{2}+8 B a b d e -3 B \,b^{2} d^{2}\right ) \ln \left (b x +a \right )}{b^{6}}\) | \(419\) |
| risch | \(\frac {B \,e^{4} x^{2}}{2 b^{4}}+\frac {e^{4} A x}{b^{4}}-\frac {4 e^{4} B a x}{b^{5}}+\frac {4 e^{3} B d x}{b^{4}}+\frac {\left (-6 A \,a^{2} b^{2} e^{4}+12 A a \,b^{3} d \,e^{3}-6 A \,b^{4} d^{2} e^{2}+10 B \,a^{3} b \,e^{4}-24 B \,a^{2} b^{2} d \,e^{3}+18 B a \,b^{3} d^{2} e^{2}-4 b^{4} B \,d^{3} e \right ) x^{2}+\left (-10 A \,a^{3} b \,e^{4}+18 A \,a^{2} b^{2} d \,e^{3}-6 A a \,b^{3} d^{2} e^{2}-2 A \,b^{4} d^{3} e +\frac {35}{2} B \,a^{4} e^{4}-40 B \,a^{3} b d \,e^{3}+27 B \,a^{2} b^{2} d^{2} e^{2}-4 B a \,b^{3} d^{3} e -\frac {1}{2} b^{4} B \,d^{4}\right ) x -\frac {26 A \,a^{4} b \,e^{4}-44 A \,a^{3} b^{2} d \,e^{3}+12 A \,a^{2} b^{3} d^{2} e^{2}+4 A \,b^{4} d^{3} e a +2 A \,b^{5} d^{4}-47 B \,a^{5} e^{4}+104 B \,a^{4} b d \,e^{3}-66 B \,a^{3} b^{2} d^{2} e^{2}+8 B \,a^{2} b^{3} d^{3} e +B \,b^{4} d^{4} a}{6 b}}{b^{5} \left (b x +a \right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )}-\frac {4 e^{4} \ln \left (b x +a \right ) A a}{b^{5}}+\frac {4 e^{3} \ln \left (b x +a \right ) A d}{b^{4}}+\frac {10 e^{4} \ln \left (b x +a \right ) a^{2} B}{b^{6}}-\frac {16 e^{3} \ln \left (b x +a \right ) B a d}{b^{5}}+\frac {6 e^{2} \ln \left (b x +a \right ) B \,d^{2}}{b^{4}}\) | \(471\) |
| parallelrisch | \(-\frac {432 B \,a^{3} b^{2} d \,e^{3} x -36 B \ln \left (b x +a \right ) a^{3} b^{2} d^{2} e^{2}-108 B \,x^{2} a \,b^{4} d^{2} e^{2}-72 A \,x^{2} a \,b^{4} d \,e^{3}+288 B \,x^{2} a^{2} b^{3} d \,e^{3}-180 B \ln \left (b x +a \right ) x^{2} a^{3} b^{2} e^{4}+36 A x a \,b^{4} d^{2} e^{2}-162 B x \,a^{2} b^{3} d^{2} e^{2}+24 B x a \,b^{4} d^{3} e -108 A x \,a^{2} b^{3} d \,e^{3}+72 A \ln \left (b x +a \right ) x \,a^{3} b^{2} e^{4}-180 B \ln \left (b x +a \right ) x \,a^{4} b \,e^{4}-24 A \ln \left (b x +a \right ) a^{3} b^{2} d \,e^{3}+96 B \ln \left (b x +a \right ) a^{4} b d \,e^{3}+24 A \ln \left (b x +a \right ) x^{3} a \,b^{4} e^{4}-24 A \ln \left (b x +a \right ) x^{3} b^{5} d \,e^{3}-60 B \ln \left (b x +a \right ) x^{3} a^{2} b^{3} e^{4}-36 B \ln \left (b x +a \right ) x^{3} b^{5} d^{2} e^{2}+72 A \ln \left (b x +a \right ) x^{2} a^{2} b^{3} e^{4}-3 B \,x^{5} e^{4} b^{5}-6 A \,x^{4} b^{5} e^{4}-60 B \ln \left (b x +a \right ) a^{5} e^{4}+B \,b^{4} d^{4} a +44 A \,a^{4} b \,e^{4}+3 B \,b^{5} d^{4} x -110 B \,a^{5} e^{4}+2 A \,b^{5} d^{4}+288 B \ln \left (b x +a \right ) x \,a^{3} b^{2} d \,e^{3}-108 B \ln \left (b x +a \right ) x \,a^{2} b^{3} d^{2} e^{2}-44 A \,a^{3} b^{2} d \,e^{3}+12 A \,a^{2} b^{3} d^{2} e^{2}+176 B \,a^{4} b d \,e^{3}+108 A \,a^{3} b^{2} e^{4} x +12 A \,b^{5} d^{3} e x -270 B \,a^{4} b \,e^{4} x +96 B \ln \left (b x +a \right ) x^{3} a \,b^{4} d \,e^{3}-72 A \ln \left (b x +a \right ) x^{2} a \,b^{4} d \,e^{3}+288 