Integrand size = 15, antiderivative size = 200 \[ \int (a+b x)^9 (c+d x)^7 \, dx=\frac {(b c-a d)^7 (a+b x)^{10}}{10 b^8}+\frac {7 d (b c-a d)^6 (a+b x)^{11}}{11 b^8}+\frac {7 d^2 (b c-a d)^5 (a+b x)^{12}}{4 b^8}+\frac {35 d^3 (b c-a d)^4 (a+b x)^{13}}{13 b^8}+\frac {5 d^4 (b c-a d)^3 (a+b x)^{14}}{2 b^8}+\frac {7 d^5 (b c-a d)^2 (a+b x)^{15}}{5 b^8}+\frac {7 d^6 (b c-a d) (a+b x)^{16}}{16 b^8}+\frac {d^7 (a+b x)^{17}}{17 b^8} \] Output:
1/10*(-a*d+b*c)^7*(b*x+a)^10/b^8+7/11*d*(-a*d+b*c)^6*(b*x+a)^11/b^8+7/4*d^ 2*(-a*d+b*c)^5*(b*x+a)^12/b^8+35/13*d^3*(-a*d+b*c)^4*(b*x+a)^13/b^8+5/2*d^ 4*(-a*d+b*c)^3*(b*x+a)^14/b^8+7/5*d^5*(-a*d+b*c)^2*(b*x+a)^15/b^8+7/16*d^6 *(-a*d+b*c)*(b*x+a)^16/b^8+1/17*d^7*(b*x+a)^17/b^8
Leaf count is larger than twice the leaf count of optimal. \(993\) vs. \(2(200)=400\).
Time = 0.08 (sec) , antiderivative size = 993, normalized size of antiderivative = 4.96 \[ \int (a+b x)^9 (c+d x)^7 \, dx=a^9 c^7 x+\frac {1}{2} a^8 c^6 (9 b c+7 a d) x^2+a^7 c^5 \left (12 b^2 c^2+21 a b c d+7 a^2 d^2\right ) x^3+\frac {7}{4} a^6 c^4 \left (12 b^3 c^3+36 a b^2 c^2 d+27 a^2 b c d^2+5 a^3 d^3\right ) x^4+\frac {7}{5} a^5 c^3 \left (18 b^4 c^4+84 a b^3 c^3 d+108 a^2 b^2 c^2 d^2+45 a^3 b c d^3+5 a^4 d^4\right ) x^5+\frac {7}{2} a^4 c^2 \left (6 b^5 c^5+42 a b^4 c^4 d+84 a^2 b^3 c^3 d^2+60 a^3 b^2 c^2 d^3+15 a^4 b c d^4+a^5 d^5\right ) x^6+a^3 c \left (12 b^6 c^6+126 a b^5 c^5 d+378 a^2 b^4 c^4 d^2+420 a^3 b^3 c^3 d^3+180 a^4 b^2 c^2 d^4+27 a^5 b c d^5+a^6 d^6\right ) x^7+\frac {1}{8} a^2 \left (36 b^7 c^7+588 a b^6 c^6 d+2646 a^2 b^5 c^5 d^2+4410 a^3 b^4 c^4 d^3+2940 a^4 b^3 c^3 d^4+756 a^5 b^2 c^2 d^5+63 a^6 b c d^6+a^7 d^7\right ) x^8+a b \left (b^7 c^7+28 a b^6 c^6 d+196 a^2 b^5 c^5 d^2+490 a^3 b^4 c^4 d^3+490 a^4 b^3 c^3 d^4+196 a^5 b^2 c^2 d^5+28 a^6 b c d^6+a^7 d^7\right ) x^9+\frac {1}{10} b^2 \left (b^7 c^7+63 a b^6 c^6 d+756 a^2 b^5 c^5 d^2+2940 a^3 b^4 c^4 d^3+4410 a^4 b^3 c^3 d^4+2646 a^5 b^2 c^2 d^5+588 a^6 b c d^6+36 a^7 d^7\right ) x^{10}+\frac {7}{11} b^3 d \left (b^6 c^6+27 a b^5 c^5 d+180 a^2 b^4 c^4 d^2+420 a^3 b^3 c^3 d^3+378 a^4 b^2 c^2 d^4+126 a^5 b c d^5+12 a^6 