Integrand size = 15, antiderivative size = 250 \[ \int (a+b x)^9 (c+d x)^{10} \, dx=-\frac {(b c-a d)^9 (c+d x)^{11}}{11 d^{10}}+\frac {3 b (b c-a d)^8 (c+d x)^{12}}{4 d^{10}}-\frac {36 b^2 (b c-a d)^7 (c+d x)^{13}}{13 d^{10}}+\frac {6 b^3 (b c-a d)^6 (c+d x)^{14}}{d^{10}}-\frac {42 b^4 (b c-a d)^5 (c+d x)^{15}}{5 d^{10}}+\frac {63 b^5 (b c-a d)^4 (c+d x)^{16}}{8 d^{10}}-\frac {84 b^6 (b c-a d)^3 (c+d x)^{17}}{17 d^{10}}+\frac {2 b^7 (b c-a d)^2 (c+d x)^{18}}{d^{10}}-\frac {9 b^8 (b c-a d) (c+d x)^{19}}{19 d^{10}}+\frac {b^9 (c+d x)^{20}}{20 d^{10}} \] Output:
-1/11*(-a*d+b*c)^9*(d*x+c)^11/d^10+3/4*b*(-a*d+b*c)^8*(d*x+c)^12/d^10-36/1 3*b^2*(-a*d+b*c)^7*(d*x+c)^13/d^10+6*b^3*(-a*d+b*c)^6*(d*x+c)^14/d^10-42/5 *b^4*(-a*d+b*c)^5*(d*x+c)^15/d^10+63/8*b^5*(-a*d+b*c)^4*(d*x+c)^16/d^10-84 /17*b^6*(-a*d+b*c)^3*(d*x+c)^17/d^10+2*b^7*(-a*d+b*c)^2*(d*x+c)^18/d^10-9/ 19*b^8*(-a*d+b*c)*(d*x+c)^19/d^10+1/20*b^9*(d*x+c)^20/d^10
Leaf count is larger than twice the leaf count of optimal. \(1397\) vs. \(2(250)=500\).
Time = 0.10 (sec) , antiderivative size = 1397, normalized size of antiderivative = 5.59 \[ \int (a+b x)^9 (c+d x)^{10} \, dx =\text {Too large to display} \] Input:
Integrate[(a + b*x)^9*(c + d*x)^10,x]
Output:
a^9*c^10*x + (a^8*c^9*(9*b*c + 10*a*d)*x^2)/2 + 3*a^7*c^8*(4*b^2*c^2 + 10* a*b*c*d + 5*a^2*d^2)*x^3 + (3*a^6*c^7*(28*b^3*c^3 + 120*a*b^2*c^2*d + 135* a^2*b*c*d^2 + 40*a^3*d^3)*x^4)/4 + (6*a^5*c^6*(21*b^4*c^4 + 140*a*b^3*c^3* d + 270*a^2*b^2*c^2*d^2 + 180*a^3*b*c*d^3 + 35*a^4*d^4)*x^5)/5 + 3*a^4*c^5 *(7*b^5*c^5 + 70*a*b^4*c^4*d + 210*a^2*b^3*c^3*d^2 + 240*a^3*b^2*c^2*d^3 + 105*a^4*b*c*d^4 + 14*a^5*d^5)*x^6 + 6*a^3*c^4*(2*b^6*c^6 + 30*a*b^5*c^5*d + 135*a^2*b^4*c^4*d^2 + 240*a^3*b^3*c^3*d^3 + 180*a^4*b^2*c^2*d^4 + 54*a^ 5*b*c*d^5 + 5*a^6*d^6)*x^7 + (3*a^2*c^3*(6*b^7*c^7 + 140*a*b^6*c^6*d + 945 *a^2*b^5*c^5*d^2 + 2520*a^3*b^4*c^4*d^3 + 2940*a^4*b^3*c^3*d^4 + 1512*a^5* b^2*c^2*d^5 + 315*a^6*b*c*d^6 + 20*a^7*d^7)*x^8)/4 + a*c^2*(b^8*c^8 + 40*a *b^7*c^7*d + 420*a^2*b^6*c^6*d^2 + 1680*a^3*b^5*c^5*d^3 + 2940*a^4*b^4*c^4 *d^4 + 2352*a^5*b^3*c^3*d^5 + 840*a^6*b^2*c^2*d^6 + 120*a^7*b*c*d^7 + 5*a^ 8*d^8)*x^9 + (c*(b^9*c^9 + 90*a*b^8*c^8*d + 1620*a^2*b^7*c^7*d^2 + 10080*a ^3*b^6*c^6*d^3 + 