Integrand size = 15, antiderivative size = 200 \[ \int (a+b x)^7 (c+d x)^{10} \, dx=-\frac {(b c-a d)^7 (c+d x)^{11}}{11 d^8}+\frac {7 b (b c-a d)^6 (c+d x)^{12}}{12 d^8}-\frac {21 b^2 (b c-a d)^5 (c+d x)^{13}}{13 d^8}+\frac {5 b^3 (b c-a d)^4 (c+d x)^{14}}{2 d^8}-\frac {7 b^4 (b c-a d)^3 (c+d x)^{15}}{3 d^8}+\frac {21 b^5 (b c-a d)^2 (c+d x)^{16}}{16 d^8}-\frac {7 b^6 (b c-a d) (c+d x)^{17}}{17 d^8}+\frac {b^7 (c+d x)^{18}}{18 d^8} \] Output:
-1/11*(-a*d+b*c)^7*(d*x+c)^11/d^8+7/12*b*(-a*d+b*c)^6*(d*x+c)^12/d^8-21/13 *b^2*(-a*d+b*c)^5*(d*x+c)^13/d^8+5/2*b^3*(-a*d+b*c)^4*(d*x+c)^14/d^8-7/3*b ^4*(-a*d+b*c)^3*(d*x+c)^15/d^8+21/16*b^5*(-a*d+b*c)^2*(d*x+c)^16/d^8-7/17* b^6*(-a*d+b*c)*(d*x+c)^17/d^8+1/18*b^7*(d*x+c)^18/d^8
Leaf count is larger than twice the leaf count of optimal. \(1105\) vs. \(2(200)=400\).
Time = 0.07 (sec) , antiderivative size = 1105, normalized size of antiderivative = 5.52 \[ \int (a+b x)^7 (c+d x)^{10} \, dx=a^7 c^{10} x+\frac {1}{2} a^6 c^9 (7 b c+10 a d) x^2+\frac {1}{3} a^5 c^8 \left (21 b^2 c^2+70 a b c d+45 a^2 d^2\right ) x^3+\frac {5}{4} a^4 c^7 \left (7 b^3 c^3+42 a b^2 c^2 d+63 a^2 b c d^2+24 a^3 d^3\right ) x^4+7 a^3 c^6 \left (b^4 c^4+10 a b^3 c^3 d+27 a^2 b^2 c^2 d^2+24 a^3 b c d^3+6 a^4 d^4\right ) x^5+\frac {7}{6} a^2 c^5 \left (3 b^5 c^5+50 a b^4 c^4 d+225 a^2 b^3 c^3 d^2+360 a^3 b^2 c^2 d^3+210 a^4 b c d^4+36 a^5 d^5\right ) x^6+a c^4 \left (b^6 c^6+30 a b^5 c^5 d+225 a^2 b^4 c^4 d^2+600 a^3 b^3 c^3 d^3+630 a^4 b^2 c^2 d^4+252 a^5 b c d^5+30 a^6 d^6\right ) x^7+\frac {1}{8} c^3 \left (b^7 c^7+70 a b^6 c^6 d+945 a^2 b^5 c^5 d^2+4200 a^3 b^4 c^4 d^3+7350 a^4 b^3 c^3 d^4+5292 a^5 b^2 c^2 d^5+1470 a^6 b c d^6+120 a^7 d^7\right ) x^8+\frac {5}{9} c^2 d \left (2 b^7 c^7+63 a b^6 c^6 d+504 a^2 b^5 c^5 d^2+1470 a^3 b^4 c^4 d^3+1764 a^4 b^3 c^3 d^4+882 a^5 b^2 c^2 d^5+168 a^6 b c d^6+9 a^7 d^7\right ) x^9+\frac {1}{2} c d^2 \left (9 b^7 c^7+168 a b^6 c^6 d+882 a^2 b^5 c^5 d^2+1764 a^3 b^4 c^4 d^3+1470 a^4 b^3 c^3 d^4+504 a^5 b^2 c^2 d^5+63 a^6 b c d^6+2 a^7 d^7\right ) x^{10}+\frac {1}{11} d^3 \left (120 b^7 c^7+1470 a b^6 c^6 d+5292 a^2 b^5 c^5 d^2+7350 a^3 b^4 c^4 d^3+4200 a^4 b^3 c^3 d^4+945 a^5 b^2 c^2 d^5+70 a^6 b c d^6+a^7 d^7\right ) x^{11}+\frac {7}{12} b d^4 \left (30 b^6 c^6+252 a b^5 c^5 d+630 a^2 b^4 c^4 d^2+600 a^3 b^3 c^3 d^3+225 a^4 b^2 c^2 d^4+30 a^5 b c d^5+a^6 d^6\right ) x^{12}+\frac {7}{13} b^2 d^5 \left (36 b^5 c^5+210 a b^4 c^4 d+360 a^2 b^3 c^3 d^2+225 a^3 b^2 c^2 d^3+50 a^4 b c d^4+3 a^5 d^5\right ) x^{13}+\frac {5}{2} b^3 d^6 \left (6 b^4 c^4+24 a b^3 c^3 d+27 a^2 b^2 c^2 d^2+10 a^3 b c d^3+a^4 d^4\right ) x^{14}+\frac {1}{3} b^4 d^7 \left (24 b^3 c^3+63 a b^2 c^2 d+42 a^2 b c d^2+7 a^3 d^3\right ) x^{15}+\frac {1}{16} b^5 d^8 \left (45 b^2 c^2+70 a b c d+21 a^2 d^2\right ) x^{16}+\frac {1}{17} b^6 d^9 (10 b c+7 a d) x^{17}+\frac {1}{18} b^7 d^{10} x^{18} \] Input:
Integrate[(a + b*x)^7*(c + d*x)^10,x]
Output:
a^7*c^10*x + (a^6*c^9*(7*b*c + 10*a*d)*x^2)/2 + (a^5*c^8*(21*b^2*c^2 + 70* a*b*c*d + 45*a^2*d^2)*x^3)/3 + (5*a^4*c^7*(7*b^3*c^3 + 42*a*b^2*c^2*d + 63 *a^2*b*c*d^2 + 24*a^3*d^3)*x^4)/4 + 7*a^3*c^6*(b^4*c^4 + 10*a*b^3*c^3*d + 27*a^2*b^2*c^2*d^2 + 24*a^3*b*c*d^3 + 6*a^4*d^4)*x^5 + (7*a^2*c^5*(3*b^5*c ^5 + 50*a*b^4*c^4*d + 225*a^2*b^3*c^3*d^2 + 360*a^3*b^2*c^2*d^3 + 210*a^4* b*c*d^4 + 36*a^5*d^5)*x^6)/6 + a*c^4*(b^6*c^6 + 30*a*b^5*c^5*d + 225*a^2*b ^4*c^4*d^2 + 600*a^3*b^3*c^3*d^3 + 630*a^4*b^2*c^2*d^4 + 252*a^5*b*c*d^5 + 30*a^6*d^6)*x^7 + (c^3*(b^7*c^7 + 70*a*b^6*c^6*d + 945*a^2*b^5*c^5*d^2 + 4200*a^3*b^4*c^4*d^3 + 7350*a^4*b^3*c^3*d^4 + 5292*a^5*b^2*c^2*d^5 + 1470* a^6*b*c*d^6 + 120*a^7*d^7)*x^8)/8 + (5*c^2*d*(2*b^7*c^7 + 63*a*b^6*c^6*d + 504*a^2*b^5*c^5*d^2 + 1470*a^3*b^4*c^4*d^3 + 1764*a^4*b^3*c^3*d^4 + 882*a ^5*b^2*c^2*d^5 + 168*a^6*b*c*d^6 + 9*a^7*d^7)*x^9)/9 + (c*d^2*(9*b^7*c^7 + 168*a*b^6*c^6*d + 882*a^2*b^5*c^5*d^2 + 1764*a^3*b^4*c^4*d^3 + 1470*a^4*b ^3*c^3*d^4 + 504*a^5*b^2*c^2*d^5 + 63*a^6*b*c*d^6 + 2*a^7*d^7)*x^10)/2 + ( d^3*(120*b^7*c^7 + 1470*a*b^6*c^6*d + 5292*a^2*b^5*c^5*d^2 + 7350*a^3*b^4* c^4*d^3 + 4200*a^4*b^3*c^3*d^4 + 945*a^5*b^2*c^2*d^5 + 70*a^6*b*c*d^6 + a^ 7*d^7)*x^11)/11 + (7*b*d^4*(30*b^6*c^6 + 252*a*b^5*c^5*d + 630*a^2*b^4*c^4 *d^2 + 600*a^3*b^3*c^3*d^3 + 225*a^4*b^2*c^2*d^4 + 30*a^5*b*c*d^5 + a^6*d^ 6)*x^12)/12 + (7*b^2*d^5*(36*b^5*c^5 + 210*a*b^4*c^4*d + 360*a^2*b^3*c^3*d ^2 + 225*a^3*b^2*c^2*d^3 + 50*a^4*b*c*d^4 + 3*a^5*d^5)*x^13)/13 + (5*b^...
