Integrand size = 26, antiderivative size = 89 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{10}} \, dx=\frac {(a+b x)^7}{9 (b d-a e) (d+e x)^9}+\frac {b (a+b x)^7}{36 (b d-a e)^2 (d+e x)^8}+\frac {b^2 (a+b x)^7}{252 (b d-a e)^3 (d+e x)^7} \]
1/9*(b*x+a)^7/(-a*e+b*d)/(e*x+d)^9+1/36*b*(b*x+a)^7/(-a*e+b*d)^2/(e*x+d)^8 +1/252*b^2*(b*x+a)^7/(-a*e+b*d)^3/(e*x+d)^7
Leaf count is larger than twice the leaf count of optimal. \(277\) vs. \(2(89)=178\).
Time = 0.06 (sec) , antiderivative size = 277, normalized size of antiderivative = 3.11 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{10}} \, dx=-\frac {28 a^6 e^6+21 a^5 b e^5 (d+9 e x)+15 a^4 b^2 e^4 \left (d^2+9 d e x+36 e^2 x^2\right )+10 a^3 b^3 e^3 \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+6 a^2 b^4 e^2 \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )+3 a b^5 e \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )+b^6 \left (d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right )}{252 e^7 (d+e x)^9} \]
-1/252*(28*a^6*e^6 + 21*a^5*b*e^5*(d + 9*e*x) + 15*a^4*b^2*e^4*(d^2 + 9*d* e*x + 36*e^2*x^2) + 10*a^3*b^3*e^3*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^ 3*x^3) + 6*a^2*b^4*e^2*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4) + 3*a*b^5*e*(d^5 + 9*d^4*e*x + 36*d^3*e^2*x^2 + 84*d^2*e^3*x^ 3 + 126*d*e^4*x^4 + 126*e^5*x^5) + b^6*(d^6 + 9*d^5*e*x + 36*d^4*e^2*x^2 + 84*d^3*e^3*x^3 + 126*d^2*e^4*x^4 + 126*d*e^5*x^5 + 84*e^6*x^6))/(e^7*(d + e*x)^9)
Time = 0.21 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.15, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1098, 27, 55, 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.
| \(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{10}} \, dx\) |
| \(\Big \downarrow \) 1098 |
| \(\displaystyle \frac {\int \frac {b^6 (a+b x)^6}{(d+e x)^{10}}dx}{b^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(a+b x)^6}{(d+e x)^{10}}dx\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {2 b \int \frac {(a+b x)^6}{(d+e x)^9}dx}{9 (b d-a e)}+\frac {(a+b x)^7}{9 (d+e x)^9 (b d-a e)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {2 b \left (\frac {b \int \frac {(a+b x)^6}{(d+e x)^8}dx}{8 (b d-a e)}+\frac {(a+b x)^7}{8 (d+e x)^8 (b d-a e)}\right )}{9 (b d-a e)}+\frac {(a+b x)^7}{9 (d+e x)^9 (b d-a e)}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {(a+b x)^7}{9 (d+e x)^9 (b d-a e)}+\frac {2 b \left (\frac {b (a+b x)^7}{56 (d+e x)^7 (b d-a e)^2}+\frac {(a+b x)^7}{8 (d+e x)^8 (b d-a e)}\right )}{9 (b d-a e)}\) |
(a + b*x)^7/(9*(b*d - a*e)*(d + e*x)^9) + (2*b*((a + b*x)^7/(8*(b*d - a*e) *(d + e*x)^8) + (b*(a + b*x)^7)/(56*(b*d - a*e)^2*(d + e*x)^7)))/(9*(b*d - a*e))
3.15.99.3.1 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. \(2(83)=166\).
