Integrand size = 31, antiderivative size = 189 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^4} \, dx=-\frac {b^3 (4 b B d-A b e-4 a B e) x}{e^5}+\frac {b^4 B x^2}{2 e^4}+\frac {(b d-a e)^4 (B d-A e)}{3 e^6 (d+e x)^3}-\frac {(b d-a e)^3 (5 b B d-4 A b e-a B e)}{2 e^6 (d+e x)^2}+\frac {2 b (b d-a e)^2 (5 b B d-3 A b e-2 a B e)}{e^6 (d+e x)}+\frac {2 b^2 (b d-a e) (5 b B d-2 A b e-3 a B e) \log (d+e x)}{e^6} \]
-b^3*(-A*b*e-4*B*a*e+4*B*b*d)*x/e^5+1/2*b^4*B*x^2/e^4+1/3*(-a*e+b*d)^4*(-A *e+B*d)/e^6/(e*x+d)^3-1/2*(-a*e+b*d)^3*(-4*A*b*e-B*a*e+5*B*b*d)/e^6/(e*x+d )^2+2*b*(-a*e+b*d)^2*(-3*A*b*e-2*B*a*e+5*B*b*d)/e^6/(e*x+d)+2*b^2*(-a*e+b* d)*(-2*A*b*e-3*B*a*e+5*B*b*d)*ln(e*x+d)/e^6
Time = 0.11 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.86 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^4} \, dx=\frac {-a^4 e^4 (2 A e+B (d+3 e x))-4 a^3 b e^3 \left (A e (d+3 e x)+2 B \left (d^2+3 d e x+3 e^2 x^2\right )\right )+6 a^2 b^2 e^2 \left (-2 A e \left (d^2+3 d e x+3 e^2 x^2\right )+B d \left (11 d^2+27 d e x+18 e^2 x^2\right )\right )+4 a b^3 e \left (A d e \left (11 d^2+27 d e x+18 e^2 x^2\right )-2 B \left (13 d^4+27 d^3 e x+9 d^2 e^2 x^2-9 d e^3 x^3-3 e^4 x^4\right )\right )+b^4 \left (2 A e \left (-13 d^4-27 d^3 e x-9 d^2 e^2 x^2+9 d e^3 x^3+3 e^4 x^4\right )+B \left (47 d^5+81 d^4 e x-9 d^3 e^2 x^2-63 d^2 e^3 x^3-15 d e^4 x^4+3 e^5 x^5\right )\right )+12 b^2 (b d-a e) (5 b B d-2 A b e-3 a B e) (d+e x)^3 \log (d+e x)}{6 e^6 (d+e x)^3} \]
(-(a^4*e^4*(2*A*e + B*(d + 3*e*x))) - 4*a^3*b*e^3*(A*e*(d + 3*e*x) + 2*B*( d^2 + 3*d*e*x + 3*e^2*x^2)) + 6*a^2*b^2*e^2*(-2*A*e*(d^2 + 3*d*e*x + 3*e^2 *x^2) + B*d*(11*d^2 + 27*d*e*x + 18*e^2*x^2)) + 4*a*b^3*e*(A*d*e*(11*d^2 + 27*d*e*x + 18*e^2*x^2) - 2*B*(13*d^4 + 27*d^3*e*x + 9*d^2*e^2*x^2 - 9*d*e ^3*x^3 - 3*e^4*x^4)) + b^4*(2*A*e*(-13*d^4 - 27*d^3*e*x - 9*d^2*e^2*x^2 + 9*d*e^3*x^3 + 3*e^4*x^4) + B*(47*d^5 + 81*d^4*e*x - 9*d^3*e^2*x^2 - 63*d^2 *e^3*x^3 - 15*d*e^4*x^4 + 3*e^5*x^5)) + 12*b^2*(b*d - a*e)*(5*b*B*d - 2*A* b*e - 3*a*B*e)*(d + e*x)^3*Log[d + e*x])/(6*e^6*(d + e*x)^3)
Time = 0.46 (sec) , antiderivative size = 189, 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.
