Integrand size = 31, antiderivative size = 92 \[ \int (a+b x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {(b d-a e)^3 (a+b x)^8}{8 b^4}+\frac {e (b d-a e)^2 (a+b x)^9}{3 b^4}+\frac {3 e^2 (b d-a e) (a+b x)^{10}}{10 b^4}+\frac {e^3 (a+b x)^{11}}{11 b^4} \] Output:
1/8*(-a*e+b*d)^3*(b*x+a)^8/b^4+1/3*e*(-a*e+b*d)^2*(b*x+a)^9/b^4+3/10*e^2*( -a*e+b*d)*(b*x+a)^10/b^4+1/11*e^3*(b*x+a)^11/b^4
Leaf count is larger than twice the leaf count of optimal. \(360\) vs. \(2(92)=184\).
Time = 0.07 (sec) , antiderivative size = 360, normalized size of antiderivative = 3.91 \[ \int (a+b x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=a^7 d^3 x+\frac {1}{2} a^6 d^2 (7 b d+3 a e) x^2+a^5 d \left (7 b^2 d^2+7 a b d e+a^2 e^2\right ) x^3+\frac {1}{4} a^4 \left (35 b^3 d^3+63 a b^2 d^2 e+21 a^2 b d e^2+a^3 e^3\right ) x^4+\frac {7}{5} a^3 b \left (5 b^3 d^3+15 a b^2 d^2 e+9 a^2 b d e^2+a^3 e^3\right ) x^5+\frac {7}{2} a^2 b^2 \left (b^3 d^3+5 a b^2 d^2 e+5 a^2 b d e^2+a^3 e^3\right ) x^6+a b^3 \left (b^3 d^3+9 a b^2 d^2 e+15 a^2 b d e^2+5 a^3 e^3\right ) x^7+\frac {1}{8} b^4 \left (b^3 d^3+21 a b^2 d^2 e+63 a^2 b d e^2+35 a^3 e^3\right ) x^8+\frac {1}{3} b^5 e \left (b^2 d^2+7 a b d e+7 a^2 e^2\right ) x^9+\frac {1}{10} b^6 e^2 (3 b d+7 a e) x^{10}+\frac {1}{11} b^7 e^3 x^{11} \] Input:
Integrate[(a + b*x)*(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
Output:
a^7*d^3*x + (a^6*d^2*(7*b*d + 3*a*e)*x^2)/2 + a^5*d*(7*b^2*d^2 + 7*a*b*d*e + a^2*e^2)*x^3 + (a^4*(35*b^3*d^3 + 63*a*b^2*d^2*e + 21*a^2*b*d*e^2 + a^3 *e^3)*x^4)/4 + (7*a^3*b*(5*b^3*d^3 + 15*a*b^2*d^2*e + 9*a^2*b*d*e^2 + a^3* e^3)*x^5)/5 + (7*a^2*b^2*(b^3*d^3 + 5*a*b^2*d^2*e + 5*a^2*b*d*e^2 + a^3*e^ 3)*x^6)/2 + a*b^3*(b^3*d^3 + 9*a*b^2*d^2*e + 15*a^2*b*d*e^2 + 5*a^3*e^3)*x ^7 + (b^4*(b^3*d^3 + 21*a*b^2*d^2*e + 63*a^2*b*d*e^2 + 35*a^3*e^3)*x^8)/8 + (b^5*e*(b^2*d^2 + 7*a*b*d*e + 7*a^2*e^2)*x^9)/3 + (b^6*e^2*(3*b*d + 7*a* e)*x^10)/10 + (b^7*e^3*x^11)/11
Time = 0.65 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, 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.
| \(\displaystyle \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3 (d+e x)^3 \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int b^6 (a+b x)^7 (d+e x)^3dx}{b^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int (a+b x)^7 (d+e x)^3dx\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \int \left (\frac {3 e^2 (a+b x)^9 (b d-a e)}{b^3}+\frac {3 e (a+b x)^8 (b d-a e)^2}{b^3}+\frac {(a+b x)^7 (b d-a e)^3}{b^3}+\frac {e^3 (a+b x)^{10}}{b^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 e^2 (a+b x)^{10} (b d-a e)}{10 b^4}+\frac {e (a+b x)^9 (b d-a e)^2}{3 b^4}+\frac {(a+b x)^8 (b d-a e)^3}{8 b^4}+\frac {e^3 (a+b x)^{11}}{11 b^4}\) |
Input:
Int[(a + b*x)*(d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
Output:
((b*d - a*e)^3*(a + b*x)^8)/(8*b^4) + (e*(b*d - a*e)^2*(a + b*x)^9)/(3*b^4 ) + (3*e^2*(b*d - a*e)*(a + b*x)^10)/(10*b^4) + (e^3*(a + b*x)^11)/(11*b^4 )
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. \(375\) vs. \(2(84)=168\).
