Integrand size = 26, antiderivative size = 58 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^9} \, dx=\frac {(a+b x)^7}{8 (b d-a e) (d+e x)^8}+\frac {b (a+b x)^7}{56 (b d-a e)^2 (d+e x)^7} \]
Leaf count is larger than twice the leaf count of optimal. \(277\) vs. \(2(58)=116\).
Time = 0.06 (sec) , antiderivative size = 277, normalized size of antiderivative = 4.78 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^9} \, dx=-\frac {7 a^6 e^6+6 a^5 b e^5 (d+8 e x)+5 a^4 b^2 e^4 \left (d^2+8 d e x+28 e^2 x^2\right )+4 a^3 b^3 e^3 \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+3 a^2 b^4 e^2 \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )+2 a b^5 e \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )+b^6 \left (d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right )}{56 e^7 (d+e x)^8} \]
-1/56*(7*a^6*e^6 + 6*a^5*b*e^5*(d + 8*e*x) + 5*a^4*b^2*e^4*(d^2 + 8*d*e*x + 28*e^2*x^2) + 4*a^3*b^3*e^3*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3 ) + 3*a^2*b^4*e^2*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^ 4*x^4) + 2*a*b^5*e*(d^5 + 8*d^4*e*x + 28*d^3*e^2*x^2 + 56*d^2*e^3*x^3 + 70 *d*e^4*x^4 + 56*e^5*x^5) + b^6*(d^6 + 8*d^5*e*x + 28*d^4*e^2*x^2 + 56*d^3* e^3*x^3 + 70*d^2*e^4*x^4 + 56*d*e^5*x^5 + 28*e^6*x^6))/(e^7*(d + e*x)^8)
Time = 0.18 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1098, 27, 55, 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.
| \(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^9} \, dx\) |
| \(\Big \downarrow \) 1098 |
| \(\displaystyle \frac {\int \frac {b^6 (a+b x)^6}{(d+e x)^9}dx}{b^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(a+b x)^6}{(d+e x)^9}dx\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {b \int \frac {(a+b x)^6}{(d+e x)^8}dx}{8 (b d-a e)}+\frac {(a+b x)^7}{8 (d+e x)^8 (b d-a e)}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {b (a+b x)^7}{56 (d+e x)^7 (b d-a e)^2}+\frac {(a+b x)^7}{8 (d+e x)^8 (b d-a e)}\) |
3.15.98.3.1 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_))^(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] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[ {a, b, c, d, e, m}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(334\) vs. \(2(54)=108\).
Time = 2.25 (sec) , antiderivative size = 335, normalized size of antiderivative = 5.78
| method | result | size |