Time = 2.25 (sec) , antiderivative size = 335, normalized size of antiderivative = 3.76
| method | result | size |
| risch | \(\frac {-\frac {b^{6} x^{6}}{3 e}-\frac {b^{5} \left (3 a e +b d \right ) x^{5}}{2 e^{2}}-\frac {b^{4} \left (6 a^{2} e^{2}+3 a b d e +b^{2} d^{2}\right ) x^{4}}{2 e^{3}}-\frac {b^{3} \left (10 a^{3} e^{3}+6 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{3}}{3 e^{4}}-\frac {b^{2} \left (15 e^{4} a^{4}+10 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}+3 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x^{2}}{7 e^{5}}-\frac {b \left (21 a^{5} e^{5}+15 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}+6 a^{2} b^{3} d^{3} e^{2}+3 a \,b^{4} d^{4} e +b^{5} d^{5}\right ) x}{28 e^{6}}-\frac {28 a^{6} e^{6}+21 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}+10 a^{3} b^{3} d^{3} e^{3}+6 a^{2} b^{4} d^{4} e^{2}+3 a \,b^{5} d^{5} e +b^{6} d^{6}}{252 e^{7}}}{\left (e x +d \right )^{9}}\) | \(335\) |
| default | \(-\frac {3 b^{4} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{e^{7} \left (e x +d \right )^{5}}-\frac {3 b \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}{4 e^{7} \left (e x +d \right )^{8}}-\frac {b^{6}}{3 e^{7} \left (e x +d \right )^{3}}-\frac {15 b^{2} \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{7 e^{7} \left (e x +d \right )^{7}}-\frac {3 b^{5} \left (a e -b d \right )}{2 e^{7} \left (e x +d \right )^{4}}-\frac {10 b^{3} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{3 e^{7} \left (e x +d \right )^{6}}-\frac {a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}}{9 e^{7} \left (e x +d \right )^{9}}\) | \(357\) |
| norman | \(\frac {-\frac {b^{6} x^{6}}{3 e}-\frac {\left (3 e^{3} a \,b^{5}+d \,e^{2} b^{6}\right ) x^{5}}{2 e^{4}}-\frac {\left (6 e^{4} a^{2} b^{4}+3 d \,e^{3} a \,b^{5}+d^{2} e^{2} b^{6}\right ) x^{4}}{2 e^{5}}-\frac {\left (10 e^{5} a^{3} b^{3}+6 a^{2} b^{4} d \,e^{4}+3 d^{2} e^{3} a \,b^{5}+d^{3} e^{2} b^{6}\right ) x^{3}}{3 e^{6}}-\frac {\left (15 a^{4} b^{2} e^{6}+10 a^{3} b^{3} d \,e^{5}+6 a^{2} b^{4} d^{2} e^{4}+3 a \,b^{5} d^{3} e^{3}+b^{6} d^{4} e^{2}\right ) x^{2}}{7 e^{7}}-\frac {\left (21 a^{5} b \,e^{7}+15 a^{4} b^{2} d \,e^{6}+10 a^{3} b^{3} d^{2} e^{5}+6 a^{2} b^{4} d^{3} e^{4}+3 a \,b^{5} d^{4} e^{3}+b^{6} d^{5} e^{2}\right ) x}{28 e^{8}}-\frac {28 a^{6} e^{8}+21 a^{5} b d \,e^{7}+15 a^{4} b^{2} d^{2} e^{6}+10 a^{3} b^{3} d^{3} e^{5}+6 a^{2} b^{4} d^{4} e^{4}+3 a \,b^{5} d^{5} e^{3}+b^{6} d^{6} e^{2}}{252 e^{9}}}{\left (e x +d \right )^{9}}\) | \(375\) |