Time = 1.19 (sec) , antiderivative size = 376, normalized size of antiderivative = 4.09
| method | result | size |
| norman | \(\frac {b^{7} e^{3} x^{11}}{11}+\left (\frac {7}{10} e^{3} a \,b^{6}+\frac {3}{10} b^{7} d \,e^{2}\right ) x^{10}+\left (\frac {7}{3} e^{3} a^{2} b^{5}+\frac {7}{3} d \,e^{2} a \,b^{6}+\frac {1}{3} d^{2} e \,b^{7}\right ) x^{9}+\left (\frac {35}{8} e^{3} a^{3} b^{4}+\frac {63}{8} d \,e^{2} a^{2} b^{5}+\frac {21}{8} d^{2} e a \,b^{6}+\frac {1}{8} d^{3} b^{7}\right ) x^{8}+\left (5 e^{3} a^{4} b^{3}+15 d \,e^{2} a^{3} b^{4}+9 d^{2} e \,a^{2} b^{5}+d^{3} a \,b^{6}\right ) x^{7}+\left (\frac {7}{2} e^{3} a^{5} b^{2}+\frac {35}{2} d \,e^{2} a^{4} b^{3}+\frac {35}{2} d^{2} e \,a^{3} b^{4}+\frac {7}{2} d^{3} a^{2} b^{5}\right ) x^{6}+\left (\frac {7}{5} e^{3} a^{6} b +\frac {63}{5} d \,e^{2} a^{5} b^{2}+21 d^{2} e \,a^{4} b^{3}+7 d^{3} a^{3} b^{4}\right ) x^{5}+\left (\frac {1}{4} e^{3} a^{7}+\frac {21}{4} d \,e^{2} a^{6} b +\frac {63}{4} d^{2} e \,a^{5} b^{2}+\frac {35}{4} d^{3} a^{4} b^{3}\right ) x^{4}+\left (d \,e^{2} a^{7}+7 d^{2} e \,a^{6} b +7 d^{3} a^{5} b^{2}\right ) x^{3}+\left (\frac {3}{2} d^{2} e \,a^{7}+\frac {7}{2} d^{3} a^{6} b \right ) x^{2}+d^{3} a^{7} x\) | \(376\) |
| risch | \(\frac {7}{10} x^{10} e^{3} a \,b^{6}+\frac {3}{10} x^{10} b^{7} d \,e^{2}+\frac {7}{3} x^{9} e^{3} a^{2} b^{5}+\frac {1}{3} x^{9} d^{2} e \,b^{7}+\frac {35}{8} x^{8} e^{3} a^{3} b^{4}+\frac {7}{2} x^{6} e^{3} a^{5} b^{2}+\frac {7}{2} x^{6} d^{3} a^{2} b^{5}+\frac {7}{5} x^{5} e^{3} a^{6} b +7 x^{5} d^{3} a^{3} b^{4}+\frac {35}{4} x^{4} d^{3} a^{4} b^{3}+\frac {3}{2} x^{2} d^{2} e \,a^{7}+\frac {7}{2} x^{2} d^{3} a^{6} b +5 a^{4} b^{3} e^{3} x^{7}+a \,b^{6} d^{3} x^{7}+a^{7} d \,e^{2} x^{3}+\frac {35}{2} x^{6} d \,e^{2} a^{4} b^{3}+\frac {35}{2} x^{6} d^{2} e \,a^{3} b^{4}+\frac {63}{5} x^{5} d \,e^{2} a^{5} b^{2}+21 x^{5} d^{2} e \,a^{4} b^{3}+\frac {21}{4} x^{4} d \,e^{2} a^{6} b +\frac {63}{4} x^{4} d^{2} e \,a^{5} b^{2}+9 a^{2} b^{5} d^{2} e \,x^{7}+7 a^{6} b \,d^{2} e \,x^{3}+7 a^{5} b^{2} d^{3} x^{3}+15 a^{3} b^{4} d \,e^{2} x^{7}+\frac {7}{3} x^{9} d \,e^{2} a \,b^{6}+\frac {63}{8} x^{8} d \,e^{2} a^{2} b^{5}+\frac {21}{8} x^{8} d^{2} e a \,b^{6}+d^{3} a^{7} x +\frac {1}{8} x^{8} d^{3} b^{7}+\frac {1}{4} x^{4} e^{3} a^{7}+\frac {1}{11} b^{7} e^{3} x^{11}\) | \(421\) |