| risch | \(\frac {-\frac {b^{6} x^{6}}{2 e}-\frac {b^{5} \left (2 a e +b d \right ) x^{5}}{e^{2}}-\frac {5 b^{4} \left (3 a^{2} e^{2}+2 a b d e +b^{2} d^{2}\right ) x^{4}}{4 e^{3}}-\frac {b^{3} \left (4 a^{3} e^{3}+3 a^{2} b d \,e^{2}+2 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{3}}{e^{4}}-\frac {b^{2} \left (5 e^{4} a^{4}+4 b \,e^{3} d \,a^{3}+3 b^{2} e^{2} d^{2} a^{2}+2 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x^{2}}{2 e^{5}}-\frac {b \left (6 a^{5} e^{5}+5 a^{4} b d \,e^{4}+4 a^{3} b^{2} d^{2} e^{3}+3 a^{2} b^{3} d^{3} e^{2}+2 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) x}{7 e^{6}}-\frac {7 a^{6} e^{6}+6 a^{5} b d \,e^{5}+5 a^{4} b^{2} d^{2} e^{4}+4 a^{3} b^{3} d^{3} e^{3}+3 a^{2} b^{4} d^{4} e^{2}+2 a \,b^{5} d^{5} e +b^{6} d^{6}}{56 e^{7}}}{\left (e x +d \right )^{8}}\) | \(335\) |
| default | \(-\frac {4 b^{3} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{e^{7} \left (e x +d \right )^{5}}-\frac {a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}}{8 e^{7} \left (e x +d \right )^{8}}-\frac {2 b^{5} \left (a e -b d \right )}{e^{7} \left (e x +d \right )^{3}}-\frac {6 b \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}{7 e^{7} \left (e x +d \right )^{7}}-\frac {15 b^{4} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{4 e^{7} \left (e x +d \right )^{4}}-\frac {b^{6}}{2 e^{7} \left (e x +d \right )^{2}}-\frac {5 b^{2} \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{2 e^{7} \left (e x +d \right )^{6}}\) | \(357\) |
| norman | \(\frac {-\frac {b^{6} x^{6}}{2 e}-\frac {\left (2 e^{2} a \,b^{5}+d e \,b^{6}\right ) x^{5}}{e^{3}}-\frac {5 \left (3 e^{3} a^{2} b^{4}+2 d \,e^{2} a \,b^{5}+d^{2} e \,b^{6}\right ) x^{4}}{4 e^{4}}-\frac {\left (4 e^{4} a^{3} b^{3}+3 d \,e^{3} a^{2} b^{4}+2 d^{2} e^{2} a \,b^{5}+d^{3} e \,b^{6}\right ) x^{3}}{e^{5}}-\frac {\left (5 e^{5} a^{4} b^{2}+4 d \,e^{4} a^{3} b^{3}+3 d^{2} e^{3} a^{2} b^{4}+2 d^{3} e^{2} a \,b^{5}+d^{4} e \,b^{6}\right ) x^{2}}{2 e^{6}}-\frac {\left (6 a^{5} b \,e^{6}+5 a^{4} b^{2} d \,e^{5}+4 a^{3} b^{3} d^{2} e^{4}+3 a^{2} b^{4} d^{3} e^{3}+2 a \,b^{5} d^{4} e^{2}+b^{6} d^{5} e \right ) x}{7 e^{7}}-\frac {7 a^{6} e^{7}+6 a^{5} b d \,e^{6}+5 a^{4} b^{2} d^{2} e^{5}+4 a^{3} b^{3} d^{3} e^{4}+3 a^{2} b^{4} d^{4} e^{3}+2 a \,b^{5} d^{5} e^{2}+b^{6} d^{6} e}{56 e^{8}}}{\left (e x +d \right )^{8}}\) | \(363\) |
