Integrand size = 29, antiderivative size = 92 \[ \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right ) \, dx=-\frac {(b d-a e)^3 (d+e x)^6}{6 e^4}+\frac {3 b (b d-a e)^2 (d+e x)^7}{7 e^4}-\frac {3 b^2 (b d-a e) (d+e x)^8}{8 e^4}+\frac {b^3 (d+e x)^9}{9 e^4} \]
-1/6*(-a*e+b*d)^3*(e*x+d)^6/e^4+3/7*b*(-a*e+b*d)^2*(e*x+d)^7/e^4-3/8*b^2*( -a*e+b*d)*(e*x+d)^8/e^4+1/9*b^3*(e*x+d)^9/e^4
Leaf count is larger than twice the leaf count of optimal. \(267\) vs. \(2(92)=184\).
Time = 0.03 (sec) , antiderivative size = 267, normalized size of antiderivative = 2.90 \[ \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right ) \, dx=a^3 d^5 x+\frac {1}{2} a^2 d^4 (3 b d+5 a e) x^2+\frac {1}{3} a d^3 \left (3 b^2 d^2+15 a b d e+10 a^2 e^2\right ) x^3+\frac {1}{4} d^2 \left (b^3 d^3+15 a b^2 d^2 e+30 a^2 b d e^2+10 a^3 e^3\right ) x^4+d e \left (b^3 d^3+6 a b^2 d^2 e+6 a^2 b d e^2+a^3 e^3\right ) x^5+\frac {1}{6} e^2 \left (10 b^3 d^3+30 a b^2 d^2 e+15 a^2 b d e^2+a^3 e^3\right ) x^6+\frac {1}{7} b e^3 \left (10 b^2 d^2+15 a b d e+3 a^2 e^2\right ) x^7+\frac {1}{8} b^2 e^4 (5 b d+3 a e) x^8+\frac {1}{9} b^3 e^5 x^9 \]
a^3*d^5*x + (a^2*d^4*(3*b*d + 5*a*e)*x^2)/2 + (a*d^3*(3*b^2*d^2 + 15*a*b*d *e + 10*a^2*e^2)*x^3)/3 + (d^2*(b^3*d^3 + 15*a*b^2*d^2*e + 30*a^2*b*d*e^2 + 10*a^3*e^3)*x^4)/4 + d*e*(b^3*d^3 + 6*a*b^2*d^2*e + 6*a^2*b*d*e^2 + a^3* e^3)*x^5 + (e^2*(10*b^3*d^3 + 30*a*b^2*d^2*e + 15*a^2*b*d*e^2 + a^3*e^3)*x ^6)/6 + (b*e^3*(10*b^2*d^2 + 15*a*b*d*e + 3*a^2*e^2)*x^7)/7 + (b^2*e^4*(5* b*d + 3*a*e)*x^8)/8 + (b^3*e^5*x^9)/9
Time = 0.35 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1184, 27, 49, 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.
| \(\displaystyle \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right ) (d+e x)^5 \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int b^2 (a+b x)^3 (d+e x)^5dx}{b^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int (a+b x)^3 (d+e x)^5dx\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \int \left (-\frac {3 b^2 (d+e x)^7 (b d-a e)}{e^3}+\frac {3 b (d+e x)^6 (b d-a e)^2}{e^3}+\frac {(d+e x)^5 (a e-b d)^3}{e^3}+\frac {b^3 (d+e x)^8}{e^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 b^2 (d+e x)^8 (b d-a e)}{8 e^4}+\frac {3 b (d+e x)^7 (b d-a e)^2}{7 e^4}-\frac {(d+e x)^6 (b d-a e)^3}{6 e^4}+\frac {b^3 (d+e x)^9}{9 e^4}\) |
-1/6*((b*d - a*e)^3*(d + e*x)^6)/e^4 + (3*b*(b*d - a*e)^2*(d + e*x)^7)/(7* e^4) - (3*b^2*(b*d - a*e)*(d + e*x)^8)/(8*e^4) + (b^3*(d + e*x)^9)/(9*e^4)
3.19.95.3.1 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] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
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. \(272\) vs. \(2(84)=168\).