| \(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2 (A+B x)}{(d+e x)^4} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int \frac {b^4 (a+b x)^4 (A+B x)}{(d+e x)^4}dx}{b^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(a+b x)^4 (A+B x)}{(d+e x)^4}dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {b^3 (4 a B e+A b e-4 b B d)}{e^5}-\frac {2 b^2 (b d-a e) (3 a B e+2 A b e-5 b B d)}{e^5 (d+e x)}+\frac {2 b (b d-a e)^2 (2 a B e+3 A b e-5 b B d)}{e^5 (d+e x)^2}+\frac {(a e-b d)^3 (a B e+4 A b e-5 b B d)}{e^5 (d+e x)^3}+\frac {(a e-b d)^4 (A e-B d)}{e^5 (d+e x)^4}+\frac {b^4 B x}{e^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {b^3 x (-4 a B e-A b e+4 b B d)}{e^5}+\frac {2 b^2 (b d-a e) \log (d+e x) (-3 a B e-2 A b e+5 b B d)}{e^6}+\frac {2 b (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{e^6 (d+e x)}-\frac {(b d-a e)^3 (-a B e-4 A b e+5 b B d)}{2 e^6 (d+e x)^2}+\frac {(b d-a e)^4 (B d-A e)}{3 e^6 (d+e x)^3}+\frac {b^4 B x^2}{2 e^4}\) |
-((b^3*(4*b*B*d - A*b*e - 4*a*B*e)*x)/e^5) + (b^4*B*x^2)/(2*e^4) + ((b*d - a*e)^4*(B*d - A*e))/(3*e^6*(d + e*x)^3) - ((b*d - a*e)^3*(5*b*B*d - 4*A*b *e - a*B*e))/(2*e^6*(d + e*x)^2) + (2*b*(b*d - a*e)^2*(5*b*B*d - 3*A*b*e - 2*a*B*e))/(e^6*(d + e*x)) + (2*b^2*(b*d - a*e)*(5*b*B*d - 2*A*b*e - 3*a*B *e)*Log[d + e*x])/e^6
3.17.84.3.1 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. \(2(183)=366\).
Time = 0.24 (sec) , antiderivative size = 416, normalized size of antiderivative = 2.20
| method | result | size |
| default | \(\frac {b^{3} \left (\frac {1}{2} B b e \,x^{2}+A b e x +4 B a e x -4 B b d x \right )}{e^{5}}-\frac {A \,a^{4} e^{5}-4 A \,a^{3} b d \,e^{4}+6 A \,a^{2} b^{2} d^{2} e^{3}-4 A a \,b^{3} d^{3} e^{2}+A \,b^{4} d^{4} e -B \,a^{4} d \,e^{4}+4 B \,a^{3} b \,d^{2} e^{3}-6 B \,a^{2} b^{2} d^{3} e^{2}+4 B a \,b^{3} d^{4} e -b^{4} B \,d^{5}}{3 e^{6} \left (e x +d \right )^{3}}-\frac {2 b \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +2 B \,e^{3} a^{3}-9 B \,a^{2} b d \,e^{2}+12 B a \,b^{2} d^{2} e -5 B \,b^{3} d^{3}\right )}{e^{6} \left (e x +d \right )}+\frac {2 b^{2} \left (2 A a b \,e^{2}-2 A \,b^{2} d e +3 a^{2} B \,e^{2}-8 B a b d e +5 B \,b^{2} d^{2}\right ) \ln \left (e x +d \right )}{e^{6}}-\frac {4 A \,a^{3} b \,e^{4}-12 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e +B \,a^{4} e^{4}-8 B \,a^{3} b d \,e^{3}+18 B \,a^{2} b^{2} d^{2} e^{2}-16 B a \,b^{3} d^{3} e +5 b^{4} B \,d^{4}}{2 e^{6} \left (e x +d \right )^{2}}\) | \(416\) |