| gosper | \(-\frac {28 x^{6} b^{6} e^{6}+112 x^{5} a \,b^{5} e^{6}+56 x^{5} b^{6} d \,e^{5}+210 x^{4} a^{2} b^{4} e^{6}+140 x^{4} a \,b^{5} d \,e^{5}+70 x^{4} b^{6} d^{2} e^{4}+224 x^{3} a^{3} b^{3} e^{6}+168 x^{3} a^{2} b^{4} d \,e^{5}+112 x^{3} a \,b^{5} d^{2} e^{4}+56 x^{3} b^{6} d^{3} e^{3}+140 x^{2} a^{4} b^{2} e^{6}+112 x^{2} a^{3} b^{3} d \,e^{5}+84 x^{2} a^{2} b^{4} d^{2} e^{4}+56 x^{2} a \,b^{5} d^{3} e^{3}+28 x^{2} b^{6} d^{4} e^{2}+48 x \,a^{5} b \,e^{6}+40 x \,a^{4} b^{2} d \,e^{5}+32 x \,a^{3} b^{3} d^{2} e^{4}+24 x \,a^{2} b^{4} d^{3} e^{3}+16 x a \,b^{5} d^{4} e^{2}+8 x \,b^{6} d^{5} e +7 a^{6} e^{6}+6 a^{5} b d \,e^{5}+5 a^{4} b^{2} d^{2} e^{4}+4 a^{3} b^{3} d^{3} e^{3}+3 a^{2} b^{4} d^{4} e^{2}+2 a \,b^{5} d^{5} e +b^{6} d^{6}}{56 e^{7} \left (e x +d \right )^{8}}\) | \(376\) |
| parallelrisch | \(\frac {-28 b^{6} x^{6} e^{7}-112 a \,b^{5} e^{7} x^{5}-56 b^{6} d \,e^{6} x^{5}-210 a^{2} b^{4} e^{7} x^{4}-140 a \,b^{5} d \,e^{6} x^{4}-70 b^{6} d^{2} e^{5} x^{4}-224 a^{3} b^{3} e^{7} x^{3}-168 a^{2} b^{4} d \,e^{6} x^{3}-112 a \,b^{5} d^{2} e^{5} x^{3}-56 b^{6} d^{3} e^{4} x^{3}-140 a^{4} b^{2} e^{7} x^{2}-112 a^{3} b^{3} d \,e^{6} x^{2}-84 a^{2} b^{4} d^{2} e^{5} x^{2}-56 a \,b^{5} d^{3} e^{4} x^{2}-28 b^{6} d^{4} e^{3} x^{2}-48 a^{5} b \,e^{7} x -40 a^{4} b^{2} d \,e^{6} x -32 a^{3} b^{3} d^{2} e^{5} x -24 a^{2} b^{4} d^{3} e^{4} x -16 a \,b^{5} d^{4} e^{3} x -8 b^{6} d^{5} e^{2} x -7 a^{6} e^{7}-6 a^{5} b d \,e^{6}-5 a^{4} b^{2} d^{2} e^{5}-4 a^{3} b^{3} d^{3} e^{4}-3 a^{2} b^{4} d^{4} e^{3}-2 a \,b^{5} d^{5} e^{2}-b^{6} d^{6} e}{56 e^{8} \left (e x +d \right )^{8}}\) | \(382\) |
(-1/2*b^6/e*x^6-b^5/e^2*(2*a*e+b*d)*x^5-5/4/e^3*b^4*(3*a^2*e^2+2*a*b*d*e+b ^2*d^2)*x^4-b^3/e^4*(4*a^3*e^3+3*a^2*b*d*e^2+2*a*b^2*d^2*e+b^3*d^3)*x^3-1/ 2*b^2/e^5*(5*a^4*e^4+4*a^3*b*d*e^3+3*a^2*b^2*d^2*e^2+2*a*b^3*d^3*e+b^4*d^4 )*x^2-1/7*b/e^6*(6*a^5*e^5+5*a^4*b*d*e^4+4*a^3*b^2*d^2*e^3+3*a^2*b^3*d^3*e ^2+2*a*b^4*d^4*e+b^5*d^5)*x-1/56/e^7*(7*a^6*e^6+6*a^5*b*d*e^5+5*a^4*b^2*d^ 2*e^4+4*a^3*b^3*d^3*e^3+3*a^2*b^4*d^4*e^2+2*a*b^5*d^5*e+b^6*d^6))/(e*x+d)^ 8
Leaf count of result is larger than twice the leaf count of optimal. 430 vs. \(2 (54) = 108\).