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}-66 B \,a^{3} b^{2} d^{2} e^{2}+8 B \,a^{2} b^{3} d^{3} e +72 A \,x^{2} a^{2} b^{3} e^{4}+36 A \,x^{2} b^{5} d^{2} e^{2}-180 B \,x^{2} a^{3} b^{2} e^{4}+24 B \,x^{2} b^{5} d^{3} e +15 B \,x^{4} a \,b^{4} e^{4}-24 B \,x^{4} b^{5} d \,e^{3}+24 A \ln \left (b x +a \right ) a^{4} b \,e^{4}+4 A \,b^{4} d^{3} e a -72 A \ln \left (b x +a \right ) x \,a^{2} b^{3} d \,e^{3}}{6 b^{6} \left (b^{2} x^{2}+2 a b x +a^{2}\right ) \left (b x +a \right )}\) | \(808\) |
e^3/b^5*(1/2*B*b*e*x^2+A*b*e*x-4*B*a*e*x+4*B*b*d*x)-1/3/b^6*(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)/(b*x+a)^ 3-2/b^6*e^2*(2*A*a*b*e^2-2*A*b^2*d*e-5*B*a^2*e^2+8*B*a*b*d*e-3*B*b^2*d^2)* ln(b*x+a)-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)/(b*x+a)-1/2*(-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/(b*x+a)^2
Leaf count of result is larger than twice the leaf count of optimal. 671 vs. \(2 (180) = 360\).
Time = 0.32 (sec) , antiderivative size = 671, normalized size of antiderivative = 3.61 \[ \int \frac {(A+B x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {3 \, B b^{5} e^{4} x^{5} - {\left (B a b^{4} + 2 \, A b^{5}\right )} d^{4} - 4 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e + 6 \, {\left (11 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} d^{2} e^{2} - 4 \, {\left (26 \, B a^{4} b - 11 \, A a^{3} b^{2}\right )} d e^{3} + {\left (47 \, B a^{5} - 26 \, A a^{4} b\right )} e^{4} + 3 \, {\left (8 \, B b^{5} d e^{3} - {\left (5 \, B a b^{4} - 2 \, A b^{5}\right )} e^{4}\right )} x^{4} + 9 \, {\left (8 \, B a b^{4} d e^{3} - {\left (7 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} e^{4}\right )} x^{3} - 3 \, {\left (8 \, B b^{5} d^{3} e - 12 \, {\left (3 \, B a b^{4} - A b^{5}\right )} d^{2} e^{2} + 24 \, {\left (B a^{2} b^{3} - A a b^{4}\right )} d e^{3} + 3 \, {\left (B a^{3} b^{2} + 2 \, A a^{2} b^{3}\right )} e^{4}\right )} x^{2} - 3 \, {\left (B b^{5} d^{4} + 4 \, {\left (2 \, B a b^{4} + A b^{5}\right )} d^{3} e - 6 \, {\left (9 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} d^{2} e^{2} + 36 \, {\left (2 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} d e^{3} - 9 \, {\left (3 \, B a^{4} b - 2 \, A a^{3} b^{2}\right )} e^{4}\right )} x + 12 \, {\left (3 \, B a^{3} b^{2} d^{2} e^{2} - 2 \, {\left (4 \, B a^{4} b - A a^{3} b^{2}\right )} d e^{3} + {\left (5 \, B a^{5} - 2 \, A a^{4} b\right )} e^{4} + {\left (3 \, B b^{5} d^{2} e^{2} - 2 \, {\left (4 \, B a b^{4} - A b^{5}\right )} d e^{3} + {\left (5 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} e^{4}\right )} x^{3} + 3 \, {\left (3 \, B a b^{4} d^{2} e^{2} - 2 \, {\left (4 \, B a^{2} b^{3} - A a b^{4}\right )} d e^{3} + {\left (5 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} e^{4}\right )} x^{2} + 3 \, {\left (3 \, B a^{2} b^{3} d^{2} e^{2} - 2 \, {\left (4 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} d e^{3} + {\left (5 \, B a^{4} b - 2 \, A a^{3} b^{2}\right )} e^{4}\right )} x\right )} \log \left (b x + a\right )}{6 \, {\left (b^{9} x^{3} + 3 \, a b^{8} x^{2} + 3 \, a^{2} b^{7} x + a^{3} b^{6}\right )}} \]
