Integrand size = 31, antiderivative size = 86 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^7} \, dx=-\frac {(B d-A e) (a+b x)^5}{6 e (b d-a e) (d+e x)^6}+\frac {(5 b B d+A b e-6 a B e) (a+b x)^5}{30 e (b d-a e)^2 (d+e x)^5} \]
-1/6*(-A*e+B*d)*(b*x+a)^5/e/(-a*e+b*d)/(e*x+d)^6+1/30*(A*b*e-6*B*a*e+5*B*b *d)*(b*x+a)^5/e/(-a*e+b*d)^2/(e*x+d)^5
Leaf count is larger than twice the leaf count of optimal. \(317\) vs. \(2(86)=172\).
Time = 0.09 (sec) , antiderivative size = 317, normalized size of antiderivative = 3.69 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^7} \, dx=-\frac {a^4 e^4 (5 A e+B (d+6 e x))+2 a^3 b e^3 \left (2 A e (d+6 e x)+B \left (d^2+6 d e x+15 e^2 x^2\right )\right )+3 a^2 b^2 e^2 \left (A e \left (d^2+6 d e x+15 e^2 x^2\right )+B \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )\right )+2 a b^3 e \left (A e \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+2 B \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )\right )+b^4 \left (A e \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )+5 B \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )\right )}{30 e^6 (d+e x)^6} \]
-1/30*(a^4*e^4*(5*A*e + B*(d + 6*e*x)) + 2*a^3*b*e^3*(2*A*e*(d + 6*e*x) + B*(d^2 + 6*d*e*x + 15*e^2*x^2)) + 3*a^2*b^2*e^2*(A*e*(d^2 + 6*d*e*x + 15*e ^2*x^2) + B*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3)) + 2*a*b^3*e*(A* e*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3) + 2*B*(d^4 + 6*d^3*e*x + 1 5*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4)) + b^4*(A*e*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4) + 5*B*(d^5 + 6*d^4*e*x + 15*d^ 3*e^2*x^2 + 20*d^2*e^3*x^3 + 15*d*e^4*x^4 + 6*e^5*x^5)))/(e^6*(d + e*x)^6)
Time = 0.22 (sec) , antiderivative size = 86, 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, 87, 48}
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.
| \(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2 (A+B x)}{(d+e x)^7} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int \frac {b^4 (a+b x)^4 (A+B x)}{(d+e x)^7}dx}{b^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(a+b x)^4 (A+B x)}{(d+e x)^7}dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(-6 a B e+A b e+5 b B d) \int \frac {(a+b x)^4}{(d+e x)^6}dx}{6 e (b d-a e)}-\frac {(a+b x)^5 (B d-A e)}{6 e (d+e x)^6 (b d-a e)}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {(a+b x)^5 (-6 a B e+A b e+5 b B d)}{30 e (d+e x)^5 (b d-a e)^2}-\frac {(a+b x)^5 (B d-A e)}{6 e (d+e x)^6 (b d-a e)}\) |
-1/6*((B*d - A*e)*(a + b*x)^5)/(e*(b*d - a*e)*(d + e*x)^6) + ((5*b*B*d + A *b*e - 6*a*B*e)*(a + b*x)^5)/(30*e*(b*d - a*e)^2*(d + e*x)^5)
3.17.87.3.1 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_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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. \(405\) vs. \(2(82)=164\).
