3.1299 \(\int (a+b x)^{12} (c+d x)^{10} \, dx\)

Optimal. Leaf size=275 \[ \frac{5 d^9 (a+b x)^{22} (b c-a d)}{11 b^{11}}+\frac{15 d^8 (a+b x)^{21} (b c-a d)^2}{7 b^{11}}+\frac{6 d^7 (a+b x)^{20} (b c-a d)^3}{b^{11}}+\frac{210 d^6 (a+b x)^{19} (b c-a d)^4}{19 b^{11}}+\frac{14 d^5 (a+b x)^{18} (b c-a d)^5}{b^{11}}+\frac{210 d^4 (a+b x)^{17} (b c-a d)^6}{17 b^{11}}+\frac{15 d^3 (a+b x)^{16} (b c-a d)^7}{2 b^{11}}+\frac{3 d^2 (a+b x)^{15} (b c-a d)^8}{b^{11}}+\frac{5 d (a+b x)^{14} (b c-a d)^9}{7 b^{11}}+\frac{(a+b x)^{13} (b c-a d)^{10}}{13 b^{11}}+\frac{d^{10} (a+b x)^{23}}{23 b^{11}} \]

[Out]

((b*c - a*d)^10*(a + b*x)^13)/(13*b^11) + (5*d*(b*c - a*d)^9*(a + b*x)^14)/(7*b^11) + (3*d^2*(b*c - a*d)^8*(a
+ b*x)^15)/b^11 + (15*d^3*(b*c - a*d)^7*(a + b*x)^16)/(2*b^11) + (210*d^4*(b*c - a*d)^6*(a + b*x)^17)/(17*b^11
) + (14*d^5*(b*c - a*d)^5*(a + b*x)^18)/b^11 + (210*d^6*(b*c - a*d)^4*(a + b*x)^19)/(19*b^11) + (6*d^7*(b*c -
a*d)^3*(a + b*x)^20)/b^11 + (15*d^8*(b*c - a*d)^2*(a + b*x)^21)/(7*b^11) + (5*d^9*(b*c - a*d)*(a + b*x)^22)/(1
1*b^11) + (d^10*(a + b*x)^23)/(23*b^11)

________________________________________________________________________________________

Rubi [A]  time = 1.46517, antiderivative size = 275, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {43} \[ \frac{5 d^9 (a+b x)^{22} (b c-a d)}{11 b^{11}}+\frac{15 d^8 (a+b x)^{21} (b c-a d)^2}{7 b^{11}}+\frac{6 d^7 (a+b x)^{20} (b c-a d)^3}{b^{11}}+\frac{210 d^6 (a+b x)^{19} (b c-a d)^4}{19 b^{11}}+\frac{14 d^5 (a+b x)^{18} (b c-a d)^5}{b^{11}}+\frac{210 d^4 (a+b x)^{17} (b c-a d)^6}{17 b^{11}}+\frac{15 d^3 (a+b x)^{16} (b c-a d)^7}{2 b^{11}}+\frac{3 d^2 (a+b x)^{15} (b c-a d)^8}{b^{11}}+\frac{5 d (a+b x)^{14} (b c-a d)^9}{7 b^{11}}+\frac{(a+b x)^{13} (b c-a d)^{10}}{13 b^{11}}+\frac{d^{10} (a+b x)^{23}}{23 b^{11}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^12*(c + d*x)^10,x]

[Out]

((b*c - a*d)^10*(a + b*x)^13)/(13*b^11) + (5*d*(b*c - a*d)^9*(a + b*x)^14)/(7*b^11) + (3*d^2*(b*c - a*d)^8*(a
+ b*x)^15)/b^11 + (15*d^3*(b*c - a*d)^7*(a + b*x)^16)/(2*b^11) + (210*d^4*(b*c - a*d)^6*(a + b*x)^17)/(17*b^11
) + (14*d^5*(b*c - a*d)^5*(a + b*x)^18)/b^11 + (210*d^6*(b*c - a*d)^4*(a + b*x)^19)/(19*b^11) + (6*d^7*(b*c -
a*d)^3*(a + b*x)^20)/b^11 + (15*d^8*(b*c - a*d)^2*(a + b*x)^21)/(7*b^11) + (5*d^9*(b*c - a*d)*(a + b*x)^22)/(1
1*b^11) + (d^10*(a + b*x)^23)/(23*b^11)

