Integrand size = 15, antiderivative size = 65 \[ \int (a+b x)^2 (c+d x)^{10} \, dx=\frac {(b c-a d)^2 (c+d x)^{11}}{11 d^3}-\frac {b (b c-a d) (c+d x)^{12}}{6 d^3}+\frac {b^2 (c+d x)^{13}}{13 d^3} \] Output:
1/11*(-a*d+b*c)^2*(d*x+c)^11/d^3-1/6*b*(-a*d+b*c)*(d*x+c)^12/d^3+1/13*b^2* (d*x+c)^13/d^3
Leaf count is larger than twice the leaf count of optimal. \(358\) vs. \(2(65)=130\).
Time = 0.02 (sec) , antiderivative size = 358, normalized size of antiderivative = 5.51 \[ \int (a+b x)^2 (c+d x)^{10} \, dx=a^2 c^{10} x+a c^9 (b c+5 a d) x^2+\frac {1}{3} c^8 \left (b^2 c^2+20 a b c d+45 a^2 d^2\right ) x^3+\frac {5}{2} c^7 d \left (b^2 c^2+9 a b c d+12 a^2 d^2\right ) x^4+3 c^6 d^2 \left (3 b^2 c^2+16 a b c d+14 a^2 d^2\right ) x^5+2 c^5 d^3 \left (10 b^2 c^2+35 a b c d+21 a^2 d^2\right ) x^6+6 c^4 d^4 \left (5 b^2 c^2+12 a b c d+5 a^2 d^2\right ) x^7+\frac {3}{2} c^3 d^5 \left (21 b^2 c^2+35 a b c d+10 a^2 d^2\right ) x^8+\frac {5}{3} c^2 d^6 \left (14 b^2 c^2+16 a b c d+3 a^2 d^2\right ) x^9+c d^7 \left (12 b^2 c^2+9 a b c d+a^2 d^2\right ) x^{10}+\frac {1}{11} d^8 \left (45 b^2 c^2+20 a b c d+a^2 d^2\right ) x^{11}+\frac {1}{6} b d^9 (5 b c+a d) x^{12}+\frac {1}{13} b^2 d^{10} x^{13} \] Input:
Integrate[(a + b*x)^2*(c + d*x)^10,x]
Output:
a^2*c^10*x + a*c^9*(b*c + 5*a*d)*x^2 + (c^8*(b^2*c^2 + 20*a*b*c*d + 45*a^2 *d^2)*x^3)/3 + (5*c^7*d*(b^2*c^2 + 9*a*b*c*d + 12*a^2*d^2)*x^4)/2 + 3*c^6* d^2*(3*b^2*c^2 + 16*a*b*c*d + 14*a^2*d^2)*x^5 + 2*c^5*d^3*(10*b^2*c^2 + 35 *a*b*c*d + 21*a^2*d^2)*x^6 + 6*c^4*d^4*(5*b^2*c^2 + 12*a*b*c*d + 5*a^2*d^2 )*x^7 + (3*c^3*d^5*(21*b^2*c^2 + 35*a*b*c*d + 10*a^2*d^2)*x^8)/2 + (5*c^2* d^6*(14*b^2*c^2 + 16*a*b*c*d + 3*a^2*d^2)*x^9)/3 + c*d^7*(12*b^2*c^2 + 9*a *b*c*d + a^2*d^2)*x^10 + (d^8*(45*b^2*c^2 + 20*a*b*c*d + a^2*d^2)*x^11)/11 + (b*d^9*(5*b*c + a*d)*x^12)/6 + (b^2*d^10*x^13)/13
Time = 0.35 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {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.
| \(\displaystyle \int (a+b x)^2 (c+d x)^{10} \, dx\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \int \left (-\frac {2 b (c+d x)^{11} (b c-a d)}{d^2}+\frac {(c+d x)^{10} (a d-b c)^2}{d^2}+\frac {b^2 (c+d x)^{12}}{d^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {b (c+d x)^{12} (b c-a d)}{6 d^3}+\frac {(c+d x)^{11} (b c-a d)^2}{11 d^3}+\frac {b^2 (c+d x)^{13}}{13 d^3}\) |
Input:
Int[(a + b*x)^2*(c + d*x)^10,x]
Output:
((b*c - a*d)^2*(c + d*x)^11)/(11*d^3) - (b*(b*c - a*d)*(c + d*x)^12)/(6*d^ 3) + (b^2*(c + d*x)^13)/(13*d^3)
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]
Leaf count of result is larger than twice the leaf count of optimal. \(379\) vs. \(2(59)=118\).
