Integrand size = 20, antiderivative size = 329 \[ \int (a+b x)^{10} (A+B x) (d+e x)^7 \, dx=\frac {(A b-a B) (b d-a e)^7 (a+b x)^{11}}{11 b^9}+\frac {(b d-a e)^6 (b B d+7 A b e-8 a B e) (a+b x)^{12}}{12 b^9}+\frac {7 e (b d-a e)^5 (b B d+3 A b e-4 a B e) (a+b x)^{13}}{13 b^9}+\frac {e^2 (b d-a e)^4 (3 b B d+5 A b e-8 a B e) (a+b x)^{14}}{2 b^9}+\frac {7 e^3 (b d-a e)^3 (b B d+A b e-2 a B e) (a+b x)^{15}}{3 b^9}+\frac {7 e^4 (b d-a e)^2 (5 b B d+3 A b e-8 a B e) (a+b x)^{16}}{16 b^9}+\frac {7 e^5 (b d-a e) (3 b B d+A b e-4 a B e) (a+b x)^{17}}{17 b^9}+\frac {e^6 (7 b B d+A b e-8 a B e) (a+b x)^{18}}{18 b^9}+\frac {B e^7 (a+b x)^{19}}{19 b^9} \] Output:
1/11*(A*b-B*a)*(-a*e+b*d)^7*(b*x+a)^11/b^9+1/12*(-a*e+b*d)^6*(7*A*b*e-8*B* a*e+B*b*d)*(b*x+a)^12/b^9+7/13*e*(-a*e+b*d)^5*(3*A*b*e-4*B*a*e+B*b*d)*(b*x +a)^13/b^9+1/2*e^2*(-a*e+b*d)^4*(5*A*b*e-8*B*a*e+3*B*b*d)*(b*x+a)^14/b^9+7 /3*e^3*(-a*e+b*d)^3*(A*b*e-2*B*a*e+B*b*d)*(b*x+a)^15/b^9+7/16*e^4*(-a*e+b* d)^2*(3*A*b*e-8*B*a*e+5*B*b*d)*(b*x+a)^16/b^9+7/17*e^5*(-a*e+b*d)*(A*b*e-4 *B*a*e+3*B*b*d)*(b*x+a)^17/b^9+1/18*e^6*(A*b*e-8*B*a*e+7*B*b*d)*(b*x+a)^18 /b^9+1/19*B*e^7*(b*x+a)^19/b^9
Leaf count is larger than twice the leaf count of optimal. \(2034\) vs. \(2(329)=658\).
Time = 0.47 (sec) , antiderivative size = 2034, normalized size of antiderivative = 6.18 \[ \int (a+b x)^{10} (A+B x) (d+e x)^7 \, dx=\text {Result too large to show} \] Input:
Integrate[(a + b*x)^10*(A + B*x)*(d + e*x)^7,x]
Output:
a^10*A*d^7*x + (a^9*d^6*(10*A*b*d + a*B*d + 7*a*A*e)*x^2)/2 + (a^8*d^5*(a* B*d*(10*b*d + 7*a*e) + A*(45*b^2*d^2 + 70*a*b*d*e + 21*a^2*e^2))*x^3)/3 + (a^7*d^4*(a*B*d*(45*b^2*d^2 + 70*a*b*d*e + 21*a^2*e^2) + 5*A*(24*b^3*d^3 + 63*a*b^2*d^2*e + 42*a^2*b*d*e^2 + 7*a^3*e^3))*x^4)/4 + a^6*d^3*(a*B*d*(24 *b^3*d^3 + 63*a*b^2*d^2*e + 42*a^2*b*d*e^2 + 7*a^3*e^3) + 7*A*(6*b^4*d^4 + 24*a*b^3*d^3*e + 27*a^2*b^2*d^2*e^2 + 10*a^3*b*d*e^3 + a^4*e^4))*x^5 + (7 *a^5*d^2*(5*a*B*d*(6*b^4*d^4 + 24*a*b^3*d^3*e + 27*a^2*b^2*d^2*e^2 + 10*a^ 3*b*d*e^3 + a^4*e^4) + A*(36*b^5*d^5 + 210*a*b^4*d^4*e + 360*a^2*b^3*d^3*e ^2 + 225*a^3*b^2*d^2*e^3 + 50*a^4*b*d*e^4 + 3*a^5*e^5))*x^6)/6 + a^4*d*(a* B*d*(36*b^5*d^5 + 210*a*b^4*d^4*e + 360*a^2*b^3*d^3*e^2 + 225*a^3*b^2*d^2* e^3 + 50*a^4*b*d*e^4 + 3*a^5*e^5) + A*(30*b^6*d^6 + 252*a*b^5*d^5*e + 630* a^2*b^4*d^4*e^2 + 600*a^3*b^3*d^3*e^3 + 225*a^4*b^2*d^2*e^4 + 