Time = 0.80 (sec) , antiderivative size = 200, 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.
| \(\displaystyle \int (a+b x)^7 (c+d x)^{10} \, dx\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \int \left (-\frac {7 b^6 (c+d x)^{16} (b c-a d)}{d^7}+\frac {21 b^5 (c+d x)^{15} (b c-a d)^2}{d^7}-\frac {35 b^4 (c+d x)^{14} (b c-a d)^3}{d^7}+\frac {35 b^3 (c+d x)^{13} (b c-a d)^4}{d^7}-\frac {21 b^2 (c+d x)^{12} (b c-a d)^5}{d^7}+\frac {7 b (c+d x)^{11} (b c-a d)^6}{d^7}+\frac {(c+d x)^{10} (a d-b c)^7}{d^7}+\frac {b^7 (c+d x)^{17}}{d^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {7 b^6 (c+d x)^{17} (b c-a d)}{17 d^8}+\frac {21 b^5 (c+d x)^{16} (b c-a d)^2}{16 d^8}-\frac {7 b^4 (c+d x)^{15} (b c-a d)^3}{3 d^8}+\frac {5 b^3 (c+d x)^{14} (b c-a d)^4}{2 d^8}-\frac {21 b^2 (c+d x)^{13} (b c-a d)^5}{13 d^8}+\frac {7 b (c+d x)^{12} (b c-a d)^6}{12 d^8}-\frac {(c+d x)^{11} (b c-a d)^7}{11 d^8}+\frac {b^7 (c+d x)^{18}}{18 d^8}\) |
Input:
Int[(a + b*x)^7*(c + d*x)^10,x]
Output:
-1/11*((b*c - a*d)^7*(c + d*x)^11)/d^8 + (7*b*(b*c - a*d)^6*(c + d*x)^12)/ (12*d^8) - (21*b^2*(b*c - a*d)^5*(c + d*x)^13)/(13*d^8) + (5*b^3*(b*c - a* d)^4*(c + d*x)^14)/(2*d^8) - (7*b^4*(b*c - a*d)^3*(c + d*x)^15)/(3*d^8) + (21*b^5*(b*c - a*d)^2*(c + d*x)^16)/(16*d^8) - (7*b^6*(b*c - a*d)*(c + d*x )^17)/(17*d^8) + (b^7*(c + d*x)^18)/(18*d^8)
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. \(1124\) vs. \(2(184)=368\).
Time = 0.12 (sec) , antiderivative size = 1125, normalized size of antiderivative = 5.62
| method | result | size |
| norman | \(\text {Expression too large to display}\) | \(1125\) |
| default | \(\text {Expression too large to display}\) | \(1141\) |
| gosper | \(\text {Expression too large to display}\) | \(1303\) |
| risch | \(\text {Expression too large to display}\) | \(1303\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1303\) |
| orering | \(\text {Expression too large to display}\) | \(1304\) |
Input:
int((b*x+a)^7*(d*x+c)^10,x,method=_RETURNVERBOSE)
Output:
a^7*c^10*x+(5*a^7*c^9*d+7/2*a^6*b*c^10)*x^2+(15*a^7*c^8*d^2+70/3*a^6*b*c^9 *d+7*a^5*b^2*c^10)*x^3+(30*a^7*c^7*d^3+315/4*a^6*b*c^8*d^2+105/2*a^5*b^2*c ^9*d+35/4*a^4*b^3*c^10)*x^4+(42*a^7*c^6*d^4+168*a^6*b*c^7*d^3+189*a^5*b^2* c^8*d^2+70*a^4*b^3*c^9*d+7*a^3*b^4*c^10)*x^5+(42*a^7*c^5*d^5+245*a^6*b*c^6 *d^4+420*a^5*b^2*c^7*d^3+525/2*a^4*b^3*c^8*d^2+175/3*a^3*b^4*c^9*d+7/2*a^2 *b^5*c^10)*x^6+(30*a^7*c^4*d^6+252*a^6*b*c^5*d^5+630*a^5*b^2*c^6*d^4+600*a ^4*b^3*c^7*d^3+225*a^3*b^4*c^8*d^2+30*a^2*b^5*c^9*d+a*b^6*c^10)*x^7+(15*a^ 7*c^3*d^7+735/4*a^6*b*c^4*d^6+1323/2*a^5*b^2*c^5*d^5+3675/4*a^4*b^3*c^6*d^ 