26460*a^4*b^5*c^5*d^4 + 31752*a^5*b^4*c^4*d^5 + 17640*a^6 *b^3*c^3*d^6 + 4320*a^7*b^2*c^2*d^7 + 405*a^8*b*c*d^8 + 10*a^9*d^9)*x^10)/ 10 + (d*(10*b^9*c^9 + 405*a*b^8*c^8*d + 4320*a^2*b^7*c^7*d^2 + 17640*a^3*b ^6*c^6*d^3 + 31752*a^4*b^5*c^5*d^4 + 26460*a^5*b^4*c^4*d^5 + 10080*a^6*b^3 *c^3*d^6 + 1620*a^7*b^2*c^2*d^7 + 90*a^8*b*c*d^8 + a^9*d^9)*x^11)/11 + (3* b*d^2*(5*b^8*c^8 + 120*a*b^7*c^7*d + 840*a^2*b^6*c^6*d^2 + 2352*a^3*b^5*c^ 5*d^3 + 2940*a^4*b^4*c^4*d^4 + 1680*a^5*b^3*c^3*d^5 + 420*a^6*b^2*c^2*d...
Time = 1.01 (sec) , antiderivative size = 250, 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.
\(\displaystyle \int (a+b x)^9 (c+d x)^{10} \, dx\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \int \left (-\frac {9 b^8 (c+d x)^{18} (b c-a d)}{d^9}+\frac {36 b^7 (c+d x)^{17} (b c-a d)^2}{d^9}-\frac {84 b^6 (c+d x)^{16} (b c-a d)^3}{d^9}+\frac {126 b^5 (c+d x)^{15} (b c-a d)^4}{d^9}-\frac {126 b^4 (c+d x)^{14} (b c-a d)^5}{d^9}+\frac {84 b^3 (c+d x)^{13} (b c-a d)^6}{d^9}-\frac {36 b^2 (c+d x)^{12} (b c-a d)^7}{d^9}+\frac {9 b (c+d x)^{11} (b c-a d)^8}{d^9}+\frac {(c+d x)^{10} (a d-b c)^9}{d^9}+\frac {b^9 (c+d x)^{19}}{d^9}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {9 b^8 (c+d x)^{19} (b c-a d)}{19 d^{10}}+\frac {2 b^7 (c+d x)^{18} (b c-a d)^2}{d^{10}}-\frac {84 b^6 (c+d x)^{17} (b c-a d)^3}{17 d^{10}}+\frac {63 b^5 (c+d x)^{16} (b c-a d)^4}{8 d^{10}}-\frac {42 b^4 (c+d x)^{15} (b c-a d)^5}{5 d^{10}}+\frac {6 b^3 (c+d x)^{14} (b c-a d)^6}{d^{10}}-\frac {36 b^2 (c+d x)^{13} (b c-a d)^7}{13 d^{10}}+\frac {3 b (c+d x)^{12} (b c-a d)^8}{4 d^{10}}-\frac {(c+d x)^{11} (b c-a d)^9}{11 d^{10}}+\frac {b^9 (c+d x)^{20}}{20 d^{10}}\) |
Input:
Int[(a + b*x)^9*(c + d*x)^10,x]
Output:
-1/11*((b*c - a*d)^9*(c + d*x)^11)/d^10 + (3*b*(b*c - a*d)^8*(c + d*x)^12) /(4*d^10) - (36*b^2*(b*c - a*d)^7*(c + d*x)^13)/(13*d^10) + (6*b^3*(b*c - a*d)^6*(c + d*x)^14)/d^10 - (42*b^4*(b*c - a*d)^5*(c + d*x)^15)/(5*d^10) + (63*b^5*(b*c - a*d)^4*(c + d*x)^16)/(8*d^10) - (84*b^6*(b*c - a*d)^3*(c + d*x)^17)/(17*d^10) + (2*b^7*(b*c - a*d)^2*(c + d*x)^18)/d^10 - (9*b^8*(b* c - a*d)*(c + d*x)^19)/(19*d^10) + (b^9*(c + d*x)^20)/(20*d^10)
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. \(1422\) vs. \(2(234)=468\).