30*a^5*b*d*e ^5 + a^6*e^6))*x^7 + (a^3*(7*a*B*d*(30*b^6*d^6 + 252*a*b^5*d^5*e + 630*a^2 *b^4*d^4*e^2 + 600*a^3*b^3*d^3*e^3 + 225*a^4*b^2*d^2*e^4 + 30*a^5*b*d*e^5 + a^6*e^6) + A*(120*b^7*d^7 + 1470*a*b^6*d^6*e + 5292*a^2*b^5*d^5*e^2 + 73 50*a^3*b^4*d^4*e^3 + 4200*a^4*b^3*d^3*e^4 + 945*a^5*b^2*d^2*e^5 + 70*a^6*b *d*e^6 + a^7*e^7))*x^8)/8 + (a^2*(a*B*(120*b^7*d^7 + 1470*a*b^6*d^6*e + 52 92*a^2*b^5*d^5*e^2 + 7350*a^3*b^4*d^4*e^3 + 4200*a^4*b^3*d^3*e^4 + 945*a^5 *b^2*d^2*e^5 + 70*a^6*b*d*e^6 + a^7*e^7) + 5*A*b*(9*b^7*d^7 + 168*a*b^6*d^ 6*e + 882*a^2*b^5*d^5*e^2 + 1764*a^3*b^4*d^4*e^3 + 1470*a^4*b^3*d^3*e^4...
Time = 2.09 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {86, 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.
\(\displaystyle \int (a+b x)^{10} (A+B x) (d+e x)^7 \, dx\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \int \left (\frac {e^6 (a+b x)^{17} (-8 a B e+A b e+7 b B d)}{b^8}+\frac {7 e^5 (a+b x)^{16} (b d-a e) (-4 a B e+A b e+3 b B d)}{b^8}+\frac {7 e^4 (a+b x)^{15} (b d-a e)^2 (-8 a B e+3 A b e+5 b B d)}{b^8}+\frac {35 e^3 (a+b x)^{14} (b d-a e)^3 (-2 a B e+A b e+b B d)}{b^8}+\frac {7 e^2 (a+b x)^{13} (b d-a e)^4 (-8 a B e+5 A b e+3 b B d)}{b^8}+\frac {7 e (a+b x)^{12} (b d-a e)^5 (-4 a B e+3 A b e+b B d)}{b^8}+\frac {(a+b x)^{11} (b d-a e)^6 (-8 a B e+7 A b e+b B d)}{b^8}+\frac {(a+b x)^{10} (A b-a B) (b d-a e)^7}{b^8}+\frac {B e^7 (a+b x)^{18}}{b^8}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {e^6 (a+b x)^{18} (-8 a B e+A b e+7 b B d)}{18 b^9}+\frac {7 e^5 (a+b x)^{17} (b d-a e) (-4 a B e+A b e+3 b B d)}{17 b^9}+\frac {7 e^4 (a+b x)^{16} (b d-a e)^2 (-8 a B e+3 A b e+5 b B d)}{16 b^9}+\frac {7 e^3 (a+b x)^{15} (b d-a e)^3 (-2 a B e+A b e+b B d)}{3 b^9}+\frac {e^2 (a+b x)^{14} (b d-a e)^4 (-8 a B e+5 A b e+3 b B d)}{2 b^9}+\frac {7 e (a+b x)^{13} (b d-a e)^5 (-4 a B e+3 A b e+b B d)}{13 b^9}+\frac {(a+b x)^{12} (b d-a e)^6 (-8 a B e+7 A b e+b B d)}{12 b^9}+\frac {(a+b x)^{11} (A b-a B) (b d-a e)^7}{11 b^9}+\frac {B e^7 (a+b x)^{19}}{19 b^9}\) |
Input:
Int[(a + b*x)^10*(A + B*x)*(d + e*x)^7,x]
Output:
((A*b - a*B)*(b*d - a*e)^7*(a + b*x)^11)/(11*b^9) + ((b*d - a*e)^6*(b*B*d + 7*A*b*e - 8*a*B*e)*(a + b*x)^12)/(12*b^9) + (7*e*(b*d - a*e)^5*(b*B*d + 3*A*b*e - 4*a*B*e)*(a + b*x)^13)/(13*b^9) + (e^2*(b*d - a*e)^4*(3*b*B*d + 5*A*b*e - 8*a*B*e)*(a + b*x)^14)/(2*b^9) + (7*e^3*(b*d - a*e)^3*(b*B*d + A *b*e - 2*a*B*e)*(a + b*x)^15)/(3*b^9) + (7*e^4*(b*d - a*e)^2*(5*b*B*d + 3* A*b*e - 8*a*B*e)*(a + b*x)^16)/(16*b^9) + (7*e^5*(b*d - a*e)*(3*b*B*d + A* b*e - 4*a*B*e)*(a + b*x)^17)/(17*b^9) + (e^6*(7*b*B*d + A*b*e - 8*a*B*e)*( a + b*x)^18)/(18*b^9) + (B*e^7*(a + b*x)^19)/(19*b^9)
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Leaf count of result is larger than twice the leaf count of optimal. \(2188\) vs. \(2(311)=622\).
Time = 0.25 (sec) , antiderivative size = 2189, normalized size of antiderivative = 6.65
method | result | size |
default | \(\text {Expression too large to display}\) | \(2189\) |
norman | \(\text {Expression too large to display}\) | \(2347\) |
gosper | \(\text {Expression too large to display}\) | \(2784\) |
risch | \(\text {Expression too large to display}\) | \(2784\) |
parallelrisch | \(\text {Expression too large to display}\) | \(2784\) |
orering | \(\text {Expression too large to display}\) | \(2784\) |
Input:
int((b*x+a)^10*(B*x+A)*(e*x+d)^7,x,method=_RETURNVERBOSE)
Output:
1/19*b^10*B*e^7*x^19+1/18*((A*b^10+10*B*a*b^9)*e^7+7*b^10*B*d*e^6)*x^18+1/ 17*((10*A*a*b^9+45*B*a^2*b^8)*e^7+7*(A*b^10+10*B*a*b^9)*d*e^6+21*b^10*B*d^ 2*e^5)*x^17+1/16*((45*A*a^2*b^8+120*B*a^3*b^7)*e^7+7*(10*A*a*b^9+45*B*a^2* b^8)*d*e^6+21*(A*b^10+10*B*a*b^9)*d^2*e^5+35*b^10*B*d^3*e^4)*x^16+1/15*((1 20*A*a^3*b^7+210*B*a^4*b^6)*e^7+7*(45*A*a^2*b^8+120*B*a^3*b^7)*d*e^6+21*(1 0*A*a*b^9+45*B*a^2*b^8)*d^2*e^5+35*(A*b^10+10*B*a*b^9)*d^3*e^4+35*b^10*B*d ^4*e^3)*x^15+1/14*((210*A*a^4*b^6+252*B*a^5*b^5)*e^7+7*(120*A*a^3*b^7+210* B*a^4*b^6)*d*e^6+21*(45*A*a^2*b^8+120*B*a^3*b^7)*d^2*e^5+35*(10*A*a*b^9+45 *B*a^2*b^8)*d^3*e^4+35*(A*b^10+10*B*a*b^9)*d^4*e^3+21*b^10*B*d^5*e^2)*x^14 +1/13*((252*A*a^5*b^5+210*B*a^6*b^4)*e^7+7*(210*A*a^4*b^6+252*B*a^5*b^5)*d *e^6+21*(120*A*a^3*b^7+210*B*a^4*b^6)*d^2*e^5+35*(45*A*a^2*b^8+120*B*a^3*b ^7)*d^3*e^4+35*(10*A*a*b^9+45*B*a^2*b^8)*d^4*e^3+21*(A*b^10+10*B*a*b^9)*d^ 5*e^2+7*b^10*B*d^6*e)*x^13+1/12*((210*A*a^6*b^4+120*B*a^7*b^3)*e^7+7*(252* A*a^5*b^5+210*B*a^6*b^4)*d*e^6+21*(210*A*a^4*b^6+252*B*a^5*b^5)*d^2*e^5+35 *(120*A*a^3*b^7+210*B*a^4*b^6)*d^3*e^4+35*(45*A*a^2*b^8+120*B*a^3*b^7)*d^4 *e^3+21*(10*A*a*b^9+45*B*a^2*b^8)*d^5*e^2+7*(A*b^10+10*B*a*b^9)*d^6*e+b^10 *B*d^7)*x^12+1/11*((120*A*a^7*b^3+45*B*a^8*b^2)*e^7+7*(210*A*a^6*b^4+120*B *a^7*b^3)*d*e^6+21*(252*A*a^5*b^5+210*B*a^6*b^4)*d^2*e^5+35*(210*A*a^4*b^6 +252*B*a^5*b^5)*d^3*e^4+35*(120*A*a^3*b^7+210*B*a^4*b^6)*d^4*e^3+21*(45*A* a^2*b^8+120*B*a^3*b^7)*d^5*e^2+7*(10*A*a*b^9+45*B*a^2*b^8)*d^6*e+(A*b^1...