4+525*a^3*b^4*c^7*d^3+945/8*a^2*b^5*c^8*d^2+35/4*a*b^6*c^9*d+1/8*b^7*c^10) *x^8+(5*a^7*c^2*d^8+280/3*a^6*b*c^3*d^7+490*a^5*b^2*c^4*d^6+980*a^4*b^3*c^ 5*d^5+2450/3*a^3*b^4*c^6*d^4+280*a^2*b^5*c^7*d^3+35*a*b^6*c^8*d^2+10/9*b^7 *c^9*d)*x^9+(a^7*c*d^9+63/2*a^6*b*c^2*d^8+252*a^5*b^2*c^3*d^7+735*a^4*b^3* c^4*d^6+882*a^3*b^4*c^5*d^5+441*a^2*b^5*c^6*d^4+84*a*b^6*c^7*d^3+9/2*b^7*c ^8*d^2)*x^10+(1/11*a^7*d^10+70/11*a^6*b*c*d^9+945/11*a^5*b^2*c^2*d^8+4200/ 11*a^4*b^3*c^3*d^7+7350/11*a^3*b^4*c^4*d^6+5292/11*a^2*b^5*c^5*d^5+1470/11 *a*b^6*c^6*d^4+120/11*b^7*c^7*d^3)*x^11+(7/12*a^6*b*d^10+35/2*a^5*b^2*c*d^ 9+525/4*a^4*b^3*c^2*d^8+350*a^3*b^4*c^3*d^7+735/2*a^2*b^5*c^4*d^6+147*a*b^ 6*c^5*d^5+35/2*b^7*c^6*d^4)*x^12+(21/13*a^5*b^2*d^10+350/13*a^4*b^3*c*d^9+ 1575/13*a^3*b^4*c^2*d^8+2520/13*a^2*b^5*c^3*d^7+1470/13*a*b^6*c^4*d^6+252/ 13*b^7*c^5*d^5)*x^13+(5/2*a^4*b^3*d^10+25*a^3*b^4*c*d^9+135/2*a^2*b^5*c...
Leaf count of result is larger than twice the leaf count of optimal. 1135 vs. \(2 (184) = 368\).
Time = 0.08 (sec) , antiderivative size = 1135, normalized size of antiderivative = 5.68 \[ \int (a+b x)^7 (c+d x)^{10} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^7*(d*x+c)^10,x, algorithm="fricas")
Output:
1/18*b^7*d^10*x^18 + a^7*c^10*x + 1/17*(10*b^7*c*d^9 + 7*a*b^6*d^10)*x^17 + 1/16*(45*b^7*c^2*d^8 + 70*a*b^6*c*d^9 + 21*a^2*b^5*d^10)*x^16 + 1/3*(24* b^7*c^3*d^7 + 63*a*b^6*c^2*d^8 + 42*a^2*b^5*c*d^9 + 7*a^3*b^4*d^10)*x^15 + 5/2*(6*b^7*c^4*d^6 + 24*a*b^6*c^3*d^7 + 27*a^2*b^5*c^2*d^8 + 10*a^3*b^4*c *d^9 + a^4*b^3*d^10)*x^14 + 7/13*(36*b^7*c^5*d^5 + 210*a*b^6*c^4*d^6 + 360 *a^2*b^5*c^3*d^7 + 225*a^3*b^4*c^2*d^8 + 50*a^4*b^3*c*d^9 + 3*a^5*b^2*d^10 )*x^13 + 7/12*(30*b^7*c^6*d^4 + 252*a*b^6*c^5*d^5 + 630*a^2*b^5*c^4*d^6 + 600*a^3*b^4*c^3*d^7 + 225*a^4*b^3*c^2*d^8 + 30*a^5*b^2*c*d^9 + a^6*b*d^10) *x^12 + 1/11*(120*b^7*c^7*d^3 + 1470*a*b^6*c^6*d^4 + 5292*a^2*b^5*c^5*d^5 + 7350*a^3*b^4*c^4*d^6 + 4200*a^4*b^3*c^3*d^7 + 945*a^5*b^2*c^2*d^8 + 70*a ^6*b*c*d^9 + a^7*d^10)*x^11 + 1/2*(9*b^7*c^8*d^2 + 168*a*b^6*c^7*d^3 + 882 *a^2*b^5*c^6*d^4 + 1764*a^3*b^4*c^5*d^5 + 1470*a^4*b^3*c^4*d^6 + 504*a^5*b ^2*c^3*d^7 + 63*a^6*b*c^2*d^8 + 2*a^7*c*d^9)*x^10 + 5/9*(2*b^7*c^9*d + 63* a*b^6*c^8*d^2 + 504*a^2*b^5*c^7*d^3 + 1470*a^3*b^4*c^6*d^4 + 1764*a^4*b^3* c^5*d^5 + 882*a^5*b^2*c^4*d^6 + 168*a^6*b*c^3*d^7 + 9*a^7*c^2*d^8)*x^9 + 1 /8*(b^7*c^10 + 70*a*b^6*c^9*d + 945*a^2*b^5*c^8*d^2 + 4200*a^3*b^4*c^7*d^3 + 7350*a^4*b^3*c^6*d^4 + 5292*a^5*b^2*c^5*d^5 + 1470*a^6*b*c^4*d^6 + 120* a^7*c^3*d^7)*x^8 + (a*b^6*c^10 + 30*a^2*b^5*c^9*d + 225*a^3*b^4*c^8*d^2 + 600*a^4*b^3*c^7*d^3 + 630*a^5*b^2*c^6*d^4 + 252*a^6*b*c^5*d^5 + 30*a^7*c^4 *d^6)*x^7 + 7/6*(3*a^2*b^5*c^10 + 50*a^3*b^4*c^9*d + 225*a^4*b^3*c^8*d^...