| gosper | \(-\frac {84 x^{6} b^{6} e^{6}+378 x^{5} a \,b^{5} e^{6}+126 x^{5} b^{6} d \,e^{5}+756 x^{4} a^{2} b^{4} e^{6}+378 x^{4} a \,b^{5} d \,e^{5}+126 x^{4} b^{6} d^{2} e^{4}+840 x^{3} a^{3} b^{3} e^{6}+504 x^{3} a^{2} b^{4} d \,e^{5}+252 x^{3} a \,b^{5} d^{2} e^{4}+84 x^{3} b^{6} d^{3} e^{3}+540 x^{2} a^{4} b^{2} e^{6}+360 x^{2} a^{3} b^{3} d \,e^{5}+216 x^{2} a^{2} b^{4} d^{2} e^{4}+108 x^{2} a \,b^{5} d^{3} e^{3}+36 x^{2} b^{6} d^{4} e^{2}+189 x \,a^{5} b \,e^{6}+135 x \,a^{4} b^{2} d \,e^{5}+90 x \,a^{3} b^{3} d^{2} e^{4}+54 x \,a^{2} b^{4} d^{3} e^{3}+27 x a \,b^{5} d^{4} e^{2}+9 x \,b^{6} d^{5} e +28 a^{6} e^{6}+21 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}+10 a^{3} b^{3} d^{3} e^{3}+6 a^{2} b^{4} d^{4} e^{2}+3 a \,b^{5} d^{5} e +b^{6} d^{6}}{252 e^{7} \left (e x +d \right )^{9}}\) | \(376\) |
| parallelrisch | \(\frac {-84 b^{6} x^{6} e^{8}-378 a \,b^{5} e^{8} x^{5}-126 b^{6} d \,e^{7} x^{5}-756 a^{2} b^{4} e^{8} x^{4}-378 a \,b^{5} d \,e^{7} x^{4}-126 b^{6} d^{2} e^{6} x^{4}-840 a^{3} b^{3} e^{8} x^{3}-504 a^{2} b^{4} d \,e^{7} x^{3}-252 a \,b^{5} d^{2} e^{6} x^{3}-84 b^{6} d^{3} e^{5} x^{3}-540 a^{4} b^{2} e^{8} x^{2}-360 a^{3} b^{3} d \,e^{7} x^{2}-216 a^{2} b^{4} d^{2} e^{6} x^{2}-108 a \,b^{5} d^{3} e^{5} x^{2}-36 b^{6} d^{4} e^{4} x^{2}-189 a^{5} b \,e^{8} x -135 a^{4} b^{2} d \,e^{7} x -90 a^{3} b^{3} d^{2} e^{6} x -54 a^{2} b^{4} d^{3} e^{5} x -27 a \,b^{5} d^{4} e^{4} x -9 b^{6} d^{5} e^{3} x -28 a^{6} e^{8}-21 a^{5} b d \,e^{7}-15 a^{4} b^{2} d^{2} e^{6}-10 a^{3} b^{3} d^{3} e^{5}-6 a^{2} b^{4} d^{4} e^{4}-3 a \,b^{5} d^{5} e^{3}-b^{6} d^{6} e^{2}}{252 e^{9} \left (e x +d \right )^{9}}\) | \(384\) |
(-1/3*b^6/e*x^6-1/2*b^5/e^2*(3*a*e+b*d)*x^5-1/2/e^3*b^4*(6*a^2*e^2+3*a*b*d *e+b^2*d^2)*x^4-1/3/e^4*b^3*(10*a^3*e^3+6*a^2*b*d*e^2+3*a*b^2*d^2*e+b^3*d^ 3)*x^3-1/7*b^2/e^5*(15*a^4*e^4+10*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2+3*a*b^3*d^ 3*e+b^4*d^4)*x^2-1/28/e^6*b*(21*a^5*e^5+15*a^4*b*d*e^4+10*a^3*b^2*d^2*e^3+ 6*a^2*b^3*d^3*e^2+3*a*b^4*d^4*e+b^5*d^5)*x-1/252/e^7*(28*a^6*e^6+21*a^5*b* d*e^5+15*a^4*b^2*d^2*e^4+10*a^3*b^3*d^3*e^3+6*a^2*b^4*d^4*e^2+3*a*b^5*d^5* e+b^6*d^6))/(e*x+d)^9
Leaf count of result is larger than twice the leaf count of optimal. 441 vs. \(2 (83) = 166\).