Time = 0.24 (sec) , antiderivative size = 406, normalized size of antiderivative = 4.72
| method | result | size |
| risch | \(\frac {-\frac {B \,b^{4} x^{5}}{e}-\frac {b^{3} \left (A b e +4 B a e +5 B b d \right ) x^{4}}{2 e^{2}}-\frac {2 b^{2} \left (2 A a b \,e^{2}+A \,b^{2} d e +3 a^{2} B \,e^{2}+4 B a b d e +5 B \,b^{2} d^{2}\right ) x^{3}}{3 e^{3}}-\frac {b \left (3 A \,a^{2} b \,e^{3}+2 A a \,b^{2} d \,e^{2}+A \,b^{3} d^{2} e +2 B \,e^{3} a^{3}+3 B \,a^{2} b d \,e^{2}+4 B a \,b^{2} d^{2} e +5 B \,b^{3} d^{3}\right ) x^{2}}{2 e^{4}}-\frac {\left (4 A \,a^{3} b \,e^{4}+3 A \,a^{2} b^{2} d \,e^{3}+2 A a \,b^{3} d^{2} e^{2}+A \,b^{4} d^{3} e +B \,a^{4} e^{4}+2 B \,a^{3} b d \,e^{3}+3 B \,a^{2} b^{2} d^{2} e^{2}+4 B a \,b^{3} d^{3} e +5 b^{4} B \,d^{4}\right ) x}{5 e^{5}}-\frac {5 A \,a^{4} e^{5}+4 A \,a^{3} b d \,e^{4}+3 A \,a^{2} b^{2} d^{2} e^{3}+2 A a \,b^{3} d^{3} e^{2}+A \,b^{4} d^{4} e +B \,a^{4} d \,e^{4}+2 B \,a^{3} b \,d^{2} e^{3}+3 B \,a^{2} b^{2} d^{3} e^{2}+4 B a \,b^{3} d^{4} e +5 b^{4} B \,d^{5}}{30 e^{6}}}{\left (e x +d \right )^{6}}\) | \(406\) |
| norman | \(\frac {-\frac {B \,b^{4} x^{5}}{e}-\frac {\left (A \,b^{4} e +4 B e \,b^{3} a +5 B \,b^{4} d \right ) x^{4}}{2 e^{2}}-\frac {2 \left (2 A a \,b^{3} e^{2}+A \,b^{4} d e +3 B \,e^{2} b^{2} a^{2}+4 B a \,b^{3} d e +5 B \,b^{4} d^{2}\right ) x^{3}}{3 e^{3}}-\frac {\left (3 A \,a^{2} b^{2} e^{3}+2 A a \,b^{3} d \,e^{2}+A \,b^{4} d^{2} e +2 B \,e^{3} b \,a^{3}+3 B \,a^{2} b^{2} d \,e^{2}+4 B a \,b^{3} d^{2} e +5 B \,b^{4} d^{3}\right ) x^{2}}{2 e^{4}}-\frac {\left (4 A \,a^{3} b \,e^{4}+3 A \,a^{2} b^{2} d \,e^{3}+2 A a \,b^{3} d^{2} e^{2}+A \,b^{4} d^{3} e +B \,a^{4} e^{4}+2 B \,a^{3} b d \,e^{3}+3 B \,a^{2} b^{2} d^{2} e^{2}+4 B a \,b^{3} d^{3} e +5 b^{4} B \,d^{4}\right ) x}{5 e^{5}}-\frac {5 A \,a^{4} e^{5}+4 A \,a^{3} b d \,e^{4}+3 A \,a^{2} b^{2} d^{2} e^{3}+2 A a \,b^{3} d^{3} e^{2}+A \,b^{4} d^{4} e +B \,a^{4} d \,e^{4}+2 B \,a^{3} b \,d^{2} e^{3}+3 B \,a^{2} b^{2} d^{3} e^{2}+4 B a \,b^{3} d^{4} e +5 b^{4} B \,d^{5}}{30 e^{6}}}{\left (e x +d \right )^{6}}\) | \(418\) |
| default | \(-\frac {A \,a^{4} e^{5}-4 A \,a^{3} b d \,e^{4}+6 A \,a^{2} b^{2} d^{2} e^{3}-4 A a \,b^{3} d^{3} e^{2}+A \,b^{4} d^{4} e -B \,a^{4} d \,e^{4}+4 B \,a^{3} b \,d^{2} e^{3}-6 B \,a^{2} b^{2} d^{3} e^{2}+4 B a \,b^{3} d^{4} e -b^{4} B \,d^{5}}{6 e^{6} \left (e x +d \right )^{6}}-\frac {4 A \,a^{3} b \,e^{4}-12 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e +B \,a^{4} e^{4}-8 B \,a^{3} b d \,e^{3}+18 B \,a^{2} b^{2} d^{2} e^{2}-16 B a \,b^{3} d^{3} e +5 b^{4} B \,d^{4}}{5 e^{6} \left (e x +d \right )^{5}}-\frac {2 b^{2} \left (2 A a b \,e^{2}-2 A \,b^{2} d e +3 a^{2} B \,e^{2}-8 B a b d e +5 B \,b^{2} d^{2}\right )}{3 e^{6} \left (e x +d \right )^{3}}-\frac {b^{4} B}{e^{6} \left (e x +d \right )}-\frac {b \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +2 B \,e^{3} a^{3}-9 B \,a^{2} b d \,e^{2}+12 B a \,b^{2} d^{2} e -5 B \,b^{3} d^{3}\right )}{2 e^{6} \left (e x +d \right )^{4}}-\frac {b^{3} \left (A b e +4 B a e -5 B b d \right )}{2 e^{6} \left (e x +d \right )^{2}}\) | \(430\) |
| gosper | \(-\frac {30 B \,x^{5} b^{4} e^{5}+15 A \,b^{4} e^{5} x^{4}+60 B \,x^{4} a \,b^{3} e^{5}+75 B \,x^{4} b^{4} d \,e^{4}+40 A \,x^{3} a \,b^{3} e^{5}+20 A \,x^{3} b^{4} d \,e^{4}+60 B \,x^{3} a^{2} b^{2} e^{5}+80 B \,x^{3} a \,b^{3} d \,e^{4}+100 B \,x^{3} b^{4} d^{2} e^{3}+45 A \,x^{2} a^{2} b^{2} e^{5}+30 A \,x^{2} a \,b^{3} d \,e^{4}+15 A \,x^{2} b^{4} d^{2} e^{3}+30 B \,x^{2} a^{3} b \,e^{5}+45 B \,x^{2} a^{2} b^{2} d \,e^{4}+60 B \,x^{2} a \,b^{3} d^{2} e^{3}+75 B \,x^{2} b^{4} d^{3} e^{2}+24 A x \,a^{3} b \,e^{5}+18 A x \,a^{2} b^{2} d \,e^{4}+12 A x a \,b^{3} d^{2} e^{3}+6 A x \,b^{4} d^{3} e^{2}+6 B x \,a^{4} e^{5}+12 B x \,a^{3} b d \,e^{4}+18 B x \,a^{2} b^{2} d^{2} e^{3}+24 B x a \,b^{3} d^{3} e^{2}+30 B x \,b^{4} d^{4} e +5 A \,a^{4} e^{5}+4 A \,a^{3} b d \,e^{4}+3 A \,a^{2} b^{2} d^{2} e^{3}+2 A a \,b^{3} d^{3} e^{2}+A \,b^{4} d^{4} e +B \,a^{4} d \,e^{4}+2 B \,a^{3} b \,d^{2} e^{3}+3 B \,a^{2} b^{2} d^{3} e^{2}+4 B a \,b^{3} d^{4} e +5 b^{4} B \,d^{5}}{30 e^{6} \left (e x +d \right )^{6}}\) | \(467\) |
| parallelrisch | \(-\frac {30 B \,x^{5} b^{4} e^{5}+15 A \,b^{4} e^{5} x^{4}+60 B \,x^{4} a \,b^{3} e^{5}+75 B \,x^{4} b^{4} d \,e^{4}+40 A \,x^{3} a \,b^{3} e^{5}+20 A \,x^{3} b^{4} d \,e^{4}+60 B \,x^{3} a^{2} b^{2} e^{5}+80 B \,x^{3} a \,b^{3} d \,e^{4}+100 B \,x^{3} b^{4} d^{2} e^{3}+45 A \,x^{2} a^{2} b^{2} e^{5}+30 A \,x^{2} a \,b^{3} d \,e^{4}+15 A \,x^{2} b^{4} d^{2} e^{3}+30 B \,x^{2} a^{3} b \,e^{5}+45 B \,x^{2} a^{2} b^{2} d \,e^{4}+60 B \,x^{2} a \,b^{3} d^{2} e^{3}+75 B \,x^{2} b^{4} d^{3} e^{2}+24 A x \,a^{3} b \,e^{5}+18 A x \,a^{2} b^{2} d \,e^{4}+12 A x a \,b^{3} d^{2} e^{3}+6 A x \,b^{4} d^{3} e^{2}+6 B x \,a^{4} e^{5}+12 B x \,a^{3} b d \,e^{4}+18 B x \,a^{2} b^{2} d^{2} e^{3}+24 B x a \,b^{3} d^{3} e^{2}+30 B x \,b^{4} d^{4} e +5 A \,a^{4} e^{5}+4 A \,a^{3} b d \,e^{4}+3 A \,a^{2} b^{2} d^{2} e^{3}+2 A a \,b^{3} d^{3} e^{2}+A \,b^{4} d^{4} e +B \,a^{4} d \,e^{4}+2 B \,a^{3} b \,d^{2} e^{3}+3 B \,a^{2} b^{2} d^{3} e^{2}+4 B a \,b^{3} d^{4} e +5 b^{4} B \,d^{5}}{30 e^{6} \left (e x +d \right )^{6}}\) | \(467\) |