Time = 0.24 (sec) , antiderivative size = 273, normalized size of antiderivative = 2.97
| method | result | size |
| norman | \(\frac {b^{3} e^{5} x^{9}}{9}+\left (\frac {3}{8} b^{2} a \,e^{5}+\frac {5}{8} b^{3} d \,e^{4}\right ) x^{8}+\left (\frac {3}{7} a^{2} b \,e^{5}+\frac {15}{7} a \,b^{2} d \,e^{4}+\frac {10}{7} b^{3} d^{2} e^{3}\right ) x^{7}+\left (\frac {1}{6} a^{3} e^{5}+\frac {5}{2} b \,a^{2} d \,e^{4}+5 b^{2} a \,d^{2} e^{3}+\frac {5}{3} b^{3} d^{3} e^{2}\right ) x^{6}+\left (a^{3} d \,e^{4}+6 b \,a^{2} d^{2} e^{3}+6 b^{2} a \,d^{3} e^{2}+b^{3} d^{4} e \right ) x^{5}+\left (\frac {5}{2} a^{3} d^{2} e^{3}+\frac {15}{2} b \,a^{2} d^{3} e^{2}+\frac {15}{4} b^{2} a \,d^{4} e +\frac {1}{4} b^{3} d^{5}\right ) x^{4}+\left (\frac {10}{3} a^{3} d^{3} e^{2}+5 b \,a^{2} d^{4} e +b^{2} a \,d^{5}\right ) x^{3}+\left (\frac {5}{2} a^{3} d^{4} e +\frac {3}{2} b \,a^{2} d^{5}\right ) x^{2}+a^{3} d^{5} x\) | \(273\) |
| risch | \(\frac {1}{9} b^{3} e^{5} x^{9}+\frac {3}{8} x^{8} b^{2} a \,e^{5}+\frac {5}{8} x^{8} b^{3} d \,e^{4}+\frac {3}{7} x^{7} a^{2} b \,e^{5}+\frac {15}{7} x^{7} a \,b^{2} d \,e^{4}+\frac {10}{7} x^{7} b^{3} d^{2} e^{3}+\frac {1}{6} x^{6} a^{3} e^{5}+\frac {5}{2} x^{6} b \,a^{2} d \,e^{4}+5 x^{6} b^{2} a \,d^{2} e^{3}+\frac {5}{3} x^{6} b^{3} d^{3} e^{2}+a^{3} d \,e^{4} x^{5}+6 a^{2} b \,d^{2} e^{3} x^{5}+6 a \,b^{2} d^{3} e^{2} x^{5}+b^{3} d^{4} e \,x^{5}+\frac {5}{2} x^{4} a^{3} d^{2} e^{3}+\frac {15}{2} x^{4} b \,a^{2} d^{3} e^{2}+\frac {15}{4} x^{4} b^{2} a \,d^{4} e +\frac {1}{4} x^{4} b^{3} d^{5}+\frac {10}{3} x^{3} a^{3} d^{3} e^{2}+5 x^{3} b \,a^{2} d^{4} e +x^{3} b^{2} a \,d^{5}+\frac {5}{2} x^{2} a^{3} d^{4} e +\frac {3}{2} x^{2} b \,a^{2} d^{5}+a^{3} d^{5} x\) | \(304\) |
| parallelrisch | \(\frac {1}{9} b^{3} e^{5} x^{9}+\frac {3}{8} x^{8} b^{2} a \,e^{5}+\frac {5}{8} x^{8} b^{3} d \,e^{4}+\frac {3}{7} x^{7} a^{2} b \,e^{5}+\frac {15}{7} x^{7} a \,b^{2} d \,e^{4}+\frac {10}{7} x^{7} b^{3} d^{2} e^{3}+\frac {1}{6} x^{6} a^{3} e^{5}+\frac {5}{2} x^{6} b \,a^{2} d \,e^{4}+5 x^{6} b^{2} a \,d^{2} e^{3}+\frac {5}{3} x^{6} b^{3} d^{3} e^{2}+a^{3} d \,e^{4} x^{5}+6 a^{2} b \,d^{2} e^{3} x^{5}+6 a \,b^{2} d^{3} e^{2} x^{5}+b^{3} d^{4} e \,x^{5}+\frac {5}{2} x^{4} a^{3} d^{2} e^{3}+\frac {15}{2} x^{4} b \,a^{2} d^{3} e^{2}+\frac {15}{4} x^{4} b^{2} a \,d^{4} e +\frac {1}{4} x^{4} b^{3} d^{5}+\frac {10}{3} x^{3} a^{3} d^{3} e^{2}+5 x^{3} b \,a^{2} d^{4} e +x^{3} b^{2} a \,d^{5}+\frac {5}{2} x^{2} a^{3} d^{4} e +\frac {3}{2} x^{2} b \,a^{2} d^{5}+a^{3} d^{5} x\) | \(304\) |