Time = 0.32 (sec) , antiderivative size = 441, normalized size of antiderivative = 4.96 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{10}} \, dx=-\frac {84 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 3 \, a b^{5} d^{5} e + 6 \, a^{2} b^{4} d^{4} e^{2} + 10 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} + 21 \, a^{5} b d e^{5} + 28 \, a^{6} e^{6} + 126 \, {\left (b^{6} d e^{5} + 3 \, a b^{5} e^{6}\right )} x^{5} + 126 \, {\left (b^{6} d^{2} e^{4} + 3 \, a b^{5} d e^{5} + 6 \, a^{2} b^{4} e^{6}\right )} x^{4} + 84 \, {\left (b^{6} d^{3} e^{3} + 3 \, a b^{5} d^{2} e^{4} + 6 \, a^{2} b^{4} d e^{5} + 10 \, a^{3} b^{3} e^{6}\right )} x^{3} + 36 \, {\left (b^{6} d^{4} e^{2} + 3 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} + 10 \, a^{3} b^{3} d e^{5} + 15 \, a^{4} b^{2} e^{6}\right )} x^{2} + 9 \, {\left (b^{6} d^{5} e + 3 \, a b^{5} d^{4} e^{2} + 6 \, a^{2} b^{4} d^{3} e^{3} + 10 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5} + 21 \, a^{5} b e^{6}\right )} x}{252 \, {\left (e^{16} x^{9} + 9 \, d e^{15} x^{8} + 36 \, d^{2} e^{14} x^{7} + 84 \, d^{3} e^{13} x^{6} + 126 \, d^{4} e^{12} x^{5} + 126 \, d^{5} e^{11} x^{4} + 84 \, d^{6} e^{10} x^{3} + 36 \, d^{7} e^{9} x^{2} + 9 \, d^{8} e^{8} x + d^{9} e^{7}\right )}} \]
-1/252*(84*b^6*e^6*x^6 + b^6*d^6 + 3*a*b^5*d^5*e + 6*a^2*b^4*d^4*e^2 + 10* a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 + 21*a^5*b*d*e^5 + 28*a^6*e^6 + 126*( b^6*d*e^5 + 3*a*b^5*e^6)*x^5 + 126*(b^6*d^2*e^4 + 3*a*b^5*d*e^5 + 6*a^2*b^ 4*e^6)*x^4 + 84*(b^6*d^3*e^3 + 3*a*b^5*d^2*e^4 + 6*a^2*b^4*d*e^5 + 10*a^3* b^3*e^6)*x^3 + 36*(b^6*d^4*e^2 + 3*a*b^5*d^3*e^3 + 6*a^2*b^4*d^2*e^4 + 10* a^3*b^3*d*e^5 + 15*a^4*b^2*e^6)*x^2 + 9*(b^6*d^5*e + 3*a*b^5*d^4*e^2 + 6*a ^2*b^4*d^3*e^3 + 10*a^3*b^3*d^2*e^4 + 15*a^4*b^2*d*e^5 + 21*a^5*b*e^6)*x)/ (e^16*x^9 + 9*d*e^15*x^8 + 36*d^2*e^14*x^7 + 84*d^3*e^13*x^6 + 126*d^4*e^1 2*x^5 + 126*d^5*e^11*x^4 + 84*d^6*e^10*x^3 + 36*d^7*e^9*x^2 + 9*d^8*e^8*x + d^9*e^7)
Timed out. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{10}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 441 vs. \(2 (83) = 166\).
Time = 0.22 (sec) , antiderivative size = 441, normalized size of antiderivative = 4.96 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{10}} \, dx=-\frac {84 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 3 \, a b^{5} d^{5} e + 6 \, a^{2} b^{4} d^{4} e^{2} + 10 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} + 21 \, a^{5} b d e^{5} + 28 \, a^{6} e^{6} + 126 \, {\left (b^{6} d e^{5} + 3 \, a