Integrand size = 20, antiderivative size = 464 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{22}} \, dx=\frac {(b d-a e)^{10} (B d-A e)}{21 e^{12} (d+e x)^{21}}-\frac {(b d-a e)^9 (11 b B d-10 A b e-a B e)}{20 e^{12} (d+e x)^{20}}+\frac {5 b (b d-a e)^8 (11 b B d-9 A b e-2 a B e)}{19 e^{12} (d+e x)^{19}}-\frac {5 b^2 (b d-a e)^7 (11 b B d-8 A b e-3 a B e)}{6 e^{12} (d+e x)^{18}}+\frac {30 b^3 (b d-a e)^6 (11 b B d-7 A b e-4 a B e)}{17 e^{12} (d+e x)^{17}}-\frac {21 b^4 (b d-a e)^5 (11 b B d-6 A b e-5 a B e)}{8 e^{12} (d+e x)^{16}}+\frac {14 b^5 (b d-a e)^4 (11 b B d-5 A b e-6 a B e)}{5 e^{12} (d+e x)^{15}}-\frac {15 b^6 (b d-a e)^3 (11 b B d-4 A b e-7 a B e)}{7 e^{12} (d+e x)^{14}}+\frac {15 b^7 (b d-a e)^2 (11 b B d-3 A b e-8 a B e)}{13 e^{12} (d+e x)^{13}}-\frac {5 b^8 (b d-a e) (11 b B d-2 A b e-9 a B e)}{12 e^{12} (d+e x)^{12}}+\frac {b^9 (11 b B d-A b e-10 a B e)}{11 e^{12} (d+e x)^{11}}-\frac {b^{10} B}{10 e^{12} (d+e x)^{10}} \] Output:
1/21*(-a*e+b*d)^10*(-A*e+B*d)/e^12/(e*x+d)^21-1/20*(-a*e+b*d)^9*(-10*A*b*e -B*a*e+11*B*b*d)/e^12/(e*x+d)^20+5/19*b*(-a*e+b*d)^8*(-9*A*b*e-2*B*a*e+11* B*b*d)/e^12/(e*x+d)^19-5/6*b^2*(-a*e+b*d)^7*(-8*A*b*e-3*B*a*e+11*B*b*d)/e^ 12/(e*x+d)^18+30/17*b^3*(-a*e+b*d)^6*(-7*A*b*e-4*B*a*e+11*B*b*d)/e^12/(e*x +d)^17-21/8*b^4*(-a*e+b*d)^5*(-6*A*b*e-5*B*a*e+11*B*b*d)/e^12/(e*x+d)^16+1 4/5*b^5*(-a*e+b*d)^4*(-5*A*b*e-6*B*a*e+11*B*b*d)/e^12/(e*x+d)^15-15/7*b^6* (-a*e+b*d)^3*(-4*A*b*e-7*B*a*e+11*B*b*d)/e^12/(e*x+d)^14+15/13*b^7*(-a*e+b *d)^2*(-3*A*b*e-8*B*a*e+11*B*b*d)/e^12/(e*x+d)^13-5/12*b^8*(-a*e+b*d)*(-2* A*b*e-9*B*a*e+11*B*b*d)/e^12/(e*x+d)^12+1/11*b^9*(-A*b*e-10*B*a*e+11*B*b*d )/e^12/(e*x+d)^11-1/10*b^10*B/e^12/(e*x+d)^10
Leaf count is larger than twice the leaf count of optimal. \(1431\) vs. \(2(464)=928\).