d^6\right ) x^{11}+\frac {7}{4} b^4 d^2 \left (b^5 c^5+15 a b^4 c^4 d+60 a^2 b^3 c^3 d^2+84 a^3 b^2 c^2 d^3+42 a^4 b c d^4+6 a^5 d^5\right ) x^{12}+\frac {7}{13} b^5 d^3 \left (5 b^4 c^4+45 a b^3 c^3 d+108 a^2 b^2 c^2 d^2+84 a^3 b c d^3+18 a^4 d^4\right ) x^{13}+\frac {1}{2} b^6 d^4 \left (5 b^3 c^3+27 a b^2 c^2 d+36 a^2 b c d^2+12 a^3 d^3\right ) x^{14}+\frac {1}{5} b^7 d^5 \left (7 b^2 c^2+21 a b c d+12 a^2 d^2\right ) x^{15}+\frac {1}{16} b^8 d^6 (7 b c+9 a d) x^{16}+\frac {1}{17} b^9 d^7 x^{17} \] Input:
Integrate[(a + b*x)^9*(c + d*x)^7,x]
Output:
a^9*c^7*x + (a^8*c^6*(9*b*c + 7*a*d)*x^2)/2 + a^7*c^5*(12*b^2*c^2 + 21*a*b *c*d + 7*a^2*d^2)*x^3 + (7*a^6*c^4*(12*b^3*c^3 + 36*a*b^2*c^2*d + 27*a^2*b *c*d^2 + 5*a^3*d^3)*x^4)/4 + (7*a^5*c^3*(18*b^4*c^4 + 84*a*b^3*c^3*d + 108 *a^2*b^2*c^2*d^2 + 45*a^3*b*c*d^3 + 5*a^4*d^4)*x^5)/5 + (7*a^4*c^2*(6*b^5* c^5 + 42*a*b^4*c^4*d + 84*a^2*b^3*c^3*d^2 + 60*a^3*b^2*c^2*d^3 + 15*a^4*b* c*d^4 + a^5*d^5)*x^6)/2 + a^3*c*(12*b^6*c^6 + 126*a*b^5*c^5*d + 378*a^2*b^ 4*c^4*d^2 + 420*a^3*b^3*c^3*d^3 + 180*a^4*b^2*c^2*d^4 + 27*a^5*b*c*d^5 + a ^6*d^6)*x^7 + (a^2*(36*b^7*c^7 + 588*a*b^6*c^6*d + 2646*a^2*b^5*c^5*d^2 + 4410*a^3*b^4*c^4*d^3 + 2940*a^4*b^3*c^3*d^4 + 756*a^5*b^2*c^2*d^5 + 63*a^6 *b*c*d^6 + a^7*d^7)*x^8)/8 + a*b*(b^7*c^7 + 28*a*b^6*c^6*d + 196*a^2*b^5*c ^5*d^2 + 490*a^3*b^4*c^4*d^3 + 490*a^4*b^3*c^3*d^4 + 196*a^5*b^2*c^2*d^5 + 28*a^6*b*c*d^6 + a^7*d^7)*x^9 + (b^2*(b^7*c^7 + 63*a*b^6*c^6*d + 756*a^2* b^5*c^5*d^2 + 2940*a^3*b^4*c^4*d^3 + 4410*a^4*b^3*c^3*d^4 + 2646*a^5*b^2*c ^2*d^5 + 588*a^6*b*c*d^6 + 36*a^7*d^7)*x^10)/10 + (7*b^3*d*(b^6*c^6 + 27*a *b^5*c^5*d + 180*a^2*b^4*c^4*d^2 + 420*a^3*b^3*c^3*d^3 + 378*a^4*b^2*c^2*d ^4 + 126*a^5*b*c*d^5 + 12*a^6*d^6)*x^11)/11 + (7*b^4*d^2*(b^5*c^5 + 15*a*b ^4*c^4*d + 60*a^2*b^3*c^3*d^2 + 84*a^3*b^2*c^2*d^3 + 42*a^4*b*c*d^4 + 6*a^ 5*d^5)*x^12)/4 + (7*b^5*d^3*(5*b^4*c^4 + 45*a*b^3*c^3*d + 108*a^2*b^2*c^2* d^2 + 84*a^3*b*c*d^3 + 18*a^4*d^4)*x^13)/13 + (b^6*d^4*(5*b^3*c^3 + 27*a*b ^2*c^2*d + 36*a^2*b*c*d^2 + 12*a^3*d^3)*x^14)/2 + (b^7*d^5*(7*b^2*c^2 +...
Time = 0.73 (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.