Leaf count of result is larger than twice the leaf count of optimal. 1280 vs. \(2 (184) = 368\).
Time = 0.08 (sec) , antiderivative size = 1280, normalized size of antiderivative = 6.40 \[ \int (a+b x)^7 (c+d x)^{10} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)**7*(d*x+c)**10,x)
Output:
a**7*c**10*x + b**7*d**10*x**18/18 + x**17*(7*a*b**6*d**10/17 + 10*b**7*c* d**9/17) + x**16*(21*a**2*b**5*d**10/16 + 35*a*b**6*c*d**9/8 + 45*b**7*c** 2*d**8/16) + x**15*(7*a**3*b**4*d**10/3 + 14*a**2*b**5*c*d**9 + 21*a*b**6* c**2*d**8 + 8*b**7*c**3*d**7) + x**14*(5*a**4*b**3*d**10/2 + 25*a**3*b**4* c*d**9 + 135*a**2*b**5*c**2*d**8/2 + 60*a*b**6*c**3*d**7 + 15*b**7*c**4*d* *6) + x**13*(21*a**5*b**2*d**10/13 + 350*a**4*b**3*c*d**9/13 + 1575*a**3*b **4*c**2*d**8/13 + 2520*a**2*b**5*c**3*d**7/13 + 1470*a*b**6*c**4*d**6/13 + 252*b**7*c**5*d**5/13) + x**12*(7*a**6*b*d**10/12 + 35*a**5*b**2*c*d**9/ 2 + 525*a**4*b**3*c**2*d**8/4 + 350*a**3*b**4*c**3*d**7 + 735*a**2*b**5*c* *4*d**6/2 + 147*a*b**6*c**5*d**5 + 35*b**7*c**6*d**4/2) + x**11*(a**7*d**1 0/11 + 70*a**6*b*c*d**9/11 + 945*a**5*b**2*c**2*d**8/11 + 4200*a**4*b**3*c **3*d**7/11 + 7350*a**3*b**4*c**4*d**6/11 + 5292*a**2*b**5*c**5*d**5/11 + 1470*a*b**6*c**6*d**4/11 + 120*b**7*c**7*d**3/11) + x**10*(a**7*c*d**9 + 6 3*a**6*b*c**2*d**8/2 + 252*a**5*b**2*c**3*d**7 + 735*a**4*b**3*c**4*d**6 + 882*a**3*b**4*c**5*d**5 + 441*a**2*b**5*c**6*d**4 + 84*a*b**6*c**7*d**3 + 9*b**7*c**8*d**2/2) + x**9*(5*a**7*c**2*d**8 + 280*a**6*b*c**3*d**7/3 + 4 90*a**5*b**2*c**4*d**6 + 980*a**4*b**3*c**5*d**5 + 2450*a**3*b**4*c**6*d** 4/3 + 280*a**2*b**5*c**7*d**3 + 35*a*b**6*c**8*d**2 + 10*b**7*c**9*d/9) + x**8*(15*a**7*c**3*d**7 + 735*a**6*b*c**4*d**6/4 + 1323*a**5*b**2*c**5*d** 5/2 + 3675*a**4*b**3*c**6*d**4/4 + 525*a**3*b**4*c**7*d**3 + 945*a**2*b...
Leaf count of result is larger than twice the leaf count of optimal. 1135 vs. \(2 (184) = 368\).
