\(\int (a+b x)^9 (c+d x)^{10} \, dx\) [1302]

Optimal result

Integrand size = 15, antiderivative size = 250 \[ \int (a+b x)^9 (c+d x)^{10} \, dx=-\frac {(b c-a d)^9 (c+d x)^{11}}{11 d^{10}}+\frac {3 b (b c-a d)^8 (c+d x)^{12}}{4 d^{10}}-\frac {36 b^2 (b c-a d)^7 (c+d x)^{13}}{13 d^{10}}+\frac {6 b^3 (b c-a d)^6 (c+d x)^{14}}{d^{10}}-\frac {42 b^4 (b c-a d)^5 (c+d x)^{15}}{5 d^{10}}+\frac {63 b^5 (b c-a d)^4 (c+d x)^{16}}{8 d^{10}}-\frac {84 b^6 (b c-a d)^3 (c+d x)^{17}}{17 d^{10}}+\frac {2 b^7 (b c-a d)^2 (c+d x)^{18}}{d^{10}}-\frac {9 b^8 (b c-a d) (c+d x)^{19}}{19 d^{10}}+\frac {b^9 (c+d x)^{20}}{20 d^{10}} \]

[Out]

-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^10-36/13*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

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45} \[ \int (a+b x)^9 (c+d x)^{10} \, dx=-\frac {9 b^8 (c+d x)^{19} (b c-a d)}{19 d^{10}}+\frac {2 b^7 (c+d x)^{18} (b c-a d)^2}{d^{10}}-\frac {84 b^6 (c+d x)^{17} (b c-a d)^3}{17 d^{10}}+\frac {63 b^5 (c+d x)^{16} (b c-a d)^4}{8 d^{10}}-\frac {42 b^4 (c+d x)^{15} (b c-a d)^5}{5 d^{10}}+\frac {6 b^3 (c+d x)^{14} (b c-a d)^6}{d^{10}}-\frac {36 b^2 (c+d x)^{13} (b c-a d)^7}{13 d^{10}}+\frac {3 b (c+d x)^{12} (b c-a d)^8}{4 d^{10}}-\frac {(c+d x)^{11} (b c-a d)^9}{11 d^{10}}+\frac {b^9 (c+d x)^{20}}{20 d^{10}} \]

[In]

Int[(a + b*x)^9*(c + d*x)^10,x]

[Out]

-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)

Rule 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b c+a d)^9 (c+d x)^{10}}{d^9}+\frac {9 b (b c-a d)^8 (c+d x)^{11}}{d^9}-\frac {36 b^2 (b c-a d)^7 (c+d x)^{12}}{d^9}+\frac {84 b^3 (b c-a d)^6 (c+d x)^{13}}{d^9}-\frac {126 b^4 (b c-a d)^5 (c+d x)^{14}}{d^9}+\frac {126 b^5 (b c-a d)^4 (c+d x)^{15}}{d^9}-\frac {84 b^6 (b c-a d)^3 (c+d x)^{16}}{d^9}+\frac {36 b^7 (b c-a d)^2 (c+d x)^{17}}{d^9}-\frac {9 b^8 (b c-a d) (c+d x)^{18}}{d^9}+\frac {b^9 (c+d x)^{19}}{d^9}\right ) \, 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}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1397\) vs. \(2(250)=500\).