| norman | \(\frac {-\frac {2 A \,a^{4} e^{5}+4 A \,a^{3} b d \,e^{4}+12 A \,a^{2} b^{2} d^{2} e^{3}-44 A a \,b^{3} d^{3} e^{2}+44 A \,b^{4} d^{4} e +B \,a^{4} d \,e^{4}+8 B \,a^{3} b \,d^{2} e^{3}-66 B \,a^{2} b^{2} d^{3} e^{2}+176 B a \,b^{3} d^{4} e -110 b^{4} B \,d^{5}}{6 e^{6}}-\frac {\left (6 A \,a^{2} b^{2} e^{3}-12 A a \,b^{3} d \,e^{2}+12 A \,b^{4} d^{2} e +4 B \,e^{3} b \,a^{3}-18 B \,a^{2} b^{2} d \,e^{2}+48 B a \,b^{3} d^{2} e -30 B \,b^{4} d^{3}\right ) x^{2}}{e^{4}}-\frac {\left (4 A \,a^{3} b \,e^{4}+12 A \,a^{2} b^{2} d \,e^{3}-36 A a \,b^{3} d^{2} e^{2}+36 A \,b^{4} d^{3} e +B \,a^{4} e^{4}+8 B \,a^{3} b d \,e^{3}-54 B \,a^{2} b^{2} d^{2} e^{2}+144 B a \,b^{3} d^{3} e -90 b^{4} B \,d^{4}\right ) x}{2 e^{5}}+\frac {B \,b^{4} x^{5}}{2 e}+\frac {b^{3} \left (2 A b e +8 B a e -5 B b d \right ) x^{4}}{2 e^{2}}}{\left (e x +d \right )^{3}}+\frac {2 b^{2} \left (2 A a b \,e^{2}-2 A \,b^{2} d e +3 a^{2} B \,e^{2}-8 B a b d e +5 B \,b^{2} d^{2}\right ) \ln \left (e x +d \right )}{e^{6}}\) | \(419\) |
| risch | \(\frac {b^{4} B \,x^{2}}{2 e^{4}}+\frac {b^{4} A x}{e^{4}}+\frac {4 b^{3} B a x}{e^{4}}-\frac {4 b^{4} B d x}{e^{5}}+\frac {\left (-6 A \,a^{2} b^{2} e^{4}+12 A a \,b^{3} d \,e^{3}-6 A \,b^{4} d^{2} e^{2}-4 B \,a^{3} b \,e^{4}+18 B \,a^{2} b^{2} d \,e^{3}-24 B a \,b^{3} d^{2} e^{2}+10 b^{4} B \,d^{3} e \right ) x^{2}+\left (-2 A \,a^{3} b \,e^{4}-6 A \,a^{2} b^{2} d \,e^{3}+18 A a \,b^{3} d^{2} e^{2}-10 A \,b^{4} d^{3} e -\frac {1}{2} B \,a^{4} e^{4}-4 B \,a^{3} b d \,e^{3}+27 B \,a^{2} b^{2} d^{2} e^{2}-40 B a \,b^{3} d^{3} e +\frac {35}{2} b^{4} B \,d^{4}\right ) x -\frac {2 A \,a^{4} e^{5}+4 A \,a^{3} b d \,e^{4}+12 A \,a^{2} b^{2} d^{2} e^{3}-44 A a \,b^{3} d^{3} e^{2}+26 A \,b^{4} d^{4} e +B \,a^{4} d \,e^{4}+8 B \,a^{3} b \,d^{2} e^{3}-66 B \,a^{2} b^{2} d^{3} e^{2}+104 B a \,b^{3} d^{4} e -47 b^{4} B \,d^{5}}{6 e}}{e^{5} \left (e x +d \right )^{3}}+\frac {4 b^{3} \ln \left (e x +d \right ) A a}{e^{4}}-\frac {4 b^{4} \ln \left (e x +d \right ) A d}{e^{5}}+\frac {6 b^{2} \ln \left (e x +d \right ) a^{2} B}{e^{4}}-\frac {16 b^{3} \ln \left (e x +d \right ) B a d}{e^{5}}+\frac {10 b^{4} \ln \left (e x +d \right ) B \,d^{2}}{e^{6}}\) | \(453\) |