(-B*b^4/e*x^5-1/2*b^3*(A*b*e+4*B*a*e+5*B*b*d)/e^2*x^4-2/3*b^2*(2*A*a*b*e^2 +A*b^2*d*e+3*B*a^2*e^2+4*B*a*b*d*e+5*B*b^2*d^2)/e^3*x^3-1/2*b*(3*A*a^2*b*e ^3+2*A*a*b^2*d*e^2+A*b^3*d^2*e+2*B*a^3*e^3+3*B*a^2*b*d*e^2+4*B*a*b^2*d^2*e +5*B*b^3*d^3)/e^4*x^2-1/5*(4*A*a^3*b*e^4+3*A*a^2*b^2*d*e^3+2*A*a*b^3*d^2*e ^2+A*b^4*d^3*e+B*a^4*e^4+2*B*a^3*b*d*e^3+3*B*a^2*b^2*d^2*e^2+4*B*a*b^3*d^3 *e+5*B*b^4*d^4)/e^5*x-1/30*(5*A*a^4*e^5+4*A*a^3*b*d*e^4+3*A*a^2*b^2*d^2*e^ 3+2*A*a*b^3*d^3*e^2+A*b^4*d^4*e+B*a^4*d*e^4+2*B*a^3*b*d^2*e^3+3*B*a^2*b^2* d^3*e^2+4*B*a*b^3*d^4*e+5*B*b^4*d^5)/e^6)/(e*x+d)^6
Leaf count of result is larger than twice the leaf count of optimal. 453 vs. \(2 (82) = 164\).
Time = 0.30 (sec) , antiderivative size = 453, normalized size of antiderivative = 5.27 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^7} \, dx=-\frac {30 \, B b^{4} e^{5} x^{5} + 5 \, B b^{4} d^{5} + 5 \, A a^{4} e^{5} + {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 15 \, {\left (5 \, B b^{4} d e^{4} + {\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 20 \, {\left (5 \, B b^{4} d^{2} e^{3} + {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 15 \, {\left (5 \, B b^{4} d^{3} e^{2} + {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} + {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 6 \, {\left (5 \, B b^{4} d^{4} e + {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{30 \, {\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} e^{6}\right )}} \]
-1/30*(30*B*b^4*e^5*x^5 + 5*B*b^4*d^5 + 5*A*a^4*e^5 + (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 + (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 + 15*(5*B*b^4*d*e^4 + (4*B*a*b^3 + A*b^4)* e^5)*x^4 + 20*(5*B*b^4*d^2*e^3 + (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 + 15*(5*B*b^4*d^3*e^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 + (2*B*a^3*b + 3*A*a^2*b^2)*e^5)*x^2 + 6 *(5*B*b^4*d^4*e + (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 + (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 ^12*x^6 + 6*d*e^11*x^5 + 15*d^2*e^10*x^4 + 20*d^3*e^9*x^3 + 15*d^4*e^8*x^2 + 6*d^5*e^7*x + d^6*e^6)
Timed out. \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^7} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 453 vs. \(2 (82) = 164\).