| gosper | \(\frac {x \left (56 b^{3} e^{5} x^{8}+189 x^{7} b^{2} a \,e^{5}+315 x^{7} b^{3} d \,e^{4}+216 x^{6} a^{2} b \,e^{5}+1080 x^{6} a \,b^{2} d \,e^{4}+720 x^{6} b^{3} d^{2} e^{3}+84 x^{5} a^{3} e^{5}+1260 x^{5} b \,a^{2} d \,e^{4}+2520 x^{5} b^{2} a \,d^{2} e^{3}+840 x^{5} b^{3} d^{3} e^{2}+504 a^{3} d \,e^{4} x^{4}+3024 a^{2} b \,d^{2} e^{3} x^{4}+3024 a \,b^{2} d^{3} e^{2} x^{4}+504 b^{3} d^{4} e \,x^{4}+1260 x^{3} a^{3} d^{2} e^{3}+3780 x^{3} b \,a^{2} d^{3} e^{2}+1890 x^{3} b^{2} a \,d^{4} e +126 x^{3} b^{3} d^{5}+1680 x^{2} a^{3} d^{3} e^{2}+2520 x^{2} b \,a^{2} d^{4} e +504 x^{2} b^{2} a \,d^{5}+1260 x \,a^{3} d^{4} e +756 x b \,a^{2} d^{5}+504 a^{3} d^{5}\right )}{504}\) | \(306\) |
| default | \(\frac {b^{3} e^{5} x^{9}}{9}+\frac {\left (\left (a \,e^{5}+5 b d \,e^{4}\right ) b^{2}+2 b^{2} a \,e^{5}\right ) x^{8}}{8}+\frac {\left (\left (5 a d \,e^{4}+10 b \,d^{2} e^{3}\right ) b^{2}+2 \left (a \,e^{5}+5 b d \,e^{4}\right ) b a +a^{2} b \,e^{5}\right ) x^{7}}{7}+\frac {\left (\left (10 a \,d^{2} e^{3}+10 b \,d^{3} e^{2}\right ) b^{2}+2 \left (5 a d \,e^{4}+10 b \,d^{2} e^{3}\right ) b a +\left (a \,e^{5}+5 b d \,e^{4}\right ) a^{2}\right ) x^{6}}{6}+\frac {\left (\left (10 a \,d^{3} e^{2}+5 b \,d^{4} e \right ) b^{2}+2 \left (10 a \,d^{2} e^{3}+10 b \,d^{3} e^{2}\right ) b a +\left (5 a d \,e^{4}+10 b \,d^{2} e^{3}\right ) a^{2}\right ) x^{5}}{5}+\frac {\left (\left (5 a \,d^{4} e +b \,d^{5}\right ) b^{2}+2 \left (10 a \,d^{3} e^{2}+5 b \,d^{4} e \right ) b a +\left (10 a \,d^{2} e^{3}+10 b \,d^{3} e^{2}\right ) a^{2}\right ) x^{4}}{4}+\frac {\left (b^{2} a \,d^{5}+2 \left (5 a \,d^{4} e +b \,d^{5}\right ) b a +\left (10 a \,d^{3} e^{2}+5 b \,d^{4} e \right ) a^{2}\right ) x^{3}}{3}+\frac {\left (2 b \,a^{2} d^{5}+\left (5 a \,d^{4} e +b \,d^{5}\right ) a^{2}\right ) x^{2}}{2}+a^{3} d^{5} x\) | \(394\) |
1/9*b^3*e^5*x^9+(3/8*b^2*a*e^5+5/8*b^3*d*e^4)*x^8+(3/7*a^2*b*e^5+15/7*a*b^ 2*d*e^4+10/7*b^3*d^2*e^3)*x^7+(1/6*a^3*e^5+5/2*b*a^2*d*e^4+5*b^2*a*d^2*e^3 +5/3*b^3*d^3*e^2)*x^6+(a^3*d*e^4+6*a^2*b*d^2*e^3+6*a*b^2*d^3*e^2+b^3*d^4*e )*x^5+(5/2*a^3*d^2*e^3+15/2*b*a^2*d^3*e^2+15/4*b^2*a*d^4*e+1/4*b^3*d^5)*x^ 4+(10/3*a^3*d^3*e^2+5*b*a^2*d^4*e+b^2*a*d^5)*x^3+(5/2*a^3*d^4*e+3/2*b*a^2* d^5)*x^2+a^3*d^5*x
Leaf count of result is larger than twice the leaf count of optimal. 277 vs. \(2 (84) = 168\).