Time = 0.14 (sec) , antiderivative size = 1423, normalized size of antiderivative = 5.69
method | result | size |
norman | \(\text {Expression too large to display}\) | \(1423\) |
default | \(\text {Expression too large to display}\) | \(1441\) |
gosper | \(\text {Expression too large to display}\) | \(1657\) |
risch | \(\text {Expression too large to display}\) | \(1657\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1657\) |
orering | \(\text {Expression too large to display}\) | \(1658\) |
Input:
int((b*x+a)^9*(d*x+c)^10,x,method=_RETURNVERBOSE)
Output:
a^9*c^10*x+(5*a^9*c^9*d+9/2*a^8*b*c^10)*x^2+(15*a^9*c^8*d^2+30*a^8*b*c^9*d +12*a^7*b^2*c^10)*x^3+(30*a^9*c^7*d^3+405/4*a^8*b*c^8*d^2+90*a^7*b^2*c^9*d +21*a^6*b^3*c^10)*x^4+(42*a^9*c^6*d^4+216*a^8*b*c^7*d^3+324*a^7*b^2*c^8*d^ 2+168*a^6*b^3*c^9*d+126/5*a^5*b^4*c^10)*x^5+(42*a^9*c^5*d^5+315*a^8*b*c^6* d^4+720*a^7*b^2*c^7*d^3+630*a^6*b^3*c^8*d^2+210*a^5*b^4*c^9*d+21*a^4*b^5*c ^10)*x^6+(30*a^9*c^4*d^6+324*a^8*b*c^5*d^5+1080*a^7*b^2*c^6*d^4+1440*a^6*b ^3*c^7*d^3+810*a^5*b^4*c^8*d^2+180*a^4*b^5*c^9*d+12*a^3*b^6*c^10)*x^7+(15* a^9*c^3*d^7+945/4*a^8*b*c^4*d^6+1134*a^7*b^2*c^5*d^5+2205*a^6*b^3*c^6*d^4+ 1890*a^5*b^4*c^7*d^3+2835/4*a^4*b^5*c^8*d^2+105*a^3*b^6*c^9*d+9/2*a^2*b^7* c^10)*x^8+(5*a^9*c^2*d^8+120*a^8*b*c^3*d^7+840*a^7*b^2*c^4*d^6+2352*a^6*b^ 3*c^5*d^5+2940*a^5*b^4*c^6*d^4+1680*a^4*b^5*c^7*d^3+420*a^3*b^6*c^8*d^2+40 *a^2*b^7*c^9*d+a*b^8*c^10)*x^9+(a^9*c*d^9+81/2*a^8*b*c^2*d^8+432*a^7*b^2*c ^3*d^7+1764*a^6*b^3*c^4*d^6+15876/5*a^5*b^4*c^5*d^5+2646*a^4*b^5*c^6*d^4+1 008*a^3*b^6*c^7*d^3+162*a^2*b^7*c^8*d^2+9*a*b^8*c^9*d+1/10*b^9*c^10)*x^10+ (1/11*a^9*d^10+90/11*a^8*b*c*d^9+1620/11*a^7*b^2*c^2*d^8+10080/11*a^6*b^3* c^3*d^7+26460/11*a^5*b^4*c^4*d^6+31752/11*a^4*b^5*c^5*d^5+17640/11*a^3*b^6 *c^6*d^4+4320/11*a^2*b^7*c^7*d^3+405/11*a*b^8*c^8*d^2+10/11*b^9*c^9*d)*x^1 1+(3/4*a^8*b*d^10+30*a^7*b^2*c*d^9+315*a^6*b^3*c^2*d^8+1260*a^5*b^4*c^3*d^ 7+2205*a^4*b^5*c^4*d^6+1764*a^3*b^6*c^5*d^5+630*a^2*b^7*c^6*d^4+90*a*b^8*c ^7*d^3+15/4*b^9*c^8*d^2)*x^12+(36/13*a^7*b^2*d^10+840/13*a^6*b^3*c*d^9+...