Rule 43

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*} \int (a+b x)^{12} (c+d x)^{10} \, dx &=\int \left (\frac{(b c-a d)^{10} (a+b x)^{12}}{b^{10}}+\frac{10 d (b c-a d)^9 (a+b x)^{13}}{b^{10}}+\frac{45 d^2 (b c-a d)^8 (a+b x)^{14}}{b^{10}}+\frac{120 d^3 (b c-a d)^7 (a+b x)^{15}}{b^{10}}+\frac{210 d^4 (b c-a d)^6 (a+b x)^{16}}{b^{10}}+\frac{252 d^5 (b c-a d)^5 (a+b x)^{17}}{b^{10}}+\frac{210 d^6 (b c-a d)^4 (a+b x)^{18}}{b^{10}}+\frac{120 d^7 (b c-a d)^3 (a+b x)^{19}}{b^{10}}+\frac{45 d^8 (b c-a d)^2 (a+b x)^{20}}{b^{10}}+\frac{10 d^9 (b c-a d) (a+b x)^{21}}{b^{10}}+\frac{d^{10} (a+b x)^{22}}{b^{10}}\right ) \, dx\\ &=\frac{(b c-a d)^{10} (a+b x)^{13}}{13 b^{11}}+\frac{5 d (b c-a d)^9 (a+b x)^{14}}{7 b^{11}}+\frac{3 d^2 (b c-a d)^8 (a+b x)^{15}}{b^{11}}+\frac{15 d^3 (b c-a d)^7 (a+b x)^{16}}{2 b^{11}}+\frac{210 d^4 (b c-a d)^6 (a+b x)^{17}}{17 b^{11}}+\frac{14 d^5 (b c-a d)^5 (a+b x)^{18}}{b^{11}}+\frac{210 d^6 (b c-a d)^4 (a+b x)^{19}}{19 b^{11}}+\frac{6 d^7 (b c-a d)^3 (a+b x)^{20}}{b^{11}}+\frac{15 d^8 (b c-a d)^2 (a+b x)^{21}}{7 b^{11}}+\frac{5 d^9 (b c-a d) (a+b x)^{22}}{11 b^{11}}+\frac{d^{10} (a+b x)^{23}}{23 b^{11}}\\ \end{align*}

Mathematica [B]  time = 0.261506, size = 1817, normalized size = 6.61 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^12*(c + d*x)^10,x]

[Out]