Time = 0.11 (sec) , antiderivative size = 1397, normalized size of antiderivative = 5.59 \[ \int (a+b x)^9 (c+d x)^{10} \, dx=a^9 c^{10} x+\frac {1}{2} a^8 c^9 (9 b c+10 a d) x^2+3 a^7 c^8 \left (4 b^2 c^2+10 a b c d+5 a^2 d^2\right ) x^3+\frac {3}{4} a^6 c^7 \left (28 b^3 c^3+120 a b^2 c^2 d+135 a^2 b c d^2+40 a^3 d^3\right ) x^4+\frac {6}{5} a^5 c^6 \left (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\right ) x^5+3 a^4 c^5 \left (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\right ) x^6+6 a^3 c^4 \left (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\right ) x^7+\frac {3}{4} a^2 c^3 \left (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\right ) x^8+a c^2 \left (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\right ) x^9+\frac {1}{10} c \left (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\right ) x^{10}+\frac {1}{11} d \left (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\right ) x^{11}+\frac {3}{4} b d^2 \left (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^6+40 a^7 b c d^7+a^8 d^8\right ) x^{12}+\frac {6}{13} b^2 d^3 \left (20 b^7 c^7+315 a b^6 c^6 d+1512 a^2 b^5 c^5 d^2+2940 a^3 b^4 c^4 d^3+2520 a^4 b^3 c^3 d^4+945 a^5 b^2 c^2 d^5+140 a^6 b c d^6+6 a^7 d^7\right ) x^{13}+3 b^3 d^4 \left (5 b^6 c^6+54 a b^5 c^5 d+180 a^2 b^4 c^4 d^2+240 a^3 b^3 c^3 d^3+135 a^4 b^2 c^2 d^4+30 a^5 b c d^5+2 a^6 d^6\right ) x^{14}+\frac {6}{5} b^4 d^5 \left (14 b^5 c^5+105 a b^4 c^4 d+240 a^2 b^3 c^3 d^2+210 a^3 b^2 c^2 d^3+70 a^4 b c d^4+7 a^5 d^5\right ) x^{15}+\frac {3}{8} b^5 d^6 \left (35 b^4 c^4+180 a b^3 c^3 d+270 a^2 b^2 c^2 d^2+140 a^3 b c d^3+21 a^4 d^4\right ) x^{16}+\frac {3}{17} b^6 d^7 \left (40 b^3 c^3+135 a b^2 c^2 d+120 a^2 b c d^2+28 a^3 d^3\right ) x^{17}+\frac {1}{2} b^7 d^8 \left (5 b^2 c^2+10 a b c d+4 a^2 d^2\right ) x^{18}+\frac {1}{19} b^8 d^9 (10 b c+9 a d) x^{19}+\frac {1}{20} b^9 d^{10} x^{20} \]

[In]

Integrate[(a + b*x)^9*(c + d*x)^10,x]

[Out]

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 + 151
2*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^6 + 40*a^7*b*c*d^7 + a^8*d^8)*x^12)/4 + (6*b^2*d^3*(20*b^7*c^7
+ 315*a*b^6*c^6*d + 1512*a^2*b^5*c^5*d^2 + 2940*a^3*b^4*c^4*d^3 + 2520*a^4*b^3*c^3*d^4 + 945*a^5*b^2*c^2*d^5
+ 140*a^6*b*c*d^6 + 6*a^7*d^7)*x^13)/13 + 3*b^3*d^4*(5*b^6*c^6 + 54*a*b^5*c^5*d + 180*a^2*b^4*c^4*d^2 + 240*a^
3*b^3*c^3*d^3 + 135*a^4*b^2*c^2*d^4 + 30*a^5*b*c*d^5 + 2*a^6*d^6)*x^14 + (6*b^4*d^5*(14*b^5*c^5 + 105*a*b^4*c^
4*d + 240*a^2*b^3*c^3*d^2 + 210*a^3*b^2*c^2*d^3 + 70*a^4*b*c*d^4 + 7*a^5*d^5)*x^15)/5 + (3*b^5*d^6*(35*b^4*c^4
+ 180*a*b^3*c^3*d + 270*a^2*b^2*c^2*d^2 + 140*a^3*b*c*d^3 + 21*a^4*d^4)*x^16)/8 + (3*b^6*d^7*(40*b^3*c^3 + 13
5*a*b^2*c^2*d + 120*a^2*b*c*d^2 + 28*a^3*d^3)*x^17)/17 + (b^7*d^8*(5*b^2*c^2 + 10*a*b*c*d + 4*a^2*d^2)*x^18)/2
+ (b^8*d^9*(10*b*c + 9*a*d)*x^19)/19 + (b^9*d^10*x^20)/20

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1422\) vs. \(2(234)=468\).