| parallelrisch | \(\frac {7}{10} x^{10} e^{3} a \,b^{6}+\frac {3}{10} x^{10} b^{7} d \,e^{2}+\frac {7}{3} x^{9} e^{3} a^{2} b^{5}+\frac {1}{3} x^{9} d^{2} e \,b^{7}+\frac {35}{8} x^{8} e^{3} a^{3} b^{4}+\frac {7}{2} x^{6} e^{3} a^{5} b^{2}+\frac {7}{2} x^{6} d^{3} a^{2} b^{5}+\frac {7}{5} x^{5} e^{3} a^{6} b +7 x^{5} d^{3} a^{3} b^{4}+\frac {35}{4} x^{4} d^{3} a^{4} b^{3}+\frac {3}{2} x^{2} d^{2} e \,a^{7}+\frac {7}{2} x^{2} d^{3} a^{6} b +5 a^{4} b^{3} e^{3} x^{7}+a \,b^{6} d^{3} x^{7}+a^{7} d \,e^{2} x^{3}+\frac {35}{2} x^{6} d \,e^{2} a^{4} b^{3}+\frac {35}{2} x^{6} d^{2} e \,a^{3} b^{4}+\frac {63}{5} x^{5} d \,e^{2} a^{5} b^{2}+21 x^{5} d^{2} e \,a^{4} b^{3}+\frac {21}{4} x^{4} d \,e^{2} a^{6} b +\frac {63}{4} x^{4} d^{2} e \,a^{5} b^{2}+9 a^{2} b^{5} d^{2} e \,x^{7}+7 a^{6} b \,d^{2} e \,x^{3}+7 a^{5} b^{2} d^{3} x^{3}+15 a^{3} b^{4} d \,e^{2} x^{7}+\frac {7}{3} x^{9} d \,e^{2} a \,b^{6}+\frac {63}{8} x^{8} d \,e^{2} a^{2} b^{5}+\frac {21}{8} x^{8} d^{2} e a \,b^{6}+d^{3} a^{7} x +\frac {1}{8} x^{8} d^{3} b^{7}+\frac {1}{4} x^{4} e^{3} a^{7}+\frac {1}{11} b^{7} e^{3} x^{11}\) | \(421\) |
| gosper | \(\frac {x \left (120 b^{7} e^{3} x^{10}+924 x^{9} e^{3} a \,b^{6}+396 x^{9} b^{7} d \,e^{2}+3080 x^{8} e^{3} a^{2} b^{5}+3080 x^{8} d \,e^{2} a \,b^{6}+440 x^{8} d^{2} e \,b^{7}+5775 x^{7} e^{3} a^{3} b^{4}+10395 x^{7} d \,e^{2} a^{2} b^{5}+3465 x^{7} d^{2} e a \,b^{6}+165 x^{7} d^{3} b^{7}+6600 a^{4} b^{3} e^{3} x^{6}+19800 a^{3} b^{4} d \,e^{2} x^{6}+11880 a^{2} b^{5} d^{2} e \,x^{6}+1320 a \,b^{6} d^{3} x^{6}+4620 x^{5} e^{3} a^{5} b^{2}+23100 x^{5} d \,e^{2} a^{4} b^{3}+23100 x^{5} d^{2} e \,a^{3} b^{4}+4620 x^{5} d^{3} a^{2} b^{5}+1848 x^{4} e^{3} a^{6} b +16632 x^{4} d \,e^{2} a^{5} b^{2}+27720 x^{4} d^{2} e \,a^{4} b^{3}+9240 x^{4} d^{3} a^{3} b^{4}+330 x^{3} e^{3} a^{7}+6930 x^{3} d \,e^{2} a^{6} b +20790 x^{3} d^{2} e \,a^{5} b^{2}+11550 x^{3} d^{3} a^{4} b^{3}+1320 a^{7} d \,e^{2} x^{2}+9240 a^{6} b \,d^{2} e \,x^{2}+9240 a^{5} b^{2} d^{3} x^{2}+1980 x \,d^{2} e \,a^{7}+4620 x \,d^{3} a^{6} b +1320 d^{3} a^{7}\right )}{1320}\) | \(422\) |
| orering | \(\frac {x \left (120 b^{7} e^{3} x^{10}+924 x^{9} e^{3} a \,b^{6}+396 x^{9} b^{7} d \,e^{2}+3080 x^{8} e^{3} a^{2} b^{5}+3080 x^{8} d \,e^{2} a \,b^{6}+440 x^{8} d^{2} e \,b^{7}+5775 x^{7} e^{3} a^{3} b^{4}+10395 x^{7} d \,e^{2} a^{2} b^{5}+3465 x^{7} d^{2} e a \,b^{6}+165 x^{7} d^{3} b^{7}+6600 a^{4} b^{3} e^{3} x^{6}+19800 a^{3} b^{4} d \,e^{2} x^{6}+11880 a^{2} b^{5} d^{2} e \,x^{6}+1320 a \,b^{6} d^{3} x^{6}+4620 x^{5} e^{3} a^{5} b^{2}+23100 x^{5} d \,e^{2} a^{4} b^{3}+23100 x^{5} d^{2} e \,a^{3} b^{4}+4620 x^{5} d^{3} a^{2} b^{5}+1848 x^{4} e^{3} a^{6} b +16632 x^{4} d \,e^{2} a^{5} b^{2}+27720 x^{4} d^{2} e \,a^{4} b^{3}+9240 x^{4} d^{3} a^{3} b^{4}+330 x^{3} e^{3} a^{7}+6930 x^{3} d \,e^{2} a^{6} b +20790 x^{3} d^{2} e \,a^{5} b^{2}+11550 x^{3} d^{3} a^{4} b^{3}+1320 a^{7} d \,e^{2} x^{2}+9240 a^{6} b \,d^{2} e \,x^{2}+9240 a^{5} b^{2} d^{3} x^{2}+1980 x \,d^{2} e \,a^{7}+4620 x \,d^{3} a^{6} b +1320 d^{3} a^{7}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{3}}{1320 \left (b x +a \right )^{6}}\) | \(447\) |