Time = 0.27 (sec) , antiderivative size = 453, normalized size of antiderivative = 5.27 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^7} \, dx=-\frac {30 \, B b^{4} e^{5} x^{5} + 5 \, B b^{4} d^{5} + 5 \, A a^{4} e^{5} + {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 15 \, {\left (5 \, B b^{4} d e^{4} + {\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 20 \, {\left (5 \, B b^{4} d^{2} e^{3} + {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 15 \, {\left (5 \, B b^{4} d^{3} e^{2} + {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} + {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 6 \, {\left (5 \, B b^{4} d^{4} e + {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{30 \, {\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} e^{6}\right )}} \]
-1/30*(30*B*b^4*e^5*x^5 + 5*B*b^4*d^5 + 5*A*a^4*e^5 + (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 + (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 + 15*(5*B*b^4*d*e^4 + (4*B*a*b^3 + A*b^4)* e^5)*x^4 + 20*(5*B*b^4*d^2*e^3 + (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 + 15*(5*B*b^4*d^3*e^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 + (2*B*a^3*b + 3*A*a^2*b^2)*e^5)*x^2 + 6 *(5*B*b^4*d^4*e + (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 + (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 ^12*x^6 + 6*d*e^11*x^5 + 15*d^2*e^10*x^4 + 20*d^3*e^9*x^3 + 15*d^4*e^8*x^2 + 6*d^5*e^7*x + d^6*e^6)
Leaf count of result is larger than twice the leaf count of optimal. 466 vs. \(2 (82) = 164\).
Time = 0.29 (sec) , antiderivative size = 466, normalized size of antiderivative = 5.42 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^7} \, dx=-\frac {30 \, B b^{4} e^{5} x^{5} + 75 \, B b^{4} d e^{4} x^{4} + 60 \, B a b^{3} e^{5} x^{4} + 15 \, A b^{4} e^{5} x^{4} + 100 \, B b^{4} d^{2} e^{3} x^{3} + 80 \, B a b^{3} d e^{4} x^{3} + 20 \, A b^{4} d e^{4} x^{3} + 60 \, B a^{2} b^{2} e^{5} x^{3} + 40 \, A a b^{3} e^{5} x^{3} + 75 \, B b^{4} d^{3} e^{2} x^{2} + 60 \, B a b^{3} d^{2} e^{3} x^{2} + 15 \, A b^{4} d^{2} e^{3} x^{2} + 45 \, B a^{2} b^{2} d e^{4} x^{2} + 30 \, A a b^{3} d e^{4} x^{2} + 30 \, B a^{3} b e^{5} x^{2} + 45 \, A a^{2} b^{2} e^{5} x^{2} + 30 \, B b^{4} d^{4} e x + 24 \, B a b^{3} d^{3} e^{2} x + 6 \, A b^{4} d^{3} e^{2} x + 18 \, B a^{2} b^{2} d^{2} e^{3} x + 12 \, A a b^{3} d^{2} e^{3} x + 12 \, B a^{3} b d e^{4} x + 18 \, A a^{2} b^{2} d e^{4} x + 6 \, B a^{4} e^{5} x + 24 \, A a^{3} b e^{5} x + 5 \, B b^{4} d^{5} + 4 \, B a b^{3} d^{4} e + A b^{4} d^{4} e + 3 \, B a^{2} b^{2} d^{3} e^{2} + 2 \, A a b^{3} d^{3} e^{2} + 2 \, B a^{3} b d^{2} e^{3} + 3 \, A a^{2} b^{2} d^{2} e^{3} + B a^{4} d e^{4} + 4 \, A a^{3} b d e^{4} + 5 \, A a^{4} e^{5}}{30 \, {\left (e x + d\right )}^{6} e^{6}} \]