Time = 0.61 (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	\(\text {Expression too large to display}\)	\(1657\)

[In]

int((b*x+a)^9*(d*x+c)^10,x,method=_RETURNVERBOSE)

[Out]

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+1008*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+10080/1
1*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^11+(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+5670/13*a^5*b^4*c^2*d^8+15120/13*a^4*b^5*c^3*d^7+17640/13*a^3*b
^6*c^4*d^6+9072/13*a^2*b^7*c^5*d^5+1890/13*a*b^8*c^6*d^4+120/13*b^9*c^7*d^3)*x^13+(6*a^6*b^3*d^10+90*a^5*b^4*c
*d^9+405*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^14+(42/5*
a^5*b^4*d^10+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+84/5*b^9*c^5*d^5)*x^15
+(63/8*a^4*b^5*d^10+105/2*a^3*b^6*c*d^9+405/4*a^2*b^7*c^2*d^8+135/2*a*b^8*c^3*d^7+105/8*b^9*c^4*d^6)*x^16+(84/
17*a^3*b^6*d^10+360/17*a^2*b^7*c*d^9+405/17*a*b^8*c^2*d^8+120/17*b^9*c^3*d^7)*x^17+(2*a^2*b^7*d^10+5*a*b^8*c*d
^9+5/2*b^9*c^2*d^8)*x^18+(9/19*a*b^8*d^10+10/19*b^9*c*d^9)*x^19+1/20*b^9*d^10*x^20

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1437 vs. \(2 (234) = 468\).

Time = 0.23 (sec) , antiderivative size = 1437, normalized size of antiderivative = 5.75 \[ \int (a+b x)^9 (c+d x)^{10} \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)^9*(d*x+c)^10,x, algorithm="fricas")

[Out]

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^1
7 + 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 + 9
0*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*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 + 120*a^8*b*c^3*d^7 + 5*a^9*c^2*d^8)*x^9 + 3/4*(6*a^2*b^7*c^10 + 140*a^3*b^6*c^9*d + 945*a^4*b^5*
c^8*d^2 + 2520*a^5*b^4*c^7*d^3 + 2940*a^6*b^3*c^6*d^4 + 1512*a^7*b^2*c^5*d^5 + 315*a^8*b*c^4*d^6 + 20*a^9*c^3*
d^7)*x^8 + 6*(2*a^3*b^6*c^10 + 30*a^4*b^5*c^9*d + 135*a^5*b^4*c^8*d^2 + 240*a^6*b^3*c^7*d^3 + 180*a^7*b^2*c^6*
d^4 + 54*a^8*b*c^5*d^5 + 5*a^9*c^4*d^6)*x^7 + 3*(7*a^4*b^5*c^10 + 70*a^5*b^4*c^9*d + 210*a^6*b^3*c^8*d^2 + 240
*a^7*b^2*c^7*d^3 + 105*a^8*b*c^6*d^4 + 14*a^9*c^5*d^5)*x^6 + 6/5*(21*a^5*b^4*c^10 + 140*a^6*b^3*c^9*d + 270*a^
7*b^2*c^8*d^2 + 180*a^8*b*c^7*d^3 + 35*a^9*c^6*d^4)*x^5 + 3/4*(28*a^6*b^3*c^10 + 120*a^7*b^2*c^9*d + 135*a^8*b
*c^8*d^2 + 40*a^9*c^7*d^3)*x^4 + 3*(4*a^7*b^2*c^10 + 10*a^8*b*c^9*d + 5*a^9*c^8*d^2)*x^3 + 1/2*(9*a^8*b*c^10 +
10*a^9*c^9*d)*x^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1598 vs. \(2 (231) = 462\).

Time = 0.12 (sec) , antiderivative size = 1598, normalized size of antiderivative = 6.39 \[ \int (a+b x)^9 (c+d x)^{10} \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)**9*(d*x+c)**10,x)

[Out]

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 + 105*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 + 405*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**10/13 + 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**5/13 + 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 + 6
30*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*c**2*d**8/2 + 432*a**7*b**2*c**3*d**7 + 1764*a**6*b**3*c**4*d**
6 + 15876*a**5*b**4*c**5*d**5/5 + 2646*a**4*b**5*c**6*d**4 + 1008*a**3*b**6*c**7*d**3 + 162*a**2*b**7*c**8*d**
2 + 9*a*b**8*c**9*d + b**9*c**10/10) + x**9*(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**8*(15*a**9*c**3*d**7 + 945*a**8*b*c**4*d**6/4 + 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*a**4*b**5*c**8*d**2/4 + 105*a**3*b**6*c**9*d
+ 9*a**2*b**7*c**10/2) + x**7*(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**6*(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**5*(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*a**5*b**4*c**10/5) + x**4*(30*a**9*c**7*d**3 + 405*a**8*b*c**8*d**2/4 + 90*a**7*b**2*c**9*d + 21*a**6*b
**3*c**10) + x**3*(15*a**9*c**8*d**2 + 30*a**8*b*c**9*d + 12*a**7*b**2*c**10) + x**2*(5*a**9*c**9*d + 9*a**8*b
*c**10/2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1437 vs. \(2 (234) = 468\).