Leaf count of result is larger than twice the leaf count of optimal. 2198 vs. \(2 (311) = 622\).
Time = 0.09 (sec) , antiderivative size = 2198, normalized size of antiderivative = 6.68 \[ \int (a+b x)^{10} (A+B x) (d+e x)^7 \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^10*(B*x+A)*(e*x+d)^7,x, algorithm="fricas")
Output:
1/19*B*b^10*e^7*x^19 + A*a^10*d^7*x + 1/18*(7*B*b^10*d*e^6 + (10*B*a*b^9 + A*b^10)*e^7)*x^18 + 1/17*(21*B*b^10*d^2*e^5 + 7*(10*B*a*b^9 + A*b^10)*d*e ^6 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*e^7)*x^17 + 1/16*(35*B*b^10*d^3*e^4 + 21* (10*B*a*b^9 + A*b^10)*d^2*e^5 + 35*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^6 + 15*(8 *B*a^3*b^7 + 3*A*a^2*b^8)*e^7)*x^16 + 1/3*(7*B*b^10*d^4*e^3 + 7*(10*B*a*b^ 9 + A*b^10)*d^3*e^4 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^5 + 21*(8*B*a^3*b ^7 + 3*A*a^2*b^8)*d*e^6 + 6*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^7)*x^15 + 1/2*(3 *B*b^10*d^5*e^2 + 5*(10*B*a*b^9 + A*b^10)*d^4*e^3 + 25*(9*B*a^2*b^8 + 2*A* a*b^9)*d^3*e^4 + 45*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^5 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^6 + 6*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^7)*x^14 + 7/13*(B*b ^10*d^6*e + 3*(10*B*a*b^9 + A*b^10)*d^5*e^2 + 25*(9*B*a^2*b^8 + 2*A*a*b^9) *d^4*e^3 + 75*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^4 + 90*(7*B*a^4*b^6 + 4*A* a^3*b^7)*d^2*e^5 + 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^6 + 6*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^7)*x^13 + 1/12*(B*b^10*d^7 + 7*(10*B*a*b^9 + A*b^10)*d^6*e + 105*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^2 + 525*(8*B*a^3*b^7 + 3*A*a^2*b^8) *d^4*e^3 + 1050*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^4 + 882*(6*B*a^5*b^5 + 5 *A*a^4*b^6)*d^2*e^5 + 294*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^6 + 30*(4*B*a^7* b^3 + 7*A*a^6*b^4)*e^7)*x^12 + 1/11*((10*B*a*b^9 + A*b^10)*d^7 + 35*(9*B*a ^2*b^8 + 2*A*a*b^9)*d^6*e + 315*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^2 + 1050 *(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^3 + 1470*(6*B*a^5*b^5 + 5*A*a^4*b^6)...