Leaf count of result is larger than twice the leaf count of optimal. 1437 vs. \(2 (234) = 468\).
Time = 0.08 (sec) , antiderivative size = 1437, normalized size of antiderivative = 5.75 \[ \int (a+b x)^9 (c+d x)^{10} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^9*(d*x+c)^10,x, algorithm="fricas")
Output:
1/20*b^9*d^10*x^20 + a^9*c^10*x + 1/19*(10*b^9*c*d^9 + 9*a*b^8*d^10)*x^19 + 1/2*(5*b^9*c^2*d^8 + 10*a*b^8*c*d^9 + 4*a^2*b^7*d^10)*x^18 + 3/17*(40*b^ 9*c^3*d^7 + 135*a*b^8*c^2*d^8 + 120*a^2*b^7*c*d^9 + 28*a^3*b^6*d^10)*x^17 + 3/8*(35*b^9*c^4*d^6 + 180*a*b^8*c^3*d^7 + 270*a^2*b^7*c^2*d^8 + 140*a^3* b^6*c*d^9 + 21*a^4*b^5*d^10)*x^16 + 6/5*(14*b^9*c^5*d^5 + 105*a*b^8*c^4*d^ 6 + 240*a^2*b^7*c^3*d^7 + 210*a^3*b^6*c^2*d^8 + 70*a^4*b^5*c*d^9 + 7*a^5*b ^4*d^10)*x^15 + 3*(5*b^9*c^6*d^4 + 54*a*b^8*c^5*d^5 + 180*a^2*b^7*c^4*d^6 + 240*a^3*b^6*c^3*d^7 + 135*a^4*b^5*c^2*d^8 + 30*a^5*b^4*c*d^9 + 2*a^6*b^3 *d^10)*x^14 + 6/13*(20*b^9*c^7*d^3 + 315*a*b^8*c^6*d^4 + 1512*a^2*b^7*c^5* d^5 + 2940*a^3*b^6*c^4*d^6 + 2520*a^4*b^5*c^3*d^7 + 945*a^5*b^4*c^2*d^8 + 140*a^6*b^3*c*d^9 + 6*a^7*b^2*d^10)*x^13 + 3/4*(5*b^9*c^8*d^2 + 120*a*b^8* c^7*d^3 + 840*a^2*b^7*c^6*d^4 + 2352*a^3*b^6*c^5*d^5 + 2940*a^4*b^5*c^4*d^ 6 + 1680*a^5*b^4*c^3*d^7 + 420*a^6*b^3*c^2*d^8 + 40*a^7*b^2*c*d^9 + a^8*b* d^10)*x^12 + 1/11*(10*b^9*c^9*d + 405*a*b^8*c^8*d^2 + 4320*a^2*b^7*c^7*d^3 + 17640*a^3*b^6*c^6*d^4 + 31752*a^4*b^5*c^5*d^5 + 26460*a^5*b^4*c^4*d^6 + 10080*a^6*b^3*c^3*d^7 + 1620*a^7*b^2*c^2*d^8 + 90*a^8*b*c*d^9 + a^9*d^10) *x^11 + 1/10*(b^9*c^10 + 90*a*b^8*c^9*d + 1620*a^2*b^7*c^8*d^2 + 10080*a^3 *b^6*c^7*d^3 + 26460*a^4*b^5*c^6*d^4 + 31752*a^5*b^4*c^5*d^5 + 17640*a^6*b ^3*c^4*d^6 + 4320*a^7*b^2*c^3*d^7 + 405*a^8*b*c^2*d^8 + 10*a^9*c*d^9)*x^10 + (a*b^8*c^10 + 40*a^2*b^7*c^9*d + 420*a^3*b^6*c^8*d^2 + 1680*a^4*b^5*...
Leaf count of result is larger than twice the leaf count of optimal. 1598 vs. \(2 (231) = 462\).