| default | \(\frac {b^{7} e^{3} x^{11}}{11}+\frac {\left (\left (a \,e^{3}+3 b d \,e^{2}\right ) b^{6}+6 e^{3} a \,b^{6}\right ) x^{10}}{10}+\frac {\left (\left (3 a d \,e^{2}+3 b \,d^{2} e \right ) b^{6}+6 \left (a \,e^{3}+3 b d \,e^{2}\right ) a \,b^{5}+15 e^{3} a^{2} b^{5}\right ) x^{9}}{9}+\frac {\left (\left (3 a \,d^{2} e +b \,d^{3}\right ) b^{6}+6 \left (3 a d \,e^{2}+3 b \,d^{2} e \right ) a \,b^{5}+15 \left (a \,e^{3}+3 b d \,e^{2}\right ) a^{2} b^{4}+20 e^{3} a^{3} b^{4}\right ) x^{8}}{8}+\frac {\left (d^{3} a \,b^{6}+6 \left (3 a \,d^{2} e +b \,d^{3}\right ) a \,b^{5}+15 \left (3 a d \,e^{2}+3 b \,d^{2} e \right ) a^{2} b^{4}+20 \left (a \,e^{3}+3 b d \,e^{2}\right ) a^{3} b^{3}+15 e^{3} a^{4} b^{3}\right ) x^{7}}{7}+\frac {\left (6 d^{3} a^{2} b^{5}+15 \left (3 a \,d^{2} e +b \,d^{3}\right ) a^{2} b^{4}+20 \left (3 a d \,e^{2}+3 b \,d^{2} e \right ) a^{3} b^{3}+15 \left (a \,e^{3}+3 b d \,e^{2}\right ) a^{4} b^{2}+6 e^{3} a^{5} b^{2}\right ) x^{6}}{6}+\frac {\left (15 d^{3} a^{3} b^{4}+20 \left (3 a \,d^{2} e +b \,d^{3}\right ) a^{3} b^{3}+15 \left (3 a d \,e^{2}+3 b \,d^{2} e \right ) a^{4} b^{2}+6 \left (a \,e^{3}+3 b d \,e^{2}\right ) a^{5} b +e^{3} a^{6} b \right ) x^{5}}{5}+\frac {\left (20 d^{3} a^{4} b^{3}+15 \left (3 a \,d^{2} e +b \,d^{3}\right ) a^{4} b^{2}+6 \left (3 a d \,e^{2}+3 b \,d^{2} e \right ) a^{5} b +\left (a \,e^{3}+3 b d \,e^{2}\right ) a^{6}\right ) x^{4}}{4}+\frac {\left (15 d^{3} a^{5} b^{2}+6 \left (3 a \,d^{2} e +b \,d^{3}\right ) a^{5} b +\left (3 a d \,e^{2}+3 b \,d^{2} e \right ) a^{6}\right ) x^{3}}{3}+\frac {\left (6 d^{3} a^{6} b +\left (3 a \,d^{2} e +b \,d^{3}\right ) a^{6}\right ) x^{2}}{2}+d^{3} a^{7} x\) | \(616\) |
Input:
int((b*x+a)*(e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^3,x,method=_RETURNVERBOSE)
Output:
1/11*b^7*e^3*x^11+(7/10*e^3*a*b^6+3/10*b^7*d*e^2)*x^10+(7/3*e^3*a^2*b^5+7/ 3*d*e^2*a*b^6+1/3*d^2*e*b^7)*x^9+(35/8*e^3*a^3*b^4+63/8*d*e^2*a^2*b^5+21/8 *d^2*e*a*b^6+1/8*d^3*b^7)*x^8+(5*a^4*b^3*e^3+15*a^3*b^4*d*e^2+9*a^2*b^5*d^ 2*e+a*b^6*d^3)*x^7+(7/2*e^3*a^5*b^2+35/2*d*e^2*a^4*b^3+35/2*d^2*e*a^3*b^4+ 7/2*d^3*a^2*b^5)*x^6+(7/5*e^3*a^6*b+63/5*d*e^2*a^5*b^2+21*d^2*e*a^4*b^3+7* d^3*a^3*b^4)*x^5+(1/4*e^3*a^7+21/4*d*e^2*a^6*b+63/4*d^2*e*a^5*b^2+35/4*d^3 *a^4*b^3)*x^4+(a^7*d*e^2+7*a^6*b*d^2*e+7*a^5*b^2*d^3)*x^3+(3/2*d^2*e*a^7+7 /2*d^3*a^6*b)*x^2+d^3*a^7*x
Leaf count of result is larger than twice the leaf count of optimal. 376 vs. \(2 (84) = 168\).