a^12*c^10*x + a^11*c^9*(6*b*c + 5*a*d)*x^2 + a^10*c^8*(22*b^2*c^2 + 40*a*b*c*d + 15*a^2*d^2)*x^3 + 5*a^9*c^7*(
11*b^3*c^3 + 33*a*b^2*c^2*d + 27*a^2*b*c*d^2 + 6*a^3*d^3)*x^4 + a^8*c^6*(99*b^4*c^4 + 440*a*b^3*c^3*d + 594*a^
2*b^2*c^2*d^2 + 288*a^3*b*c*d^3 + 42*a^4*d^4)*x^5 + 3*a^7*c^5*(44*b^5*c^5 + 275*a*b^4*c^4*d + 550*a^2*b^3*c^3*
d^2 + 440*a^3*b^2*c^2*d^3 + 140*a^4*b*c*d^4 + 14*a^5*d^5)*x^6 + (3*a^6*c^4*(308*b^6*c^6 + 2640*a*b^5*c^5*d + 7
425*a^2*b^4*c^4*d^2 + 8800*a^3*b^3*c^3*d^3 + 4620*a^4*b^2*c^2*d^4 + 1008*a^5*b*c*d^5 + 70*a^6*d^6)*x^7)/7 + 3*
a^5*c^3*(33*b^7*c^7 + 385*a*b^6*c^6*d + 1485*a^2*b^5*c^5*d^2 + 2475*a^3*b^4*c^4*d^3 + 1925*a^4*b^3*c^3*d^4 + 6
93*a^5*b^2*c^2*d^5 + 105*a^6*b*c*d^6 + 5*a^7*d^7)*x^8 + 5*a^4*c^2*(11*b^8*c^8 + 176*a*b^7*c^7*d + 924*a^2*b^6*
c^6*d^2 + 2112*a^3*b^5*c^5*d^3 + 2310*a^4*b^4*c^4*d^4 + 1232*a^5*b^3*c^3*d^5 + 308*a^6*b^2*c^2*d^6 + 32*a^7*b*
c*d^7 + a^8*d^8)*x^9 + a^3*c*(22*b^9*c^9 + 495*a*b^8*c^8*d + 3564*a^2*b^7*c^7*d^2 + 11088*a^3*b^6*c^6*d^3 + 16
632*a^4*b^5*c^5*d^4 + 12474*a^5*b^4*c^4*d^5 + 4620*a^6*b^3*c^3*d^6 + 792*a^7*b^2*c^2*d^7 + 54*a^8*b*c*d^8 + a^
9*d^9)*x^10 + (a^2*(66*b^10*c^10 + 2200*a*b^9*c^9*d + 22275*a^2*b^8*c^8*d^2 + 95040*a^3*b^7*c^7*d^3 + 194040*a
^4*b^6*c^6*d^4 + 199584*a^5*b^5*c^5*d^5 + 103950*a^6*b^4*c^4*d^6 + 26400*a^7*b^3*c^3*d^7 + 2970*a^8*b^2*c^2*d^
8 + 120*a^9*b*c*d^9 + a^10*d^10)*x^11)/11 + a*b*(b^10*c^10 + 55*a*b^9*c^9*d + 825*a^2*b^8*c^8*d^2 + 4950*a^3*b
^7*c^7*d^3 + 13860*a^4*b^6*c^6*d^4 + 19404*a^5*b^5*c^5*d^5 + 13860*a^6*b^4*c^4*d^6 + 4950*a^7*b^3*c^3*d^7 + 82
5*a^8*b^2*c^2*d^8 + 55*a^9*b*c*d^9 + a^10*d^10)*x^12 + (b^2*(b^10*c^10 + 120*a*b^9*c^9*d + 2970*a^2*b^8*c^8*d^
2 + 26400*a^3*b^7*c^7*d^3 + 103950*a^4*b^6*c^6*d^4 + 199584*a^5*b^5*c^5*d^5 + 194040*a^6*b^4*c^4*d^6 + 95040*a
^7*b^3*c^3*d^7 + 22275*a^8*b^2*c^2*d^8 + 2200*a^9*b*c*d^9 + 66*a^10*d^10)*x^13)/13 + (5*b^3*d*(b^9*c^9 + 54*a*
b^8*c^8*d + 792*a^2*b^7*c^7*d^2 + 4620*a^3*b^6*c^6*d^3 + 12474*a^4*b^5*c^5*d^4 + 16632*a^5*b^4*c^4*d^5 + 11088
*a^6*b^3*c^3*d^6 + 3564*a^7*b^2*c^2*d^7 + 495*a^8*b*c*d^8 + 22*a^9*d^9)*x^14)/7 + 3*b^4*d^2*(b^8*c^8 + 32*a*b^
7*c^7*d + 308*a^2*b^6*c^6*d^2 + 1232*a^3*b^5*c^5*d^3 + 2310*a^4*b^4*c^4*d^4 + 2112*a^5*b^3*c^3*d^5 + 924*a^6*b
^2*c^2*d^6 + 176*a^7*b*c*d^7 + 11*a^8*d^8)*x^15 + (3*b^5*d^3*(5*b^7*c^7 + 105*a*b^6*c^6*d + 693*a^2*b^5*c^5*d^
2 + 1925*a^3*b^4*c^4*d^3 + 2475*a^4*b^3*c^3*d^4 + 1485*a^5*b^2*c^2*d^5 + 385*a^6*b*c*d^6 + 33*a^7*d^7)*x^16)/2
+ (3*b^6*d^4*(70*b^6*c^6 + 1008*a*b^5*c^5*d + 4620*a^2*b^4*c^4*d^2 + 8800*a^3*b^3*c^3*d^3 + 7425*a^4*b^2*c^2*
d^4 + 2640*a^5*b*c*d^5 + 308*a^6*d^6)*x^17)/17 + b^7*d^5*(14*b^5*c^5 + 140*a*b^4*c^4*d + 440*a^2*b^3*c^3*d^2 +
550*a^3*b^2*c^2*d^3 + 275*a^4*b*c*d^4 + 44*a^5*d^5)*x^18 + (5*b^8*d^6*(42*b^4*c^4 + 288*a*b^3*c^3*d + 594*a^2
*b^2*c^2*d^2 + 440*a^3*b*c*d^3 + 99*a^4*d^4)*x^19)/19 + b^9*d^7*(6*b^3*c^3 + 27*a*b^2*c^2*d + 33*a^2*b*c*d^2 +
11*a^3*d^3)*x^20 + (b^10*d^8*(15*b^2*c^2 + 40*a*b*c*d + 22*a^2*d^2)*x^21)/7 + (b^11*d^9*(5*b*c + 6*a*d)*x^22)
/11 + (b^12*d^10*x^23)/23

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Maple [B]  time = 0.004, size = 1891, normalized size = 6.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^12*(d*x+c)^10,x)

[Out]