Time = 0.09 (sec) , antiderivative size = 380, normalized size of antiderivative = 5.85
| method | result | size |
| norman | \(\frac {b^{2} d^{10} x^{13}}{13}+\left (\frac {1}{6} a b \,d^{10}+\frac {5}{6} b^{2} c \,d^{9}\right ) x^{12}+\left (\frac {1}{11} a^{2} d^{10}+\frac {20}{11} a b c \,d^{9}+\frac {45}{11} b^{2} c^{2} d^{8}\right ) x^{11}+\left (a^{2} c \,d^{9}+9 a b \,c^{2} d^{8}+12 b^{2} c^{3} d^{7}\right ) x^{10}+\left (5 a^{2} c^{2} d^{8}+\frac {80}{3} a b \,c^{3} d^{7}+\frac {70}{3} b^{2} c^{4} d^{6}\right ) x^{9}+\left (15 a^{2} c^{3} d^{7}+\frac {105}{2} a b \,c^{4} d^{6}+\frac {63}{2} b^{2} c^{5} d^{5}\right ) x^{8}+\left (30 a^{2} c^{4} d^{6}+72 a b \,c^{5} d^{5}+30 b^{2} c^{6} d^{4}\right ) x^{7}+\left (42 a^{2} c^{5} d^{5}+70 a b \,c^{6} d^{4}+20 b^{2} c^{7} d^{3}\right ) x^{6}+\left (42 a^{2} c^{6} d^{4}+48 a b \,c^{7} d^{3}+9 b^{2} c^{8} d^{2}\right ) x^{5}+\left (30 a^{2} c^{7} d^{3}+\frac {45}{2} a b \,c^{8} d^{2}+\frac {5}{2} b^{2} c^{9} d \right ) x^{4}+\left (15 a^{2} c^{8} d^{2}+\frac {20}{3} a b \,c^{9} d +\frac {1}{3} b^{2} c^{10}\right ) x^{3}+\left (5 a^{2} c^{9} d +a b \,c^{10}\right ) x^{2}+a^{2} c^{10} x\) | \(380\) |
| default | \(\frac {b^{2} d^{10} x^{13}}{13}+\frac {\left (2 a b \,d^{10}+10 b^{2} c \,d^{9}\right ) x^{12}}{12}+\frac {\left (a^{2} d^{10}+20 a b c \,d^{9}+45 b^{2} c^{2} d^{8}\right ) x^{11}}{11}+\frac {\left (10 a^{2} c \,d^{9}+90 a b \,c^{2} d^{8}+120 b^{2} c^{3} d^{7}\right ) x^{10}}{10}+\frac {\left (45 a^{2} c^{2} d^{8}+240 a b \,c^{3} d^{7}+210 b^{2} c^{4} d^{6}\right ) x^{9}}{9}+\frac {\left (120 a^{2} c^{3} d^{7}+420 a b \,c^{4} d^{6}+252 b^{2} c^{5} d^{5}\right ) x^{8}}{8}+\frac {\left (210 a^{2} c^{4} d^{6}+504 a b \,c^{5} d^{5}+210 b^{2} c^{6} d^{4}\right ) x^{7}}{7}+\frac {\left (252 a^{2} c^{5} d^{5}+420 a b \,c^{6} d^{4}+120 b^{2} c^{7} d^{3}\right ) x^{6}}{6}+\frac {\left (210 a^{2} c^{6} d^{4}+240 a b \,c^{7} d^{3}+45 b^{2} c^{8} d^{2}\right ) x^{5}}{5}+\frac {\left (120 a^{2} c^{7} d^{3}+90 a b \,c^{8} d^{2}+10 b^{2} c^{9} d \right ) x^{4}}{4}+\frac {\left (45 a^{2} c^{8} d^{2}+20 a b \,c^{9} d +b^{2} c^{10}\right ) x^{3}}{3}+\frac {\left (10 a^{2} c^{9} d +2 a b \,c^{10}\right ) x^{2}}{2}+a^{2} c^{10} x\) | \(391\) |
| gosper | \(\frac {1}{3} x^{3} b^{2} c^{10}+\frac {1}{13} b^{2} d^{10} x^{13}+a^{2} c^{10} x +\frac {1}{11} x^{11} a^{2} d^{10}+30 a^{2} c^{4} d^{6} x^{7}+30 b^{2} c^{6} d^{4} x^{7}+42 a^{2} c^{5} d^{5} x^{6}+20 b^{2} c^{7} d^{3} x^{6}+42 a^{2} c^{6} d^{4} x^{5}+9 b^{2} c^{8} d^{2} x^{5}+5 a^{2} c^{9} d \,x^{2}+a b \,c^{10} x^{2}+\frac {1}{6} x^{12} a b \,d^{10}+\frac {5}{6} x^{12} b^{2} c \,d^{9}+\frac {45}{11} x^{11} b^{2} c^{2} d^{8}+5 x^{9} a^{2} c^{2} d^{8}+\frac {70}{3} x^{9} b^{2} c^{4} d^{6}+15 x^{8} a^{2} c^{3} d^{7}+\frac {63}{2} x^{8} b^{2} c^{5} d^{5}+30 x^{4} a^{2} c^{7} d^{3}+\frac {5}{2} x^{4} b^{2} c^{9} d +15 x^{3} a^{2} c^{8} d^{2}+a^{2} c \,d^{9} x^{10}+12 b^{2} c^{3} d^{7} x^{10}+\frac {80}{3} x^{9} a b \,c^{3} d^{7}+\frac {105}{2} x^{8} a b \,c^{4} d^{6}+\frac {45}{2} x^{4} a b \,c^{8} d^{2}+\frac {20}{3} x^{3} a b \,c^{9} d +9 a b \,c^{2} d^{8} x^{10}+72 a b \,c^{5} d^{5} x^{7}+70 a b \,c^{6} d^{4} x^{6}+48 a b \,c^{7} d^{3} x^{5}+\frac {20}{11} x^{11} a b c \,d^{9}\) | \(418\) |
| risch | \(\frac {1}{3} x^{3} b^{2} c^{10}+\frac {1}{13} b^{2} d^{10} x^{13}+a^{2} c^{10} x +\frac {1}{11} x^{11} a^{2} d^{10}+30 a^{2} c^{4} d^{6} x^{7}+30 b^{2} c^{6} d^{4} x^{7}+42 a^{2} c^{5} d^{5} x^{6}+20 b^{2} c^{7} d^{3} x^{6}+42 a^{2} c^{6} d^{4} x^{5}+9 b^{2} c^{8} d^{2} x^{5}+5 a^{2} c^{9} d \,x^{2}+a b \,c^{10} x^{2}+\frac {1}{6} x^{12} a b \,d^{10}+\frac {5}{6} x^{12} b^{2} c \,d^{9}+\frac {45}{11} x^{11} b^{2} c^{2} d^{8}+5 x^{9} a^{2} c^{2} d^{8}+\frac {70}{3} x^{9} b^{2} c^{4} d^{6}+15 x^{8} a^{2} c^{3} d^{7}+\frac {63}{2} x^{8} b^{2} c^{5} d^{5}+30 x^{4} a^{2} c^{7} d^{3}+\frac {5}{2} x^{4} b^{2} c^{9} d +15 x^{3} a^{2} c^{8} d^{2}+a^{2} c \,d^{9} x^{10}+12 b^{2} c^{3} d^{7} x^{10}+\frac {80}{3} x^{9} a b \,c^{3} d^{7}+\frac {105}{2} x^{8} a b \,c^{4} d^{6}+\frac {45}{2} x^{4} a b \,c^{8} d^{2}+\frac {20}{3} x^{3} a b \,c^{9} d +9 a b \,c^{2} d^{8} x^{10}+72 a b \,c^{5} d^{5} x^{7}+70 a b \,c^{6} d^{4} x^{6}+48 a b \,c^{7} d^{3} x^{5}+\frac {20}{11} x^{11} a b c \,d^{9}\) | \(418\) |
| parallelrisch | \(\frac {1}{3} x^{3} b^{2} c^{10}+\frac {1}{13} b^{2} d^{10} x^{13}+a^{2} c^{10} x +\frac {1}{11} x^{11} a^{2} d^{10}+30 a^{2} c^{4} d^{6} x^{7}+30 b^{2} c^{6} d^{4} x^{7}+42 a^{2} c^{5} d^{5} x^{6}+20 b^{2} c^{7} d^{3} x^{6}+42 a^{2} c^{6} d^{4} x^{5}+9 b^{2} c^{8} d^{2} x^{5}+5 a^{2} c^{9} d \,x^{2}+a b \,c^{10} x^{2}+\frac {1}{6} x^{12} a b \,d^{10}+\frac {5}{6} x^{12} b^{2} c \,d^{9}+\frac {45}{11} x^{11} b^{2} c^{2} d^{8}+5 x^{9} a^{2} c^{2} d^{8}+\frac {70}{3} x^{9} b^{2} c^{4} d^{6}+15 x^{8} a^{2} c^{3} d^{7}+\frac {63}{2} x^{8} b^{2} c^{5} d^{5}+30 x^{4} a^{2} c^{7} d^{3}+\frac {5}{2} x^{4} b^{2} c^{9} d +15 x^{3} a^{2} c^{8} d^{2}+a^{2} c \,d^{9} x^{10}+12 b^{2} c^{3} d^{7} x^{10}+\frac {80}{3} x^{9} a b \,c^{3} d^{7}+\frac {105}{2} x^{8} a b \,c^{4} d^{6}+\frac {45}{2} x^{4} a b \,c^{8} d^{2}+\frac {20}{3} x^{3} a b \,c^{9} d +9 a b \,c^{2} d^{8} x^{10}+72 a b \,c^{5} d^{5} x^{7}+70 a b \,c^{6} d^{4} x^{6}+48 a b \,c^{7} d^{3} x^{5}+\frac {20}{11} x^{11} a b c \,d^{9}\) | \(418\) |