Leaf count of result is larger than twice the leaf count of optimal. 2824 vs. \(2 (335) = 670\).
Time = 0.25 (sec) , antiderivative size = 2824, normalized size of antiderivative = 8.58 \[ \int (a+b x)^{10} (A+B x) (d+e x)^7 \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)**10*(B*x+A)*(e*x+d)**7,x)
Output:
A*a**10*d**7*x + B*b**10*e**7*x**19/19 + x**18*(A*b**10*e**7/18 + 5*B*a*b* *9*e**7/9 + 7*B*b**10*d*e**6/18) + x**17*(10*A*a*b**9*e**7/17 + 7*A*b**10* d*e**6/17 + 45*B*a**2*b**8*e**7/17 + 70*B*a*b**9*d*e**6/17 + 21*B*b**10*d* *2*e**5/17) + x**16*(45*A*a**2*b**8*e**7/16 + 35*A*a*b**9*d*e**6/8 + 21*A* b**10*d**2*e**5/16 + 15*B*a**3*b**7*e**7/2 + 315*B*a**2*b**8*d*e**6/16 + 1 05*B*a*b**9*d**2*e**5/8 + 35*B*b**10*d**3*e**4/16) + x**15*(8*A*a**3*b**7* e**7 + 21*A*a**2*b**8*d*e**6 + 14*A*a*b**9*d**2*e**5 + 7*A*b**10*d**3*e**4 /3 + 14*B*a**4*b**6*e**7 + 56*B*a**3*b**7*d*e**6 + 63*B*a**2*b**8*d**2*e** 5 + 70*B*a*b**9*d**3*e**4/3 + 7*B*b**10*d**4*e**3/3) + x**14*(15*A*a**4*b* *6*e**7 + 60*A*a**3*b**7*d*e**6 + 135*A*a**2*b**8*d**2*e**5/2 + 25*A*a*b** 9*d**3*e**4 + 5*A*b**10*d**4*e**3/2 + 18*B*a**5*b**5*e**7 + 105*B*a**4*b** 6*d*e**6 + 180*B*a**3*b**7*d**2*e**5 + 225*B*a**2*b**8*d**3*e**4/2 + 25*B* a*b**9*d**4*e**3 + 3*B*b**10*d**5*e**2/2) + x**13*(252*A*a**5*b**5*e**7/13 + 1470*A*a**4*b**6*d*e**6/13 + 2520*A*a**3*b**7*d**2*e**5/13 + 1575*A*a** 2*b**8*d**3*e**4/13 + 350*A*a*b**9*d**4*e**3/13 + 21*A*b**10*d**5*e**2/13 + 210*B*a**6*b**4*e**7/13 + 1764*B*a**5*b**5*d*e**6/13 + 4410*B*a**4*b**6* d**2*e**5/13 + 4200*B*a**3*b**7*d**3*e**4/13 + 1575*B*a**2*b**8*d**4*e**3/ 13 + 210*B*a*b**9*d**5*e**2/13 + 7*B*b**10*d**6*e/13) + x**12*(35*A*a**6*b **4*e**7/2 + 147*A*a**5*b**5*d*e**6 + 735*A*a**4*b**6*d**2*e**5/2 + 350*A* a**3*b**7*d**3*e**4 + 525*A*a**2*b**8*d**4*e**3/4 + 35*A*a*b**9*d**5*e*...
Leaf count of result is larger than twice the leaf count of optimal. 2198 vs. \(2 (311) = 622\).