1/23*b^12*d^10*x^23+1/22*(12*a*b^11*d^10+10*b^12*c*d^9)*x^22+1/21*(66*a^2*b^10*d^10+120*a*b^11*c*d^9+45*b^12*c
^2*d^8)*x^21+1/20*(220*a^3*b^9*d^10+660*a^2*b^10*c*d^9+540*a*b^11*c^2*d^8+120*b^12*c^3*d^7)*x^20+1/19*(495*a^4
*b^8*d^10+2200*a^3*b^9*c*d^9+2970*a^2*b^10*c^2*d^8+1440*a*b^11*c^3*d^7+210*b^12*c^4*d^6)*x^19+1/18*(792*a^5*b^
7*d^10+4950*a^4*b^8*c*d^9+9900*a^3*b^9*c^2*d^8+7920*a^2*b^10*c^3*d^7+2520*a*b^11*c^4*d^6+252*b^12*c^5*d^5)*x^1
8+1/17*(924*a^6*b^6*d^10+7920*a^5*b^7*c*d^9+22275*a^4*b^8*c^2*d^8+26400*a^3*b^9*c^3*d^7+13860*a^2*b^10*c^4*d^6
+3024*a*b^11*c^5*d^5+210*b^12*c^6*d^4)*x^17+1/16*(792*a^7*b^5*d^10+9240*a^6*b^6*c*d^9+35640*a^5*b^7*c^2*d^8+59
400*a^4*b^8*c^3*d^7+46200*a^3*b^9*c^4*d^6+16632*a^2*b^10*c^5*d^5+2520*a*b^11*c^6*d^4+120*b^12*c^7*d^3)*x^16+1/
15*(495*a^8*b^4*d^10+7920*a^7*b^5*c*d^9+41580*a^6*b^6*c^2*d^8+95040*a^5*b^7*c^3*d^7+103950*a^4*b^8*c^4*d^6+554
40*a^3*b^9*c^5*d^5+13860*a^2*b^10*c^6*d^4+1440*a*b^11*c^7*d^3+45*b^12*c^8*d^2)*x^15+1/14*(220*a^9*b^3*d^10+495
0*a^8*b^4*c*d^9+35640*a^7*b^5*c^2*d^8+110880*a^6*b^6*c^3*d^7+166320*a^5*b^7*c^4*d^6+124740*a^4*b^8*c^5*d^5+462
00*a^3*b^9*c^6*d^4+7920*a^2*b^10*c^7*d^3+540*a*b^11*c^8*d^2+10*b^12*c^9*d)*x^14+1/13*(66*a^10*b^2*d^10+2200*a^
9*b^3*c*d^9+22275*a^8*b^4*c^2*d^8+95040*a^7*b^5*c^3*d^7+194040*a^6*b^6*c^4*d^6+199584*a^5*b^7*c^5*d^5+103950*a
^4*b^8*c^6*d^4+26400*a^3*b^9*c^7*d^3+2970*a^2*b^10*c^8*d^2+120*a*b^11*c^9*d+b^12*c^10)*x^13+1/12*(12*a^11*b*d^
10+660*a^10*b^2*c*d^9+9900*a^9*b^3*c^2*d^8+59400*a^8*b^4*c^3*d^7+166320*a^7*b^5*c^4*d^6+232848*a^6*b^6*c^5*d^5
+166320*a^5*b^7*c^6*d^4+59400*a^4*b^8*c^7*d^3+9900*a^3*b^9*c^8*d^2+660*a^2*b^10*c^9*d+12*a*b^11*c^10)*x^12+1/1
1*(a^12*d^10+120*a^11*b*c*d^9+2970*a^10*b^2*c^2*d^8+26400*a^9*b^3*c^3*d^7+103950*a^8*b^4*c^4*d^6+199584*a^7*b^
5*c^5*d^5+194040*a^6*b^6*c^6*d^4+95040*a^5*b^7*c^7*d^3+22275*a^4*b^8*c^8*d^2+2200*a^3*b^9*c^9*d+66*a^2*b^10*c^
10)*x^11+1/10*(10*a^12*c*d^9+540*a^11*b*c^2*d^8+7920*a^10*b^2*c^3*d^7+46200*a^9*b^3*c^4*d^6+124740*a^8*b^4*c^5
*d^5+166320*a^7*b^5*c^6*d^4+110880*a^6*b^6*c^7*d^3+35640*a^5*b^7*c^8*d^2+4950*a^4*b^8*c^9*d+220*a^3*b^9*c^10)*
x^10+1/9*(45*a^12*c^2*d^8+1440*a^11*b*c^3*d^7+13860*a^10*b^2*c^4*d^6+55440*a^9*b^3*c^5*d^5+103950*a^8*b^4*c^6*
d^4+95040*a^7*b^5*c^7*d^3+41580*a^6*b^6*c^8*d^2+7920*a^5*b^7*c^9*d+495*a^4*b^8*c^10)*x^9+1/8*(120*a^12*c^3*d^7
+2520*a^11*b*c^4*d^6+16632*a^10*b^2*c^5*d^5+46200*a^9*b^3*c^6*d^4+59400*a^8*b^4*c^7*d^3+35640*a^7*b^5*c^8*d^2+
9240*a^6*b^6*c^9*d+792*a^5*b^7*c^10)*x^8+1/7*(210*a^12*c^4*d^6+3024*a^11*b*c^5*d^5+13860*a^10*b^2*c^6*d^4+2640
0*a^9*b^3*c^7*d^3+22275*a^8*b^4*c^8*d^2+7920*a^7*b^5*c^9*d+924*a^6*b^6*c^10)*x^7+1/6*(252*a^12*c^5*d^5+2520*a^
11*b*c^6*d^4+7920*a^10*b^2*c^7*d^3+9900*a^9*b^3*c^8*d^2+4950*a^8*b^4*c^9*d+792*a^7*b^5*c^10)*x^6+1/5*(210*a^12
*c^6*d^4+1440*a^11*b*c^7*d^3+2970*a^10*b^2*c^8*d^2+2200*a^9*b^3*c^9*d+495*a^8*b^4*c^10)*x^5+1/4*(120*a^12*c^7*
d^3+540*a^11*b*c^8*d^2+660*a^10*b^2*c^9*d+220*a^9*b^3*c^10)*x^4+1/3*(45*a^12*c^8*d^2+120*a^11*b*c^9*d+66*a^10*
b^2*c^10)*x^3+1/2*(10*a^12*c^9*d+12*a^11*b*c^10)*x^2+a^12*c^10*x