Time = 0.05 (sec) , antiderivative size = 1135, normalized size of antiderivative = 5.68 \[ \int (a+b x)^7 (c+d x)^{10} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^7*(d*x+c)^10,x, algorithm="maxima")
Output:
1/18*b^7*d^10*x^18 + a^7*c^10*x + 1/17*(10*b^7*c*d^9 + 7*a*b^6*d^10)*x^17 + 1/16*(45*b^7*c^2*d^8 + 70*a*b^6*c*d^9 + 21*a^2*b^5*d^10)*x^16 + 1/3*(24* b^7*c^3*d^7 + 63*a*b^6*c^2*d^8 + 42*a^2*b^5*c*d^9 + 7*a^3*b^4*d^10)*x^15 + 5/2*(6*b^7*c^4*d^6 + 24*a*b^6*c^3*d^7 + 27*a^2*b^5*c^2*d^8 + 10*a^3*b^4*c *d^9 + a^4*b^3*d^10)*x^14 + 7/13*(36*b^7*c^5*d^5 + 210*a*b^6*c^4*d^6 + 360 *a^2*b^5*c^3*d^7 + 225*a^3*b^4*c^2*d^8 + 50*a^4*b^3*c*d^9 + 3*a^5*b^2*d^10 )*x^13 + 7/12*(30*b^7*c^6*d^4 + 252*a*b^6*c^5*d^5 + 630*a^2*b^5*c^4*d^6 + 600*a^3*b^4*c^3*d^7 + 225*a^4*b^3*c^2*d^8 + 30*a^5*b^2*c*d^9 + a^6*b*d^10) *x^12 + 1/11*(120*b^7*c^7*d^3 + 1470*a*b^6*c^6*d^4 + 5292*a^2*b^5*c^5*d^5 + 7350*a^3*b^4*c^4*d^6 + 4200*a^4*b^3*c^3*d^7 + 945*a^5*b^2*c^2*d^8 + 70*a ^6*b*c*d^9 + a^7*d^10)*x^11 + 1/2*(9*b^7*c^8*d^2 + 168*a*b^6*c^7*d^3 + 882 *a^2*b^5*c^6*d^4 + 1764*a^3*b^4*c^5*d^5 + 1470*a^4*b^3*c^4*d^6 + 504*a^5*b ^2*c^3*d^7 + 63*a^6*b*c^2*d^8 + 2*a^7*c*d^9)*x^10 + 5/9*(2*b^7*c^9*d + 63* a*b^6*c^8*d^2 + 504*a^2*b^5*c^7*d^3 + 1470*a^3*b^4*c^6*d^4 + 1764*a^4*b^3* c^5*d^5 + 882*a^5*b^2*c^4*d^6 + 168*a^6*b*c^3*d^7 + 9*a^7*c^2*d^8)*x^9 + 1 /8*(b^7*c^10 + 70*a*b^6*c^9*d + 945*a^2*b^5*c^8*d^2 + 4200*a^3*b^4*c^7*d^3 + 7350*a^4*b^3*c^6*d^4 + 5292*a^5*b^2*c^5*d^5 + 1470*a^6*b*c^4*d^6 + 120* a^7*c^3*d^7)*x^8 + (a*b^6*c^10 + 30*a^2*b^5*c^9*d + 225*a^3*b^4*c^8*d^2 + 600*a^4*b^3*c^7*d^3 + 630*a^5*b^2*c^6*d^4 + 252*a^6*b*c^5*d^5 + 30*a^7*c^4 *d^6)*x^7 + 7/6*(3*a^2*b^5*c^10 + 50*a^3*b^4*c^9*d + 225*a^4*b^3*c^8*d^...
Leaf count of result is larger than twice the leaf count of optimal. 1302 vs. \(2 (184) = 368\).