1/6*(3*B*b^5*e^4*x^5 - (B*a*b^4 + 2*A*b^5)*d^4 - 4*(2*B*a^2*b^3 + A*a*b^4) *d^3*e + 6*(11*B*a^3*b^2 - 2*A*a^2*b^3)*d^2*e^2 - 4*(26*B*a^4*b - 11*A*a^3 *b^2)*d*e^3 + (47*B*a^5 - 26*A*a^4*b)*e^4 + 3*(8*B*b^5*d*e^3 - (5*B*a*b^4 - 2*A*b^5)*e^4)*x^4 + 9*(8*B*a*b^4*d*e^3 - (7*B*a^2*b^3 - 2*A*a*b^4)*e^4)* x^3 - 3*(8*B*b^5*d^3*e - 12*(3*B*a*b^4 - A*b^5)*d^2*e^2 + 24*(B*a^2*b^3 - A*a*b^4)*d*e^3 + 3*(B*a^3*b^2 + 2*A*a^2*b^3)*e^4)*x^2 - 3*(B*b^5*d^4 + 4*( 2*B*a*b^4 + A*b^5)*d^3*e - 6*(9*B*a^2*b^3 - 2*A*a*b^4)*d^2*e^2 + 36*(2*B*a ^3*b^2 - A*a^2*b^3)*d*e^3 - 9*(3*B*a^4*b - 2*A*a^3*b^2)*e^4)*x + 12*(3*B*a ^3*b^2*d^2*e^2 - 2*(4*B*a^4*b - A*a^3*b^2)*d*e^3 + (5*B*a^5 - 2*A*a^4*b)*e ^4 + (3*B*b^5*d^2*e^2 - 2*(4*B*a*b^4 - A*b^5)*d*e^3 + (5*B*a^2*b^3 - 2*A*a *b^4)*e^4)*x^3 + 3*(3*B*a*b^4*d^2*e^2 - 2*(4*B*a^2*b^3 - A*a*b^4)*d*e^3 + (5*B*a^3*b^2 - 2*A*a^2*b^3)*e^4)*x^2 + 3*(3*B*a^2*b^3*d^2*e^2 - 2*(4*B*a^3 *b^2 - A*a^2*b^3)*d*e^3 + (5*B*a^4*b - 2*A*a^3*b^2)*e^4)*x)*log(b*x + a))/ (b^9*x^3 + 3*a*b^8*x^2 + 3*a^2*b^7*x + a^3*b^6)
Leaf count of result is larger than twice the leaf count of optimal. 486 vs. \(2 (190) = 380\).
Time = 13.93 (sec) , antiderivative size = 486, normalized size of antiderivative = 2.61 \[ \int \frac {(A+B x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {B e^{4} x^{2}}{2 b^{4}} + x \left (\frac {A e^{4}}{b^{4}} - \frac {4 B a e^{4}}{b^{5}} + \frac {4 B d e^{3}}{b^{4}}\right ) + \frac {- 26 A a^{4} b e^{4} + 44 A a^{3} b^{2} d e^{3} - 12 A a^{2} b^{3} d^{2} e^{2} - 4 A a b^{4} d^{3} e - 2 A b^{5} d^{4} + 47 B a^{5} e^{4} - 104 B a^{4} b d e^{3} + 66 B a^{3} b^{2} d^{2} e^{2} - 8 B a^{2} b^{3} d^{3} e - B a b^{4} d^{4} + x^{2} \left (- 36 A a^{2} b^{3} e^{4} + 72 A a b^{4} d e^{3} - 36 A b^{5} d^{2} e^{2} + 60 B a^{3} b^{2} e^{4} - 144 B a^{2} b^{3} d e^{3} + 108 B a b^{4} d^{2} e^{2} - 24 B b^{5} d^{3} e\right ) + x \left (- 60 A a^{3} b^{2} e^{4} + 108 A a^{2} b^{3} d e^{3} - 36 A a b^{4} d^{2} e^{2} - 12 A b^{5} d^{3} e + 105 B a^{4} b e^{4} - 240 B a^{3} b^{2} d e^{3} + 162 B a^{2} b^{3} d^{2} e^{2} - 24 B a b^{4} d^{3} e - 3 B b^{5} d^{4}\right )}{6 a^{3} b^{6} + 18 a^{2} b^{7} x + 18 a b^{8} x^{2} + 6 b^{9} x^{3}} + \frac {2 e^{2} \left (a e - b d\right ) \left (- 2 A b e + 5 B a e - 3 B b d\right ) \log {\left (a + b x \right )}}{b^{6}} \]