b^{5} e^{6}\right )} x^{5} + 126 \, {\left (b^{6} d^{2} e^{4} + 3 \, a b^{5} d e^{5} + 6 \, a^{2} b^{4} e^{6}\right )} x^{4} + 84 \, {\left (b^{6} d^{3} e^{3} + 3 \, a b^{5} d^{2} e^{4} + 6 \, a^{2} b^{4} d e^{5} + 10 \, a^{3} b^{3} e^{6}\right )} x^{3} + 36 \, {\left (b^{6} d^{4} e^{2} + 3 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} + 10 \, a^{3} b^{3} d e^{5} + 15 \, a^{4} b^{2} e^{6}\right )} x^{2} + 9 \, {\left (b^{6} d^{5} e + 3 \, a b^{5} d^{4} e^{2} + 6 \, a^{2} b^{4} d^{3} e^{3} + 10 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5} + 21 \, a^{5} b e^{6}\right )} x}{252 \, {\left (e^{16} x^{9} + 9 \, d e^{15} x^{8} + 36 \, d^{2} e^{14} x^{7} + 84 \, d^{3} e^{13} x^{6} + 126 \, d^{4} e^{12} x^{5} + 126 \, d^{5} e^{11} x^{4} + 84 \, d^{6} e^{10} x^{3} + 36 \, d^{7} e^{9} x^{2} + 9 \, d^{8} e^{8} x + d^{9} e^{7}\right )}} \]
-1/252*(84*b^6*e^6*x^6 + b^6*d^6 + 3*a*b^5*d^5*e + 6*a^2*b^4*d^4*e^2 + 10* a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 + 21*a^5*b*d*e^5 + 28*a^6*e^6 + 126*( b^6*d*e^5 + 3*a*b^5*e^6)*x^5 + 126*(b^6*d^2*e^4 + 3*a*b^5*d*e^5 + 6*a^2*b^ 4*e^6)*x^4 + 84*(b^6*d^3*e^3 + 3*a*b^5*d^2*e^4 + 6*a^2*b^4*d*e^5 + 10*a^3* b^3*e^6)*x^3 + 36*(b^6*d^4*e^2 + 3*a*b^5*d^3*e^3 + 6*a^2*b^4*d^2*e^4 + 10* a^3*b^3*d*e^5 + 15*a^4*b^2*e^6)*x^2 + 9*(b^6*d^5*e + 3*a*b^5*d^4*e^2 + 6*a ^2*b^4*d^3*e^3 + 10*a^3*b^3*d^2*e^4 + 15*a^4*b^2*d*e^5 + 21*a^5*b*e^6)*x)/ (e^16*x^9 + 9*d*e^15*x^8 + 36*d^2*e^14*x^7 + 84*d^3*e^13*x^6 + 126*d^4*e^1 2*x^5 + 126*d^5*e^11*x^4 + 84*d^6*e^10*x^3 + 36*d^7*e^9*x^2 + 9*d^8*e^8*x + d^9*e^7)
Leaf count of result is larger than twice the leaf count of optimal. 375 vs. \(2 (83) = 166\).
Time = 0.26 (sec) , antiderivative size = 375, normalized size of antiderivative = 4.21 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{10}} \, dx=-\frac {84 \, b^{6} e^{6} x^{6} + 126 \, b^{6} d e^{5} x^{5} + 378 \, a b^{5} e^{6} x^{5} + 126 \, b^{6} d^{2} e^{4} x^{4} + 378 \, a b^{5} d e^{5} x^{4} + 756 \, a^{2} b^{4} e^{6} x^{4} + 84 \, b^{6} d^{3} e^{3} x^{3} + 252 \, a b^{5} d^{2} e^{4} x^{3} + 504 \, a^{2} b^{4} d e^{5} x^{3} + 840 \, a^{3} b^{3} e^{6} x^{3} + 36 \, b^{6} d^{4} e^{2} x^{2} + 108 \, a b^{5} d^{3} e^{3} x^{2} + 216 \, a^{2} b^{4} d^{2} e^{4} x^{2} + 360 \, a^{3} b^{3} d e^{5} x^{2} + 540 \, a^{4} b^{2} e^{6} x^{2} + 9 \, b^{6} d^{5} e x + 27 \, a b^{5} d^{4} e^{2} x + 54 \, a^{2} b^{4} d^{3} e^{3} x + 90 \, a^{3} b^{3} d^{2} e^{4} x + 135 \, a^{4} b^{2} d e^{5} x + 189 \, a^{5} b e^{6} x + b^{6} d^{6} + 3 \, a b^{5} d^{5} e + 6 \, a^{2} b^{4} d^{4} e^{2} + 10 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} + 21 \, a^{5} b d e^{5} + 28 \, a^{6} e^{6}}{252 \, {\left (e x + d\right )}^{9} e^{7}} \]