| orering | \(\frac {x \left (66 b^{2} d^{10} x^{12}+143 a b \,d^{10} x^{11}+715 b^{2} c \,d^{9} x^{11}+78 a^{2} d^{10} x^{10}+1560 a b c \,d^{9} x^{10}+3510 b^{2} c^{2} d^{8} x^{10}+858 a^{2} c \,d^{9} x^{9}+7722 a b \,c^{2} d^{8} x^{9}+10296 b^{2} c^{3} d^{7} x^{9}+4290 a^{2} c^{2} d^{8} x^{8}+22880 a b \,c^{3} d^{7} x^{8}+20020 b^{2} c^{4} d^{6} x^{8}+12870 a^{2} c^{3} d^{7} x^{7}+45045 a b \,c^{4} d^{6} x^{7}+27027 b^{2} c^{5} d^{5} x^{7}+25740 a^{2} c^{4} d^{6} x^{6}+61776 a b \,c^{5} d^{5} x^{6}+25740 b^{2} c^{6} d^{4} x^{6}+36036 a^{2} c^{5} d^{5} x^{5}+60060 a b \,c^{6} d^{4} x^{5}+17160 b^{2} c^{7} d^{3} x^{5}+36036 a^{2} c^{6} d^{4} x^{4}+41184 a b \,c^{7} d^{3} x^{4}+7722 b^{2} c^{8} d^{2} x^{4}+25740 a^{2} c^{7} d^{3} x^{3}+19305 a b \,c^{8} d^{2} x^{3}+2145 b^{2} c^{9} d \,x^{3}+12870 a^{2} c^{8} d^{2} x^{2}+5720 a b \,c^{9} d \,x^{2}+286 b^{2} c^{10} x^{2}+4290 a^{2} c^{9} d x +858 a b \,c^{10} x +858 a^{2} c^{10}\right )}{858}\) | \(419\) |
Input:
int((b*x+a)^2*(d*x+c)^10,x,method=_RETURNVERBOSE)
Output:
1/13*b^2*d^10*x^13+(1/6*a*b*d^10+5/6*b^2*c*d^9)*x^12+(1/11*a^2*d^10+20/11* a*b*c*d^9+45/11*b^2*c^2*d^8)*x^11+(a^2*c*d^9+9*a*b*c^2*d^8+12*b^2*c^3*d^7) *x^10+(5*a^2*c^2*d^8+80/3*a*b*c^3*d^7+70/3*b^2*c^4*d^6)*x^9+(15*a^2*c^3*d^ 7+105/2*a*b*c^4*d^6+63/2*b^2*c^5*d^5)*x^8+(30*a^2*c^4*d^6+72*a*b*c^5*d^5+3 0*b^2*c^6*d^4)*x^7+(42*a^2*c^5*d^5+70*a*b*c^6*d^4+20*b^2*c^7*d^3)*x^6+(42* a^2*c^6*d^4+48*a*b*c^7*d^3+9*b^2*c^8*d^2)*x^5+(30*a^2*c^7*d^3+45/2*a*b*c^8 *d^2+5/2*b^2*c^9*d)*x^4+(15*a^2*c^8*d^2+20/3*a*b*c^9*d+1/3*b^2*c^10)*x^3+( 5*a^2*c^9*d+a*b*c^10)*x^2+a^2*c^10*x
Leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (59) = 118\).
Time = 0.09 (sec) , antiderivative size = 384, normalized size of antiderivative = 5.91 \[ \int (a+b x)^2 (c+d x)^{10} \, dx=\frac {1}{13} \, b^{2} d^{10} x^{13} + a^{2} c^{10} x + \frac {1}{6} \, {\left (5 \, b^{2} c d^{9} + a b d^{10}\right )} x^{12} + \frac {1}{11} \, {\left (45 \, b^{2} c^{2} d^{8} + 20 \, a b c d^{9} + a^{2} d^{10}\right )} x^{11} + {\left (12 \, b^{2} c^{3} d^{7} + 9 \, a b c^{2} d^{8} + a^{2} c d^{9}\right )} x^{10} + \frac {5}{3} \, {\left (14 \, b^{2} c^{4} d^{6} + 16 \, a b c^{3} d^{7} + 3 \, a^{2} c^{2} d^{8}\right )} x^{9} + \frac {3}{2} \, {\left (21 \, b^{2} c^{5} d^{5} + 35 \, a b c^{4} d^{6} + 10 \, a^{2} c^{3} d^{7}\right )} x^{8} + 6 \, {\left (5 \, b^{2} c^{6} d^{4} + 12 \, a b c^{5} d^{5} + 5 \, a^{2} c^{4} d^{6}\right )} x^{7} + 2 \, {\left (10 \, b^{2} c^{7} d^{3} + 35 \, a b c^{6} d^{4} + 21 \, a^{2} c^{5} d^{5}\right )} x^{6} + 3 \, {\left (3 \, b^{2} c^{8} d^{2} + 16 \, a b c^{7} d^{3} + 14 \, a^{2} c^{6} d^{4}\right )} x^{5} + \frac {5}{2} \, {\left (b^{2} c^{9} d + 9 \, a b c^{8} d^{2} + 12 \, a^{2} c^{7} d^{3}\right )} x^{4} + \frac {1}{3} \, {\left (b^{2} c^{10} + 20 \, a b c^{9} d + 45 \, a^{2} c^{8} d^{2}\right )} x^{3} + {\left (a b c^{10} + 5 \, a^{2} c^{9} d\right )} x^{2} \] Input:
integrate((b*x+a)^2*(d*x+c)^10,x, algorithm="fricas")