\(\displaystyle \int (a+b x)^9 (c+d x)^7 \, dx\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \int \left (\frac {7 d^6 (a+b x)^{15} (b c-a d)}{b^7}+\frac {21 d^5 (a+b x)^{14} (b c-a d)^2}{b^7}+\frac {35 d^4 (a+b x)^{13} (b c-a d)^3}{b^7}+\frac {35 d^3 (a+b x)^{12} (b c-a d)^4}{b^7}+\frac {21 d^2 (a+b x)^{11} (b c-a d)^5}{b^7}+\frac {7 d (a+b x)^{10} (b c-a d)^6}{b^7}+\frac {(a+b x)^9 (b c-a d)^7}{b^7}+\frac {d^7 (a+b x)^{16}}{b^7}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {7 d^6 (a+b x)^{16} (b c-a d)}{16 b^8}+\frac {7 d^5 (a+b x)^{15} (b c-a d)^2}{5 b^8}+\frac {5 d^4 (a+b x)^{14} (b c-a d)^3}{2 b^8}+\frac {35 d^3 (a+b x)^{13} (b c-a d)^4}{13 b^8}+\frac {7 d^2 (a+b x)^{12} (b c-a d)^5}{4 b^8}+\frac {7 d (a+b x)^{11} (b c-a d)^6}{11 b^8}+\frac {(a+b x)^{10} (b c-a d)^7}{10 b^8}+\frac {d^7 (a+b x)^{17}}{17 b^8}\) |
Input:
Int[(a + b*x)^9*(c + d*x)^7,x]
Output:
((b*c - a*d)^7*(a + b*x)^10)/(10*b^8) + (7*d*(b*c - a*d)^6*(a + b*x)^11)/( 11*b^8) + (7*d^2*(b*c - a*d)^5*(a + b*x)^12)/(4*b^8) + (35*d^3*(b*c - a*d) ^4*(a + b*x)^13)/(13*b^8) + (5*d^4*(b*c - a*d)^3*(a + b*x)^14)/(2*b^8) + ( 7*d^5*(b*c - a*d)^2*(a + b*x)^15)/(5*b^8) + (7*d^6*(b*c - a*d)*(a + b*x)^1 6)/(16*b^8) + (d^7*(a + b*x)^17)/(17*b^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. \(1016\) vs. \(2(184)=368\).
Time = 0.11 (sec) , antiderivative size = 1017, normalized size of antiderivative = 5.08
method | result | size |
norman | \(\text {Expression too large to display}\) | \(1017\) |
default | \(\text {Expression too large to display}\) | \(1033\) |
gosper | \(\text {Expression too large to display}\) | \(1176\) |
risch | \(\text {Expression too large to display}\) | \(1176\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1176\) |
orering | \(\text {Expression too large to display}\) | \(1178\) |
Input:
int((b*x+a)^9*(d*x+c)^7,x,method=_RETURNVERBOSE)
Output:
a^9*c^7*x+(7/2*a^9*c^6*d+9/2*a^8*b*c^7)*x^2+(7*a^9*c^5*d^2+21*a^8*b*c^6*d+ 12*a^7*b^2*c^7)*x^3+(35/4*a^9*c^4*d^3+189/4*a^8*b*c^5*d^2+63*a^7*b^2*c^6*d +21*a^6*b^3*c^7)*x^4+(7*a^9*c^3*d^4+63*a^8*b*c^4*d^3+756/5*a^7*b^2*c^5*d^2 +588/5*a^6*b^3*c^6*d+126/5*a^5*b^4*c^7)*x^5+(7/2*a^9*c^2*d^5+105/2*a^8*b*c ^3*d^4+210*a^7*b^2*c^4*d^3+294*a^6*b^3*c^5*d^2+147*a^5*b^4*c^6*d+21*a^4*b^ 5*c^7)*x^6+(a^9*c*d^6+27*a^8*b*c^2*d^5+180*a^7*b^2*c^3*d^4+420*a^6*b^3*c^4 *d^3+378*a^5*b^4*c^5*d^2+126*a^4*b^5*c^6*d+12*a^3*b^6*c^7)*x^7+(1/8*a^9*d^ 7+63/8*a^8*b*c*d^6+189/2*a^7*b^2*c^2*d^5+735/2*a^6*b^3*c^3*d^4+2205/4*a^5* b^4*c^4*d^3+1323/4*a^4*b^5*c^5*d^2+147/2*a^3*b^6*c^6*d+9/2*a^2*b^7*c^7)*x^ 