Time = 0.12 (sec) , antiderivative size = 1302, normalized size of antiderivative = 6.51 \[ \int (a+b x)^7 (c+d x)^{10} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^7*(d*x+c)^10,x, algorithm="giac")
Output:
1/18*b^7*d^10*x^18 + 10/17*b^7*c*d^9*x^17 + 7/17*a*b^6*d^10*x^17 + 45/16*b ^7*c^2*d^8*x^16 + 35/8*a*b^6*c*d^9*x^16 + 21/16*a^2*b^5*d^10*x^16 + 8*b^7* c^3*d^7*x^15 + 21*a*b^6*c^2*d^8*x^15 + 14*a^2*b^5*c*d^9*x^15 + 7/3*a^3*b^4 *d^10*x^15 + 15*b^7*c^4*d^6*x^14 + 60*a*b^6*c^3*d^7*x^14 + 135/2*a^2*b^5*c ^2*d^8*x^14 + 25*a^3*b^4*c*d^9*x^14 + 5/2*a^4*b^3*d^10*x^14 + 252/13*b^7*c ^5*d^5*x^13 + 1470/13*a*b^6*c^4*d^6*x^13 + 2520/13*a^2*b^5*c^3*d^7*x^13 + 1575/13*a^3*b^4*c^2*d^8*x^13 + 350/13*a^4*b^3*c*d^9*x^13 + 21/13*a^5*b^2*d ^10*x^13 + 35/2*b^7*c^6*d^4*x^12 + 147*a*b^6*c^5*d^5*x^12 + 735/2*a^2*b^5* c^4*d^6*x^12 + 350*a^3*b^4*c^3*d^7*x^12 + 525/4*a^4*b^3*c^2*d^8*x^12 + 35/ 2*a^5*b^2*c*d^9*x^12 + 7/12*a^6*b*d^10*x^12 + 120/11*b^7*c^7*d^3*x^11 + 14 70/11*a*b^6*c^6*d^4*x^11 + 5292/11*a^2*b^5*c^5*d^5*x^11 + 7350/11*a^3*b^4* c^4*d^6*x^11 + 4200/11*a^4*b^3*c^3*d^7*x^11 + 945/11*a^5*b^2*c^2*d^8*x^11 + 70/11*a^6*b*c*d^9*x^11 + 1/11*a^7*d^10*x^11 + 9/2*b^7*c^8*d^2*x^10 + 84* a*b^6*c^7*d^3*x^10 + 441*a^2*b^5*c^6*d^4*x^10 + 882*a^3*b^4*c^5*d^5*x^10 + 735*a^4*b^3*c^4*d^6*x^10 + 252*a^5*b^2*c^3*d^7*x^10 + 63/2*a^6*b*c^2*d^8* x^10 + a^7*c*d^9*x^10 + 10/9*b^7*c^9*d*x^9 + 35*a*b^6*c^8*d^2*x^9 + 280*a^ 2*b^5*c^7*d^3*x^9 + 2450/3*a^3*b^4*c^6*d^4*x^9 + 980*a^4*b^3*c^5*d^5*x^9 + 490*a^5*b^2*c^4*d^6*x^9 + 280/3*a^6*b*c^3*d^7*x^9 + 5*a^7*c^2*d^8*x^9 + 1 /8*b^7*c^10*x^8 + 35/4*a*b^6*c^9*d*x^8 + 945/8*a^2*b^5*c^8*d^2*x^8 + 525*a ^3*b^4*c^7*d^3*x^8 + 3675/4*a^4*b^3*c^6*d^4*x^8 + 1323/2*a^5*b^2*c^5*d^...
Time = 0.29 (sec) , antiderivative size = 1106, normalized size of antiderivative = 5.53 \[ \int (a+b x)^7 (c+d x)^{10} \, dx =\text {Too large to display} \] Input:
int((a + b*x)^7*(c + d*x)^10,x)
Output:
x^10*(a^7*c*d^9 + (9*b^7*c^8*d^2)/2 + 84*a*b^6*c^7*d^3 + (63*a^6*b*c^2*d^8 )/2 + 441*a^2*b^5*c^6*d^4 + 882*a^3*b^4*c^5*d^5 + 735*a^4*b^3*c^4*d^6 + 25 2*a^5*b^2*c^3*d^7) + x^9*((10*b^7*c^9*d)/9 + 5*a^7*c^2*d^8 + 35*a*b^6*c^8* d^2 + (280*a^6*b*c^3*d^7)/3 + 280*a^2*b^5*c^7*d^3 + (2450*a^3*b^4*c^6*d^4) /3 + 980*a^4*b^3*c^5*d^5 + 490*a^5*b^2*c^4*d^6) + x^5*(7*a^3*b^4*c^10 + 42 *a^7*c^6*d^4 + 70*a^4*b^3*c^9*d + 168*a^6*b*c^7*d^3 + 189*a^5*b^2*c^8*d^2) + x^14*((5*a^4*b^3*d^10)/2 + 15*b^7*c^4*d^6 + 60*a*b^6*c^3*d^7 + 25*a^3*b ^4*c*d^9 + (135*a^2*b^5*c^2*d^8)/2) + x^8*((b^7*c^10)/8 + 15*a^7*c^3*d^7 + (735*a^6*b*c^4*d^6)/4 + (945*a^2*b^5*c^8*d^2)/8 + 525*a^3*b^4*c^7*d^3 + ( 3675*a^4*b^3*c^6*d^4)/4 + (1323*a^5*b^2*c^5*d^5)/2 + (35*a*b^6*c^9*d)/4) + x^11*((a^7*d^10)/11 + (120*b^7*c^7*d^3)/11 + (1470*a*b^6*c^6*d^4)/11 + (5 292*a^2*b^5*c^5*d^5)/11 + (7350*a^3*b^4*c^4*d^6)/11 + (4200*a^4*b^3*c^3*d^ 7)/11 + (945*a^5*b^2*c^2*d^8)/11 + (70*a^6*b*c*d^9)/11) + x^6*((7*a^2*b^5* c^10)/2 + 42*a^7*c^5*d^5 + (175*a^3*b^4*c^9*d)/3 + 245*a^6*b*c^6*d^4 + (52 5*a^4*b^3*c^8*d^2)/2 + 420*a^5*b^2*c^7*d^3) + x^13*((21*a^5*b^2*d^10)/13 + (252*b^7*c^5*d^5)/13 + (1470*a*b^6*c^4*d^6)/13 + (350*a^4*b^3*c*d^9)/13 + (2520*a^2*b^5*c^3*d^7)/13 + (1575*a^3*b^4*c^2*d^8)/13) + x^7*(a*b^6*c^10 + 30*a^7*c^4*d^6 + 30*a^2*b^5*c^9*d + 252*a^6*b*c^5*d^5 + 225*a^3*b^4*c^8* d^2 + 600*a^4*b^3*c^7*d^3 + 630*a^5*b^2*c^6*d^4) + x^12*((7*a^6*b*d^10)/12 + (35*b^7*c^6*d^4)/2 + 147*a*b^6*c^5*d^5 + (35*a^5*b^2*c*d^9)/2 + (735...
Time = 0.16 (sec) , antiderivative size = 1303, normalized size of antiderivative = 6.52 \[ \int (a+b x)^7 (c+d x)^{10} \, dx =\text {Too large to display} \] Input:
int((b*x+a)^7*(d*x+c)^10,x)
Output:
(x*(350064*a**7*c**10 + 1750320*a**7*c**9*d*x + 5250960*a**7*c**8*d**2*x** 2 + 10501920*a**7*c**7*d**3*x**3 + 14702688*a**7*c**6*d**4*x**4 + 14702688 *a**7*c**5*d**5*x**5 + 10501920*a**7*c**4*d**6*x**6 + 5250960*a**7*c**3*d* *7*x**7 + 1750320*a**7*c**2*d**8*x**8 + 350064*a**7*c*d**9*x**9 + 31824*a* *7*d**10*x**10 + 1225224*a**6*b*c**10*x + 8168160*a**6*b*c**9*d*x**2 + 275 67540*a**6*b*c**8*d**2*x**3 + 58810752*a**6*b*c**7*d**3*x**4 + 85765680*a* *6*b*c**6*d**4*x**5 + 88216128*a**6*b*c**5*d**5*x**6 + 64324260*a**6*b*c** 4*d**6*x**7 + 32672640*a**6*b*c**3*d**7*x**8 + 11027016*a**6*b*c**2*d**8*x **9 + 2227680*a**6*b*c*d**9*x**10 + 204204*a**6*b*d**10*x**11 + 2450448*a* *5*b**2*c**10*x**2 + 18378360*a**5*b**2*c**9*d*x**3 + 66162096*a**5*b**2*c **8*d**2*x**4 + 147026880*a**5*b**2*c**7*d**3*x**5 + 220540320*a**5*b**2*c **6*d**4*x**6 + 231567336*a**5*b**2*c**5*d**5*x**7 + 171531360*a**5*b**2*c **4*d**6*x**8 + 88216128*a**5*b**2*c**3*d**7*x**9 + 30073680*a**5*b**2*c** 2*d**8*x**10 + 6126120*a**5*b**2*c*d**9*x**11 + 565488*a**5*b**2*d**10*x** 12 + 3063060*a**4*b**3*c**10*x**3 + 24504480*a**4*b**3*c**9*d*x**4 + 91891 800*a**4*b**3*c**8*d**2*x**5 + 210038400*a**4*b**3*c**7*d**3*x**6 + 321621 300*a**4*b**3*c**6*d**4*x**7 + 343062720*a**4*b**3*c**5*d**5*x**8 + 257297 040*a**4*b**3*c**4*d**6*x**9 + 133660800*a**4*b**3*c**3*d**7*x**10 + 45945 900*a**4*b**3*c**2*d**8*x**11 + 9424800*a**4*b**3*c*d**9*x**12 + 875160*a* *4*b**3*d**10*x**13 + 2450448*a**3*b**4*c**10*x**4 + 20420400*a**3*b**4...