Time = 0.08 (sec) , antiderivative size = 376, normalized size of antiderivative = 4.09 \[ \int (a+b x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{11} \, b^{7} e^{3} x^{11} + a^{7} d^{3} x + \frac {1}{10} \, {\left (3 \, b^{7} d e^{2} + 7 \, a b^{6} e^{3}\right )} x^{10} + \frac {1}{3} \, {\left (b^{7} d^{2} e + 7 \, a b^{6} d e^{2} + 7 \, a^{2} b^{5} e^{3}\right )} x^{9} + \frac {1}{8} \, {\left (b^{7} d^{3} + 21 \, a b^{6} d^{2} e + 63 \, a^{2} b^{5} d e^{2} + 35 \, a^{3} b^{4} e^{3}\right )} x^{8} + {\left (a b^{6} d^{3} + 9 \, a^{2} b^{5} d^{2} e + 15 \, a^{3} b^{4} d e^{2} + 5 \, a^{4} b^{3} e^{3}\right )} x^{7} + \frac {7}{2} \, {\left (a^{2} b^{5} d^{3} + 5 \, a^{3} b^{4} d^{2} e + 5 \, a^{4} b^{3} d e^{2} + a^{5} b^{2} e^{3}\right )} x^{6} + \frac {7}{5} \, {\left (5 \, a^{3} b^{4} d^{3} + 15 \, a^{4} b^{3} d^{2} e + 9 \, a^{5} b^{2} d e^{2} + a^{6} b e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (35 \, a^{4} b^{3} d^{3} + 63 \, a^{5} b^{2} d^{2} e + 21 \, a^{6} b d e^{2} + a^{7} e^{3}\right )} x^{4} + {\left (7 \, a^{5} b^{2} d^{3} + 7 \, a^{6} b d^{2} e + a^{7} d e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (7 \, a^{6} b d^{3} + 3 \, a^{7} d^{2} e\right )} x^{2} \] Input:
integrate((b*x+a)*(e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")
Output:
1/11*b^7*e^3*x^11 + a^7*d^3*x + 1/10*(3*b^7*d*e^2 + 7*a*b^6*e^3)*x^10 + 1/ 3*(b^7*d^2*e + 7*a*b^6*d*e^2 + 7*a^2*b^5*e^3)*x^9 + 1/8*(b^7*d^3 + 21*a*b^ 6*d^2*e + 63*a^2*b^5*d*e^2 + 35*a^3*b^4*e^3)*x^8 + (a*b^6*d^3 + 9*a^2*b^5* d^2*e + 15*a^3*b^4*d*e^2 + 5*a^4*b^3*e^3)*x^7 + 7/2*(a^2*b^5*d^3 + 5*a^3*b ^4*d^2*e + 5*a^4*b^3*d*e^2 + a^5*b^2*e^3)*x^6 + 7/5*(5*a^3*b^4*d^3 + 15*a^ 4*b^3*d^2*e + 9*a^5*b^2*d*e^2 + a^6*b*e^3)*x^5 + 1/4*(35*a^4*b^3*d^3 + 63* a^5*b^2*d^2*e + 21*a^6*b*d*e^2 + a^7*e^3)*x^4 + (7*a^5*b^2*d^3 + 7*a^6*b*d ^2*e + a^7*d*e^2)*x^3 + 1/2*(7*a^6*b*d^3 + 3*a^7*d^2*e)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (80) = 160\).
Time = 0.06 (sec) , antiderivative size = 427, normalized size of antiderivative = 4.64 \[ \int (a+b x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=a^{7} d^{3} x + \frac {b^{7} e^{3} x^{11}}{11} + x^{10} \cdot \left (\frac {7 a b^{6} e^{3}}{10} + \frac {3 b^{7} d e^{2}}{10}\right ) + x^{9} \cdot \left (\frac {7 a^{2} b^{5} e^{3}}{3} + \frac {7 a b^{6} d e^{2}}{3} + \frac {b^{7} d^{2} e}{3}\right ) + x^{8} \cdot \left (\frac {35 a^{3} b^{4} e^{3}}{8} + \frac {63 a^{2} b^{5} d e^{2}}{8} + \frac {21 a b^{6} d^{2} e}{8} + \frac {b^{7} d^{3}}{8}\right ) + x^{7} \cdot \left (5 a^{4} b^{3} e^{3} + 15 a^{3} b^{4} d e^{2} + 9 a^{2} b^{5} d^{2} e + a b^{6} d^{3}\right ) + x^{6} \cdot \left (\frac {7 a^{5} b^{2} e^{3}}{2} + \frac {35 a^{4} b^{3} d e^{2}}{2} + \frac {35 a^{3} b^{4} d^{2} e}{2} + \frac {7 a^{2} b^{5} d^{3}}{2}\right ) + x^{5} \cdot \left (\frac {7 a^{6} b e^{3}}{5} + \frac {63 a^{5} b^{2} d e^{2}}{5} + 21 a^{4} b^{3} d^{2} e + 7 a^{3} b^{4} d^{3}\right ) + x^{4} \left (\frac {a^{7} e^{3}}{4} + \frac {21 a^{6} b d e^{2}}{4} + \frac {63 a^{5} b^{2} d^{2} e}{4} + \frac {35 a^{4} b^{3} d^{3}}{4}\right ) + x^{3} \left (a^{7} d e^{2} + 7 a^{6} b d^{2} e + 7 a^{5} b^{2} d^{3}\right ) + x^{2} \cdot \left (\frac {3 a^{7} d^{2} e}{2} + \frac {7 a^{6} b d^{3}}{2}\right ) \] Input:
integrate((b*x+a)*(e*x+d)**3*(b**2*x**2+2*a*b*x+a**2)**3,x)
Output:
a**7*d**3*x + b**7*e**3*x**11/11 + x**10*(7*a*b**6*e**3/10 + 3*b**7*d*e**2 /10) + x**9*(7*a**2*b**5*e**3/3 + 7*a*b**6*d*e**2/3 + b**7*d**2*e/3) + x** 8*(35*a**3*b**4*e**3/8 + 63*a**2*b**5*d*e**2/8 + 21*a*b**6*d**2*e/8 + b**7 *d**3/8) + x**7*(5*a**4*b**3*e**3 + 15*a**3*b**4*d*e**2 + 9*a**2*b**5*d**2 *e + a*b**6*d**3) + x**6*(7*a**5*b**2*e**3/2 + 35*a**4*b**3*d*e**2/2 + 35* a**3*b**4*d**2*e/2 + 7*a**2*b**5*d**3/2) + x**5*(7*a**6*b*e**3/5 + 63*a**5 *b**2*d*e**2/5 + 21*a**4*b**3*d**2*e + 7*a**3*b**4*d**3) + x**4*(a**7*e**3 /4 + 21*a**6*b*d*e**2/4 + 63*a**5*b**2*d**2*e/4 + 35*a**4*b**3*d**3/4) + x **3*(a**7*d*e**2 + 7*a**6*b*d**2*e + 7*a**5*b**2*d**3) + x**2*(3*a**7*d**2 *e/2 + 7*a**6*b*d**3/2)
Leaf count of result is larger than twice the leaf count of optimal. 376 vs. \(2 (84) = 168\).