Time = 0.10 (sec) , antiderivative size = 1598, normalized size of antiderivative = 6.39 \[ \int (a+b x)^9 (c+d x)^{10} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)**9*(d*x+c)**10,x)
Output:
a**9*c**10*x + b**9*d**10*x**20/20 + x**19*(9*a*b**8*d**10/19 + 10*b**9*c* d**9/19) + x**18*(2*a**2*b**7*d**10 + 5*a*b**8*c*d**9 + 5*b**9*c**2*d**8/2 ) + x**17*(84*a**3*b**6*d**10/17 + 360*a**2*b**7*c*d**9/17 + 405*a*b**8*c* *2*d**8/17 + 120*b**9*c**3*d**7/17) + x**16*(63*a**4*b**5*d**10/8 + 105*a* *3*b**6*c*d**9/2 + 405*a**2*b**7*c**2*d**8/4 + 135*a*b**8*c**3*d**7/2 + 10 5*b**9*c**4*d**6/8) + x**15*(42*a**5*b**4*d**10/5 + 84*a**4*b**5*c*d**9 + 252*a**3*b**6*c**2*d**8 + 288*a**2*b**7*c**3*d**7 + 126*a*b**8*c**4*d**6 + 84*b**9*c**5*d**5/5) + x**14*(6*a**6*b**3*d**10 + 90*a**5*b**4*c*d**9 + 4 05*a**4*b**5*c**2*d**8 + 720*a**3*b**6*c**3*d**7 + 540*a**2*b**7*c**4*d**6 + 162*a*b**8*c**5*d**5 + 15*b**9*c**6*d**4) + x**13*(36*a**7*b**2*d**10/1 3 + 840*a**6*b**3*c*d**9/13 + 5670*a**5*b**4*c**2*d**8/13 + 15120*a**4*b** 5*c**3*d**7/13 + 17640*a**3*b**6*c**4*d**6/13 + 9072*a**2*b**7*c**5*d**5/1 3 + 1890*a*b**8*c**6*d**4/13 + 120*b**9*c**7*d**3/13) + x**12*(3*a**8*b*d* *10/4 + 30*a**7*b**2*c*d**9 + 315*a**6*b**3*c**2*d**8 + 1260*a**5*b**4*c** 3*d**7 + 2205*a**4*b**5*c**4*d**6 + 1764*a**3*b**6*c**5*d**5 + 630*a**2*b* *7*c**6*d**4 + 90*a*b**8*c**7*d**3 + 15*b**9*c**8*d**2/4) + x**11*(a**9*d* *10/11 + 90*a**8*b*c*d**9/11 + 1620*a**7*b**2*c**2*d**8/11 + 10080*a**6*b* *3*c**3*d**7/11 + 26460*a**5*b**4*c**4*d**6/11 + 31752*a**4*b**5*c**5*d**5 /11 + 17640*a**3*b**6*c**6*d**4/11 + 4320*a**2*b**7*c**7*d**3/11 + 405*a*b **8*c**8*d**2/11 + 10*b**9*c**9*d/11) + x**10*(a**9*c*d**9 + 81*a**8*b*...
Leaf count of result is larger than twice the leaf count of optimal. 1437 vs. \(2 (234) = 468\).