B*e**4*x**2/(2*b**4) + x*(A*e**4/b**4 - 4*B*a*e**4/b**5 + 4*B*d*e**3/b**4) + (-26*A*a**4*b*e**4 + 44*A*a**3*b**2*d*e**3 - 12*A*a**2*b**3*d**2*e**2 - 4*A*a*b**4*d**3*e - 2*A*b**5*d**4 + 47*B*a**5*e**4 - 104*B*a**4*b*d*e**3 + 66*B*a**3*b**2*d**2*e**2 - 8*B*a**2*b**3*d**3*e - B*a*b**4*d**4 + x**2*( -36*A*a**2*b**3*e**4 + 72*A*a*b**4*d*e**3 - 36*A*b**5*d**2*e**2 + 60*B*a** 3*b**2*e**4 - 144*B*a**2*b**3*d*e**3 + 108*B*a*b**4*d**2*e**2 - 24*B*b**5* d**3*e) + x*(-60*A*a**3*b**2*e**4 + 108*A*a**2*b**3*d*e**3 - 36*A*a*b**4*d **2*e**2 - 12*A*b**5*d**3*e + 105*B*a**4*b*e**4 - 240*B*a**3*b**2*d*e**3 + 162*B*a**2*b**3*d**2*e**2 - 24*B*a*b**4*d**3*e - 3*B*b**5*d**4))/(6*a**3* b**6 + 18*a**2*b**7*x + 18*a*b**8*x**2 + 6*b**9*x**3) + 2*e**2*(a*e - b*d) *(-2*A*b*e + 5*B*a*e - 3*B*b*d)*log(a + b*x)/b**6
Leaf count of result is larger than twice the leaf count of optimal. 434 vs. \(2 (180) = 360\).
Time = 0.21 (sec) , antiderivative size = 434, normalized size of antiderivative = 2.33 \[ \int \frac {(A+B x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {{\left (B a b^{4} + 2 \, A b^{5}\right )} d^{4} + 4 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e - 6 \, {\left (11 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} d^{2} e^{2} + 4 \, {\left (26 \, B a^{4} b - 11 \, A a^{3} b^{2}\right )} d e^{3} - {\left (47 \, B a^{5} - 26 \, A a^{4} b\right )} e^{4} + 12 \, {\left (2 \, B b^{5} d^{3} e - 3 \, {\left (3 \, B a b^{4} - A b^{5}\right )} d^{2} e^{2} + 6 \, {\left (2 \, B a^{2} b^{3} - A a b^{4}\right )} d e^{3} - {\left (5 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} e^{4}\right )} x^{2} + 3 \, {\left (B b^{5} d^{4} + 4 \, {\left (2 \, B a b^{4} + A b^{5}\right )} d^{3} e - 6 \, {\left (9 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} d^{2} e^{2} + 4 \, {\left (20 \, B a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} d e^{3} - 5 \, {\left (7 \, B a^{4} b - 4 \, A a^{3} b^{2}\right )} e^{4}\right )} x}{6 \, {\left (b^{9} x^{3} + 3 \, a b^{8} x^{2} + 3 \, a^{2} b^{7} x + a^{3} b^{6}\right )}} + \frac {B b e^{4} x^{2} + 2 \, {\left (4 \, B b d e^{3} - {\left (4 \, B a - A b\right )} e^{4}\right )} x}{2 \, b^{5}} + \frac {2 \, {\left (3 \, B b^{2} d^{2} e^{2} - 2 \, {\left (4 \, B a b - A b^{2}\right )} d e^{3} + {\left (5 \, B a^{2} - 2 \, A a b\right )} e^{4}\right )} \log \left (b x + a\right )}{b^{6}} \]
-1/6*((B*a*b^4 + 2*A*b^5)*d^4 + 4*(2*B*a^2*b^3 + A*a*b^4)*d^3*e - 6*(11*B* a^3*b^2 - 2*A*a^2*b^3)*d^2*e^2 + 4*(26*B*a^4*b - 11*A*a^3*b^2)*d*e^3 - (47 *B*a^5 - 26*A*a^4*b)*e^4 + 12*(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 + 3*(B*b^5*d^4 + 4*(2*B*a*b^4 + A*b^5)*d^3*e - 6*(9*B*a^2*b^3 - 2*A*a*b^ 4)*d^2*e^2 + 4*(20*B*a^3*b^2 - 9*A*a^2*b^3)*d*e^3 - 5*(7*B*a^4*b - 4*A*a^3 *b^2)*e^4)*x)/(b^9*x^3 + 3*a*b^8*x^2 + 3*a^2*b^7*x + a^3*b^6) + 1/2*(B*b*e ^4*x^2 + 2*(4*B*b*d*e^3 - (4*B*a - A*b)*e^4)*x)/b^5 + 2*(3*B*b^2*d^2*e^2 - 2*(4*B*a*b - A*b^2)*d*e^3 + (5*B*a^2 - 2*A*a*b)*e^4)*log(b*x + a)/b^6
Leaf count of result is larger than twice the leaf count of optimal. 437 vs. \(2 (180) = 360\).