Time = 0.03 (sec) , antiderivative size = 376, normalized size of antiderivative = 4.09 \[ \int (a+b x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{11} \, b^{7} e^{3} x^{11} + a^{7} d^{3} x + \frac {1}{10} \, {\left (3 \, b^{7} d e^{2} + 7 \, a b^{6} e^{3}\right )} x^{10} + \frac {1}{3} \, {\left (b^{7} d^{2} e + 7 \, a b^{6} d e^{2} + 7 \, a^{2} b^{5} e^{3}\right )} x^{9} + \frac {1}{8} \, {\left (b^{7} d^{3} + 21 \, a b^{6} d^{2} e + 63 \, a^{2} b^{5} d e^{2} + 35 \, a^{3} b^{4} e^{3}\right )} x^{8} + {\left (a b^{6} d^{3} + 9 \, a^{2} b^{5} d^{2} e + 15 \, a^{3} b^{4} d e^{2} + 5 \, a^{4} b^{3} e^{3}\right )} x^{7} + \frac {7}{2} \, {\left (a^{2} b^{5} d^{3} + 5 \, a^{3} b^{4} d^{2} e + 5 \, a^{4} b^{3} d e^{2} + a^{5} b^{2} e^{3}\right )} x^{6} + \frac {7}{5} \, {\left (5 \, a^{3} b^{4} d^{3} + 15 \, a^{4} b^{3} d^{2} e + 9 \, a^{5} b^{2} d e^{2} + a^{6} b e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (35 \, a^{4} b^{3} d^{3} + 63 \, a^{5} b^{2} d^{2} e + 21 \, a^{6} b d e^{2} + a^{7} e^{3}\right )} x^{4} + {\left (7 \, a^{5} b^{2} d^{3} + 7 \, a^{6} b d^{2} e + a^{7} d e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (7 \, a^{6} b d^{3} + 3 \, a^{7} d^{2} e\right )} x^{2} \] Input:
integrate((b*x+a)*(e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")
Output:
1/11*b^7*e^3*x^11 + a^7*d^3*x + 1/10*(3*b^7*d*e^2 + 7*a*b^6*e^3)*x^10 + 1/ 3*(b^7*d^2*e + 7*a*b^6*d*e^2 + 7*a^2*b^5*e^3)*x^9 + 1/8*(b^7*d^3 + 21*a*b^ 6*d^2*e + 63*a^2*b^5*d*e^2 + 35*a^3*b^4*e^3)*x^8 + (a*b^6*d^3 + 9*a^2*b^5* d^2*e + 15*a^3*b^4*d*e^2 + 5*a^4*b^3*e^3)*x^7 + 7/2*(a^2*b^5*d^3 + 5*a^3*b ^4*d^2*e + 5*a^4*b^3*d*e^2 + a^5*b^2*e^3)*x^6 + 7/5*(5*a^3*b^4*d^3 + 15*a^ 4*b^3*d^2*e + 9*a^5*b^2*d*e^2 + a^6*b*e^3)*x^5 + 1/4*(35*a^4*b^3*d^3 + 63* a^5*b^2*d^2*e + 21*a^6*b*d*e^2 + a^7*e^3)*x^4 + (7*a^5*b^2*d^3 + 7*a^6*b*d ^2*e + a^7*d*e^2)*x^3 + 1/2*(7*a^6*b*d^3 + 3*a^7*d^2*e)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 420 vs. \(2 (84) = 168\).