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Maxima [B]  time = 1.01257, size = 2534, normalized size = 9.21 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^12*(d*x+c)^10,x, algorithm="maxima")

[Out]

1/23*b^12*d^10*x^23 + a^12*c^10*x + 1/11*(5*b^12*c*d^9 + 6*a*b^11*d^10)*x^22 + 1/7*(15*b^12*c^2*d^8 + 40*a*b^1
1*c*d^9 + 22*a^2*b^10*d^10)*x^21 + (6*b^12*c^3*d^7 + 27*a*b^11*c^2*d^8 + 33*a^2*b^10*c*d^9 + 11*a^3*b^9*d^10)*
x^20 + 5/19*(42*b^12*c^4*d^6 + 288*a*b^11*c^3*d^7 + 594*a^2*b^10*c^2*d^8 + 440*a^3*b^9*c*d^9 + 99*a^4*b^8*d^10
)*x^19 + (14*b^12*c^5*d^5 + 140*a*b^11*c^4*d^6 + 440*a^2*b^10*c^3*d^7 + 550*a^3*b^9*c^2*d^8 + 275*a^4*b^8*c*d^
9 + 44*a^5*b^7*d^10)*x^18 + 3/17*(70*b^12*c^6*d^4 + 1008*a*b^11*c^5*d^5 + 4620*a^2*b^10*c^4*d^6 + 8800*a^3*b^9
*c^3*d^7 + 7425*a^4*b^8*c^2*d^8 + 2640*a^5*b^7*c*d^9 + 308*a^6*b^6*d^10)*x^17 + 3/2*(5*b^12*c^7*d^3 + 105*a*b^
11*c^6*d^4 + 693*a^2*b^10*c^5*d^5 + 1925*a^3*b^9*c^4*d^6 + 2475*a^4*b^8*c^3*d^7 + 1485*a^5*b^7*c^2*d^8 + 385*a
^6*b^6*c*d^9 + 33*a^7*b^5*d^10)*x^16 + 3*(b^12*c^8*d^2 + 32*a*b^11*c^7*d^3 + 308*a^2*b^10*c^6*d^4 + 1232*a^3*b
^9*c^5*d^5 + 2310*a^4*b^8*c^4*d^6 + 2112*a^5*b^7*c^3*d^7 + 924*a^6*b^6*c^2*d^8 + 176*a^7*b^5*c*d^9 + 11*a^8*b^
4*d^10)*x^15 + 5/7*(b^12*c^9*d + 54*a*b^11*c^8*d^2 + 792*a^2*b^10*c^7*d^3 + 4620*a^3*b^9*c^6*d^4 + 12474*a^4*b
^8*c^5*d^5 + 16632*a^5*b^7*c^4*d^6 + 11088*a^6*b^6*c^3*d^7 + 3564*a^7*b^5*c^2*d^8 + 495*a^8*b^4*c*d^9 + 22*a^9
*b^3*d^10)*x^14 + 1/13*(b^12*c^10 + 120*a*b^11*c^9*d + 2970*a^2*b^10*c^8*d^2 + 26400*a^3*b^9*c^7*d^3 + 103950*
a^4*b^8*c^6*d^4 + 199584*a^5*b^7*c^5*d^5 + 194040*a^6*b^6*c^4*d^6 + 95040*a^7*b^5*c^3*d^7 + 22275*a^8*b^4*c^2*
d^8 + 2200*a^9*b^3*c*d^9 + 66*a^10*b^2*d^10)*x^13 + (a*b^11*c^10 + 55*a^2*b^10*c^9*d + 825*a^3*b^9*c^8*d^2 + 4
950*a^4*b^8*c^7*d^3 + 13860*a^5*b^7*c^6*d^4 + 19404*a^6*b^6*c^5*d^5 + 13860*a^7*b^5*c^4*d^6 + 4950*a^8*b^4*c^3
*d^7 + 825*a^9*b^3*c^2*d^8 + 55*a^10*b^2*c*d^9 + a^11*b*d^10)*x^12 + 1/11*(66*a^2*b^10*c^10 + 2200*a^3*b^9*c^9
*d + 22275*a^4*b^8*c^8*d^2 + 95040*a^5*b^7*c^7*d^3 + 194040*a^6*b^6*c^6*d^4 + 199584*a^7*b^5*c^5*d^5 + 103950*
a^8*b^4*c^4*d^6 + 26400*a^9*b^3*c^3*d^7 + 2970*a^10*b^2*c^2*d^8 + 120*a^11*b*c*d^9 + a^12*d^10)*x^11 + (22*a^3
*b^9*c^10 + 495*a^4*b^8*c^9*d + 3564*a^5*b^7*c^8*d^2 + 11088*a^6*b^6*c^7*d^3 + 16632*a^7*b^5*c^6*d^4 + 12474*a
^8*b^4*c^5*d^5 + 4620*a^9*b^3*c^4*d^6 + 792*a^10*b^2*c^3*d^7 + 54*a^11*b*c^2*d^8 + a^12*c*d^9)*x^10 + 5*(11*a^
4*b^8*c^10 + 176*a^5*b^7*c^9*d + 924*a^6*b^6*c^8*d^2 + 2112*a^7*b^5*c^7*d^3 + 2310*a^8*b^4*c^6*d^4 + 1232*a^9*
b^3*c^5*d^5 + 308*a^10*b^2*c^4*d^6 + 32*a^11*b*c^3*d^7 + a^12*c^2*d^8)*x^9 + 3*(33*a^5*b^7*c^10 + 385*a^6*b^6*
c^9*d + 1485*a^7*b^5*c^8*d^2 + 2475*a^8*b^4*c^7*d^3 + 1925*a^9*b^3*c^6*d^4 + 693*a^10*b^2*c^5*d^5 + 105*a^11*b
*c^4*d^6 + 5*a^12*c^3*d^7)*x^8 + 3/7*(308*a^6*b^6*c^10 + 2640*a^7*b^5*c^9*d + 7425*a^8*b^4*c^8*d^2 + 8800*a^9*
b^3*c^7*d^3 + 4620*a^10*b^2*c^6*d^4 + 1008*a^11*b*c^5*d^5 + 70*a^12*c^4*d^6)*x^7 + 3*(44*a^7*b^5*c^10 + 275*a^
8*b^4*c^9*d + 550*a^9*b^3*c^8*d^2 + 440*a^10*b^2*c^7*d^3 + 140*a^11*b*c^6*d^4 + 14*a^12*c^5*d^5)*x^6 + (99*a^8
*b^4*c^10 + 440*a^9*b^3*c^9*d + 594*a^10*b^2*c^8*d^2 + 288*a^11*b*c^7*d^3 + 42*a^12*c^6*d^4)*x^5 + 5*(11*a^9*b
^3*c^10 + 33*a^10*b^2*c^9*d + 27*a^11*b*c^8*d^2 + 6*a^12*c^7*d^3)*x^4 + (22*a^10*b^2*c^10 + 40*a^11*b*c^9*d +
15*a^12*c^8*d^2)*x^3 + (6*a^11*b*c^10 + 5*a^12*c^9*d)*x^2