Time = 0.05 (sec) , antiderivative size = 2198, normalized size of antiderivative = 6.68 \[ \int (a+b x)^{10} (A+B x) (d+e x)^7 \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^10*(B*x+A)*(e*x+d)^7,x, algorithm="maxima")
Output:
1/19*B*b^10*e^7*x^19 + A*a^10*d^7*x + 1/18*(7*B*b^10*d*e^6 + (10*B*a*b^9 + A*b^10)*e^7)*x^18 + 1/17*(21*B*b^10*d^2*e^5 + 7*(10*B*a*b^9 + A*b^10)*d*e ^6 + 5*(9*B*a^2*b^8 + 2*A*a*b^9)*e^7)*x^17 + 1/16*(35*B*b^10*d^3*e^4 + 21* (10*B*a*b^9 + A*b^10)*d^2*e^5 + 35*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^6 + 15*(8 *B*a^3*b^7 + 3*A*a^2*b^8)*e^7)*x^16 + 1/3*(7*B*b^10*d^4*e^3 + 7*(10*B*a*b^ 9 + A*b^10)*d^3*e^4 + 21*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^5 + 21*(8*B*a^3*b ^7 + 3*A*a^2*b^8)*d*e^6 + 6*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^7)*x^15 + 1/2*(3 *B*b^10*d^5*e^2 + 5*(10*B*a*b^9 + A*b^10)*d^4*e^3 + 25*(9*B*a^2*b^8 + 2*A* a*b^9)*d^3*e^4 + 45*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^5 + 30*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^6 + 6*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^7)*x^14 + 7/13*(B*b ^10*d^6*e + 3*(10*B*a*b^9 + A*b^10)*d^5*e^2 + 25*(9*B*a^2*b^8 + 2*A*a*b^9) *d^4*e^3 + 75*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^4 + 90*(7*B*a^4*b^6 + 4*A* a^3*b^7)*d^2*e^5 + 42*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^6 + 6*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^7)*x^13 + 1/12*(B*b^10*d^7 + 7*(10*B*a*b^9 + A*b^10)*d^6*e + 105*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^2 + 525*(8*B*a^3*b^7 + 3*A*a^2*b^8) *d^4*e^3 + 1050*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^4 + 882*(6*B*a^5*b^5 + 5 *A*a^4*b^6)*d^2*e^5 + 294*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^6 + 30*(4*B*a^7* b^3 + 7*A*a^6*b^4)*e^7)*x^12 + 1/11*((10*B*a*b^9 + A*b^10)*d^7 + 35*(9*B*a ^2*b^8 + 2*A*a*b^9)*d^6*e + 315*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^5*e^2 + 1050 *(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^3 + 1470*(6*B*a^5*b^5 + 5*A*a^4*b^6)...
Leaf count of result is larger than twice the leaf count of optimal. 2783 vs. \(2 (311) = 622\).
Time = 0.13 (sec) , antiderivative size = 2783, normalized size of antiderivative = 8.46 \[ \int (a+b x)^{10} (A+B x) (d+e x)^7 \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^10*(B*x+A)*(e*x+d)^7,x, algorithm="giac")
Output:
1/19*B*b^10*e^7*x^19 + 7/18*B*b^10*d*e^6*x^18 + 5/9*B*a*b^9*e^7*x^18 + 1/1 8*A*b^10*e^7*x^18 + 21/17*B*b^10*d^2*e^5*x^17 + 70/17*B*a*b^9*d*e^6*x^17 + 7/17*A*b^10*d*e^6*x^17 + 45/17*B*a^2*b^8*e^7*x^17 + 10/17*A*a*b^9*e^7*x^1 7 + 35/16*B*b^10*d^3*e^4*x^16 + 105/8*B*a*b^9*d^2*e^5*x^16 + 21/16*A*b^10* d^2*e^5*x^16 + 315/16*B*a^2*b^8*d*e^6*x^16 + 35/8*A*a*b^9*d*e^6*x^16 + 15/ 2*B*a^3*b^7*e^7*x^16 + 45/16*A*a^2*b^8*e^7*x^16 + 7/3*B*b^10*d^4*e^3*x^15 + 70/3*B*a*b^9*d^3*e^4*x^15 + 7/3*A*b^10*d^3*e^4*x^15 + 63*B*a^2*b^8*d^2*e ^5*x^15 + 14*A*a*b^9*d^2*e^5*x^15 + 56*B*a^3*b^7*d*e^6*x^15 + 21*A*a^2*b^8 *d*e^6*x^15 + 14*B*a^4*b^6*e^7*x^15 + 8*A*a^3*b^7*e^7*x^15 + 3/2*B*b^10*d^ 5*e^2*x^14 + 25*B*a*b^9*d^4*e^3*x^14 + 5/2*A*b^10*d^4*e^3*x^14 + 225/2*B*a ^2*b^8*d^3*e^4*x^14 + 25*A*a*b^9*d^3*e^4*x^14 + 180*B*a^3*b^7*d^2*e^5*x^14 + 135/2*A*a^2*b^8*d^2*e^5*x^14 + 105*B*a^4*b^6*d*e^6*x^14 + 60*A*a^3*b^7* d*e^6*x^14 + 18*B*a^5*b^5*e^7*x^14 + 15*A*a^4*b^6*e^7*x^14 + 7/13*B*b^10*d ^6*e*x^13 + 210/13*B*a*b^9*d^5*e^2*x^13 + 21/13*A*b^10*d^5*e^2*x^13 + 1575 /13*B*a^2*b^8*d^4*e^3*x^13 + 350/13*A*a*b^9*d^4*e^3*x^13 + 4200/13*B*a^3*b ^7*d^3*e^4*x^13 + 1575/13*A*a^2*b^8*d^3*e^4*x^13 + 4410/13*B*a^4*b^6*d^2*e ^5*x^13 + 2520/13*A*a^3*b^7*d^2*e^5*x^13 + 1764/13*B*a^5*b^5*d*e^6*x^13 + 1470/13*A*a^4*b^6*d*e^6*x^13 + 210/13*B*a^6*b^4*e^7*x^13 + 252/13*A*a^5*b^ 5*e^7*x^13 + 1/12*B*b^10*d^7*x^12 + 35/6*B*a*b^9*d^6*e*x^12 + 7/12*A*b^10* d^6*e*x^12 + 315/4*B*a^2*b^8*d^5*e^2*x^12 + 35/2*A*a*b^9*d^5*e^2*x^12 +...
Time = 1.42 (sec) , antiderivative size = 2316, normalized size of antiderivative = 7.04 \[ \int (a+b x)^{10} (A+B x) (d+e x)^7 \, dx=\text {Too large to display} \] Input:
int((A + B*x)*(a + b*x)^10*(d + e*x)^7,x)
Output:
x^5*(42*A*a^6*b^4*d^7 + 24*B*a^7*b^3*d^7 + 7*A*a^10*d^3*e^4 + 7*B*a^10*d^4 *e^3 + 168*A*a^7*b^3*d^6*e + 70*A*a^9*b*d^4*e^3 + 63*B*a^8*b^2*d^6*e + 42* B*a^9*b*d^5*e^2 + 189*A*a^8*b^2*d^5*e^2) + x^15*(8*A*a^3*b^7*e^7 + 14*B*a^ 4*b^6*e^7 + (7*A*b^10*d^3*e^4)/3 + (7*B*b^10*d^4*e^3)/3 + 14*A*a*b^9*d^2*e ^5 + 21*A*a^2*b^8*d*e^6 + (70*B*a*b^9*d^3*e^4)/3 + 56*B*a^3*b^7*d*e^6 + 63 *B*a^2*b^8*d^2*e^5) + x^9*((B*a^10*e^7)/9 + (10*A*a^9*b*e^7)/9 + 5*A*a^2*b ^8*d^7 + (40*B*a^3*b^7*d^7)/3 + (280*A*a^3*b^7*d^6*e)/3 + 35*A*a^8*b^2*d*e ^6 + (490*B*a^4*b^6*d^6*e)/3 + 490*A*a^4*b^6*d^5*e^2 + 980*A*a^5*b^5*d^4*e ^3 + (2450*A*a^6*b^4*d^3*e^4)/3 + 280*A*a^7*b^3*d^2*e^5 + 588*B*a^5*b^5*d^ 5*e^2 + (2450*B*a^6*b^4*d^4*e^3)/3 + (1400*B*a^7*b^3*d^3*e^4)/3 + 105*B*a^ 8*b^2*d^2*e^5 + (70*B*a^9*b*d*e^6)/9) + x^11*((A*b^10*d^7)/11 + (10*B*a*b^ 9*d^7)/11 + (120*A*a^7*b^3*e^7)/11 + (45*B*a^8*b^2*e^7)/11 + (1470*A*a^6*b ^4*d*e^6)/11 + (315*B*a^2*b^8*d^6*e)/11 + (840*B*a^7*b^3*d*e^6)/11 + (945* A*a^2*b^8*d^5*e^2)/11 + (4200*A*a^3*b^7*d^4*e^3)/11 + (7350*A*a^4*b^6*d^3* e^4)/11 + (5292*A*a^5*b^5*d^2*e^5)/11 + (2520*B*a^3*b^7*d^5*e^2)/11 + (735 0*B*a^4*b^6*d^4*e^3)/11 + (8820*B*a^5*b^5*d^3*e^4)/11 + (4410*B*a^6*b^4*d^ 2*e^5)/11 + (70*A*a*b^9*d^6*e)/11) + x^7*(A*a^10*d*e^6 + 30*A*a^4*b^6*d^7 + 36*B*a^5*b^5*d^7 + 3*B*a^10*d^2*e^5 + 252*A*a^5*b^5*d^6*e + 30*A*a^9*b*d ^2*e^5 + 210*B*a^6*b^4*d^6*e + 50*B*a^9*b*d^3*e^4 + 630*A*a^6*b^4*d^5*e^2 + 600*A*a^7*b^3*d^4*e^3 + 225*A*a^8*b^2*d^3*e^4 + 360*B*a^7*b^3*d^5*e^2...