| parallelrisch | \(\frac {-2 A \,a^{4} e^{5}+110 b^{4} B \,d^{5}-24 B x \,a^{3} b d \,e^{4}-72 A \ln \left (e x +d \right ) x^{2} b^{4} d^{2} e^{3}+180 B \ln \left (e x +d \right ) x^{2} b^{4} d^{3} e^{2}-72 A \ln \left (e x +d \right ) x \,b^{4} d^{3} e^{2}+180 B \ln \left (e x +d \right ) x \,b^{4} d^{4} e +24 A \ln \left (e x +d \right ) x^{3} a \,b^{3} e^{5}-24 A \ln \left (e x +d \right ) x^{3} b^{4} d \,e^{4}+36 B \ln \left (e x +d \right ) x^{3} a^{2} b^{2} e^{5}+60 B \ln \left (e x +d \right ) x^{3} b^{4} d^{2} e^{3}-288 B \,x^{2} a \,b^{3} d^{2} e^{3}-36 A x \,a^{2} b^{2} d \,e^{4}+108 A x a \,b^{3} d^{2} e^{3}-432 B x a \,b^{3} d^{3} e^{2}+72 A \,x^{2} a \,b^{3} d \,e^{4}+108 B \,x^{2} a^{2} b^{2} d \,e^{4}+24 A \ln \left (e x +d \right ) a \,b^{3} d^{3} e^{2}+36 B \ln \left (e x +d \right ) a^{2} b^{2} d^{3} e^{2}-96 B \ln \left (e x +d \right ) a \,b^{3} d^{4} e +60 B \ln \left (e x +d \right ) b^{4} d^{5}-3 B x \,a^{4} e^{5}+3 B \,x^{5} b^{4} e^{5}+6 A \,b^{4} e^{5} x^{4}-44 A \,b^{4} d^{4} e -B \,a^{4} d \,e^{4}+162 B x \,a^{2} b^{2} d^{2} e^{3}+270 B x \,b^{4} d^{4} e +24 B \,x^{4} a \,b^{3} e^{5}-15 B \,x^{4} b^{4} d \,e^{4}+44 A a \,b^{3} d^{3} e^{2}+66 B \,a^{2} b^{2} d^{3} e^{2}-176 B a \,b^{3} d^{4} e -4 A \,a^{3} b d \,e^{4}-12 A \,a^{2} b^{2} d^{2} e^{3}-8 B \,a^{3} b \,d^{2} e^{3}-96 B \ln \left (e x +d \right ) x^{3} a \,b^{3} d \,e^{4}+72 A \ln \left (e x +d \right ) x a \,b^{3} d^{2} e^{3}+108 B \ln \left (e x +d \right ) x \,a^{2} b^{2} d^{2} e^{3}-288 B \ln \left (e x +d \right ) x a \,b^{3} d^{3} e^{2}-108 A x \,b^{4} d^{3} e^{2}-288 B \ln \left (e x +d \right ) x^{2} a \,b^{3} d^{2} e^{3}+72 A \ln \left (e x +d \right ) x^{2} a \,b^{3} d \,e^{4}+108 B \ln \left (e x +d \right ) x^{2} a^{2} b^{2} d \,e^{4}-36 A \,x^{2} a^{2} b^{2} e^{5}-72 A \,x^{2} b^{4} d^{2} e^{3}-24 B \,x^{2} a^{3} b \,e^{5}+180 B \,x^{2} b^{4} d^{3} e^{2}-12 A x \,a^{3} b \,e^{5}-24 A \ln \left (e x +d \right ) b^{4} d^{4} e}{6 e^{6} \left (e x +d \right )^{3}}\) | \(791\) |
b^3/e^5*(1/2*B*b*e*x^2+A*b*e*x+4*B*a*e*x-4*B*b*d*x)-1/3*(A*a^4*e^5-4*A*a^3 *b*d*e^4+6*A*a^2*b^2*d^2*e^3-4*A*a*b^3*d^3*e^2+A*b^4*d^4*e-B*a^4*d*e^4+4*B *a^3*b*d^2*e^3-6*B*a^2*b^2*d^3*e^2+4*B*a*b^3*d^4*e-B*b^4*d^5)/e^6/(e*x+d)^ 3-2*b/e^6*(3*A*a^2*b*e^3-6*A*a*b^2*d*e^2+3*A*b^3*d^2*e+2*B*a^3*e^3-9*B*a^2 *b*d*e^2+12*B*a*b^2*d^2*e-5*B*b^3*d^3)/(e*x+d)+2*b^2/e^6*(2*A*a*b*e^2-2*A* b^2*d*e+3*B*a^2*e^2-8*B*a*b*d*e+5*B*b^2*d^2)*ln(e*x+d)-1/2/e^6*(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+B*a^4*e^4-8*B*a^3* b*d*e^3+18*B*a^2*b^2*d^2*e^2-16*B*a*b^3*d^3*e+5*B*b^4*d^4)/(e*x+d)^2
Leaf count of result is larger than twice the leaf count of optimal. 650 vs. \(2 (183) = 366\).