8+(a^8*b*d^7+28*a^7*b^2*c*d^6+196*a^6*b^3*c^2*d^5+490*a^5*b^4*c^3*d^4+490* a^4*b^5*c^4*d^3+196*a^3*b^6*c^5*d^2+28*a^2*b^7*c^6*d+a*b^8*c^7)*x^9+(18/5* a^7*b^2*d^7+294/5*a^6*b^3*c*d^6+1323/5*a^5*b^4*c^2*d^5+441*a^4*b^5*c^3*d^4 +294*a^3*b^6*c^4*d^3+378/5*a^2*b^7*c^5*d^2+63/10*a*b^8*c^6*d+1/10*b^9*c^7) *x^10+(84/11*a^6*b^3*d^7+882/11*a^5*b^4*c*d^6+2646/11*a^4*b^5*c^2*d^5+2940 /11*a^3*b^6*c^3*d^4+1260/11*a^2*b^7*c^4*d^3+189/11*a*b^8*c^5*d^2+7/11*b^9* c^6*d)*x^11+(21/2*a^5*b^4*d^7+147/2*a^4*b^5*c*d^6+147*a^3*b^6*c^2*d^5+105* a^2*b^7*c^3*d^4+105/4*a*b^8*c^4*d^3+7/4*b^9*c^5*d^2)*x^12+(126/13*a^4*b^5* d^7+588/13*a^3*b^6*c*d^6+756/13*a^2*b^7*c^2*d^5+315/13*a*b^8*c^3*d^4+35/13 *b^9*c^4*d^3)*x^13+(6*a^3*b^6*d^7+18*a^2*b^7*c*d^6+27/2*a*b^8*c^2*d^5+5/2* b^9*c^3*d^4)*x^14+(12/5*a^2*b^7*d^7+21/5*a*b^8*c*d^6+7/5*b^9*c^2*d^5)*x...
Leaf count of result is larger than twice the leaf count of optimal. 1023 vs. \(2 (184) = 368\).
Time = 0.08 (sec) , antiderivative size = 1023, normalized size of antiderivative = 5.12 \[ \int (a+b x)^9 (c+d x)^7 \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)^9*(d*x+c)^7,x, algorithm="fricas")
Output:
1/17*b^9*d^7*x^17 + a^9*c^7*x + 1/16*(7*b^9*c*d^6 + 9*a*b^8*d^7)*x^16 + 1/ 5*(7*b^9*c^2*d^5 + 21*a*b^8*c*d^6 + 12*a^2*b^7*d^7)*x^15 + 1/2*(5*b^9*c^3* d^4 + 27*a*b^8*c^2*d^5 + 36*a^2*b^7*c*d^6 + 12*a^3*b^6*d^7)*x^14 + 7/13*(5 *b^9*c^4*d^3 + 45*a*b^8*c^3*d^4 + 108*a^2*b^7*c^2*d^5 + 84*a^3*b^6*c*d^6 + 18*a^4*b^5*d^7)*x^13 + 7/4*(b^9*c^5*d^2 + 15*a*b^8*c^4*d^3 + 60*a^2*b^7*c ^3*d^4 + 84*a^3*b^6*c^2*d^5 + 42*a^4*b^5*c*d^6 + 6*a^5*b^4*d^7)*x^12 + 7/1 1*(b^9*c^6*d + 27*a*b^8*c^5*d^2 + 180*a^2*b^7*c^4*d^3 + 420*a^3*b^6*c^3*d^ 4 + 378*a^4*b^5*c^2*d^5 + 126*a^5*b^4*c*d^6 + 12*a^6*b^3*d^7)*x^11 + 1/10* (b^9*c^7 + 63*a*b^8*c^6*d + 756*a^2*b^7*c^5*d^2 + 2940*a^3*b^6*c^4*d^3 + 4 410*a^4*b^5*c^3*d^4 + 2646*a^5*b^4*c^2*d^5 + 588*a^6*b^3*c*d^6 + 36*a^7*b^ 2*d^7)*x^10 + (a*b^8*c^7 + 28*a^2*b^7*c^6*d + 196*a^3*b^6*c^5*d^2 + 490*a^ 4*b^5*c^4*d^3 + 490*a^5*b^4*c^3*d^4 + 196*a^6*b^3*c^2*d^5 + 28*a^7*b^2*c*d ^6 + a^8*b*d^7)*x^9 + 1/8*(36*a^2*b^7*c^7 + 588*a^3*b^6*c^6*d + 2646*a^4*b ^5*c^5*d^2 + 4410*a^5*b^4*c^4*d^3 + 2940*a^6*b^3*c^3*d^4 + 756*a^7*b^2*c^2 *d^5 + 63*a^8*b*c*d^6 + a^9*d^7)*x^8 + (12*a^3*b^6*c^7 + 126*a^4*b^5*c^6*d + 378*a^5*b^4*c^5*d^2 + 420*a^6*b^3*c^4*d^3 + 180*a^7*b^2*c^3*d^4 + 27*a^ 8*b*c^2*d^5 + a^9*c*d^6)*x^7 + 7/2*(6*a^4*b^5*c^7 + 42*a^5*b^4*c^6*d + 84* a^6*b^3*c^5*d^2 + 60*a^7*b^2*c^4*d^3 + 15*a^8*b*c^3*d^4 + a^9*c^2*d^5)*x^6 + 7/5*(18*a^5*b^4*c^7 + 84*a^6*b^3*c^6*d + 108*a^7*b^2*c^5*d^2 + 45*a^8*b *c^4*d^3 + 5*a^9*c^3*d^4)*x^5 + 7/4*(12*a^6*b^3*c^7 + 36*a^7*b^2*c^6*d ...