Time = 0.28 (sec) , antiderivative size = 437, normalized size of antiderivative = 2.35 \[ \int \frac {(A+B x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {2 \, {\left (3 \, B b^{2} d^{2} e^{2} - 8 \, B a b d e^{3} + 2 \, A b^{2} d e^{3} + 5 \, B a^{2} e^{4} - 2 \, A a b e^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} + \frac {B b^{4} e^{4} x^{2} + 8 \, B b^{4} d e^{3} x - 8 \, B a b^{3} e^{4} x + 2 \, A b^{4} e^{4} x}{2 \, b^{8}} - \frac {B a b^{4} d^{4} + 2 \, A b^{5} d^{4} + 8 \, B a^{2} b^{3} d^{3} e + 4 \, A a b^{4} d^{3} e - 66 \, B a^{3} b^{2} d^{2} e^{2} + 12 \, A a^{2} b^{3} d^{2} e^{2} + 104 \, B a^{4} b d e^{3} - 44 \, A a^{3} b^{2} d e^{3} - 47 \, B a^{5} e^{4} + 26 \, A a^{4} b e^{4} + 12 \, {\left (2 \, B b^{5} d^{3} e - 9 \, B a b^{4} d^{2} e^{2} + 3 \, A b^{5} d^{2} e^{2} + 12 \, B a^{2} b^{3} d e^{3} - 6 \, A a b^{4} d e^{3} - 5 \, B a^{3} b^{2} e^{4} + 3 \, A a^{2} b^{3} e^{4}\right )} x^{2} + 3 \, {\left (B b^{5} d^{4} + 8 \, B a b^{4} d^{3} e + 4 \, A b^{5} d^{3} e - 54 \, B a^{2} b^{3} d^{2} e^{2} + 12 \, A a b^{4} d^{2} e^{2} + 80 \, B a^{3} b^{2} d e^{3} - 36 \, A a^{2} b^{3} d e^{3} - 35 \, B a^{4} b e^{4} + 20 \, A a^{3} b^{2} e^{4}\right )} x}{6 \, {\left (b x + a\right )}^{3} b^{6}} \]
2*(3*B*b^2*d^2*e^2 - 8*B*a*b*d*e^3 + 2*A*b^2*d*e^3 + 5*B*a^2*e^4 - 2*A*a*b *e^4)*log(abs(b*x + a))/b^6 + 1/2*(B*b^4*e^4*x^2 + 8*B*b^4*d*e^3*x - 8*B*a *b^3*e^4*x + 2*A*b^4*e^4*x)/b^8 - 1/6*(B*a*b^4*d^4 + 2*A*b^5*d^4 + 8*B*a^2 *b^3*d^3*e + 4*A*a*b^4*d^3*e - 66*B*a^3*b^2*d^2*e^2 + 12*A*a^2*b^3*d^2*e^2 + 104*B*a^4*b*d*e^3 - 44*A*a^3*b^2*d*e^3 - 47*B*a^5*e^4 + 26*A*a^4*b*e^4 + 12*(2*B*b^5*d^3*e - 9*B*a*b^4*d^2*e^2 + 3*A*b^5*d^2*e^2 + 12*B*a^2*b^3*d *e^3 - 6*A*a*b^4*d*e^3 - 5*B*a^3*b^2*e^4 + 3*A*a^2*b^3*e^4)*x^2 + 3*(B*b^5 *d^4 + 8*B*a*b^4*d^3*e + 4*A*b^5*d^3*e - 54*B*a^2*b^3*d^2*e^2 + 12*A*a*b^4 *d^2*e^2 + 80*B*a^3*b^2*d*e^3 - 36*A*a^2*b^3*d*e^3 - 35*B*a^4*b*e^4 + 20*A *a^3*b^2*e^4)*x)/((b*x + a)^3*b^6)