Time = 0.31 (sec) , antiderivative size = 650, normalized size of antiderivative = 3.44 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^4} \, dx=\frac {3 \, B b^{4} e^{5} x^{5} + 47 \, B b^{4} d^{5} - 2 \, A a^{4} e^{5} - 26 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 22 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 4 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} - {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} - 3 \, {\left (5 \, B b^{4} d e^{4} - 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} - 9 \, {\left (7 \, B b^{4} d^{2} e^{3} - 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4}\right )} x^{3} - 3 \, {\left (3 \, B b^{4} d^{3} e^{2} + 6 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} - 12 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} + 4 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 3 \, {\left (27 \, B b^{4} d^{4} e - 18 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 18 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} - 4 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} - {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x + 12 \, {\left (5 \, B b^{4} d^{5} - 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + {\left (5 \, B b^{4} d^{2} e^{3} - 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 3 \, {\left (5 \, B b^{4} d^{3} e^{2} - 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4}\right )} x^{2} + 3 \, {\left (5 \, B b^{4} d^{4} e - 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3}\right )} x\right )} \log \left (e x + d\right )}{6 \, {\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} \]
1/6*(3*B*b^4*e^5*x^5 + 47*B*b^4*d^5 - 2*A*a^4*e^5 - 26*(4*B*a*b^3 + A*b^4) *d^4*e + 22*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 - 4*(2*B*a^3*b + 3*A*a^2*b^2 )*d^2*e^3 - (B*a^4 + 4*A*a^3*b)*d*e^4 - 3*(5*B*b^4*d*e^4 - 2*(4*B*a*b^3 + A*b^4)*e^5)*x^4 - 9*(7*B*b^4*d^2*e^3 - 2*(4*B*a*b^3 + A*b^4)*d*e^4)*x^3 - 3*(3*B*b^4*d^3*e^2 + 6*(4*B*a*b^3 + A*b^4)*d^2*e^3 - 12*(3*B*a^2*b^2 + 2*A *a*b^3)*d*e^4 + 4*(2*B*a^3*b + 3*A*a^2*b^2)*e^5)*x^2 + 3*(27*B*b^4*d^4*e - 18*(4*B*a*b^3 + A*b^4)*d^3*e^2 + 18*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^3 - 4 *(2*B*a^3*b + 3*A*a^2*b^2)*d*e^4 - (B*a^4 + 4*A*a^3*b)*e^5)*x + 12*(5*B*b^ 4*d^5 - 2*(4*B*a*b^3 + A*b^4)*d^4*e + (3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 + (5*B*b^4*d^2*e^3 - 2*(4*B*a*b^3 + A*b^4)*d*e^4 + (3*B*a^2*b^2 + 2*A*a*b^3) *e^5)*x^3 + 3*(5*B*b^4*d^3*e^2 - 2*(4*B*a*b^3 + A*b^4)*d^2*e^3 + (3*B*a^2* b^2 + 2*A*a*b^3)*d*e^4)*x^2 + 3*(5*B*b^4*d^4*e - 2*(4*B*a*b^3 + A*b^4)*d^3 *e^2 + (3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^3)*x)*log(e*x + d))/(e^9*x^3 + 3*d* e^8*x^2 + 3*d^2*e^7*x + d^3*e^6)
Leaf count of result is larger than twice the leaf count of optimal. 486 vs. \(2 (189) = 378\).