Time = 0.33 (sec) , antiderivative size = 277, normalized size of antiderivative = 3.01 \[ \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {1}{9} \, b^{3} e^{5} x^{9} + a^{3} d^{5} x + \frac {1}{8} \, {\left (5 \, b^{3} d e^{4} + 3 \, a b^{2} e^{5}\right )} x^{8} + \frac {1}{7} \, {\left (10 \, b^{3} d^{2} e^{3} + 15 \, a b^{2} d e^{4} + 3 \, a^{2} b e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (10 \, b^{3} d^{3} e^{2} + 30 \, a b^{2} d^{2} e^{3} + 15 \, a^{2} b d e^{4} + a^{3} e^{5}\right )} x^{6} + {\left (b^{3} d^{4} e + 6 \, a b^{2} d^{3} e^{2} + 6 \, a^{2} b d^{2} e^{3} + a^{3} d e^{4}\right )} x^{5} + \frac {1}{4} \, {\left (b^{3} d^{5} + 15 \, a b^{2} d^{4} e + 30 \, a^{2} b d^{3} e^{2} + 10 \, a^{3} d^{2} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, a b^{2} d^{5} + 15 \, a^{2} b d^{4} e + 10 \, a^{3} d^{3} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b d^{5} + 5 \, a^{3} d^{4} e\right )} x^{2} \]
1/9*b^3*e^5*x^9 + a^3*d^5*x + 1/8*(5*b^3*d*e^4 + 3*a*b^2*e^5)*x^8 + 1/7*(1 0*b^3*d^2*e^3 + 15*a*b^2*d*e^4 + 3*a^2*b*e^5)*x^7 + 1/6*(10*b^3*d^3*e^2 + 30*a*b^2*d^2*e^3 + 15*a^2*b*d*e^4 + a^3*e^5)*x^6 + (b^3*d^4*e + 6*a*b^2*d^ 3*e^2 + 6*a^2*b*d^2*e^3 + a^3*d*e^4)*x^5 + 1/4*(b^3*d^5 + 15*a*b^2*d^4*e + 30*a^2*b*d^3*e^2 + 10*a^3*d^2*e^3)*x^4 + 1/3*(3*a*b^2*d^5 + 15*a^2*b*d^4* e + 10*a^3*d^3*e^2)*x^3 + 1/2*(3*a^2*b*d^5 + 5*a^3*d^4*e)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 308 vs. \(2 (82) = 164\).
Time = 0.04 (sec) , antiderivative size = 308, normalized size of antiderivative = 3.35 \[ \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right ) \, dx=a^{3} d^{5} x + \frac {b^{3} e^{5} x^{9}}{9} + x^{8} \cdot \left (\frac {3 a b^{2} e^{5}}{8} + \frac {5 b^{3} d e^{4}}{8}\right ) + x^{7} \cdot \left (\frac {3 a^{2} b e^{5}}{7} + \frac {15 a b^{2} d e^{4}}{7} + \frac {10 b^{3} d^{2} e^{3}}{7}\right ) + x^{6} \left (\frac {a^{3} e^{5}}{6} + \frac {5 a^{2} b d e^{4}}{2} + 5 a b^{2} d^{2} e^{3} + \frac {5 b^{3} d^{3} e^{2}}{3}\right ) + x^{5} \left (a^{3} d e^{4} + 6 a^{2} b d^{2} e^{3} + 6 a b^{2} d^{3} e^{2} + b^{3} d^{4} e\right ) + x^{4} \cdot \left (\frac {5 a^{3} d^{2} e^{3}}{2} + \frac {15 a^{2} b d^{3} e^{2}}{2} + \frac {15 a b^{2} d^{4} e}{4} + \frac {b^{3} d^{5}}{4}\right ) + x^{3} \cdot \left (\frac {10 a^{3} d^{3} e^{2}}{3} + 5 a^{2} b d^{4} e + a b^{2} d^{5}\right ) + x^{2} \cdot \left (\frac {5 a^{3} d^{4} e}{2} + \frac {3 a^{2} b d^{5}}{2}\right ) \]
a**3*d**5*x + b**3*e**5*x**9/9 + x**8*(3*a*b**2*e**5/8 + 5*b**3*d*e**4/8) + x**7*(3*a**2*b*e**5/7 + 15*a*b**2*d*e**4/7 + 10*b**3*d**2*e**3/7) + x**6 *(a**3*e**5/6 + 5*a**2*b*d*e**4/2 + 5*a*b**2*d**2*e**3 + 5*b**3*d**3*e**2/ 3) + x**5*(a**3*d*e**4 + 6*a**2*b*d**2*e**3 + 6*a*b**2*d**3*e**2 + b**3*d* *4*e) + x**4*(5*a**3*d**2*e**3/2 + 15*a**2*b*d**3*e**2/2 + 15*a*b**2*d**4* e/4 + b**3*d**5/4) + x**3*(10*a**3*d**3*e**2/3 + 5*a**2*b*d**4*e + a*b**2* d**5) + x**2*(5*a**3*d**4*e/2 + 3*a**2*b*d**5/2)
Leaf count of result is larger than twice the leaf count of optimal. 277 vs. \(2 (84) = 168\).