Time = 0.54 (sec) , antiderivative size = 1431, normalized size of antiderivative = 3.08 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{22}} \, dx =\text {Too large to display} \] Input:
Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^22,x]
Output:
-1/38798760*(92378*a^10*e^10*(20*A*e + B*(d + 21*e*x)) + 48620*a^9*b*e^9*( 19*A*e*(d + 21*e*x) + 2*B*(d^2 + 21*d*e*x + 210*e^2*x^2)) + 72930*a^8*b^2* e^8*(6*A*e*(d^2 + 21*d*e*x + 210*e^2*x^2) + B*(d^3 + 21*d^2*e*x + 210*d*e^ 2*x^2 + 1330*e^3*x^3)) + 11440*a^7*b^3*e^7*(17*A*e*(d^3 + 21*d^2*e*x + 210 *d*e^2*x^2 + 1330*e^3*x^3) + 4*B*(d^4 + 21*d^3*e*x + 210*d^2*e^2*x^2 + 133 0*d*e^3*x^3 + 5985*e^4*x^4)) + 5005*a^6*b^4*e^6*(16*A*e*(d^4 + 21*d^3*e*x + 210*d^2*e^2*x^2 + 1330*d*e^3*x^3 + 5985*e^4*x^4) + 5*B*(d^5 + 21*d^4*e*x + 210*d^3*e^2*x^2 + 1330*d^2*e^3*x^3 + 5985*d*e^4*x^4 + 20349*e^5*x^5)) + 6006*a^5*b^5*e^5*(5*A*e*(d^5 + 21*d^4*e*x + 210*d^3*e^2*x^2 + 1330*d^2*e^ 3*x^3 + 5985*d*e^4*x^4 + 20349*e^5*x^5) + 2*B*(d^6 + 21*d^5*e*x + 210*d^4* e^2*x^2 + 1330*d^3*e^3*x^3 + 5985*d^2*e^4*x^4 + 20349*d*e^5*x^5 + 54264*e^ 6*x^6)) + 5005*a^4*b^6*e^4*(2*A*e*(d^6 + 21*d^5*e*x + 210*d^4*e^2*x^2 + 13 30*d^3*e^3*x^3 + 5985*d^2*e^4*x^4 + 20349*d*e^5*x^5 + 54264*e^6*x^6) + B*( d^7 + 21*d^6*e*x + 210*d^5*e^2*x^2 + 1330*d^4*e^3*x^3 + 5985*d^3*e^4*x^4 + 20349*d^2*e^5*x^5 + 54264*d*e^6*x^6 + 116280*e^7*x^7)) + 220*a^3*b^7*e^3* (13*A*e*(d^7 + 21*d^6*e*x + 210*d^5*e^2*x^2 + 1330*d^4*e^3*x^3 + 5985*d^3* e^4*x^4 + 20349*d^2*e^5*x^5 + 54264*d*e^6*x^6 + 116280*e^7*x^7) + 8*B*(d^8 + 21*d^7*e*x + 210*d^6*e^2*x^2 + 1330*d^5*e^3*x^3 + 5985*d^4*e^4*x^4 + 20 349*d^3*e^5*x^5 + 54264*d^2*e^6*x^6 + 116280*d*e^7*x^7 + 203490*e^8*x^8)) + 165*a^2*b^8*e^2*(4*A*e*(d^8 + 21*d^7*e*x + 210*d^6*e^2*x^2 + 1330*d^5...
Time = 1.21 (sec) , antiderivative size = 464, 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.
\(\displaystyle \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{22}} \, dx\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \int \left (\frac {b^9 (10 a B e+A b e-11 b B d)}{e^{11} (d+e x)^{12}}-\frac {5 b^8 (b d-a e) (9 a B e+2 A b e-11 b B d)}{e^{11} (d+e x)^{13}}+\frac {15 b^7 (b d-a e)^2 (8 a B e+3 A b e-11 b B d)}{e^{11} (d+e x)^{14}}-\frac {30 b^6 (b d-a e)^3 (7 a B e+4 A b e-11 b B d)}{e^{11} (d+e x)^{15}}+\frac {42 b^5 (b d-a e)^4 (6 a B e+5 A b e-11 b B d)}{e^{11} (d+e