Output:
1/13*b^2*d^10*x^13 + a^2*c^10*x + 1/6*(5*b^2*c*d^9 + a*b*d^10)*x^12 + 1/11 *(45*b^2*c^2*d^8 + 20*a*b*c*d^9 + a^2*d^10)*x^11 + (12*b^2*c^3*d^7 + 9*a*b *c^2*d^8 + a^2*c*d^9)*x^10 + 5/3*(14*b^2*c^4*d^6 + 16*a*b*c^3*d^7 + 3*a^2* c^2*d^8)*x^9 + 3/2*(21*b^2*c^5*d^5 + 35*a*b*c^4*d^6 + 10*a^2*c^3*d^7)*x^8 + 6*(5*b^2*c^6*d^4 + 12*a*b*c^5*d^5 + 5*a^2*c^4*d^6)*x^7 + 2*(10*b^2*c^7*d ^3 + 35*a*b*c^6*d^4 + 21*a^2*c^5*d^5)*x^6 + 3*(3*b^2*c^8*d^2 + 16*a*b*c^7* d^3 + 14*a^2*c^6*d^4)*x^5 + 5/2*(b^2*c^9*d + 9*a*b*c^8*d^2 + 12*a^2*c^7*d^ 3)*x^4 + 1/3*(b^2*c^10 + 20*a*b*c^9*d + 45*a^2*c^8*d^2)*x^3 + (a*b*c^10 + 5*a^2*c^9*d)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 415 vs. \(2 (54) = 108\).
Time = 0.05 (sec) , antiderivative size = 415, normalized size of antiderivative = 6.38 \[ \int (a+b x)^2 (c+d x)^{10} \, dx=a^{2} c^{10} x + \frac {b^{2} d^{10} x^{13}}{13} + x^{12} \left (\frac {a b d^{10}}{6} + \frac {5 b^{2} c d^{9}}{6}\right ) + x^{11} \left (\frac {a^{2} d^{10}}{11} + \frac {20 a b c d^{9}}{11} + \frac {45 b^{2} c^{2} d^{8}}{11}\right ) + x^{10} \left (a^{2} c d^{9} + 9 a b c^{2} d^{8} + 12 b^{2} c^{3} d^{7}\right ) + x^{9} \cdot \left (5 a^{2} c^{2} d^{8} + \frac {80 a b c^{3} d^{7}}{3} + \frac {70 b^{2} c^{4} d^{6}}{3}\right ) + x^{8} \cdot \left (15 a^{2} c^{3} d^{7} + \frac {105 a b c^{4} d^{6}}{2} + \frac {63 b^{2} c^{5} d^{5}}{2}\right ) + x^{7} \cdot \left (30 a^{2} c^{4} d^{6} + 72 a b c^{5} d^{5} + 30 b^{2} c^{6} d^{4}\right ) + x^{6} \cdot \left (42 a^{2} c^{5} d^{5} + 70 a b c^{6} d^{4} + 20 b^{2} c^{7} d^{3}\right ) + x^{5} \cdot \left (42 a^{2} c^{6} d^{4} + 48 a b c^{7} d^{3} + 9 b^{2} c^{8} d^{2}\right ) + x^{4} \cdot \left (30 a^{2} c^{7} d^{3} + \frac {45 a b c^{8} d^{2}}{2} + \frac {5 b^{2} c^{9} d}{2}\right ) + x^{3} \cdot \left (15 a^{2} c^{8} d^{2} + \frac {20 a b c^{9} d}{3} + \frac {b^{2} c^{10}}{3}\right ) + x^{2} \cdot \left (5 a^{2} c^{9} d + a b c^{10}\right ) \] Input:
integrate((b*x+a)**2*(d*x+c)**10,x)
Output:
a**2*c**10*x + b**2*d**10*x**13/13 + x**12*(a*b*d**10/6 + 5*b**2*c*d**9/6) + x**11*(a**2*d**10/11 + 20*a*b*c*d**9/11 + 45*b**2*c**2*d**8/11) + x**10 *(a**2*c*d**9 + 9*a*b*c**2*d**8 + 12*b**2*c**3*d**7) + x**9*(5*a**2*c**2*d **8 + 80*a*b*c**3*d**7/3 + 70*b**2*c**4*d**6/3) + x**8*(15*a**2*c**3*d**7 + 105*a*b*c**4*d**6/2 + 63*b**2*c**5*d**5/2) + x**7*(30*a**2*c**4*d**6 + 7 2*a*b*c**5*d**5 + 30*b**2*c**6*d**4) + x**6*(42*a**2*c**5*d**5 + 70*a*b*c* *6*d**4 + 20*b**2*c**7*d**3) + x**5*(42*a**2*c**6*d**4 + 48*a*b*c**7*d**3 + 9*b**2*c**8*d**2) + x**4*(30*a**2*c**7*d**3 + 45*a*b*c**8*d**2/2 + 5*b** 2*c**9*d/2) + x**3*(15*a**2*c**8*d**2 + 20*a*b*c**9*d/3 + b**2*c**10/3) + x**2*(5*a**2*c**9*d + a*b*c**10)
Leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (59) = 118\).