Time = 0.04 (sec) , antiderivative size = 1437, normalized size of antiderivative = 5.75 \[ \int (a+b x)^9 (c+d x)^{10} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^9*(d*x+c)^10,x, algorithm="maxima")
Output:
1/20*b^9*d^10*x^20 + a^9*c^10*x + 1/19*(10*b^9*c*d^9 + 9*a*b^8*d^10)*x^19 + 1/2*(5*b^9*c^2*d^8 + 10*a*b^8*c*d^9 + 4*a^2*b^7*d^10)*x^18 + 3/17*(40*b^ 9*c^3*d^7 + 135*a*b^8*c^2*d^8 + 120*a^2*b^7*c*d^9 + 28*a^3*b^6*d^10)*x^17 + 3/8*(35*b^9*c^4*d^6 + 180*a*b^8*c^3*d^7 + 270*a^2*b^7*c^2*d^8 + 140*a^3* b^6*c*d^9 + 21*a^4*b^5*d^10)*x^16 + 6/5*(14*b^9*c^5*d^5 + 105*a*b^8*c^4*d^ 6 + 240*a^2*b^7*c^3*d^7 + 210*a^3*b^6*c^2*d^8 + 70*a^4*b^5*c*d^9 + 7*a^5*b ^4*d^10)*x^15 + 3*(5*b^9*c^6*d^4 + 54*a*b^8*c^5*d^5 + 180*a^2*b^7*c^4*d^6 + 240*a^3*b^6*c^3*d^7 + 135*a^4*b^5*c^2*d^8 + 30*a^5*b^4*c*d^9 + 2*a^6*b^3 *d^10)*x^14 + 6/13*(20*b^9*c^7*d^3 + 315*a*b^8*c^6*d^4 + 1512*a^2*b^7*c^5* d^5 + 2940*a^3*b^6*c^4*d^6 + 2520*a^4*b^5*c^3*d^7 + 945*a^5*b^4*c^2*d^8 + 140*a^6*b^3*c*d^9 + 6*a^7*b^2*d^10)*x^13 + 3/4*(5*b^9*c^8*d^2 + 120*a*b^8* c^7*d^3 + 840*a^2*b^7*c^6*d^4 + 2352*a^3*b^6*c^5*d^5 + 2940*a^4*b^5*c^4*d^ 6 + 1680*a^5*b^4*c^3*d^7 + 420*a^6*b^3*c^2*d^8 + 40*a^7*b^2*c*d^9 + a^8*b* d^10)*x^12 + 1/11*(10*b^9*c^9*d + 405*a*b^8*c^8*d^2 + 4320*a^2*b^7*c^7*d^3 + 17640*a^3*b^6*c^6*d^4 + 31752*a^4*b^5*c^5*d^5 + 26460*a^5*b^4*c^4*d^6 + 10080*a^6*b^3*c^3*d^7 + 1620*a^7*b^2*c^2*d^8 + 90*a^8*b*c*d^9 + a^9*d^10) *x^11 + 1/10*(b^9*c^10 + 90*a*b^8*c^9*d + 1620*a^2*b^7*c^8*d^2 + 10080*a^3 *b^6*c^7*d^3 + 26460*a^4*b^5*c^6*d^4 + 31752*a^5*b^4*c^5*d^5 + 17640*a^6*b ^3*c^4*d^6 + 4320*a^7*b^2*c^3*d^7 + 405*a^8*b*c^2*d^8 + 10*a^9*c*d^9)*x^10 + (a*b^8*c^10 + 40*a^2*b^7*c^9*d + 420*a^3*b^6*c^8*d^2 + 1680*a^4*b^5*...
Leaf count of result is larger than twice the leaf count of optimal. 1656 vs. \(2 (234) = 468\).
Time = 0.12 (sec) , antiderivative size = 1656, normalized size of antiderivative = 6.62 \[ \int (a+b x)^9 (c+d x)^{10} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^9*(d*x+c)^10,x, algorithm="giac")
Output:
1/20*b^9*d^10*x^20 + 10/19*b^9*c*d^9*x^19 + 9/19*a*b^8*d^10*x^19 + 5/2*b^9 *c^2*d^8*x^18 + 5*a*b^8*c*d^9*x^18 + 2*a^2*b^7*d^10*x^18 + 120/17*b^9*c^3* d^7*x^17 + 405/17*a*b^8*c^2*d^8*x^17 + 360/17*a^2*b^7*c*d^9*x^17 + 84/17*a ^3*b^6*d^10*x^17 + 105/8*b^9*c^4*d^6*x^16 + 135/2*a*b^8*c^3*d^7*x^16 + 405 /4*a^2*b^7*c^2*d^8*x^16 + 105/2*a^3*b^6*c*d^9*x^16 + 63/8*a^4*b^5*d^10*x^1 6 + 84/5*b^9*c^5*d^5*x^15 + 126*a*b^8*c^4*d^6*x^15 + 288*a^2*b^7*c^3*d^7*x ^15 + 252*a^3*b^6*c^2*d^8*x^15 + 84*a^4*b^5*c*d^9*x^15 + 42/5*a^5*b^4*d^10 *x^15 + 15*b^9*c^6*d^4*x^14 + 162*a*b^8*c^5*d^5*x^14 + 