Time = 14.53 (sec) , antiderivative size = 486, normalized size of antiderivative = 2.57 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^4} \, dx=\frac {B b^{4} x^{2}}{2 e^{4}} + \frac {2 b^{2} \left (a e - b d\right ) \left (2 A b e + 3 B a e - 5 B b d\right ) \log {\left (d + e x \right )}}{e^{6}} + x \left (\frac {A b^{4}}{e^{4}} + \frac {4 B a b^{3}}{e^{4}} - \frac {4 B b^{4} d}{e^{5}}\right ) + \frac {- 2 A a^{4} e^{5} - 4 A a^{3} b d e^{4} - 12 A a^{2} b^{2} d^{2} e^{3} + 44 A a b^{3} d^{3} e^{2} - 26 A b^{4} d^{4} e - B a^{4} d e^{4} - 8 B a^{3} b d^{2} e^{3} + 66 B a^{2} b^{2} d^{3} e^{2} - 104 B a b^{3} d^{4} e + 47 B b^{4} d^{5} + x^{2} \left (- 36 A a^{2} b^{2} e^{5} + 72 A a b^{3} d e^{4} - 36 A b^{4} d^{2} e^{3} - 24 B a^{3} b e^{5} + 108 B a^{2} b^{2} d e^{4} - 144 B a b^{3} d^{2} e^{3} + 60 B b^{4} d^{3} e^{2}\right ) + x \left (- 12 A a^{3} b e^{5} - 36 A a^{2} b^{2} d e^{4} + 108 A a b^{3} d^{2} e^{3} - 60 A b^{4} d^{3} e^{2} - 3 B a^{4} e^{5} - 24 B a^{3} b d e^{4} + 162 B a^{2} b^{2} d^{2} e^{3} - 240 B a b^{3} d^{3} e^{2} + 105 B b^{4} d^{4} e\right )}{6 d^{3} e^{6} + 18 d^{2} e^{7} x + 18 d e^{8} x^{2} + 6 e^{9} x^{3}} \]
B*b**4*x**2/(2*e**4) + 2*b**2*(a*e - b*d)*(2*A*b*e + 3*B*a*e - 5*B*b*d)*lo g(d + e*x)/e**6 + x*(A*b**4/e**4 + 4*B*a*b**3/e**4 - 4*B*b**4*d/e**5) + (- 2*A*a**4*e**5 - 4*A*a**3*b*d*e**4 - 12*A*a**2*b**2*d**2*e**3 + 44*A*a*b**3 *d**3*e**2 - 26*A*b**4*d**4*e - B*a**4*d*e**4 - 8*B*a**3*b*d**2*e**3 + 66* B*a**2*b**2*d**3*e**2 - 104*B*a*b**3*d**4*e + 47*B*b**4*d**5 + x**2*(-36*A *a**2*b**2*e**5 + 72*A*a*b**3*d*e**4 - 36*A*b**4*d**2*e**3 - 24*B*a**3*b*e **5 + 108*B*a**2*b**2*d*e**4 - 144*B*a*b**3*d**2*e**3 + 60*B*b**4*d**3*e** 2) + x*(-12*A*a**3*b*e**5 - 36*A*a**2*b**2*d*e**4 + 108*A*a*b**3*d**2*e**3 - 60*A*b**4*d**3*e**2 - 3*B*a**4*e**5 - 24*B*a**3*b*d*e**4 + 162*B*a**2*b **2*d**2*e**3 - 240*B*a*b**3*d**3*e**2 + 105*B*b**4*d**4*e))/(6*d**3*e**6 + 18*d**2*e**7*x + 18*d*e**8*x**2 + 6*e**9*x**3)
Leaf count of result is larger than twice the leaf count of optimal. 431 vs. \(2 (183) = 366\).
Time = 0.29 (sec) , antiderivative size = 431, normalized size of antiderivative = 2.28 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^4} \, dx=\frac {47 \, B b^{4} d^{5} - 2 \, A a^{4} e^{5} - 26 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 22 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 4 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} - {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 12 \, {\left (5 \, B b^{4} d^{3} e^{2} - 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} - {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 3 \, {\left (35 \, B b^{4} d^{4} e - 20 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 18 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} - 4 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} - {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{6 \, {\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} + \frac {B b^{4} e x^{2} - 2 \, {\left (4 \, B b^{4} d - {\left (4 \, B a b^{3} + A b^{4}\right )} e\right )} x}{2 \, e^{5}} + \frac {2 \, {\left (5 \, B b^{4} d^{2} - 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e + {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{2}\right )} \log \left (e x + d\right )}{e^{6}} \]
1/6*(47*B*b^4*d^5 - 2*A*a^4*e^5 - 26*(4*B*a*b^3 + A*b^4)*d^4*e + 22*(3*B*a ^2*b^2 + 2*A*a*b^3)*d^3*e^2 - 4*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^3 - (B*a^4 + 4*A*a^3*b)*d*e^4 + 12*(5*B*b^4*d^3*e^2 - 3*(4*B*a*b^3 + A*b^4)*d^2*e^3 + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^4 - (2*B*a^3*b + 3*A*a^2*b^2)*e^5)*x^2 + 3*(35*B*b^4*d^4*e - 20*(4*B*a*b^3 + A*b^4)*d^3*e^2 + 18*(3*B*a^2*b^2 + 2* A*a*b^3)*d^2*e^3 - 4*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^4 - (B*a^4 + 4*A*a^3*b) *e^5)*x)/(e^9*x^3 + 3*d*e^8*x^2 + 3*d^2*e^7*x + d^3*e^6) + 1/2*(B*b^4*e*x^ 2 - 2*(4*B*b^4*d - (4*B*a*b^3 + A*b^4)*e)*x)/e^5 + 2*(5*B*b^4*d^2 - 2*(4*B *a*b^3 + A*b^4)*d*e + (3*B*a^2*b^2 + 2*A*a*b^3)*e^2)*log(e*x + d)/e^6
Leaf count of result is larger than twice the leaf count of optimal. 439 vs. \(2 (183) = 366\).