________________________________________________________________________________________

Fricas [B]  time = 1.60203, size = 5079, normalized size = 18.47 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^12*(d*x+c)^10,x, algorithm="fricas")

[Out]

1/23*x^23*d^10*b^12 + 5/11*x^22*d^9*c*b^12 + 6/11*x^22*d^10*b^11*a + 15/7*x^21*d^8*c^2*b^12 + 40/7*x^21*d^9*c*
b^11*a + 22/7*x^21*d^10*b^10*a^2 + 6*x^20*d^7*c^3*b^12 + 27*x^20*d^8*c^2*b^11*a + 33*x^20*d^9*c*b^10*a^2 + 11*
x^20*d^10*b^9*a^3 + 210/19*x^19*d^6*c^4*b^12 + 1440/19*x^19*d^7*c^3*b^11*a + 2970/19*x^19*d^8*c^2*b^10*a^2 + 2
200/19*x^19*d^9*c*b^9*a^3 + 495/19*x^19*d^10*b^8*a^4 + 14*x^18*d^5*c^5*b^12 + 140*x^18*d^6*c^4*b^11*a + 440*x^
18*d^7*c^3*b^10*a^2 + 550*x^18*d^8*c^2*b^9*a^3 + 275*x^18*d^9*c*b^8*a^4 + 44*x^18*d^10*b^7*a^5 + 210/17*x^17*d
^4*c^6*b^12 + 3024/17*x^17*d^5*c^5*b^11*a + 13860/17*x^17*d^6*c^4*b^10*a^2 + 26400/17*x^17*d^7*c^3*b^9*a^3 + 2
2275/17*x^17*d^8*c^2*b^8*a^4 + 7920/17*x^17*d^9*c*b^7*a^5 + 924/17*x^17*d^10*b^6*a^6 + 15/2*x^16*d^3*c^7*b^12
+ 315/2*x^16*d^4*c^6*b^11*a + 2079/2*x^16*d^5*c^5*b^10*a^2 + 5775/2*x^16*d^6*c^4*b^9*a^3 + 7425/2*x^16*d^7*c^3
*b^8*a^4 + 4455/2*x^16*d^8*c^2*b^7*a^5 + 1155/2*x^16*d^9*c*b^6*a^6 + 99/2*x^16*d^10*b^5*a^7 + 3*x^15*d^2*c^8*b
^12 + 96*x^15*d^3*c^7*b^11*a + 924*x^15*d^4*c^6*b^10*a^2 + 3696*x^15*d^5*c^5*b^9*a^3 + 6930*x^15*d^6*c^4*b^8*a
^4 + 6336*x^15*d^7*c^3*b^7*a^5 + 2772*x^15*d^8*c^2*b^6*a^6 + 528*x^15*d^9*c*b^5*a^7 + 33*x^15*d^10*b^4*a^8 + 5
/7*x^14*d*c^9*b^12 + 270/7*x^14*d^2*c^8*b^11*a + 3960/7*x^14*d^3*c^7*b^10*a^2 + 3300*x^14*d^4*c^6*b^9*a^3 + 89
10*x^14*d^5*c^5*b^8*a^4 + 11880*x^14*d^6*c^4*b^7*a^5 + 7920*x^14*d^7*c^3*b^6*a^6 + 17820/7*x^14*d^8*c^2*b^5*a^
7 + 2475/7*x^14*d^9*c*b^4*a^8 + 110/7*x^14*d^10*b^3*a^9 + 1/13*x^13*c^10*b^12 + 120/13*x^13*d*c^9*b^11*a + 297
0/13*x^13*d^2*c^8*b^10*a^2 + 26400/13*x^13*d^3*c^7*b^9*a^3 + 103950/13*x^13*d^4*c^6*b^8*a^4 + 199584/13*x^13*d
^5*c^5*b^7*a^5 + 194040/13*x^13*d^6*c^4*b^6*a^6 + 95040/13*x^13*d^7*c^3*b^5*a^7 + 22275/13*x^13*d^8*c^2*b^4*a^