Time = 0.16 (sec) , antiderivative size = 1429, normalized size of antiderivative = 4.34 \[ \int (a+b x)^{10} (A+B x) (d+e x)^7 \, dx =\text {Too large to display} \] Input:
int((b*x+a)^10*(B*x+A)*(e*x+d)^7,x)
Output:
(x*(604656*a**11*d**7 + 2116296*a**11*d**6*e*x + 4232592*a**11*d**5*e**2*x **2 + 5290740*a**11*d**4*e**3*x**3 + 4232592*a**11*d**3*e**4*x**4 + 211629 6*a**11*d**2*e**5*x**5 + 604656*a**11*d*e**6*x**6 + 75582*a**11*e**7*x**7 + 3325608*a**10*b*d**7*x + 15519504*a**10*b*d**6*e*x**2 + 34918884*a**10*b *d**5*e**2*x**3 + 46558512*a**10*b*d**4*e**3*x**4 + 38798760*a**10*b*d**3* e**4*x**5 + 19953648*a**10*b*d**2*e**5*x**6 + 5819814*a**10*b*d*e**6*x**7 + 739024*a**10*b*e**7*x**8 + 11085360*a**9*b**2*d**7*x**2 + 58198140*a**9* b**2*d**6*e*x**3 + 139675536*a**9*b**2*d**5*e**2*x**4 + 193993800*a**9*b** 2*d**4*e**3*x**5 + 166280400*a**9*b**2*d**3*e**4*x**6 + 87297210*a**9*b**2 *d**2*e**5*x**7 + 25865840*a**9*b**2*d*e**6*x**8 + 3325608*a**9*b**2*e**7* x**9 + 24942060*a**8*b**3*d**7*x**3 + 139675536*a**8*b**3*d**6*e*x**4 + 34 9188840*a**8*b**3*d**5*e**2*x**5 + 498841200*a**8*b**3*d**4*e**3*x**6 + 43 6486050*a**8*b**3*d**3*e**4*x**7 + 232792560*a**8*b**3*d**2*e**5*x**8 + 69 837768*a**8*b**3*d*e**6*x**9 + 9069840*a**8*b**3*e**7*x**10 + 39907296*a** 7*b**4*d**7*x**4 + 232792560*a**7*b**4*d**6*e*x**5 + 598609440*a**7*b**4*d **5*e**2*x**6 + 872972100*a**7*b**4*d**4*e**3*x**7 + 775975200*a**7*b**4*d **3*e**4*x**8 + 419026608*a**7*b**4*d**2*e**5*x**9 + 126977760*a**7*b**4*d *e**6*x**10 + 16628040*a**7*b**4*e**7*x**11 + 46558512*a**6*b**5*d**7*x**5 + 279351072*a**6*b**5*d**6*e*x**6 + 733296564*a**6*b**5*d**5*e**2*x**7 + 1086365280*a**6*b**5*d**4*e**3*x**8 + 977728752*a**6*b**5*d**3*e**4*x**...