Time = 0.28 (sec) , antiderivative size = 439, normalized size of antiderivative = 2.32 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^4} \, dx=\frac {2 \, {\left (5 \, B b^{4} d^{2} - 8 \, B a b^{3} d e - 2 \, A b^{4} d e + 3 \, B a^{2} b^{2} e^{2} + 2 \, A a b^{3} e^{2}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{6}} + \frac {B b^{4} e^{4} x^{2} - 8 \, B b^{4} d e^{3} x + 8 \, B a b^{3} e^{4} x + 2 \, A b^{4} e^{4} x}{2 \, e^{8}} + \frac {47 \, B b^{4} d^{5} - 104 \, B a b^{3} d^{4} e - 26 \, A b^{4} d^{4} e + 66 \, B a^{2} b^{2} d^{3} e^{2} + 44 \, A a b^{3} d^{3} e^{2} - 8 \, B a^{3} b d^{2} e^{3} - 12 \, A a^{2} b^{2} d^{2} e^{3} - B a^{4} d e^{4} - 4 \, A a^{3} b d e^{4} - 2 \, A a^{4} e^{5} + 12 \, {\left (5 \, B b^{4} d^{3} e^{2} - 12 \, B a b^{3} d^{2} e^{3} - 3 \, A b^{4} d^{2} e^{3} + 9 \, B a^{2} b^{2} d e^{4} + 6 \, A a b^{3} d e^{4} - 2 \, B a^{3} b e^{5} - 3 \, A a^{2} b^{2} e^{5}\right )} x^{2} + 3 \, {\left (35 \, B b^{4} d^{4} e - 80 \, B a b^{3} d^{3} e^{2} - 20 \, A b^{4} d^{3} e^{2} + 54 \, B a^{2} b^{2} d^{2} e^{3} + 36 \, A a b^{3} d^{2} e^{3} - 8 \, B a^{3} b d e^{4} - 12 \, A a^{2} b^{2} d e^{4} - B a^{4} e^{5} - 4 \, A a^{3} b e^{5}\right )} x}{6 \, {\left (e x + d\right )}^{3} e^{6}} \]
2*(5*B*b^4*d^2 - 8*B*a*b^3*d*e - 2*A*b^4*d*e + 3*B*a^2*b^2*e^2 + 2*A*a*b^3 *e^2)*log(abs(e*x + d))/e^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)/e^8 + 1/6*(47*B*b^4*d^5 - 104*B*a*b^3*d^4*e - 26*A*b^4*d^4*e + 66*B*a^2*b^2*d^3*e^2 + 44*A*a*b^3*d^3*e^2 - 8*B*a^3*b*d^2 *e^3 - 12*A*a^2*b^2*d^2*e^3 - B*a^4*d*e^4 - 4*A*a^3*b*d*e^4 - 2*A*a^4*e^5 + 12*(5*B*b^4*d^3*e^2 - 12*B*a*b^3*d^2*e^3 - 3*A*b^4*d^2*e^3 + 9*B*a^2*b^2 *d*e^4 + 6*A*a*b^3*d*e^4 - 2*B*a^3*b*e^5 - 3*A*a^2*b^2*e^5)*x^2 + 3*(35*B* b^4*d^4*e - 80*B*a*b^3*d^3*e^2 - 20*A*b^4*d^3*e^2 + 54*B*a^2*b^2*d^2*e^3 + 36*A*a*b^3*d^2*e^3 - 8*B*a^3*b*d*e^4 - 12*A*a^2*b^2*d*e^4 - B*a^4*e^5 - 4 *A*a^3*b*e^5)*x)/((e*x + d)^3*e^6)