Time = 0.19 (sec) , antiderivative size = 277, normalized size of antiderivative = 3.01 \[ \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {1}{9} \, b^{3} e^{5} x^{9} + a^{3} d^{5} x + \frac {1}{8} \, {\left (5 \, b^{3} d e^{4} + 3 \, a b^{2} e^{5}\right )} x^{8} + \frac {1}{7} \, {\left (10 \, b^{3} d^{2} e^{3} + 15 \, a b^{2} d e^{4} + 3 \, a^{2} b e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (10 \, b^{3} d^{3} e^{2} + 30 \, a b^{2} d^{2} e^{3} + 15 \, a^{2} b d e^{4} + a^{3} e^{5}\right )} x^{6} + {\left (b^{3} d^{4} e + 6 \, a b^{2} d^{3} e^{2} + 6 \, a^{2} b d^{2} e^{3} + a^{3} d e^{4}\right )} x^{5} + \frac {1}{4} \, {\left (b^{3} d^{5} + 15 \, a b^{2} d^{4} e + 30 \, a^{2} b d^{3} e^{2} + 10 \, a^{3} d^{2} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, a b^{2} d^{5} + 15 \, a^{2} b d^{4} e + 10 \, a^{3} d^{3} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} b d^{5} + 5 \, a^{3} d^{4} e\right )} x^{2} \]
1/9*b^3*e^5*x^9 + a^3*d^5*x + 1/8*(5*b^3*d*e^4 + 3*a*b^2*e^5)*x^8 + 1/7*(1 0*b^3*d^2*e^3 + 15*a*b^2*d*e^4 + 3*a^2*b*e^5)*x^7 + 1/6*(10*b^3*d^3*e^2 + 30*a*b^2*d^2*e^3 + 15*a^2*b*d*e^4 + a^3*e^5)*x^6 + (b^3*d^4*e + 6*a*b^2*d^ 3*e^2 + 6*a^2*b*d^2*e^3 + a^3*d*e^4)*x^5 + 1/4*(b^3*d^5 + 15*a*b^2*d^4*e + 30*a^2*b*d^3*e^2 + 10*a^3*d^2*e^3)*x^4 + 1/3*(3*a*b^2*d^5 + 15*a^2*b*d^4* e + 10*a^3*d^3*e^2)*x^3 + 1/2*(3*a^2*b*d^5 + 5*a^3*d^4*e)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 303 vs. \(2 (84) = 168\).