Leaf count of result is larger than twice the leaf count of optimal. 1163 vs. \(2 (184) = 368\).
Time = 0.08 (sec) , antiderivative size = 1163, normalized size of antiderivative = 5.82 \[ \int (a+b x)^9 (c+d x)^7 \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)**9*(d*x+c)**7,x)
Output:
a**9*c**7*x + b**9*d**7*x**17/17 + x**16*(9*a*b**8*d**7/16 + 7*b**9*c*d**6 /16) + x**15*(12*a**2*b**7*d**7/5 + 21*a*b**8*c*d**6/5 + 7*b**9*c**2*d**5/ 5) + x**14*(6*a**3*b**6*d**7 + 18*a**2*b**7*c*d**6 + 27*a*b**8*c**2*d**5/2 + 5*b**9*c**3*d**4/2) + x**13*(126*a**4*b**5*d**7/13 + 588*a**3*b**6*c*d* *6/13 + 756*a**2*b**7*c**2*d**5/13 + 315*a*b**8*c**3*d**4/13 + 35*b**9*c** 4*d**3/13) + x**12*(21*a**5*b**4*d**7/2 + 147*a**4*b**5*c*d**6/2 + 147*a** 3*b**6*c**2*d**5 + 105*a**2*b**7*c**3*d**4 + 105*a*b**8*c**4*d**3/4 + 7*b* *9*c**5*d**2/4) + x**11*(84*a**6*b**3*d**7/11 + 882*a**5*b**4*c*d**6/11 + 2646*a**4*b**5*c**2*d**5/11 + 2940*a**3*b**6*c**3*d**4/11 + 1260*a**2*b**7 *c**4*d**3/11 + 189*a*b**8*c**5*d**2/11 + 7*b**9*c**6*d/11) + x**10*(18*a* *7*b**2*d**7/5 + 294*a**6*b**3*c*d**6/5 + 1323*a**5*b**4*c**2*d**5/5 + 441 *a**4*b**5*c**3*d**4 + 294*a**3*b**6*c**4*d**3 + 378*a**2*b**7*c**5*d**2/5 + 63*a*b**8*c**6*d/10 + b**9*c**7/10) + x**9*(a**8*b*d**7 + 28*a**7*b**2* c*d**6 + 196*a**6*b**3*c**2*d**5 + 490*a**5*b**4*c**3*d**4 + 490*a**4*b**5 *c**4*d**3 + 196*a**3*b**6*c**5*d**2 + 28*a**2*b**7*c**6*d + a*b**8*c**7) + x**8*(a**9*d**7/8 + 63*a**8*b*c*d**6/8 + 189*a**7*b**2*c**2*d**5/2 + 735 *a**6*b**3*c**3*d**4/2 + 2205*a**5*b**4*c**4*d**3/4 + 1323*a**4*b**5*c**5* d**2/4 + 147*a**3*b**6*c**6*d/2 + 9*a**2*b**7*c**7/2) + x**7*(a**9*c*d**6 + 27*a**8*b*c**2*d**5 + 180*a**7*b**2*c**3*d**4 + 420*a**6*b**3*c**4*d**3 + 378*a**5*b**4*c**5*d**2 + 126*a**4*b**5*c**6*d + 12*a**3*b**6*c**7) +...
Leaf count of result is larger than twice the leaf count of optimal. 1023 vs. \(2 (184) = 368\).