x)^{16}}-\frac {42 b^4 (b d-a e)^5 (5 a B e+6 A b e-11 b B d)}{e^{11} (d+e x)^{17}}+\frac {30 b^3 (b d-a e)^6 (4 a B e+7 A b e-11 b B d)}{e^{11} (d+e x)^{18}}-\frac {15 b^2 (b d-a e)^7 (3 a B e+8 A b e-11 b B d)}{e^{11} (d+e x)^{19}}+\frac {5 b (b d-a e)^8 (2 a B e+9 A b e-11 b B d)}{e^{11} (d+e x)^{20}}+\frac {(a e-b d)^9 (a B e+10 A b e-11 b B d)}{e^{11} (d+e x)^{21}}+\frac {(a e-b d)^{10} (A e-B d)}{e^{11} (d+e x)^{22}}+\frac {b^{10} B}{e^{11} (d+e x)^{11}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b^9 (-10 a B e-A b e+11 b B d)}{11 e^{12} (d+e x)^{11}}-\frac {5 b^8 (b d-a e) (-9 a B e-2 A b e+11 b B d)}{12 e^{12} (d+e x)^{12}}+\frac {15 b^7 (b d-a e)^2 (-8 a B e-3 A b e+11 b B d)}{13 e^{12} (d+e x)^{13}}-\frac {15 b^6 (b d-a e)^3 (-7 a B e-4 A b e+11 b B d)}{7 e^{12} (d+e x)^{14}}+\frac {14 b^5 (b d-a e)^4 (-6 a B e-5 A b e+11 b B d)}{5 e^{12} (d+e x)^{15}}-\frac {21 b^4 (b d-a e)^5 (-5 a B e-6 A b e+11 b B d)}{8 e^{12} (d+e x)^{16}}+\frac {30 b^3 (b d-a e)^6 (-4 a B e-7 A b e+11 b B d)}{17 e^{12} (d+e x)^{17}}-\frac {5 b^2 (b d-a e)^7 (-3 a B e-8 A b e+11 b B d)}{6 e^{12} (d+e x)^{18}}+\frac {5 b (b d-a e)^8 (-2 a B e-9 A b e+11 b B d)}{19 e^{12} (d+e x)^{19}}-\frac {(b d-a e)^9 (-a B e-10 A b e+11 b B d)}{20 e^{12} (d+e x)^{20}}+\frac {(b d-a e)^{10} (B d-A e)}{21 e^{12} (d+e x)^{21}}-\frac {b^{10} B}{10 e^{12} (d+e x)^{10}}\) |
Input:
Int[((a + b*x)^10*(A + B*x))/(d + e*x)^22,x]
Output:
((b*d - a*e)^10*(B*d - A*e))/(21*e^12*(d + e*x)^21) - ((b*d - a*e)^9*(11*b *B*d - 10*A*b*e - a*B*e))/(20*e^12*(d + e*x)^20) + (5*b*(b*d - a*e)^8*(11* b*B*d - 9*A*b*e - 2*a*B*e))/(19*e^12*(d + e*x)^19) - (5*b^2*(b*d - a*e)^7* (11*b*B*d - 8*A*b*e - 3*a*B*e))/(6*e^12*(d + e*x)^18) + (30*b^3*(b*d - a*e )^6*(11*b*B*d - 7*A*b*e - 4*a*B*e))/(17*e^12*(d + e*x)^17) - (21*b^4*(b*d - a*e)^5*(11*b*B*d - 6*A*b*e - 5*a*B*e))/(8*e^12*(d + e*x)^16) + (14*b^5*( b*d - a*e)^4*(11*b*B*d - 5*A*b*e - 6*a*B*e))/(5*e^12*(d + e*x)^15) - (15*b ^6*(b*d - a*e)^3*(11*b*B*d - 4*A*b*e - 7*a*B*e))/(7*e^12*(d + e*x)^14) + ( 15*b^7*(b*d - a*e)^2*(11*b*B*d - 3*A*b*e - 8*a*B*e))/(13*e^12*(d + e*x)^13 ) - (5*b^8*(b*d - a*e)*(11*b*B*d - 2*A*b*e - 9*a*B*e))/(12*e^12*(d + e*x)^ 12) + (b^9*(11*b*B*d - A*b*e - 10*a*B*e))/(11*e^12*(d + e*x)^11) - (b^10*B )/(10*e^12*(d + e*x)^10)
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. \(1900\) vs. \(2(440)=880\).