-1/30*(30*B*b^4*e^5*x^5 + 75*B*b^4*d*e^4*x^4 + 60*B*a*b^3*e^5*x^4 + 15*A*b ^4*e^5*x^4 + 100*B*b^4*d^2*e^3*x^3 + 80*B*a*b^3*d*e^4*x^3 + 20*A*b^4*d*e^4 *x^3 + 60*B*a^2*b^2*e^5*x^3 + 40*A*a*b^3*e^5*x^3 + 75*B*b^4*d^3*e^2*x^2 + 60*B*a*b^3*d^2*e^3*x^2 + 15*A*b^4*d^2*e^3*x^2 + 45*B*a^2*b^2*d*e^4*x^2 + 3 0*A*a*b^3*d*e^4*x^2 + 30*B*a^3*b*e^5*x^2 + 45*A*a^2*b^2*e^5*x^2 + 30*B*b^4 *d^4*e*x + 24*B*a*b^3*d^3*e^2*x + 6*A*b^4*d^3*e^2*x + 18*B*a^2*b^2*d^2*e^3 *x + 12*A*a*b^3*d^2*e^3*x + 12*B*a^3*b*d*e^4*x + 18*A*a^2*b^2*d*e^4*x + 6* B*a^4*e^5*x + 24*A*a^3*b*e^5*x + 5*B*b^4*d^5 + 4*B*a*b^3*d^4*e + A*b^4*d^4 *e + 3*B*a^2*b^2*d^3*e^2 + 2*A*a*b^3*d^3*e^2 + 2*B*a^3*b*d^2*e^3 + 3*A*a^2 *b^2*d^2*e^3 + B*a^4*d*e^4 + 4*A*a^3*b*d*e^4 + 5*A*a^4*e^5)/((e*x + d)^6*e ^6)
Time = 10.85 (sec) , antiderivative size = 460, normalized size of antiderivative = 5.35 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^7} \, dx=-\frac {\frac {B\,a^4\,d\,e^4+5\,A\,a^4\,e^5+2\,B\,a^3\,b\,d^2\,e^3+4\,A\,a^3\,b\,d\,e^4+3\,B\,a^2\,b^2\,d^3\,e^2+3\,A\,a^2\,b^2\,d^2\,e^3+4\,B\,a\,b^3\,d^4\,e+2\,A\,a\,b^3\,d^3\,e^2+5\,B\,b^4\,d^5+A\,b^4\,d^4\,e}{30\,e^6}+\frac {x\,\left (B\,a^4\,e^4+2\,B\,a^3\,b\,d\,e^3+4\,A\,a^3\,b\,e^4+3\,B\,a^2\,b^2\,d^2\,e^2+3\,A\,a^2\,b^2\,d\,e^3+4\,B\,a\,b^3\,d^3\,e+2\,A\,a\,b^3\,d^2\,e^2+5\,B\,b^4\,d^4+A\,b^4\,d^3\,e\right )}{5\,e^5}+\frac {b^3\,x^4\,\left (A\,b\,e+4\,B\,a\,e+5\,B\,b\,d\right )}{2\,e^2}+\frac {b\,x^2\,\left (2\,B\,a^3\,e^3+3\,B\,a^2\,b\,d\,e^2+3\,A\,a^2\,b\,e^3+4\,B\,a\,b^2\,d^2\,e+2\,A\,a\,b^2\,d\,e^2+5\,B\,b^3\,d^3+A\,b^3\,d^2\,e\right )}{2\,e^4}+\frac {2\,b^2\,x^3\,\left (3\,B\,a^2\,e^2+4\,B\,a\,b\,d\,e+2\,A\,a\,b\,e^2+5\,B\,b^2\,d^2+A\,b^2\,d\,e\right )}{3\,e^3}+\frac {B\,b^4\,x^5}{e}}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6} \]
-((5*A*a^4*e^5 + 5*B*b^4*d^5 + A*b^4*d^4*e + B*a^4*d*e^4 + 2*A*a*b^3*d^3*e ^2 + 2*B*a^3*b*d^2*e^3 + 3*A*a^2*b^2*d^2*e^3 + 3*B*a^2*b^2*d^3*e^2 + 4*A*a ^3*b*d*e^4 + 4*B*a*b^3*d^4*e)/(30*e^6) + (x*(B*a^4*e^4 + 5*B*b^4*d^4 + 4*A *a^3*b*e^4 + A*b^4*d^3*e + 2*A*a*b^3*d^2*e^2 + 3*A*a^2*b^2*d*e^3 + 3*B*a^2 *b^2*d^2*e^2 + 4*B*a*b^3*d^3*e + 2*B*a^3*b*d*e^3))/(5*e^5) + (b^3*x^4*(A*b *e + 4*B*a*e + 5*B*b*d))/(2*e^2) + (b*x^2*(2*B*a^3*e^3 + 5*B*b^3*d^3 + 3*A *a^2*b*e^3 + A*b^3*d^2*e + 2*A*a*b^2*d*e^2 + 4*B*a*b^2*d^2*e + 3*B*a^2*b*d *e^2))/(2*e^4) + (2*b^2*x^3*(3*B*a^2*e^2 + 5*B*b^2*d^2 + 2*A*a*b*e^2 + A*b ^2*d*e + 4*B*a*b*d*e))/(3*e^3) + (B*b^4*x^5)/e)/(d^6 + e^6*x^6 + 6*d*e^5*x ^5 + 15*d^4*e^2*x^2 + 20*d^3*e^3*x^3 + 15*d^2*e^4*x^4 + 6*d^5*e*x)