Time = 0.22 (sec) , antiderivative size = 1437, normalized size of antiderivative = 5.75 \[ \int (a+b x)^9 (c+d x)^{10} \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)^9*(d*x+c)^10,x, algorithm="maxima")

[Out]

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^1
7 + 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 + 9
0*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*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 + 120*a^8*b*c^3*d^7 + 5*a^9*c^2*d^8)*x^9 + 3/4*(6*a^2*b^7*c^10 + 140*a^3*b^6*c^9*d + 945*a^4*b^5*
c^8*d^2 + 2520*a^5*b^4*c^7*d^3 + 2940*a^6*b^3*c^6*d^4 + 1512*a^7*b^2*c^5*d^5 + 315*a^8*b*c^4*d^6 + 20*a^9*c^3*
d^7)*x^8 + 6*(2*a^3*b^6*c^10 + 30*a^4*b^5*c^9*d + 135*a^5*b^4*c^8*d^2 + 240*a^6*b^3*c^7*d^3 + 180*a^7*b^2*c^6*
d^4 + 54*a^8*b*c^5*d^5 + 5*a^9*c^4*d^6)*x^7 + 3*(7*a^4*b^5*c^10 + 70*a^5*b^4*c^9*d + 210*a^6*b^3*c^8*d^2 + 240
*a^7*b^2*c^7*d^3 + 105*a^8*b*c^6*d^4 + 14*a^9*c^5*d^5)*x^6 + 6/5*(21*a^5*b^4*c^10 + 140*a^6*b^3*c^9*d + 270*a^
7*b^2*c^8*d^2 + 180*a^8*b*c^7*d^3 + 35*a^9*c^6*d^4)*x^5 + 3/4*(28*a^6*b^3*c^10 + 120*a^7*b^2*c^9*d + 135*a^8*b
*c^8*d^2 + 40*a^9*c^7*d^3)*x^4 + 3*(4*a^7*b^2*c^10 + 10*a^8*b*c^9*d + 5*a^9*c^8*d^2)*x^3 + 1/2*(9*a^8*b*c^10 +
10*a^9*c^9*d)*x^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1656 vs. \(2 (234) = 468\).

Time = 0.28 (sec) , antiderivative size = 1656, normalized size of antiderivative = 6.62 \[ \int (a+b x)^9 (c+d x)^{10} \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)^9*(d*x+c)^10,x, algorithm="giac")

[Out]