Time = 0.38 (sec) , antiderivative size = 1901, normalized size of antiderivative = 4.10
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1901\) |
default | \(\text {Expression too large to display}\) | \(1942\) |
norman | \(\text {Expression too large to display}\) | \(2014\) |
gosper | \(\text {Expression too large to display}\) | \(2233\) |
orering | \(\text {Expression too large to display}\) | \(2233\) |
parallelrisch | \(\text {Expression too large to display}\) | \(2242\) |
Input:
int((b*x+a)^10*(B*x+A)/(e*x+d)^22,x,method=_RETURNVERBOSE)
Output:
(-1/38798760/e^12*(1847560*A*a^10*e^11+923780*A*a^9*b*d*e^10+437580*A*a^8* b^2*d^2*e^9+194480*A*a^7*b^3*d^3*e^8+80080*A*a^6*b^4*d^4*e^7+30030*A*a^5*b ^5*d^5*e^6+10010*A*a^4*b^6*d^6*e^5+2860*A*a^3*b^7*d^7*e^4+660*A*a^2*b^8*d^ 8*e^3+110*A*a*b^9*d^9*e^2+10*A*b^10*d^10*e+92378*B*a^10*d*e^10+97240*B*a^9 *b*d^2*e^9+72930*B*a^8*b^2*d^3*e^8+45760*B*a^7*b^3*d^4*e^7+25025*B*a^6*b^4 *d^5*e^6+12012*B*a^5*b^5*d^6*e^5+5005*B*a^4*b^6*d^7*e^4+1760*B*a^3*b^7*d^8 *e^3+495*B*a^2*b^8*d^9*e^2+100*B*a*b^9*d^10*e+11*B*b^10*d^11)-1/1847560/e^ 11*(923780*A*a^9*b*e^10+437580*A*a^8*b^2*d*e^9+194480*A*a^7*b^3*d^2*e^8+80 080*A*a^6*b^4*d^3*e^7+30030*A*a^5*b^5*d^4*e^6+10010*A*a^4*b^6*d^5*e^5+2860 *A*a^3*b^7*d^6*e^4+660*A*a^2*b^8*d^7*e^3+110*A*a*b^9*d^8*e^2+10*A*b^10*d^9 *e+92378*B*a^10*e^10+97240*B*a^9*b*d*e^9+72930*B*a^8*b^2*d^2*e^8+45760*B*a ^7*b^3*d^3*e^7+25025*B*a^6*b^4*d^4*e^6+12012*B*a^5*b^5*d^5*e^5+5005*B*a^4* b^6*d^6*e^4+1760*B*a^3*b^7*d^7*e^3+495*B*a^2*b^8*d^8*e^2+100*B*a*b^9*d^9*e +11*B*b^10*d^10)*x-1/184756*b/e^10*(437580*A*a^8*b*e^9+194480*A*a^7*b^2*d* e^8+80080*A*a^6*b^3*d^2*e^7+30030*A*a^5*b^4*d^3*e^6+10010*A*a^4*b^5*d^4*e^ 5+2860*A*a^3*b^6*d^5*e^4+660*A*a^2*b^7*d^6*e^3+110*A*a*b^8*d^7*e^2+10*A*b^ 9*d^8*e+97240*B*a^9*e^9+72930*B*a^8*b*d*e^8+45760*B*a^7*b^2*d^2*e^7+25025* B*a^6*b^3*d^3*e^6+12012*B*a^5*b^4*d^4*e^5+5005*B*a^4*b^5*d^5*e^4+1760*B*a^ 3*b^6*d^6*e^3+495*B*a^2*b^7*d^7*e^2+100*B*a*b^8*d^8*e+11*B*b^9*d^9)*x^2-1/ 29172*b^2/e^9*(194480*A*a^7*b*e^8+80080*A*a^6*b^2*d*e^7+30030*A*a^5*b^3...
Leaf count of result is larger than twice the leaf count of optimal. 2039 vs. \(2 (440) = 880\).
Time = 0.26 (sec) , antiderivative size = 2039, normalized size of antiderivative = 4.39 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{22}} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^10*(B*x+A)/(e*x+d)^22,x, algorithm="fricas")
Output:
-1/38798760*(3879876*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 1847560*A*a^10*e^ 11 + 10*(10*B*a*b^9 + A*b^10)*d^10*e + 55*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^ 2 + 220*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 715*(7*B*a^4*b^6 + 4*A*a^3*b ^7)*d^7*e^4 + 2002*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 5005*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 11440*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 2431 0*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + 48620*(2*B*a^9*b + 9*A*a^8*b^2)*d^ 2*e^9 + 92378*(B*a^10 + 10*A*a^9*b)*d*e^10 + 352716*(11*B*b^10*d*e^10 + 10 *(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 293930*(11*B*b^10*d^2*e^9 + 10*(10*B*a *b^9 + A*b^10)*d*e^10 + 55*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 203490*(1 1*B*b^10*d^3*e^8 + 10*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 55*(9*B*a^2*b^8 + 2* A*a*b^9)*d*e^10 + 220*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 116280*(11*B *b^10*d^4*e^7 + 10*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 55*(9*B*a^2*b^8 + 2*A*a *b^9)*d^2*e^9 + 220*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + 715*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 54264*(11*B*b^10*d^5*e^6 + 10*(10*B*a*b^9 + A*b ^10)*d^4*e^7 + 55*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 + 220*(8*B*a^3*b^7 + 3 *A*a^2*b^8)*d^2*e^9 + 715*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 + 2002*(6*B*a ^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 20349*(11*B*b^10*d^6*e^5 + 10*(10*B*a*b^ 9 + A*b^10)*d^5*e^6 + 55*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 + 220*(8*B*a^3* b^7 + 3*A*a^2*b^8)*d^3*e^8 + 715*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + 200 2*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 5005*(5*B*a^6*b^4 + 6*A*a^5*b^5)...