Time = 0.27 (sec) , antiderivative size = 430, normalized size of antiderivative = 7.41 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^9} \, dx=-\frac {28 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 2 \, a b^{5} d^{5} e + 3 \, a^{2} b^{4} d^{4} e^{2} + 4 \, a^{3} b^{3} d^{3} e^{3} + 5 \, a^{4} b^{2} d^{2} e^{4} + 6 \, a^{5} b d e^{5} + 7 \, a^{6} e^{6} + 56 \, {\left (b^{6} d e^{5} + 2 \, a b^{5} e^{6}\right )} x^{5} + 70 \, {\left (b^{6} d^{2} e^{4} + 2 \, a b^{5} d e^{5} + 3 \, a^{2} b^{4} e^{6}\right )} x^{4} + 56 \, {\left (b^{6} d^{3} e^{3} + 2 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} + 4 \, a^{3} b^{3} e^{6}\right )} x^{3} + 28 \, {\left (b^{6} d^{4} e^{2} + 2 \, a b^{5} d^{3} e^{3} + 3 \, a^{2} b^{4} d^{2} e^{4} + 4 \, a^{3} b^{3} d e^{5} + 5 \, a^{4} b^{2} e^{6}\right )} x^{2} + 8 \, {\left (b^{6} d^{5} e + 2 \, a b^{5} d^{4} e^{2} + 3 \, a^{2} b^{4} d^{3} e^{3} + 4 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} + 6 \, a^{5} b e^{6}\right )} x}{56 \, {\left (e^{15} x^{8} + 8 \, d e^{14} x^{7} + 28 \, d^{2} e^{13} x^{6} + 56 \, d^{3} e^{12} x^{5} + 70 \, d^{4} e^{11} x^{4} + 56 \, d^{5} e^{10} x^{3} + 28 \, d^{6} e^{9} x^{2} + 8 \, d^{7} e^{8} x + d^{8} e^{7}\right )}} \]
-1/56*(28*b^6*e^6*x^6 + b^6*d^6 + 2*a*b^5*d^5*e + 3*a^2*b^4*d^4*e^2 + 4*a^ 3*b^3*d^3*e^3 + 5*a^4*b^2*d^2*e^4 + 6*a^5*b*d*e^5 + 7*a^6*e^6 + 56*(b^6*d* e^5 + 2*a*b^5*e^6)*x^5 + 70*(b^6*d^2*e^4 + 2*a*b^5*d*e^5 + 3*a^2*b^4*e^6)* x^4 + 56*(b^6*d^3*e^3 + 2*a*b^5*d^2*e^4 + 3*a^2*b^4*d*e^5 + 4*a^3*b^3*e^6) *x^3 + 28*(b^6*d^4*e^2 + 2*a*b^5*d^3*e^3 + 3*a^2*b^4*d^2*e^4 + 4*a^3*b^3*d *e^5 + 5*a^4*b^2*e^6)*x^2 + 8*(b^6*d^5*e + 2*a*b^5*d^4*e^2 + 3*a^2*b^4*d^3 *e^3 + 4*a^3*b^3*d^2*e^4 + 5*a^4*b^2*d*e^5 + 6*a^5*b*e^6)*x)/(e^15*x^8 + 8 *d*e^14*x^7 + 28*d^2*e^13*x^6 + 56*d^3*e^12*x^5 + 70*d^4*e^11*x^4 + 56*d^5 *e^10*x^3 + 28*d^6*e^9*x^2 + 8*d^7*e^8*x + d^8*e^7)
Timed out. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^9} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 430 vs. \(2 (54) = 108\).
Time = 0.26 (sec) , antiderivative size = 430, normalized size of antiderivative = 7.41 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^9} \, dx=-\frac {28 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 2 \, a b^{5} d^{5} e + 3 \, a^{2} b^{4} d^{4} e^{2} + 4 \, a^{3} b^{3} d^{3} e^{3} + 5 \, a^{4} b^{2} d^{2} e^{4} + 6 \, a^{5} b d e^{5} + 7 \, a^{6} e^{6} + 56 \, {\left (b^{6} d e^{5} + 2 \, a b^{5} e^{6}\right )} x^{5} + 70 \, {\left (b^{6} d^{2} e^{4} + 2 \, a b^{5} d e^{5} + 3 \, a^{2} b^{4} e^{6}\right )} x^{4} + 56 \, {\left (b^{6} d^{3} e^{3} + 2 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} + 4 \, a^{3} b^{3} e^{6}\right )} x^{3} + 28 \, {\left (b^{6} d^{4} e^{2} + 2 \, a b^{5} d^{3} e^{3} + 3 \, a^{2} b^{4} d^{2} e^{4} + 4 \, a^{3} b^{3} d e^{5} + 5 \, a^{4} b^{2} e^{6}\right )} x^{2} + 8 \, {\left (b^{6} d^{5} e + 2 \, a b^{5} d^{4} e^{2} + 3 \, a^{2} b^{4} d^{3} e^{3} + 4 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} + 6 \, a^{5} b e^{6}\right )} x}{56 \, {\left (e^{15} x^{8} + 8 \, d e^{14} x^{7} + 28 \, d^{2} e^{13} x^{6} + 56 \, d^{3} e^{12} x^{5} + 70 \, d^{4} e^{11} x^{4} + 56 \, d^{5} e^{10} x^{3} + 28 \, d^{6} e^{9} x^{2} + 8 \, d^{7} e^{8} x + d^{8} e^{7}\right )}} \]
-1/56*(28*b^6*e^6*x^6 + b^6*d^6 + 2*a*b^5*d^5*e + 3*a^2*b^4*d^4*e^2 + 4*a^ 3*b^3*d^3*e^3 + 5*a^4*b^2*d^2*e^4 + 6*a^5*b*d*e^5 + 7*a^6*e^6 + 56*(b^6*d* e^5 + 2*a*b^5*e^6)*x^5 + 70*(b^6*d^2*e^4 + 2*a*b^5*d*e^5 + 3*a^2*b^4*e^6)* x^4 + 56*(b^6*d^3*e^3 + 2*a*b^5*d^2*e^4 + 3*a^2*b^4*d*e^5 + 4*a^3*b^3*e^6) *x^3 + 28*(b^6*d^4*e^2 + 2*a*b^5*d^3*e^3 + 3*a^2*b^4*d^2*e^4 + 4*a^3*b^3*d *e^5 + 5*a^4*b^2*e^6)*x^2 + 8*(b^6*d^5*e + 2*a*b^5*d^4*e^2 + 3*a^2*b^4*d^3 *e^3 + 4*a^3*b^3*d^2*e^4 + 5*a^4*b^2*d*e^5 + 6*a^5*b*e^6)*x)/(e^15*x^8 + 8 *d*e^14*x^7 + 28*d^2*e^13*x^6 + 56*d^3*e^12*x^5 + 70*d^4*e^11*x^4 + 56*d^5 *e^10*x^3 + 28*d^6*e^9*x^2 + 8*d^7*e^8*x + d^8*e^7)
Leaf count of result is larger than twice the leaf count of optimal. 375 vs. \(2 (54) = 108\).
Time = 0.25 (sec) , antiderivative size = 375, normalized size of antiderivative = 6.47 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^9} \, dx=-\frac {28 \, b^{6} e^{6} x^{6} + 56 \, b^{6} d e^{5} x^{5} + 112 \, a b^{5} e^{6} x^{5} + 70 \, b^{6} d^{2} e^{4} x^{4} + 140 \, a b^{5} d e^{5} x^{4} + 210 \, a^{2} b^{4} e^{6} x^{4} + 56 \, b^{6} d^{3} e^{3} x^{3} + 112 \, a b^{5} d^{2} e^{4} x^{3} + 168 \, a^{2} b^{4} d e^{5} x^{3} + 224 \, a^{3} b^{3} e^{6} x^{3} + 28 \, b^{6} d^{4} e^{2} x^{2} + 56 \, a b^{5} d^{3} e^{3} x^{2} + 84 \, a^{2} b^{4} d^{2} e^{4} x^{2} + 112 \, a^{3} b^{3} d e^{5} x^{2} + 140 \, a^{4} b^{2} e^{6} x^{2} + 8 \, b^{6} d^{5} e x + 16 \, a b^{5} d^{4} e^{2} x + 24 \, a^{2} b^{4} d^{3} e^{3} x + 32 \, a^{3} b^{3} d^{2} e^{4} x + 40 \, a^{4} b^{2} d e^{5} x + 48 \, a^{5} b e^{6} x + b^{6} d^{6} + 2 \, a b^{5} d^{5} e + 3 \, a^{2} b^{4} d^{4} e^{2} + 4 \, a^{3} b^{3} d^{3} e^{3} + 5 \, a^{4} b^{2} d^{2} e^{4} + 6 \, a^{5} b d e^{5} + 7 \, a^{6} e^{6}}{56 \, {\left (e x + d\right )}^{8} e^{7}} \]