Time = 0.03 (sec) , antiderivative size = 384, normalized size of antiderivative = 5.91 \[ \int (a+b x)^2 (c+d x)^{10} \, dx=\frac {1}{13} \, b^{2} d^{10} x^{13} + a^{2} c^{10} x + \frac {1}{6} \, {\left (5 \, b^{2} c d^{9} + a b d^{10}\right )} x^{12} + \frac {1}{11} \, {\left (45 \, b^{2} c^{2} d^{8} + 20 \, a b c d^{9} + a^{2} d^{10}\right )} x^{11} + {\left (12 \, b^{2} c^{3} d^{7} + 9 \, a b c^{2} d^{8} + a^{2} c d^{9}\right )} x^{10} + \frac {5}{3} \, {\left (14 \, b^{2} c^{4} d^{6} + 16 \, a b c^{3} d^{7} + 3 \, a^{2} c^{2} d^{8}\right )} x^{9} + \frac {3}{2} \, {\left (21 \, b^{2} c^{5} d^{5} + 35 \, a b c^{4} d^{6} + 10 \, a^{2} c^{3} d^{7}\right )} x^{8} + 6 \, {\left (5 \, b^{2} c^{6} d^{4} + 12 \, a b c^{5} d^{5} + 5 \, a^{2} c^{4} d^{6}\right )} x^{7} + 2 \, {\left (10 \, b^{2} c^{7} d^{3} + 35 \, a b c^{6} d^{4} + 21 \, a^{2} c^{5} d^{5}\right )} x^{6} + 3 \, {\left (3 \, b^{2} c^{8} d^{2} + 16 \, a b c^{7} d^{3} + 14 \, a^{2} c^{6} d^{4}\right )} x^{5} + \frac {5}{2} \, {\left (b^{2} c^{9} d + 9 \, a b c^{8} d^{2} + 12 \, a^{2} c^{7} d^{3}\right )} x^{4} + \frac {1}{3} \, {\left (b^{2} c^{10} + 20 \, a b c^{9} d + 45 \, a^{2} c^{8} d^{2}\right )} x^{3} + {\left (a b c^{10} + 5 \, a^{2} c^{9} d\right )} x^{2} \] Input:
integrate((b*x+a)^2*(d*x+c)^10,x, algorithm="maxima")
Output:
1/13*b^2*d^10*x^13 + a^2*c^10*x + 1/6*(5*b^2*c*d^9 + a*b*d^10)*x^12 + 1/11 *(45*b^2*c^2*d^8 + 20*a*b*c*d^9 + a^2*d^10)*x^11 + (12*b^2*c^3*d^7 + 9*a*b *c^2*d^8 + a^2*c*d^9)*x^10 + 5/3*(14*b^2*c^4*d^6 + 16*a*b*c^3*d^7 + 3*a^2* c^2*d^8)*x^9 + 3/2*(21*b^2*c^5*d^5 + 35*a*b*c^4*d^6 + 10*a^2*c^3*d^7)*x^8 + 6*(5*b^2*c^6*d^4 + 12*a*b*c^5*d^5 + 5*a^2*c^4*d^6)*x^7 + 2*(10*b^2*c^7*d ^3 + 35*a*b*c^6*d^4 + 21*a^2*c^5*d^5)*x^6 + 3*(3*b^2*c^8*d^2 + 16*a*b*c^7* d^3 + 14*a^2*c^6*d^4)*x^5 + 5/2*(b^2*c^9*d + 9*a*b*c^8*d^2 + 12*a^2*c^7*d^ 3)*x^4 + 1/3*(b^2*c^10 + 20*a*b*c^9*d + 45*a^2*c^8*d^2)*x^3 + (a*b*c^10 + 5*a^2*c^9*d)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 417 vs. \(2 (59) = 118\).