Time = 0.04 (sec) , antiderivative size = 1023, normalized size of antiderivative = 5.12 \[ \int (a+b x)^9 (c+d x)^7 \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)^9*(d*x+c)^7,x, algorithm="maxima")
Output:
1/17*b^9*d^7*x^17 + a^9*c^7*x + 1/16*(7*b^9*c*d^6 + 9*a*b^8*d^7)*x^16 + 1/ 5*(7*b^9*c^2*d^5 + 21*a*b^8*c*d^6 + 12*a^2*b^7*d^7)*x^15 + 1/2*(5*b^9*c^3* d^4 + 27*a*b^8*c^2*d^5 + 36*a^2*b^7*c*d^6 + 12*a^3*b^6*d^7)*x^14 + 7/13*(5 *b^9*c^4*d^3 + 45*a*b^8*c^3*d^4 + 108*a^2*b^7*c^2*d^5 + 84*a^3*b^6*c*d^6 + 18*a^4*b^5*d^7)*x^13 + 7/4*(b^9*c^5*d^2 + 15*a*b^8*c^4*d^3 + 60*a^2*b^7*c ^3*d^4 + 84*a^3*b^6*c^2*d^5 + 42*a^4*b^5*c*d^6 + 6*a^5*b^4*d^7)*x^12 + 7/1 1*(b^9*c^6*d + 27*a*b^8*c^5*d^2 + 180*a^2*b^7*c^4*d^3 + 420*a^3*b^6*c^3*d^ 4 + 378*a^4*b^5*c^2*d^5 + 126*a^5*b^4*c*d^6 + 12*a^6*b^3*d^7)*x^11 + 1/10* (b^9*c^7 + 63*a*b^8*c^6*d + 756*a^2*b^7*c^5*d^2 + 2940*a^3*b^6*c^4*d^3 + 4 410*a^4*b^5*c^3*d^4 + 2646*a^5*b^4*c^2*d^5 + 588*a^6*b^3*c*d^6 + 36*a^7*b^ 2*d^7)*x^10 + (a*b^8*c^7 + 28*a^2*b^7*c^6*d + 196*a^3*b^6*c^5*d^2 + 490*a^ 4*b^5*c^4*d^3 + 490*a^5*b^4*c^3*d^4 + 196*a^6*b^3*c^2*d^5 + 28*a^7*b^2*c*d ^6 + a^8*b*d^7)*x^9 + 1/8*(36*a^2*b^7*c^7 + 588*a^3*b^6*c^6*d + 2646*a^4*b ^5*c^5*d^2 + 4410*a^5*b^4*c^4*d^3 + 2940*a^6*b^3*c^3*d^4 + 756*a^7*b^2*c^2 *d^5 + 63*a^8*b*c*d^6 + a^9*d^7)*x^8 + (12*a^3*b^6*c^7 + 126*a^4*b^5*c^6*d + 378*a^5*b^4*c^5*d^2 + 420*a^6*b^3*c^4*d^3 + 180*a^7*b^2*c^3*d^4 + 27*a^ 8*b*c^2*d^5 + a^9*c*d^6)*x^7 + 7/2*(6*a^4*b^5*c^7 + 42*a^5*b^4*c^6*d + 84* a^6*b^3*c^5*d^2 + 60*a^7*b^2*c^4*d^3 + 15*a^8*b*c^3*d^4 + a^9*c^2*d^5)*x^6 + 7/5*(18*a^5*b^4*c^7 + 84*a^6*b^3*c^6*d + 108*a^7*b^2*c^5*d^2 + 45*a^8*b *c^4*d^3 + 5*a^9*c^3*d^4)*x^5 + 7/4*(12*a^6*b^3*c^7 + 36*a^7*b^2*c^6*d ...
Leaf count of result is larger than twice the leaf count of optimal. 1175 vs. \(2 (184) = 368\).