540*a^2*b^7*c^4*d^6 *x^14 + 720*a^3*b^6*c^3*d^7*x^14 + 405*a^4*b^5*c^2*d^8*x^14 + 90*a^5*b^4*c *d^9*x^14 + 6*a^6*b^3*d^10*x^14 + 120/13*b^9*c^7*d^3*x^13 + 1890/13*a*b^8* c^6*d^4*x^13 + 9072/13*a^2*b^7*c^5*d^5*x^13 + 17640/13*a^3*b^6*c^4*d^6*x^1 3 + 15120/13*a^4*b^5*c^3*d^7*x^13 + 5670/13*a^5*b^4*c^2*d^8*x^13 + 840/13* a^6*b^3*c*d^9*x^13 + 36/13*a^7*b^2*d^10*x^13 + 15/4*b^9*c^8*d^2*x^12 + 90* a*b^8*c^7*d^3*x^12 + 630*a^2*b^7*c^6*d^4*x^12 + 1764*a^3*b^6*c^5*d^5*x^12 + 2205*a^4*b^5*c^4*d^6*x^12 + 1260*a^5*b^4*c^3*d^7*x^12 + 315*a^6*b^3*c^2* d^8*x^12 + 30*a^7*b^2*c*d^9*x^12 + 3/4*a^8*b*d^10*x^12 + 10/11*b^9*c^9*d*x ^11 + 405/11*a*b^8*c^8*d^2*x^11 + 4320/11*a^2*b^7*c^7*d^3*x^11 + 17640/11* a^3*b^6*c^6*d^4*x^11 + 31752/11*a^4*b^5*c^5*d^5*x^11 + 26460/11*a^5*b^4*c^ 4*d^6*x^11 + 10080/11*a^6*b^3*c^3*d^7*x^11 + 1620/11*a^7*b^2*c^2*d^8*x^11 + 90/11*a^8*b*c*d^9*x^11 + 1/11*a^9*d^10*x^11 + 1/10*b^9*c^10*x^10 + 9*...
Time = 0.47 (sec) , antiderivative size = 1404, normalized size of antiderivative = 5.62 \[ \int (a+b x)^9 (c+d x)^{10} \, dx=\text {Too large to display} \] Input:
int((a + b*x)^9*(c + d*x)^10,x)
Output:
x^7*(12*a^3*b^6*c^10 + 30*a^9*c^4*d^6 + 180*a^4*b^5*c^9*d + 324*a^8*b*c^5* d^5 + 810*a^5*b^4*c^8*d^2 + 1440*a^6*b^3*c^7*d^3 + 1080*a^7*b^2*c^6*d^4) + x^14*(6*a^6*b^3*d^10 + 15*b^9*c^6*d^4 + 162*a*b^8*c^5*d^5 + 90*a^5*b^4*c* d^9 + 540*a^2*b^7*c^4*d^6 + 720*a^3*b^6*c^3*d^7 + 405*a^4*b^5*c^2*d^8) + x ^5*((126*a^5*b^4*c^10)/5 + 42*a^9*c^6*d^4 + 168*a^6*b^3*c^9*d + 216*a^8*b* c^7*d^3 + 324*a^7*b^2*c^8*d^2) + x^16*((63*a^4*b^5*d^10)/8 + (105*b^9*c^4* d^6)/8 + (135*a*b^8*c^3*d^7)/2 + (105*a^3*b^6*c*d^9)/2 + (405*a^2*b^7*c^2* d^8)/4) + x^8*((9*a^2*b^7*c^10)/2 + 15*a^9*c^3*d^7 + 105*a^3*b^6*c^9*d + ( 945*a^8*b*c^4*d^6)/4 + (2835*a^4*b^5*c^8*d^2)/4 + 1890*a^5*b^4*c^7*d^3 + 2 205*a^6*b^3*c^6*d^4 + 1134*a^7*b^2*c^5*d^5) + x^13*((36*a^7*b^2*d^10)/13 + (120*b^9*c^7*d^3)/13 + (1890*a*b^8*c^6*d^4)/13 + (840*a^6*b^3*c*d^9)/13 + (9072*a^2*b^7*c^5*d^5)/13 + (17640*a^3*b^6*c^4*d^6)/13 + (15120*a^4*b^5*c ^3*d^7)/13 + (5670*a^5*b^4*c^2*d^8)/13) + x^9*(a*b^8*c^10 + 5*a^9*c^2*d^8 + 40*a^2*b^7*c^9*d + 120*a^8*b*c^3*d^7 + 420*a^3*b^6*c^8*d^2 + 1680*a^4*b^ 5*c^7*d^3 + 2940*a^5*b^4*c^6*d^4 + 2352*a^6*b^3*c^5*d^5 + 840*a^7*b^2*c^4* d^6) + x^12*((3*a^8*b*d^10)/4 + (15*b^9*c^8*d^2)/4 + 90*a*b^8*c^7*d^3 + 30 *a^7*b^2*c*d^9 + 630*a^2*b^7*c^6*d^4 + 1764*a^3*b^6*c^5*d^5 + 2205*a^4*b^5 *c^4*d^6 + 1260*a^5*b^4*c^3*d^7 + 315*a^6*b^3*c^2*d^8) + x^6*(21*a^4*b^5*c ^10 + 42*a^9*c^5*d^5 + 210*a^5*b^4*c^9*d + 315*a^8*b*c^6*d^4 + 630*a^6*b^3 *c^8*d^2 + 720*a^7*b^2*c^7*d^3) + x^15*((42*a^5*b^4*d^10)/5 + (84*b^9*c...