Time = 0.27 (sec) , antiderivative size = 303, normalized size of antiderivative = 3.29 \[ \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {1}{9} \, b^{3} e^{5} x^{9} + \frac {5}{8} \, b^{3} d e^{4} x^{8} + \frac {3}{8} \, a b^{2} e^{5} x^{8} + \frac {10}{7} \, b^{3} d^{2} e^{3} x^{7} + \frac {15}{7} \, a b^{2} d e^{4} x^{7} + \frac {3}{7} \, a^{2} b e^{5} x^{7} + \frac {5}{3} \, b^{3} d^{3} e^{2} x^{6} + 5 \, a b^{2} d^{2} e^{3} x^{6} + \frac {5}{2} \, a^{2} b d e^{4} x^{6} + \frac {1}{6} \, a^{3} e^{5} x^{6} + b^{3} d^{4} e x^{5} + 6 \, a b^{2} d^{3} e^{2} x^{5} + 6 \, a^{2} b d^{2} e^{3} x^{5} + a^{3} d e^{4} x^{5} + \frac {1}{4} \, b^{3} d^{5} x^{4} + \frac {15}{4} \, a b^{2} d^{4} e x^{4} + \frac {15}{2} \, a^{2} b d^{3} e^{2} x^{4} + \frac {5}{2} \, a^{3} d^{2} e^{3} x^{4} + a b^{2} d^{5} x^{3} + 5 \, a^{2} b d^{4} e x^{3} + \frac {10}{3} \, a^{3} d^{3} e^{2} x^{3} + \frac {3}{2} \, a^{2} b d^{5} x^{2} + \frac {5}{2} \, a^{3} d^{4} e x^{2} + a^{3} d^{5} x \]
1/9*b^3*e^5*x^9 + 5/8*b^3*d*e^4*x^8 + 3/8*a*b^2*e^5*x^8 + 10/7*b^3*d^2*e^3 *x^7 + 15/7*a*b^2*d*e^4*x^7 + 3/7*a^2*b*e^5*x^7 + 5/3*b^3*d^3*e^2*x^6 + 5* a*b^2*d^2*e^3*x^6 + 5/2*a^2*b*d*e^4*x^6 + 1/6*a^3*e^5*x^6 + b^3*d^4*e*x^5 + 6*a*b^2*d^3*e^2*x^5 + 6*a^2*b*d^2*e^3*x^5 + a^3*d*e^4*x^5 + 1/4*b^3*d^5* x^4 + 15/4*a*b^2*d^4*e*x^4 + 15/2*a^2*b*d^3*e^2*x^4 + 5/2*a^3*d^2*e^3*x^4 + a*b^2*d^5*x^3 + 5*a^2*b*d^4*e*x^3 + 10/3*a^3*d^3*e^2*x^3 + 3/2*a^2*b*d^5 *x^2 + 5/2*a^3*d^4*e*x^2 + a^3*d^5*x
Time = 0.11 (sec) , antiderivative size = 261, normalized size of antiderivative = 2.84 \[ \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right ) \, dx=x^5\,\left (a^3\,d\,e^4+6\,a^2\,b\,d^2\,e^3+6\,a\,b^2\,d^3\,e^2+b^3\,d^4\,e\right )+x^4\,\left (\frac {5\,a^3\,d^2\,e^3}{2}+\frac {15\,a^2\,b\,d^3\,e^2}{2}+\frac {15\,a\,b^2\,d^4\,e}{4}+\frac {b^3\,d^5}{4}\right )+x^6\,\left (\frac {a^3\,e^5}{6}+\frac {5\,a^2\,b\,d\,e^4}{2}+5\,a\,b^2\,d^2\,e^3+\frac {5\,b^3\,d^3\,e^2}{3}\right )+a^3\,d^5\,x+\frac {b^3\,e^5\,x^9}{9}+\frac {a^2\,d^4\,x^2\,\left (5\,a\,e+3\,b\,d\right )}{2}+\frac {b^2\,e^4\,x^8\,\left (3\,a\,e+5\,b\,d\right )}{8}+\frac {a\,d^3\,x^3\,\left (10\,a^2\,e^2+15\,a\,b\,d\,e+3\,b^2\,d^2\right )}{3}+\frac {b\,e^3\,x^7\,\left (3\,a^2\,e^2+15\,a\,b\,d\,e+10\,b^2\,d^2\right )}{7} \]
x^5*(a^3*d*e^4 + b^3*d^4*e + 6*a*b^2*d^3*e^2 + 6*a^2*b*d^2*e^3) + x^4*((b^ 3*d^5)/4 + (5*a^3*d^2*e^3)/2 + (15*a^2*b*d^3*e^2)/2 + (15*a*b^2*d^4*e)/4) + x^6*((a^3*e^5)/6 + (5*b^3*d^3*e^2)/3 + 5*a*b^2*d^2*e^3 + (5*a^2*b*d*e^4) /2) + a^3*d^5*x + (b^3*e^5*x^9)/9 + (a^2*d^4*x^2*(5*a*e + 3*b*d))/2 + (b^2 *e^4*x^8*(3*a*e + 5*b*d))/8 + (a*d^3*x^3*(10*a^2*e^2 + 3*b^2*d^2 + 15*a*b* d*e))/3 + (b*e^3*x^7*(3*a^2*e^2 + 10*b^2*d^2 + 15*a*b*d*e))/7