Time = 0.18 (sec) , antiderivative size = 420, normalized size of antiderivative = 4.57 \[ \int (a+b x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{11} \, b^{7} e^{3} x^{11} + \frac {3}{10} \, b^{7} d e^{2} x^{10} + \frac {7}{10} \, a b^{6} e^{3} x^{10} + \frac {1}{3} \, b^{7} d^{2} e x^{9} + \frac {7}{3} \, a b^{6} d e^{2} x^{9} + \frac {7}{3} \, a^{2} b^{5} e^{3} x^{9} + \frac {1}{8} \, b^{7} d^{3} x^{8} + \frac {21}{8} \, a b^{6} d^{2} e x^{8} + \frac {63}{8} \, a^{2} b^{5} d e^{2} x^{8} + \frac {35}{8} \, a^{3} b^{4} e^{3} x^{8} + a b^{6} d^{3} x^{7} + 9 \, a^{2} b^{5} d^{2} e x^{7} + 15 \, a^{3} b^{4} d e^{2} x^{7} + 5 \, a^{4} b^{3} e^{3} x^{7} + \frac {7}{2} \, a^{2} b^{5} d^{3} x^{6} + \frac {35}{2} \, a^{3} b^{4} d^{2} e x^{6} + \frac {35}{2} \, a^{4} b^{3} d e^{2} x^{6} + \frac {7}{2} \, a^{5} b^{2} e^{3} x^{6} + 7 \, a^{3} b^{4} d^{3} x^{5} + 21 \, a^{4} b^{3} d^{2} e x^{5} + \frac {63}{5} \, a^{5} b^{2} d e^{2} x^{5} + \frac {7}{5} \, a^{6} b e^{3} x^{5} + \frac {35}{4} \, a^{4} b^{3} d^{3} x^{4} + \frac {63}{4} \, a^{5} b^{2} d^{2} e x^{4} + \frac {21}{4} \, a^{6} b d e^{2} x^{4} + \frac {1}{4} \, a^{7} e^{3} x^{4} + 7 \, a^{5} b^{2} d^{3} x^{3} + 7 \, a^{6} b d^{2} e x^{3} + a^{7} d e^{2} x^{3} + \frac {7}{2} \, a^{6} b d^{3} x^{2} + \frac {3}{2} \, a^{7} d^{2} e x^{2} + a^{7} d^{3} x \] Input:
integrate((b*x+a)*(e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")
Output:
1/11*b^7*e^3*x^11 + 3/10*b^7*d*e^2*x^10 + 7/10*a*b^6*e^3*x^10 + 1/3*b^7*d^ 2*e*x^9 + 7/3*a*b^6*d*e^2*x^9 + 7/3*a^2*b^5*e^3*x^9 + 1/8*b^7*d^3*x^8 + 21 /8*a*b^6*d^2*e*x^8 + 63/8*a^2*b^5*d*e^2*x^8 + 35/8*a^3*b^4*e^3*x^8 + a*b^6 *d^3*x^7 + 9*a^2*b^5*d^2*e*x^7 + 15*a^3*b^4*d*e^2*x^7 + 5*a^4*b^3*e^3*x^7 + 7/2*a^2*b^5*d^3*x^6 + 35/2*a^3*b^4*d^2*e*x^6 + 35/2*a^4*b^3*d*e^2*x^6 + 7/2*a^5*b^2*e^3*x^6 + 7*a^3*b^4*d^3*x^5 + 21*a^4*b^3*d^2*e*x^5 + 63/5*a^5* b^2*d*e^2*x^5 + 7/5*a^6*b*e^3*x^5 + 35/4*a^4*b^3*d^3*x^4 + 63/4*a^5*b^2*d^ 2*e*x^4 + 21/4*a^6*b*d*e^2*x^4 + 1/4*a^7*e^3*x^4 + 7*a^5*b^2*d^3*x^3 + 7*a ^6*b*d^2*e*x^3 + a^7*d*e^2*x^3 + 7/2*a^6*b*d^3*x^2 + 3/2*a^7*d^2*e*x^2 + a ^7*d^3*x
Time = 0.16 (sec) , antiderivative size = 356, normalized size of antiderivative = 3.87 \[ \int (a+b x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=x^7\,\left (5\,a^4\,b^3\,e^3+15\,a^3\,b^4\,d\,e^2+9\,a^2\,b^5\,d^2\,e+a\,b^6\,d^3\right )+x^5\,\left (\frac {7\,a^6\,b\,e^3}{5}+\frac {63\,a^5\,b^2\,d\,e^2}{5}+21\,a^4\,b^3\,d^2\,e+7\,a^3\,b^4\,d^3\right )+x^4\,\left (\frac {a^7\,e^3}{4}+\frac {21\,a^6\,b\,d\,e^2}{4}+\frac {63\,a^5\,b^2\,d^2\,e}{4}+\frac {35\,a^4\,b^3\,d^3}{4}\right )+x^8\,\left (\frac {35\,a^3\,b^4\,e^3}{8}+\frac {63\,a^2\,b^5\,d\,e^2}{8}+\frac {21\,a\,b^6\,d^2\,e}{8}+\frac {b^7\,d^3}{8}\right )+a^7\,d^3\,x+\frac {b^7\,e^3\,x^{11}}{11}+\frac {7\,a^2\,b^2\,x^6\,\left (a^3\,e^3+5\,a^2\,b\,d\,e^2+5\,a\,b^2\,d^2\,e+b^3\,d^3\right )}{2}+\frac {a^6\,d^2\,x^2\,\left (3\,a\,e+7\,b\,d\right )}{2}+\frac {b^6\,e^2\,x^{10}\,\left (7\,a\,e+3\,b\,d\right )}{10}+a^5\,d\,x^3\,\left (a^2\,e^2+7\,a\,b\,d\,e+7\,b^2\,d^2\right )+\frac {b^5\,e\,x^9\,\left (7\,a^2\,e^2+7\,a\,b\,d\,e+b^2\,d^2\right )}{3} \] Input:
int((a + b*x)*(d + e*x)^3*(a^2 + b^2*x^2 + 2*a*b*x)^3,x)
Output:
x^7*(a*b^6*d^3 + 5*a^4*b^3*e^3 + 9*a^2*b^5*d^2*e + 15*a^3*b^4*d*e^2) + x^5 *((7*a^6*b*e^3)/5 + 7*a^3*b^4*d^3 + 21*a^4*b^3*d^2*e + (63*a^5*b^2*d*e^2)/ 5) + x^4*((a^7*e^3)/4 + (35*a^4*b^3*d^3)/4 + (63*a^5*b^2*d^2*e)/4 + (21*a^ 6*b*d*e^2)/4) + x^8*((b^7*d^3)/8 + (35*a^3*b^4*e^3)/8 + (63*a^2*b^5*d*e^2) /8 + (21*a*b^6*d^2*e)/8) + a^7*d^3*x + (b^7*e^3*x^11)/11 + (7*a^2*b^2*x^6* (a^3*e^3 + b^3*d^3 + 5*a*b^2*d^2*e + 5*a^2*b*d*e^2))/2 + (a^6*d^2*x^2*(3*a *e + 7*b*d))/2 + (b^6*e^2*x^10*(7*a*e + 3*b*d))/10 + a^5*d*x^3*(a^2*e^2 + 7*b^2*d^2 + 7*a*b*d*e) + (b^5*e*x^9*(7*a^2*e^2 + b^2*d^2 + 7*a*b*d*e))/3
Time = 0.27 (sec) , antiderivative size = 421, normalized size of antiderivative = 4.58 \[ \int (a+b x) (d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {x \left (120 b^{7} e^{3} x^{10}+924 a \,b^{6} e^{3} x^{9}+396 b^{7} d \,e^{2} x^{9}+3080 a^{2} b^{5} e^{3} x^{8}+3080 a \,b^{6} d \,e^{2} x^{8}+440 b^{7} d^{2} e \,x^{8}+5775 a^{3} b^{4} e^{3} x^{7}+10395 a^{2} b^{5} d \,e^{2} x^{7}+3465 a \,b^{6} d^{2} e \,x^{7}+165 b^{7} d^{3} x^{7}+6600 a^{4} b^{3} e^{3} x^{6}+19800 a^{3} b^{4} d \,e^{2} x^{6}+11880 a^{2} b^{5} d^{2} e \,x^{6}+1320 a \,b^{6} d^{3} x^{6}+4620 a^{5} b^{2} e^{3} x^{5}+23100 a^{4} b^{3} d \,e^{2} x^{5}+23100 a^{3} b^{4} d^{2} e \,x^{5}+4620 a^{2} b^{5} d^{3} x^{5}+1848 a^{6} b \,e^{3} x^{4}+16632 a^{5} b^{2} d \,e^{2} x^{4}+27720 a^{4} b^{3} d^{2} e \,x^{4}+9240 a^{3} b^{4} d^{3} x^{4}+330 a^{7} e^{3} x^{3}+6930 a^{6} b d \,e^{2} x^{3}+20790 a^{5} b^{2} d^{2} e \,x^{3}+11550 a^{4} b^{3} d^{3} x^{3}+1320 a^{7} d \,e^{2} x^{2}+9240 a^{6} b \,d^{2} e \,x^{2}+9240 a^{5} b^{2} d^{3} x^{2}+1980 a^{7} d^{2} e x +4620 a^{6} b \,d^{3} x +1320 a^{7} d^{3}\right )}{1320} \] Input:
int((b*x+a)*(e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^3,x)
Output:
(x*(1320*a**7*d**3 + 1980*a**7*d**2*e*x + 1320*a**7*d*e**2*x**2 + 330*a**7 *e**3*x**3 + 4620*a**6*b*d**3*x + 9240*a**6*b*d**2*e*x**2 + 6930*a**6*b*d* e**2*x**3 + 1848*a**6*b*e**3*x**4 + 9240*a**5*b**2*d**3*x**2 + 20790*a**5* b**2*d**2*e*x**3 + 16632*a**5*b**2*d*e**2*x**4 + 4620*a**5*b**2*e**3*x**5 + 11550*a**4*b**3*d**3*x**3 + 27720*a**4*b**3*d**2*e*x**4 + 23100*a**4*b** 3*d*e**2*x**5 + 6600*a**4*b**3*e**3*x**6 + 9240*a**3*b**4*d**3*x**4 + 2310 0*a**3*b**4*d**2*e*x**5 + 19800*a**3*b**4*d*e**2*x**6 + 5775*a**3*b**4*e** 3*x**7 + 4620*a**2*b**5*d**3*x**5 + 11880*a**2*b**5*d**2*e*x**6 + 10395*a* *2*b**5*d*e**2*x**7 + 3080*a**2*b**5*e**3*x**8 + 1320*a*b**6*d**3*x**6 + 3 465*a*b**6*d**2*e*x**7 + 3080*a*b**6*d*e**2*x**8 + 924*a*b**6*e**3*x**9 + 165*b**7*d**3*x**7 + 440*b**7*d**2*e*x**8 + 396*b**7*d*e**2*x**9 + 120*b** 7*e**3*x**10))/1320