Time = 0.13 (sec) , antiderivative size = 1175, normalized size of antiderivative = 5.88 \[ \int (a+b x)^9 (c+d x)^7 \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^9*(d*x+c)^7,x, algorithm="giac")
Output:
1/17*b^9*d^7*x^17 + 7/16*b^9*c*d^6*x^16 + 9/16*a*b^8*d^7*x^16 + 7/5*b^9*c^ 2*d^5*x^15 + 21/5*a*b^8*c*d^6*x^15 + 12/5*a^2*b^7*d^7*x^15 + 5/2*b^9*c^3*d ^4*x^14 + 27/2*a*b^8*c^2*d^5*x^14 + 18*a^2*b^7*c*d^6*x^14 + 6*a^3*b^6*d^7* x^14 + 35/13*b^9*c^4*d^3*x^13 + 315/13*a*b^8*c^3*d^4*x^13 + 756/13*a^2*b^7 *c^2*d^5*x^13 + 588/13*a^3*b^6*c*d^6*x^13 + 126/13*a^4*b^5*d^7*x^13 + 7/4* b^9*c^5*d^2*x^12 + 105/4*a*b^8*c^4*d^3*x^12 + 105*a^2*b^7*c^3*d^4*x^12 + 1 47*a^3*b^6*c^2*d^5*x^12 + 147/2*a^4*b^5*c*d^6*x^12 + 21/2*a^5*b^4*d^7*x^12 + 7/11*b^9*c^6*d*x^11 + 189/11*a*b^8*c^5*d^2*x^11 + 1260/11*a^2*b^7*c^4*d ^3*x^11 + 2940/11*a^3*b^6*c^3*d^4*x^11 + 2646/11*a^4*b^5*c^2*d^5*x^11 + 88 2/11*a^5*b^4*c*d^6*x^11 + 84/11*a^6*b^3*d^7*x^11 + 1/10*b^9*c^7*x^10 + 63/ 10*a*b^8*c^6*d*x^10 + 378/5*a^2*b^7*c^5*d^2*x^10 + 294*a^3*b^6*c^4*d^3*x^1 0 + 441*a^4*b^5*c^3*d^4*x^10 + 1323/5*a^5*b^4*c^2*d^5*x^10 + 294/5*a^6*b^3 *c*d^6*x^10 + 18/5*a^7*b^2*d^7*x^10 + a*b^8*c^7*x^9 + 28*a^2*b^7*c^6*d*x^9 + 196*a^3*b^6*c^5*d^2*x^9 + 490*a^4*b^5*c^4*d^3*x^9 + 490*a^5*b^4*c^3*d^4 *x^9 + 196*a^6*b^3*c^2*d^5*x^9 + 28*a^7*b^2*c*d^6*x^9 + a^8*b*d^7*x^9 + 9/ 2*a^2*b^7*c^7*x^8 + 147/2*a^3*b^6*c^6*d*x^8 + 1323/4*a^4*b^5*c^5*d^2*x^8 + 2205/4*a^5*b^4*c^4*d^3*x^8 + 735/2*a^6*b^3*c^3*d^4*x^8 + 189/2*a^7*b^2*c^ 2*d^5*x^8 + 63/8*a^8*b*c*d^6*x^8 + 1/8*a^9*d^7*x^8 + 12*a^3*b^6*c^7*x^7 + 126*a^4*b^5*c^6*d*x^7 + 378*a^5*b^4*c^5*d^2*x^7 + 420*a^6*b^3*c^4*d^3*x^7 + 180*a^7*b^2*c^3*d^4*x^7 + 27*a^8*b*c^2*d^5*x^7 + a^9*c*d^6*x^7 + 21*a...
Time = 0.35 (sec) , antiderivative size = 997, normalized size of antiderivative = 4.98 \[ \int (a+b x)^9 (c+d x)^7 \, dx =\text {Too large to display} \] Input:
int((a + b*x)^9*(c + d*x)^7,x)
Output:
x^5*((126*a^5*b^4*c^7)/5 + 7*a^9*c^3*d^4 + (588*a^6*b^3*c^6*d)/5 + 63*a^8* b*c^4*d^3 + (756*a^7*b^2*c^5*d^2)/5) + x^13*((126*a^4*b^5*d^7)/13 + (35*b^ 9*c^4*d^3)/13 + (315*a*b^8*c^3*d^4)/13 + (588*a^3*b^6*c*d^6)/13 + (756*a^2 *b^7*c^2*d^5)/13) + x^8*((a^9*d^7)/8 + (9*a^2*b^7*c^7)/2 + (147*a^3*b^6*c^ 6*d)/2 + (1323*a^4*b^5*c^5*d^2)/4 + (2205*a^5*b^4*c^4*d^3)/4 + (735*a^6*b^ 3*c^3*d^4)/2 + (189*a^7*b^2*c^2*d^5)/2 + (63*a^8*b*c*d^6)/8) + x^10*((b^9* c^7)/10 + (18*a^7*b^2*d^7)/5 + (294*a^6*b^3*c*d^6)/5 + (378*a^2*b^7*c^5*d^ 2)/5 + 294*a^3*b^6*c^4*d^3 + 441*a^4*b^5*c^3*d^4 + (1323*a^5*b^4*c^2*d^5)/ 5 + (63*a*b^8*c^6*d)/10) + x^6*(21*a^4*b^5*c^7 + (7*a^9*c^2*d^5)/2 + 147*a ^5*b^4*c^6*d + (105*a^8*b*c^3*d^4)/2 + 294*a^6*b^3*c^5*d^2 + 210*a^7*b^2*c ^4*d^3) + x^12*((21*a^5*b^4*d^7)/2 + (7*b^9*c^5*d^2)/4 + (105*a*b^8*c^4*d^ 3)/4 + (147*a^4*b^5*c*d^6)/2 + 105*a^2*b^7*c^3*d^4 + 147*a^3*b^6*c^2*d^5) + x^7*(a^9*c*d^6 + 12*a^3*b^6*c^7 + 126*a^4*b^5*c^6*d + 27*a^8*b*c^2*d^5 + 378*a^5*b^4*c^5*d^2 + 420*a^6*b^3*c^4*d^3 + 180*a^7*b^2*c^3*d^4) + x^11*( (7*b^9*c^6*d)/11 + (84*a^6*b^3*d^7)/11 + (189*a*b^8*c^5*d^2)/11 + (882*a^5 *b^4*c*d^6)/11 + (1260*a^2*b^7*c^4*d^3)/11 + (2940*a^3*b^6*c^3*d^4)/11 + ( 2646*a^4*b^5*c^2*d^5)/11) + x^9*(a*b^8*c^7 + a^8*b*d^7 + 28*a^2*b^7*c^6*d + 28*a^7*b^2*c*d^6 + 196*a^3*b^6*c^5*d^2 + 490*a^4*b^5*c^4*d^3 + 490*a^5*b ^4*c^3*d^4 + 196*a^6*b^3*c^2*d^5) + a^9*c^7*x + (b^9*d^7*x^17)/17 + (7*a^6 *c^4*x^4*(5*a^3*d^3 + 12*b^3*c^3 + 36*a*b^2*c^2*d + 27*a^2*b*c*d^2))/4 ...
Time = 0.17 (sec) , antiderivative size = 1177, normalized size of antiderivative = 5.88 \[ \int (a+b x)^9 (c+d x)^7 \, dx =\text {Too large to display} \] Input:
int((b*x+a)^9*(d*x+c)^7,x)
Output:
(x*(194480*a**9*c**7 + 680680*a**9*c**6*d*x + 1361360*a**9*c**5*d**2*x**2 + 1701700*a**9*c**4*d**3*x**3 + 1361360*a**9*c**3*d**4*x**4 + 680680*a**9* c**2*d**5*x**5 + 194480*a**9*c*d**6*x**6 + 24310*a**9*d**7*x**7 + 875160*a **8*b*c**7*x + 4084080*a**8*b*c**6*d*x**2 + 9189180*a**8*b*c**5*d**2*x**3 + 12252240*a**8*b*c**4*d**3*x**4 + 10210200*a**8*b*c**3*d**4*x**5 + 525096 0*a**8*b*c**2*d**5*x**6 + 1531530*a**8*b*c*d**6*x**7 + 194480*a**8*b*d**7* x**8 + 2333760*a**7*b**2*c**7*x**2 + 12252240*a**7*b**2*c**6*d*x**3 + 2940 5376*a**7*b**2*c**5*d**2*x**4 + 40840800*a**7*b**2*c**4*d**3*x**5 + 350064 00*a**7*b**2*c**3*d**4*x**6 + 18378360*a**7*b**2*c**2*d**5*x**7 + 5445440* a**7*b**2*c*d**6*x**8 + 700128*a**7*b**2*d**7*x**9 + 4084080*a**6*b**3*c** 7*x**3 + 22870848*a**6*b**3*c**6*d*x**4 + 57177120*a**6*b**3*c**5*d**2*x** 5 + 81681600*a**6*b**3*c**4*d**3*x**6 + 71471400*a**6*b**3*c**3*d**4*x**7 + 38118080*a**6*b**3*c**2*d**5*x**8 + 11435424*a**6*b**3*c*d**6*x**9 + 148 5120*a**6*b**3*d**7*x**10 + 4900896*a**5*b**4*c**7*x**4 + 28588560*a**5*b* *4*c**6*d*x**5 + 73513440*a**5*b**4*c**5*d**2*x**6 + 107207100*a**5*b**4*c **4*d**3*x**7 + 95295200*a**5*b**4*c**3*d**4*x**8 + 51459408*a**5*b**4*c** 2*d**5*x**9 + 15593760*a**5*b**4*c*d**6*x**10 + 2042040*a**5*b**4*d**7*x** 11 + 4084080*a**4*b**5*c**7*x**5 + 24504480*a**4*b**5*c**6*d*x**6 + 643242 60*a**4*b**5*c**5*d**2*x**7 + 95295200*a**4*b**5*c**4*d**3*x**8 + 85765680 *a**4*b**5*c**3*d**4*x**9 + 46781280*a**4*b**5*c**2*d**5*x**10 + 142942...