Time = 0.17 (sec) , antiderivative size = 1657, normalized size of antiderivative = 6.63 \[ \int (a+b x)^9 (c+d x)^{10} \, dx =\text {Too large to display} \] Input:
int((b*x+a)^9*(d*x+c)^10,x)
Output:
(x*(1847560*a**9*c**10 + 9237800*a**9*c**9*d*x + 27713400*a**9*c**8*d**2*x **2 + 55426800*a**9*c**7*d**3*x**3 + 77597520*a**9*c**6*d**4*x**4 + 775975 20*a**9*c**5*d**5*x**5 + 55426800*a**9*c**4*d**6*x**6 + 27713400*a**9*c**3 *d**7*x**7 + 9237800*a**9*c**2*d**8*x**8 + 1847560*a**9*c*d**9*x**9 + 1679 60*a**9*d**10*x**10 + 8314020*a**8*b*c**10*x + 55426800*a**8*b*c**9*d*x**2 + 187065450*a**8*b*c**8*d**2*x**3 + 399072960*a**8*b*c**7*d**3*x**4 + 581 981400*a**8*b*c**6*d**4*x**5 + 598609440*a**8*b*c**5*d**5*x**6 + 436486050 *a**8*b*c**4*d**6*x**7 + 221707200*a**8*b*c**3*d**7*x**8 + 74826180*a**8*b *c**2*d**8*x**9 + 15116400*a**8*b*c*d**9*x**10 + 1385670*a**8*b*d**10*x**1 1 + 22170720*a**7*b**2*c**10*x**2 + 166280400*a**7*b**2*c**9*d*x**3 + 5986 09440*a**7*b**2*c**8*d**2*x**4 + 1330243200*a**7*b**2*c**7*d**3*x**5 + 199 5364800*a**7*b**2*c**6*d**4*x**6 + 2095133040*a**7*b**2*c**5*d**5*x**7 + 1 551950400*a**7*b**2*c**4*d**6*x**8 + 798145920*a**7*b**2*c**3*d**7*x**9 + 272095200*a**7*b**2*c**2*d**8*x**10 + 55426800*a**7*b**2*c*d**9*x**11 + 51 16320*a**7*b**2*d**10*x**12 + 38798760*a**6*b**3*c**10*x**3 + 310390080*a* *6*b**3*c**9*d*x**4 + 1163962800*a**6*b**3*c**8*d**2*x**5 + 2660486400*a** 6*b**3*c**7*d**3*x**6 + 4073869800*a**6*b**3*c**6*d**4*x**7 + 4345461120*a **6*b**3*c**5*d**5*x**8 + 3259095840*a**6*b**3*c**4*d**6*x**9 + 1693036800 *a**6*b**3*c**3*d**7*x**10 + 581981400*a**6*b**3*c**2*d**8*x**11 + 1193808 00*a**6*b**3*c*d**9*x**12 + 11085360*a**6*b**3*d**10*x**13 + 46558512*a...