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^16 + 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 + 16
2*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^13 + 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 + 26460/1
1*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*a*b^8*c^9*d*x^10 + 162*a^2*b^7*c^8*d^2*x^10 + 1008*a^3*b^6*c^7*
d^3*x^10 + 2646*a^4*b^5*c^6*d^4*x^10 + 15876/5*a^5*b^4*c^5*d^5*x^10 + 1764*a^6*b^3*c^4*d^6*x^10 + 432*a^7*b^2*
c^3*d^7*x^10 + 81/2*a^8*b*c^2*d^8*x^10 + a^9*c*d^9*x^10 + a*b^8*c^10*x^9 + 40*a^2*b^7*c^9*d*x^9 + 420*a^3*b^6*
c^8*d^2*x^9 + 1680*a^4*b^5*c^7*d^3*x^9 + 2940*a^5*b^4*c^6*d^4*x^9 + 2352*a^6*b^3*c^5*d^5*x^9 + 840*a^7*b^2*c^4
*d^6*x^9 + 120*a^8*b*c^3*d^7*x^9 + 5*a^9*c^2*d^8*x^9 + 9/2*a^2*b^7*c^10*x^8 + 105*a^3*b^6*c^9*d*x^8 + 2835/4*a
^4*b^5*c^8*d^2*x^8 + 1890*a^5*b^4*c^7*d^3*x^8 + 2205*a^6*b^3*c^6*d^4*x^8 + 1134*a^7*b^2*c^5*d^5*x^8 + 945/4*a^
8*b*c^4*d^6*x^8 + 15*a^9*c^3*d^7*x^8 + 12*a^3*b^6*c^10*x^7 + 180*a^4*b^5*c^9*d*x^7 + 810*a^5*b^4*c^8*d^2*x^7 +
1440*a^6*b^3*c^7*d^3*x^7 + 1080*a^7*b^2*c^6*d^4*x^7 + 324*a^8*b*c^5*d^5*x^7 + 30*a^9*c^4*d^6*x^7 + 21*a^4*b^5
*c^10*x^6 + 210*a^5*b^4*c^9*d*x^6 + 630*a^6*b^3*c^8*d^2*x^6 + 720*a^7*b^2*c^7*d^3*x^6 + 315*a^8*b*c^6*d^4*x^6
+ 42*a^9*c^5*d^5*x^6 + 126/5*a^5*b^4*c^10*x^5 + 168*a^6*b^3*c^9*d*x^5 + 324*a^7*b^2*c^8*d^2*x^5 + 216*a^8*b*c^
7*d^3*x^5 + 42*a^9*c^6*d^4*x^5 + 21*a^6*b^3*c^10*x^4 + 90*a^7*b^2*c^9*d*x^4 + 405/4*a^8*b*c^8*d^2*x^4 + 30*a^9
*c^7*d^3*x^4 + 12*a^7*b^2*c^10*x^3 + 30*a^8*b*c^9*d*x^3 + 15*a^9*c^8*d^2*x^3 + 9/2*a^8*b*c^10*x^2 + 5*a^9*c^9*
d*x^2 + a^9*c^10*x

Mupad [B] (verification not implemented)

Time = 1.05 (sec) , antiderivative size = 1404, normalized size of antiderivative = 5.62 \[ \int (a+b x)^9 (c+d x)^{10} \, dx=\text {Too large to display} \]

[In]

int((a + b*x)^9*(c + d*x)^10,x)

[Out]

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 + 2205*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^5*d^5)/5 + 126*a*b^8*c^4*d^6 + 84*a^4*b^5*c*d^9 + 288*a^2*b^7*
c^3*d^7 + 252*a^3*b^6*c^2*d^8) + x^10*((b^9*c^10)/10 + a^9*c*d^9 + (81*a^8*b*c^2*d^8)/2 + 162*a^2*b^7*c^8*d^2
+ 1008*a^3*b^6*c^7*d^3 + 2646*a^4*b^5*c^6*d^4 + (15876*a^5*b^4*c^5*d^5)/5 + 1764*a^6*b^3*c^4*d^6 + 432*a^7*b^2
*c^3*d^7 + 9*a*b^8*c^9*d) + x^11*((a^9*d^10)/11 + (10*b^9*c^9*d)/11 + (405*a*b^8*c^8*d^2)/11 + (4320*a^2*b^7*c
^7*d^3)/11 + (17640*a^3*b^6*c^6*d^4)/11 + (31752*a^4*b^5*c^5*d^5)/11 + (26460*a^5*b^4*c^4*d^6)/11 + (10080*a^6
*b^3*c^3*d^7)/11 + (1620*a^7*b^2*c^2*d^8)/11 + (90*a^8*b*c*d^9)/11) + a^9*c^10*x + (b^9*d^10*x^20)/20 + (3*a^6
*c^7*x^4*(40*a^3*d^3 + 28*b^3*c^3 + 120*a*b^2*c^2*d + 135*a^2*b*c*d^2))/4 + (3*b^6*d^7*x^17*(28*a^3*d^3 + 40*b
^3*c^3 + 135*a*b^2*c^2*d + 120*a^2*b*c*d^2))/17 + (a^8*c^9*x^2*(10*a*d + 9*b*c))/2 + (b^8*d^9*x^19*(9*a*d + 10
*b*c))/19 + 3*a^7*c^8*x^3*(5*a^2*d^2 + 4*b^2*c^2 + 10*a*b*c*d) + (b^7*d^8*x^18*(4*a^2*d^2 + 5*b^2*c^2 + 10*a*b
*c*d))/2