Timed out. \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{22}} \, dx=\text {Timed out} \] Input:
integrate((b*x+a)**10*(B*x+A)/(e*x+d)**22,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 2039 vs. \(2 (440) = 880\).
Time = 0.16 (sec) , antiderivative size = 2039, normalized size of antiderivative = 4.39 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{22}} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^10*(B*x+A)/(e*x+d)^22,x, algorithm="maxima")
Output:
-1/38798760*(3879876*B*b^10*e^11*x^11 + 11*B*b^10*d^11 + 1847560*A*a^10*e^ 11 + 10*(10*B*a*b^9 + A*b^10)*d^10*e + 55*(9*B*a^2*b^8 + 2*A*a*b^9)*d^9*e^ 2 + 220*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 + 715*(7*B*a^4*b^6 + 4*A*a^3*b ^7)*d^7*e^4 + 2002*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^6*e^5 + 5005*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 + 11440*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e^7 + 2431 0*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 + 48620*(2*B*a^9*b + 9*A*a^8*b^2)*d^ 2*e^9 + 92378*(B*a^10 + 10*A*a^9*b)*d*e^10 + 352716*(11*B*b^10*d*e^10 + 10 *(10*B*a*b^9 + A*b^10)*e^11)*x^10 + 293930*(11*B*b^10*d^2*e^9 + 10*(10*B*a *b^9 + A*b^10)*d*e^10 + 55*(9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 203490*(1 1*B*b^10*d^3*e^8 + 10*(10*B*a*b^9 + A*b^10)*d^2*e^9 + 55*(9*B*a^2*b^8 + 2* A*a*b^9)*d*e^10 + 220*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 116280*(11*B *b^10*d^4*e^7 + 10*(10*B*a*b^9 + A*b^10)*d^3*e^8 + 55*(9*B*a^2*b^8 + 2*A*a *b^9)*d^2*e^9 + 220*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d*e^10 + 715*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 54264*(11*B*b^10*d^5*e^6 + 10*(10*B*a*b^9 + A*b ^10)*d^4*e^7 + 55*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 + 220*(8*B*a^3*b^7 + 3 *A*a^2*b^8)*d^2*e^9 + 715*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d*e^10 + 2002*(6*B*a ^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 20349*(11*B*b^10*d^6*e^5 + 10*(10*B*a*b^ 9 + A*b^10)*d^5*e^6 + 55*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 + 220*(8*B*a^3* b^7 + 3*A*a^2*b^8)*d^3*e^8 + 715*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^2*e^9 + 200 2*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 + 5005*(5*B*a^6*b^4 + 6*A*a^5*b^5)...
Leaf count of result is larger than twice the leaf count of optimal. 2232 vs. \(2 (440) = 880\).