8 + 2200/13*x^13*d^9*c*b^3*a^9 + 66/13*x^13*d^10*b^2*a^10 + x^12*c^10*b^11*a + 55*x^12*d*c^9*b^10*a^2 + 825*x^
12*d^2*c^8*b^9*a^3 + 4950*x^12*d^3*c^7*b^8*a^4 + 13860*x^12*d^4*c^6*b^7*a^5 + 19404*x^12*d^5*c^5*b^6*a^6 + 138
60*x^12*d^6*c^4*b^5*a^7 + 4950*x^12*d^7*c^3*b^4*a^8 + 825*x^12*d^8*c^2*b^3*a^9 + 55*x^12*d^9*c*b^2*a^10 + x^12
*d^10*b*a^11 + 6*x^11*c^10*b^10*a^2 + 200*x^11*d*c^9*b^9*a^3 + 2025*x^11*d^2*c^8*b^8*a^4 + 8640*x^11*d^3*c^7*b
^7*a^5 + 17640*x^11*d^4*c^6*b^6*a^6 + 18144*x^11*d^5*c^5*b^5*a^7 + 9450*x^11*d^6*c^4*b^4*a^8 + 2400*x^11*d^7*c
^3*b^3*a^9 + 270*x^11*d^8*c^2*b^2*a^10 + 120/11*x^11*d^9*c*b*a^11 + 1/11*x^11*d^10*a^12 + 22*x^10*c^10*b^9*a^3
+ 495*x^10*d*c^9*b^8*a^4 + 3564*x^10*d^2*c^8*b^7*a^5 + 11088*x^10*d^3*c^7*b^6*a^6 + 16632*x^10*d^4*c^6*b^5*a^
7 + 12474*x^10*d^5*c^5*b^4*a^8 + 4620*x^10*d^6*c^4*b^3*a^9 + 792*x^10*d^7*c^3*b^2*a^10 + 54*x^10*d^8*c^2*b*a^1
1 + x^10*d^9*c*a^12 + 55*x^9*c^10*b^8*a^4 + 880*x^9*d*c^9*b^7*a^5 + 4620*x^9*d^2*c^8*b^6*a^6 + 10560*x^9*d^3*c
^7*b^5*a^7 + 11550*x^9*d^4*c^6*b^4*a^8 + 6160*x^9*d^5*c^5*b^3*a^9 + 1540*x^9*d^6*c^4*b^2*a^10 + 160*x^9*d^7*c^
3*b*a^11 + 5*x^9*d^8*c^2*a^12 + 99*x^8*c^10*b^7*a^5 + 1155*x^8*d*c^9*b^6*a^6 + 4455*x^8*d^2*c^8*b^5*a^7 + 7425
*x^8*d^3*c^7*b^4*a^8 + 5775*x^8*d^4*c^6*b^3*a^9 + 2079*x^8*d^5*c^5*b^2*a^10 + 315*x^8*d^6*c^4*b*a^11 + 15*x^8*
d^7*c^3*a^12 + 132*x^7*c^10*b^6*a^6 + 7920/7*x^7*d*c^9*b^5*a^7 + 22275/7*x^7*d^2*c^8*b^4*a^8 + 26400/7*x^7*d^3
*c^7*b^3*a^9 + 1980*x^7*d^4*c^6*b^2*a^10 + 432*x^7*d^5*c^5*b*a^11 + 30*x^7*d^6*c^4*a^12 + 132*x^6*c^10*b^5*a^7
+ 825*x^6*d*c^9*b^4*a^8 + 1650*x^6*d^2*c^8*b^3*a^9 + 1320*x^6*d^3*c^7*b^2*a^10 + 420*x^6*d^4*c^6*b*a^11 + 42*
x^6*d^5*c^5*a^12 + 99*x^5*c^10*b^4*a^8 + 440*x^5*d*c^9*b^3*a^9 + 594*x^5*d^2*c^8*b^2*a^10 + 288*x^5*d^3*c^7*b*
a^11 + 42*x^5*d^4*c^6*a^12 + 55*x^4*c^10*b^3*a^9 + 165*x^4*d*c^9*b^2*a^10 + 135*x^4*d^2*c^8*b*a^11 + 30*x^4*d^
3*c^7*a^12 + 22*x^3*c^10*b^2*a^10 + 40*x^3*d*c^9*b*a^11 + 15*x^3*d^2*c^8*a^12 + 6*x^2*c^10*b*a^11 + 5*x^2*d*c^
9*a^12 + x*c^10*a^12