-1/56*(28*b^6*e^6*x^6 + 56*b^6*d*e^5*x^5 + 112*a*b^5*e^6*x^5 + 70*b^6*d^2* e^4*x^4 + 140*a*b^5*d*e^5*x^4 + 210*a^2*b^4*e^6*x^4 + 56*b^6*d^3*e^3*x^3 + 112*a*b^5*d^2*e^4*x^3 + 168*a^2*b^4*d*e^5*x^3 + 224*a^3*b^3*e^6*x^3 + 28* b^6*d^4*e^2*x^2 + 56*a*b^5*d^3*e^3*x^2 + 84*a^2*b^4*d^2*e^4*x^2 + 112*a^3* b^3*d*e^5*x^2 + 140*a^4*b^2*e^6*x^2 + 8*b^6*d^5*e*x + 16*a*b^5*d^4*e^2*x + 24*a^2*b^4*d^3*e^3*x + 32*a^3*b^3*d^2*e^4*x + 40*a^4*b^2*d*e^5*x + 48*a^5 *b*e^6*x + b^6*d^6 + 2*a*b^5*d^5*e + 3*a^2*b^4*d^4*e^2 + 4*a^3*b^3*d^3*e^3 + 5*a^4*b^2*d^2*e^4 + 6*a^5*b*d*e^5 + 7*a^6*e^6)/((e*x + d)^8*e^7)
Time = 0.13 (sec) , antiderivative size = 410, normalized size of antiderivative = 7.07 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^9} \, dx=-\frac {\frac {7\,a^6\,e^6+6\,a^5\,b\,d\,e^5+5\,a^4\,b^2\,d^2\,e^4+4\,a^3\,b^3\,d^3\,e^3+3\,a^2\,b^4\,d^4\,e^2+2\,a\,b^5\,d^5\,e+b^6\,d^6}{56\,e^7}+\frac {b^6\,x^6}{2\,e}+\frac {b^3\,x^3\,\left (4\,a^3\,e^3+3\,a^2\,b\,d\,e^2+2\,a\,b^2\,d^2\,e+b^3\,d^3\right )}{e^4}+\frac {b\,x\,\left (6\,a^5\,e^5+5\,a^4\,b\,d\,e^4+4\,a^3\,b^2\,d^2\,e^3+3\,a^2\,b^3\,d^3\,e^2+2\,a\,b^4\,d^4\,e+b^5\,d^5\right )}{7\,e^6}+\frac {b^5\,x^5\,\left (2\,a\,e+b\,d\right )}{e^2}+\frac {b^2\,x^2\,\left (5\,a^4\,e^4+4\,a^3\,b\,d\,e^3+3\,a^2\,b^2\,d^2\,e^2+2\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{2\,e^5}+\frac {5\,b^4\,x^4\,\left (3\,a^2\,e^2+2\,a\,b\,d\,e+b^2\,d^2\right )}{4\,e^3}}{d^8+8\,d^7\,e\,x+28\,d^6\,e^2\,x^2+56\,d^5\,e^3\,x^3+70\,d^4\,e^4\,x^4+56\,d^3\,e^5\,x^5+28\,d^2\,e^6\,x^6+8\,d\,e^7\,x^7+e^8\,x^8} \]
-((7*a^6*e^6 + b^6*d^6 + 3*a^2*b^4*d^4*e^2 + 4*a^3*b^3*d^3*e^3 + 5*a^4*b^2 *d^2*e^4 + 2*a*b^5*d^5*e + 6*a^5*b*d*e^5)/(56*e^7) + (b^6*x^6)/(2*e) + (b^ 3*x^3*(4*a^3*e^3 + b^3*d^3 + 2*a*b^2*d^2*e + 3*a^2*b*d*e^2))/e^4 + (b*x*(6 *a^5*e^5 + b^5*d^5 + 3*a^2*b^3*d^3*e^2 + 4*a^3*b^2*d^2*e^3 + 2*a*b^4*d^4*e + 5*a^4*b*d*e^4))/(7*e^6) + (b^5*x^5*(2*a*e + b*d))/e^2 + (b^2*x^2*(5*a^4 *e^4 + b^4*d^4 + 3*a^2*b^2*d^2*e^2 + 2*a*b^3*d^3*e + 4*a^3*b*d*e^3))/(2*e^ 5) + (5*b^4*x^4*(3*a^2*e^2 + b^2*d^2 + 2*a*b*d*e))/(4*e^3))/(d^8 + e^8*x^8 + 8*d*e^7*x^7 + 28*d^6*e^2*x^2 + 56*d^5*e^3*x^3 + 70*d^4*e^4*x^4 + 56*d^3 *e^5*x^5 + 28*d^2*e^6*x^6 + 8*d^7*e*x)