Time = 0.13 (sec) , antiderivative size = 2232, normalized size of antiderivative = 4.81 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{22}} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)^10*(B*x+A)/(e*x+d)^22,x, algorithm="giac")
Output:
-1/38798760*(3879876*B*b^10*e^11*x^11 + 3879876*B*b^10*d*e^10*x^10 + 35271 600*B*a*b^9*e^11*x^10 + 3527160*A*b^10*e^11*x^10 + 3233230*B*b^10*d^2*e^9* x^9 + 29393000*B*a*b^9*d*e^10*x^9 + 2939300*A*b^10*d*e^10*x^9 + 145495350* B*a^2*b^8*e^11*x^9 + 32332300*A*a*b^9*e^11*x^9 + 2238390*B*b^10*d^3*e^8*x^ 8 + 20349000*B*a*b^9*d^2*e^9*x^8 + 2034900*A*b^10*d^2*e^9*x^8 + 100727550* B*a^2*b^8*d*e^10*x^8 + 22383900*A*a*b^9*d*e^10*x^8 + 358142400*B*a^3*b^7*e ^11*x^8 + 134303400*A*a^2*b^8*e^11*x^8 + 1279080*B*b^10*d^4*e^7*x^7 + 1162 8000*B*a*b^9*d^3*e^8*x^7 + 1162800*A*b^10*d^3*e^8*x^7 + 57558600*B*a^2*b^8 *d^2*e^9*x^7 + 12790800*A*a*b^9*d^2*e^9*x^7 + 204652800*B*a^3*b^7*d*e^10*x ^7 + 76744800*A*a^2*b^8*d*e^10*x^7 + 581981400*B*a^4*b^6*e^11*x^7 + 332560 800*A*a^3*b^7*e^11*x^7 + 596904*B*b^10*d^5*e^6*x^6 + 5426400*B*a*b^9*d^4*e ^7*x^6 + 542640*A*b^10*d^4*e^7*x^6 + 26860680*B*a^2*b^8*d^3*e^8*x^6 + 5969 040*A*a*b^9*d^3*e^8*x^6 + 95504640*B*a^3*b^7*d^2*e^9*x^6 + 35814240*A*a^2* b^8*d^2*e^9*x^6 + 271591320*B*a^4*b^6*d*e^10*x^6 + 155195040*A*a^3*b^7*d*e ^10*x^6 + 651819168*B*a^5*b^5*e^11*x^6 + 543182640*A*a^4*b^6*e^11*x^6 + 22 3839*B*b^10*d^6*e^5*x^5 + 2034900*B*a*b^9*d^5*e^6*x^5 + 203490*A*b^10*d^5* e^6*x^5 + 10072755*B*a^2*b^8*d^4*e^7*x^5 + 2238390*A*a*b^9*d^4*e^7*x^5 + 3 5814240*B*a^3*b^7*d^3*e^8*x^5 + 13430340*A*a^2*b^8*d^3*e^8*x^5 + 101846745 *B*a^4*b^6*d^2*e^9*x^5 + 58198140*A*a^3*b^7*d^2*e^9*x^5 + 244432188*B*a^5* b^5*d*e^10*x^5 + 203693490*A*a^4*b^6*d*e^10*x^5 + 509233725*B*a^6*b^4*e...
Time = 1.63 (sec) , antiderivative size = 2121, normalized size of antiderivative = 4.57 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{22}} \, dx=\text {Too large to display} \] Input:
int(((A + B*x)*(a + b*x)^10)/(d + e*x)^22,x)
Output:
-((1847560*A*a^10*e^11 + 11*B*b^10*d^11 + 10*A*b^10*d^10*e + 92378*B*a^10* d*e^10 + 110*A*a*b^9*d^9*e^2 + 97240*B*a^9*b*d^2*e^9 + 660*A*a^2*b^8*d^8*e ^3 + 2860*A*a^3*b^7*d^7*e^4 + 10010*A*a^4*b^6*d^6*e^5 + 30030*A*a^5*b^5*d^ 5*e^6 + 80080*A*a^6*b^4*d^4*e^7 + 194480*A*a^7*b^3*d^3*e^8 + 437580*A*a^8* b^2*d^2*e^9 + 495*B*a^2*b^8*d^9*e^2 + 1760*B*a^3*b^7*d^8*e^3 + 5005*B*a^4* b^6*d^7*e^4 + 12012*B*a^5*b^5*d^6*e^5 + 25025*B*a^6*b^4*d^5*e^6 + 45760*B* a^7*b^3*d^4*e^7 + 72930*B*a^8*b^2*d^3*e^8 + 923780*A*a^9*b*d*e^10 + 100*B* a*b^9*d^10*e)/(38798760*e^12) + (x*(92378*B*a^10*e^10 + 11*B*b^10*d^10 + 9 23780*A*a^9*b*e^10 + 10*A*b^10*d^9*e + 110*A*a*b^9*d^8*e^2 + 437580*A*a^8* b^2*d*e^9 + 660*A*a^2*b^8*d^7*e^3 + 2860*A*a^3*b^7*d^6*e^4 + 10010*A*a^4*b ^6*d^5*e^5 + 30030*A*a^5*b^5*d^4*e^6 + 80080*A*a^6*b^4*d^3*e^7 + 194480*A* a^7*b^3*d^2*e^8 + 495*B*a^2*b^8*d^8*e^2 + 1760*B*a^3*b^7*d^7*e^3 + 5005*B* a^4*b^6*d^6*e^4 + 12012*B*a^5*b^5*d^5*e^5 + 25025*B*a^6*b^4*d^4*e^6 + 4576 0*B*a^7*b^3*d^3*e^7 + 72930*B*a^8*b^2*d^2*e^8 + 100*B*a*b^9*d^9*e + 97240* B*a^9*b*d*e^9))/(1847560*e^11) + (3*b^7*x^8*(1760*B*a^3*e^3 + 11*B*b^3*d^3 + 660*A*a^2*b*e^3 + 10*A*b^3*d^2*e + 110*A*a*b^2*d*e^2 + 100*B*a*b^2*d^2* e + 495*B*a^2*b*d*e^2))/(572*e^4) + (3*b^4*x^5*(25025*B*a^6*e^6 + 11*B*b^6 *d^6 + 30030*A*a^5*b*e^6 + 10*A*b^6*d^5*e + 110*A*a*b^5*d^4*e^2 + 10010*A* a^4*b^2*d*e^5 + 660*A*a^2*b^4*d^3*e^3 + 2860*A*a^3*b^3*d^2*e^4 + 495*B*a^2 *b^4*d^4*e^2 + 1760*B*a^3*b^3*d^3*e^3 + 5005*B*a^4*b^2*d^2*e^4 + 100*B*...
Time = 0.17 (sec) , antiderivative size = 1371, normalized size of antiderivative = 2.95 \[ \int \frac {(a+b x)^{10} (A+B x)}{(d+e x)^{22}} \, dx =\text {Too large to display} \] Input:
int((b*x+a)^10*(B*x+A)/(e*x+d)^22,x)
Output:
( - 167960*a**11*e**11 - 92378*a**10*b*d*e**10 - 1939938*a**10*b*e**11*x - 48620*a**9*b**2*d**2*e**9 - 1021020*a**9*b**2*d*e**10*x - 10210200*a**9*b **2*e**11*x**2 - 24310*a**8*b**3*d**3*e**8 - 510510*a**8*b**3*d**2*e**9*x - 5105100*a**8*b**3*d*e**10*x**2 - 32332300*a**8*b**3*e**11*x**3 - 11440*a **7*b**4*d**4*e**7 - 240240*a**7*b**4*d**3*e**8*x - 2402400*a**7*b**4*d**2 *e**9*x**2 - 15215200*a**7*b**4*d*e**10*x**3 - 68468400*a**7*b**4*e**11*x* *4 - 5005*a**6*b**5*d**5*e**6 - 105105*a**6*b**5*d**4*e**7*x - 1051050*a** 6*b**5*d**3*e**8*x**2 - 6656650*a**6*b**5*d**2*e**9*x**3 - 29954925*a**6*b **5*d*e**10*x**4 - 101846745*a**6*b**5*e**11*x**5 - 2002*a**5*b**6*d**6*e* *5 - 42042*a**5*b**6*d**5*e**6*x - 420420*a**5*b**6*d**4*e**7*x**2 - 26626 60*a**5*b**6*d**3*e**8*x**3 - 11981970*a**5*b**6*d**2*e**9*x**4 - 40738698 *a**5*b**6*d*e**10*x**5 - 108636528*a**5*b**6*e**11*x**6 - 715*a**4*b**7*d **7*e**4 - 15015*a**4*b**7*d**6*e**5*x - 150150*a**4*b**7*d**5*e**6*x**2 - 950950*a**4*b**7*d**4*e**7*x**3 - 4279275*a**4*b**7*d**3*e**8*x**4 - 1454 9535*a**4*b**7*d**2*e**9*x**5 - 38798760*a**4*b**7*d*e**10*x**6 - 83140200 *a**4*b**7*e**11*x**7 - 220*a**3*b**8*d**8*e**3 - 4620*a**3*b**8*d**7*e**4 *x - 46200*a**3*b**8*d**6*e**5*x**2 - 292600*a**3*b**8*d**5*e**6*x**3 - 13 16700*a**3*b**8*d**4*e**7*x**4 - 4476780*a**3*b**8*d**3*e**8*x**5 - 119380 80*a**3*b**8*d**2*e**9*x**6 - 25581600*a**3*b**8*d*e**10*x**7 - 44767800*a **3*b**8*e**11*x**8 - 55*a**2*b**9*d**9*e**2 - 1155*a**2*b**9*d**8*e**3...