Time = 0.19 (sec) , antiderivative size = 451, normalized size of antiderivative = 2.42 \[ \int \frac {(A+B x) (d+e x)^4}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=x\,\left (\frac {A\,e^4+4\,B\,d\,e^3}{b^4}-\frac {4\,B\,a\,e^4}{b^5}\right )-\frac {\frac {-47\,B\,a^5\,e^4+104\,B\,a^4\,b\,d\,e^3+26\,A\,a^4\,b\,e^4-66\,B\,a^3\,b^2\,d^2\,e^2-44\,A\,a^3\,b^2\,d\,e^3+8\,B\,a^2\,b^3\,d^3\,e+12\,A\,a^2\,b^3\,d^2\,e^2+B\,a\,b^4\,d^4+4\,A\,a\,b^4\,d^3\,e+2\,A\,b^5\,d^4}{6\,b}+x\,\left (-\frac {35\,B\,a^4\,e^4}{2}+40\,B\,a^3\,b\,d\,e^3+10\,A\,a^3\,b\,e^4-27\,B\,a^2\,b^2\,d^2\,e^2-18\,A\,a^2\,b^2\,d\,e^3+4\,B\,a\,b^3\,d^3\,e+6\,A\,a\,b^3\,d^2\,e^2+\frac {B\,b^4\,d^4}{2}+2\,A\,b^4\,d^3\,e\right )+x^2\,\left (-10\,B\,a^3\,b\,e^4+24\,B\,a^2\,b^2\,d\,e^3+6\,A\,a^2\,b^2\,e^4-18\,B\,a\,b^3\,d^2\,e^2-12\,A\,a\,b^3\,d\,e^3+4\,B\,b^4\,d^3\,e+6\,A\,b^4\,d^2\,e^2\right )}{a^3\,b^5+3\,a^2\,b^6\,x+3\,a\,b^7\,x^2+b^8\,x^3}+\frac {\ln \left (a+b\,x\right )\,\left (10\,B\,a^2\,e^4-16\,B\,a\,b\,d\,e^3-4\,A\,a\,b\,e^4+6\,B\,b^2\,d^2\,e^2+4\,A\,b^2\,d\,e^3\right )}{b^6}+\frac {B\,e^4\,x^2}{2\,b^4} \]
x*((A*e^4 + 4*B*d*e^3)/b^4 - (4*B*a*e^4)/b^5) - ((2*A*b^5*d^4 - 47*B*a^5*e ^4 + 26*A*a^4*b*e^4 + B*a*b^4*d^4 - 44*A*a^3*b^2*d*e^3 + 8*B*a^2*b^3*d^3*e + 12*A*a^2*b^3*d^2*e^2 - 66*B*a^3*b^2*d^2*e^2 + 4*A*a*b^4*d^3*e + 104*B*a ^4*b*d*e^3)/(6*b) + x*((B*b^4*d^4)/2 - (35*B*a^4*e^4)/2 + 10*A*a^3*b*e^4 + 2*A*b^4*d^3*e + 6*A*a*b^3*d^2*e^2 - 18*A*a^2*b^2*d*e^3 - 27*B*a^2*b^2*d^2 *e^2 + 4*B*a*b^3*d^3*e + 40*B*a^3*b*d*e^3) + x^2*(4*B*b^4*d^3*e - 10*B*a^3 *b*e^4 + 6*A*a^2*b^2*e^4 + 6*A*b^4*d^2*e^2 - 18*B*a*b^3*d^2*e^2 + 24*B*a^2 *b^2*d*e^3 - 12*A*a*b^3*d*e^3))/(a^3*b^5 + b^8*x^3 + 3*a^2*b^6*x + 3*a*b^7 *x^2) + (log(a + b*x)*(10*B*a^2*e^4 - 4*A*a*b*e^4 + 4*A*b^2*d*e^3 + 6*B*b^ 2*d^2*e^2 - 16*B*a*b*d*e^3))/b^6 + (B*e^4*x^2)/(2*b^4)