-1/252*(84*b^6*e^6*x^6 + 126*b^6*d*e^5*x^5 + 378*a*b^5*e^6*x^5 + 126*b^6*d ^2*e^4*x^4 + 378*a*b^5*d*e^5*x^4 + 756*a^2*b^4*e^6*x^4 + 84*b^6*d^3*e^3*x^ 3 + 252*a*b^5*d^2*e^4*x^3 + 504*a^2*b^4*d*e^5*x^3 + 840*a^3*b^3*e^6*x^3 + 36*b^6*d^4*e^2*x^2 + 108*a*b^5*d^3*e^3*x^2 + 216*a^2*b^4*d^2*e^4*x^2 + 360 *a^3*b^3*d*e^5*x^2 + 540*a^4*b^2*e^6*x^2 + 9*b^6*d^5*e*x + 27*a*b^5*d^4*e^ 2*x + 54*a^2*b^4*d^3*e^3*x + 90*a^3*b^3*d^2*e^4*x + 135*a^4*b^2*d*e^5*x + 189*a^5*b*e^6*x + b^6*d^6 + 3*a*b^5*d^5*e + 6*a^2*b^4*d^4*e^2 + 10*a^3*b^3 *d^3*e^3 + 15*a^4*b^2*d^2*e^4 + 21*a^5*b*d*e^5 + 28*a^6*e^6)/((e*x + d)^9* e^7)
Time = 9.74 (sec) , antiderivative size = 423, normalized size of antiderivative = 4.75 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{10}} \, dx=-\frac {\frac {28\,a^6\,e^6+21\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4+10\,a^3\,b^3\,d^3\,e^3+6\,a^2\,b^4\,d^4\,e^2+3\,a\,b^5\,d^5\,e+b^6\,d^6}{252\,e^7}+\frac {b^6\,x^6}{3\,e}+\frac {b^3\,x^3\,\left (10\,a^3\,e^3+6\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e+b^3\,d^3\right )}{3\,e^4}+\frac {b\,x\,\left (21\,a^5\,e^5+15\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3+6\,a^2\,b^3\,d^3\,e^2+3\,a\,b^4\,d^4\,e+b^5\,d^5\right )}{28\,e^6}+\frac {b^5\,x^5\,\left (3\,a\,e+b\,d\right )}{2\,e^2}+\frac {b^2\,x^2\,\left (15\,a^4\,e^4+10\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2+3\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{7\,e^5}+\frac {b^4\,x^4\,\left (6\,a^2\,e^2+3\,a\,b\,d\,e+b^2\,d^2\right )}{2\,e^3}}{d^9+9\,d^8\,e\,x+36\,d^7\,e^2\,x^2+84\,d^6\,e^3\,x^3+126\,d^5\,e^4\,x^4+126\,d^4\,e^5\,x^5+84\,d^3\,e^6\,x^6+36\,d^2\,e^7\,x^7+9\,d\,e^8\,x^8+e^9\,x^9} \]
-((28*a^6*e^6 + b^6*d^6 + 6*a^2*b^4*d^4*e^2 + 10*a^3*b^3*d^3*e^3 + 15*a^4* b^2*d^2*e^4 + 3*a*b^5*d^5*e + 21*a^5*b*d*e^5)/(252*e^7) + (b^6*x^6)/(3*e) + (b^3*x^3*(10*a^3*e^3 + b^3*d^3 + 3*a*b^2*d^2*e + 6*a^2*b*d*e^2))/(3*e^4) + (b*x*(21*a^5*e^5 + b^5*d^5 + 6*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2*e^3 + 3 *a*b^4*d^4*e + 15*a^4*b*d*e^4))/(28*e^6) + (b^5*x^5*(3*a*e + b*d))/(2*e^2) + (b^2*x^2*(15*a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 + 3*a*b^3*d^3*e + 10 *a^3*b*d*e^3))/(7*e^5) + (b^4*x^4*(6*a^2*e^2 + b^2*d^2 + 3*a*b*d*e))/(2*e^ 3))/(d^9 + e^9*x^9 + 9*d*e^8*x^8 + 36*d^7*e^2*x^2 + 84*d^6*e^3*x^3 + 126*d ^5*e^4*x^4 + 126*d^4*e^5*x^5 + 84*d^3*e^6*x^6 + 36*d^2*e^7*x^7 + 9*d^8*e*x )