Time = 0.18 (sec) , antiderivative size = 451, normalized size of antiderivative = 2.39 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^4} \, dx=x\,\left (\frac {A\,b^4+4\,B\,a\,b^3}{e^4}-\frac {4\,B\,b^4\,d}{e^5}\right )-\frac {\frac {B\,a^4\,d\,e^4+2\,A\,a^4\,e^5+8\,B\,a^3\,b\,d^2\,e^3+4\,A\,a^3\,b\,d\,e^4-66\,B\,a^2\,b^2\,d^3\,e^2+12\,A\,a^2\,b^2\,d^2\,e^3+104\,B\,a\,b^3\,d^4\,e-44\,A\,a\,b^3\,d^3\,e^2-47\,B\,b^4\,d^5+26\,A\,b^4\,d^4\,e}{6\,e}+x\,\left (\frac {B\,a^4\,e^4}{2}+4\,B\,a^3\,b\,d\,e^3+2\,A\,a^3\,b\,e^4-27\,B\,a^2\,b^2\,d^2\,e^2+6\,A\,a^2\,b^2\,d\,e^3+40\,B\,a\,b^3\,d^3\,e-18\,A\,a\,b^3\,d^2\,e^2-\frac {35\,B\,b^4\,d^4}{2}+10\,A\,b^4\,d^3\,e\right )+x^2\,\left (4\,B\,a^3\,b\,e^4-18\,B\,a^2\,b^2\,d\,e^3+6\,A\,a^2\,b^2\,e^4+24\,B\,a\,b^3\,d^2\,e^2-12\,A\,a\,b^3\,d\,e^3-10\,B\,b^4\,d^3\,e+6\,A\,b^4\,d^2\,e^2\right )}{d^3\,e^5+3\,d^2\,e^6\,x+3\,d\,e^7\,x^2+e^8\,x^3}+\frac {\ln \left (d+e\,x\right )\,\left (6\,B\,a^2\,b^2\,e^2-16\,B\,a\,b^3\,d\,e+4\,A\,a\,b^3\,e^2+10\,B\,b^4\,d^2-4\,A\,b^4\,d\,e\right )}{e^6}+\frac {B\,b^4\,x^2}{2\,e^4} \]
x*((A*b^4 + 4*B*a*b^3)/e^4 - (4*B*b^4*d)/e^5) - ((2*A*a^4*e^5 - 47*B*b^4*d ^5 + 26*A*b^4*d^4*e + B*a^4*d*e^4 - 44*A*a*b^3*d^3*e^2 + 8*B*a^3*b*d^2*e^3 + 12*A*a^2*b^2*d^2*e^3 - 66*B*a^2*b^2*d^3*e^2 + 4*A*a^3*b*d*e^4 + 104*B*a *b^3*d^4*e)/(6*e) + x*((B*a^4*e^4)/2 - (35*B*b^4*d^4)/2 + 2*A*a^3*b*e^4 + 10*A*b^4*d^3*e - 18*A*a*b^3*d^2*e^2 + 6*A*a^2*b^2*d*e^3 - 27*B*a^2*b^2*d^2 *e^2 + 40*B*a*b^3*d^3*e + 4*B*a^3*b*d*e^3) + x^2*(4*B*a^3*b*e^4 - 10*B*b^4 *d^3*e + 6*A*a^2*b^2*e^4 + 6*A*b^4*d^2*e^2 + 24*B*a*b^3*d^2*e^2 - 18*B*a^2 *b^2*d*e^3 - 12*A*a*b^3*d*e^3))/(d^3*e^5 + e^8*x^3 + 3*d^2*e^6*x + 3*d*e^7 *x^2) + (log(d + e*x)*(10*B*b^4*d^2 - 4*A*b^4*d*e + 4*A*a*b^3*e^2 + 6*B*a^ 2*b^2*e^2 - 16*B*a*b^3*d*e))/e^6 + (B*b^4*x^2)/(2*e^4)