Time = 0.13 (sec) , antiderivative size = 417, normalized size of antiderivative = 6.42 \[ \int (a+b x)^2 (c+d x)^{10} \, dx=\frac {1}{13} \, b^{2} d^{10} x^{13} + \frac {5}{6} \, b^{2} c d^{9} x^{12} + \frac {1}{6} \, a b d^{10} x^{12} + \frac {45}{11} \, b^{2} c^{2} d^{8} x^{11} + \frac {20}{11} \, a b c d^{9} x^{11} + \frac {1}{11} \, a^{2} d^{10} x^{11} + 12 \, b^{2} c^{3} d^{7} x^{10} + 9 \, a b c^{2} d^{8} x^{10} + a^{2} c d^{9} x^{10} + \frac {70}{3} \, b^{2} c^{4} d^{6} x^{9} + \frac {80}{3} \, a b c^{3} d^{7} x^{9} + 5 \, a^{2} c^{2} d^{8} x^{9} + \frac {63}{2} \, b^{2} c^{5} d^{5} x^{8} + \frac {105}{2} \, a b c^{4} d^{6} x^{8} + 15 \, a^{2} c^{3} d^{7} x^{8} + 30 \, b^{2} c^{6} d^{4} x^{7} + 72 \, a b c^{5} d^{5} x^{7} + 30 \, a^{2} c^{4} d^{6} x^{7} + 20 \, b^{2} c^{7} d^{3} x^{6} + 70 \, a b c^{6} d^{4} x^{6} + 42 \, a^{2} c^{5} d^{5} x^{6} + 9 \, b^{2} c^{8} d^{2} x^{5} + 48 \, a b c^{7} d^{3} x^{5} + 42 \, a^{2} c^{6} d^{4} x^{5} + \frac {5}{2} \, b^{2} c^{9} d x^{4} + \frac {45}{2} \, a b c^{8} d^{2} x^{4} + 30 \, a^{2} c^{7} d^{3} x^{4} + \frac {1}{3} \, b^{2} c^{10} x^{3} + \frac {20}{3} \, a b c^{9} d x^{3} + 15 \, a^{2} c^{8} d^{2} x^{3} + a b c^{10} x^{2} + 5 \, a^{2} c^{9} d x^{2} + a^{2} c^{10} x \] Input:
integrate((b*x+a)^2*(d*x+c)^10,x, algorithm="giac")
Output:
1/13*b^2*d^10*x^13 + 5/6*b^2*c*d^9*x^12 + 1/6*a*b*d^10*x^12 + 45/11*b^2*c^ 2*d^8*x^11 + 20/11*a*b*c*d^9*x^11 + 1/11*a^2*d^10*x^11 + 12*b^2*c^3*d^7*x^ 10 + 9*a*b*c^2*d^8*x^10 + a^2*c*d^9*x^10 + 70/3*b^2*c^4*d^6*x^9 + 80/3*a*b *c^3*d^7*x^9 + 5*a^2*c^2*d^8*x^9 + 63/2*b^2*c^5*d^5*x^8 + 105/2*a*b*c^4*d^ 6*x^8 + 15*a^2*c^3*d^7*x^8 + 30*b^2*c^6*d^4*x^7 + 72*a*b*c^5*d^5*x^7 + 30* a^2*c^4*d^6*x^7 + 20*b^2*c^7*d^3*x^6 + 70*a*b*c^6*d^4*x^6 + 42*a^2*c^5*d^5 *x^6 + 9*b^2*c^8*d^2*x^5 + 48*a*b*c^7*d^3*x^5 + 42*a^2*c^6*d^4*x^5 + 5/2*b ^2*c^9*d*x^4 + 45/2*a*b*c^8*d^2*x^4 + 30*a^2*c^7*d^3*x^4 + 1/3*b^2*c^10*x^ 3 + 20/3*a*b*c^9*d*x^3 + 15*a^2*c^8*d^2*x^3 + a*b*c^10*x^2 + 5*a^2*c^9*d*x ^2 + a^2*c^10*x
Time = 0.20 (sec) , antiderivative size = 348, normalized size of antiderivative = 5.35 \[ \int (a+b x)^2 (c+d x)^{10} \, dx=x^3\,\left (15\,a^2\,c^8\,d^2+\frac {20\,a\,b\,c^9\,d}{3}+\frac {b^2\,c^{10}}{3}\right )+x^{11}\,\left (\frac {a^2\,d^{10}}{11}+\frac {20\,a\,b\,c\,d^9}{11}+\frac {45\,b^2\,c^2\,d^8}{11}\right )+a^2\,c^{10}\,x+\frac {b^2\,d^{10}\,x^{13}}{13}+a\,c^9\,x^2\,\left (5\,a\,d+b\,c\right )+\frac {b\,d^9\,x^{12}\,\left (a\,d+5\,b\,c\right )}{6}+\frac {5\,c^7\,d\,x^4\,\left (12\,a^2\,d^2+9\,a\,b\,c\,d+b^2\,c^2\right )}{2}+c\,d^7\,x^{10}\,\left (a^2\,d^2+9\,a\,b\,c\,d+12\,b^2\,c^2\right )+6\,c^4\,d^4\,x^7\,\left (5\,a^2\,d^2+12\,a\,b\,c\,d+5\,b^2\,c^2\right )+3\,c^6\,d^2\,x^5\,\left (14\,a^2\,d^2+16\,a\,b\,c\,d+3\,b^2\,c^2\right )+\frac {5\,c^2\,d^6\,x^9\,\left (3\,a^2\,d^2+16\,a\,b\,c\,d+14\,b^2\,c^2\right )}{3}+2\,c^5\,d^3\,x^6\,\left (21\,a^2\,d^2+35\,a\,b\,c\,d+10\,b^2\,c^2\right )+\frac {3\,c^3\,d^5\,x^8\,\left (10\,a^2\,d^2+35\,a\,b\,c\,d+21\,b^2\,c^2\right )}{2} \] Input:
int((a + b*x)^2*(c + d*x)^10,x)
Output:
x^3*((b^2*c^10)/3 + 15*a^2*c^8*d^2 + (20*a*b*c^9*d)/3) + x^11*((a^2*d^10)/ 11 + (45*b^2*c^2*d^8)/11 + (20*a*b*c*d^9)/11) + a^2*c^10*x + (b^2*d^10*x^1 3)/13 + a*c^9*x^2*(5*a*d + b*c) + (b*d^9*x^12*(a*d + 5*b*c))/6 + (5*c^7*d* x^4*(12*a^2*d^2 + b^2*c^2 + 9*a*b*c*d))/2 + c*d^7*x^10*(a^2*d^2 + 12*b^2*c ^2 + 9*a*b*c*d) + 6*c^4*d^4*x^7*(5*a^2*d^2 + 5*b^2*c^2 + 12*a*b*c*d) + 3*c ^6*d^2*x^5*(14*a^2*d^2 + 3*b^2*c^2 + 16*a*b*c*d) + (5*c^2*d^6*x^9*(3*a^2*d ^2 + 14*b^2*c^2 + 16*a*b*c*d))/3 + 2*c^5*d^3*x^6*(21*a^2*d^2 + 10*b^2*c^2 + 35*a*b*c*d) + (3*c^3*d^5*x^8*(10*a^2*d^2 + 21*b^2*c^2 + 35*a*b*c*d))/2
Time = 0.17 (sec) , antiderivative size = 418, normalized size of antiderivative = 6.43 \[ \int (a+b x)^2 (c+d x)^{10} \, dx=\frac {x \left (66 b^{2} d^{10} x^{12}+143 a b \,d^{10} x^{11}+715 b^{2} c \,d^{9} x^{11}+78 a^{2} d^{10} x^{10}+1560 a b c \,d^{9} x^{10}+3510 b^{2} c^{2} d^{8} x^{10}+858 a^{2} c \,d^{9} x^{9}+7722 a b \,c^{2} d^{8} x^{9}+10296 b^{2} c^{3} d^{7} x^{9}+4290 a^{2} c^{2} d^{8} x^{8}+22880 a b \,c^{3} d^{7} x^{8}+20020 b^{2} c^{4} d^{6} x^{8}+12870 a^{2} c^{3} d^{7} x^{7}+45045 a b \,c^{4} d^{6} x^{7}+27027 b^{2} c^{5} d^{5} x^{7}+25740 a^{2} c^{4} d^{6} x^{6}+61776 a b \,c^{5} d^{5} x^{6}+25740 b^{2} c^{6} d^{4} x^{6}+36036 a^{2} c^{5} d^{5} x^{5}+60060 a b \,c^{6} d^{4} x^{5}+17160 b^{2} c^{7} d^{3} x^{5}+36036 a^{2} c^{6} d^{4} x^{4}+41184 a b \,c^{7} d^{3} x^{4}+7722 b^{2} c^{8} d^{2} x^{4}+25740 a^{2} c^{7} d^{3} x^{3}+19305 a b \,c^{8} d^{2} x^{3}+2145 b^{2} c^{9} d \,x^{3}+12870 a^{2} c^{8} d^{2} x^{2}+5720 a b \,c^{9} d \,x^{2}+286 b^{2} c^{10} x^{2}+4290 a^{2} c^{9} d x +858 a b \,c^{10} x +858 a^{2} c^{10}\right )}{858} \] Input:
int((b*x+a)^2*(d*x+c)^10,x)
Output:
(x*(858*a**2*c**10 + 4290*a**2*c**9*d*x + 12870*a**2*c**8*d**2*x**2 + 2574 0*a**2*c**7*d**3*x**3 + 36036*a**2*c**6*d**4*x**4 + 36036*a**2*c**5*d**5*x **5 + 25740*a**2*c**4*d**6*x**6 + 12870*a**2*c**3*d**7*x**7 + 4290*a**2*c* *2*d**8*x**8 + 858*a**2*c*d**9*x**9 + 78*a**2*d**10*x**10 + 858*a*b*c**10* x + 5720*a*b*c**9*d*x**2 + 19305*a*b*c**8*d**2*x**3 + 41184*a*b*c**7*d**3* x**4 + 60060*a*b*c**6*d**4*x**5 + 61776*a*b*c**5*d**5*x**6 + 45045*a*b*c** 4*d**6*x**7 + 22880*a*b*c**3*d**7*x**8 + 7722*a*b*c**2*d**8*x**9 + 1560*a* b*c*d**9*x**10 + 143*a*b*d**10*x**11 + 286*b**2*c**10*x**2 + 2145*b**2*c** 9*d*x**3 + 7722*b**2*c**8*d**2*x**4 + 17160*b**2*c**7*d**3*x**5 + 25740*b* *2*c**6*d**4*x**6 + 27027*b**2*c**5*d**5*x**7 + 20020*b**2*c**4*d**6*x**8 + 10296*b**2*c**3*d**7*x**9 + 3510*b**2*c**2*d**8*x**10 + 715*b**2*c*d**9* x**11 + 66*b**2*d**10*x**12))/858