________________________________________________________________________________________

Sympy [B]  time = 0.296201, size = 2088, normalized size = 7.59 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**12*(d*x+c)**10,x)

[Out]

a**12*c**10*x + b**12*d**10*x**23/23 + x**22*(6*a*b**11*d**10/11 + 5*b**12*c*d**9/11) + x**21*(22*a**2*b**10*d
**10/7 + 40*a*b**11*c*d**9/7 + 15*b**12*c**2*d**8/7) + x**20*(11*a**3*b**9*d**10 + 33*a**2*b**10*c*d**9 + 27*a
*b**11*c**2*d**8 + 6*b**12*c**3*d**7) + x**19*(495*a**4*b**8*d**10/19 + 2200*a**3*b**9*c*d**9/19 + 2970*a**2*b
**10*c**2*d**8/19 + 1440*a*b**11*c**3*d**7/19 + 210*b**12*c**4*d**6/19) + x**18*(44*a**5*b**7*d**10 + 275*a**4
*b**8*c*d**9 + 550*a**3*b**9*c**2*d**8 + 440*a**2*b**10*c**3*d**7 + 140*a*b**11*c**4*d**6 + 14*b**12*c**5*d**5
) + x**17*(924*a**6*b**6*d**10/17 + 7920*a**5*b**7*c*d**9/17 + 22275*a**4*b**8*c**2*d**8/17 + 26400*a**3*b**9*
c**3*d**7/17 + 13860*a**2*b**10*c**4*d**6/17 + 3024*a*b**11*c**5*d**5/17 + 210*b**12*c**6*d**4/17) + x**16*(99
*a**7*b**5*d**10/2 + 1155*a**6*b**6*c*d**9/2 + 4455*a**5*b**7*c**2*d**8/2 + 7425*a**4*b**8*c**3*d**7/2 + 5775*
a**3*b**9*c**4*d**6/2 + 2079*a**2*b**10*c**5*d**5/2 + 315*a*b**11*c**6*d**4/2 + 15*b**12*c**7*d**3/2) + x**15*
(33*a**8*b**4*d**10 + 528*a**7*b**5*c*d**9 + 2772*a**6*b**6*c**2*d**8 + 6336*a**5*b**7*c**3*d**7 + 6930*a**4*b
**8*c**4*d**6 + 3696*a**3*b**9*c**5*d**5 + 924*a**2*b**10*c**6*d**4 + 96*a*b**11*c**7*d**3 + 3*b**12*c**8*d**2
) + x**14*(110*a**9*b**3*d**10/7 + 2475*a**8*b**4*c*d**9/7 + 17820*a**7*b**5*c**2*d**8/7 + 7920*a**6*b**6*c**3
*d**7 + 11880*a**5*b**7*c**4*d**6 + 8910*a**4*b**8*c**5*d**5 + 3300*a**3*b**9*c**6*d**4 + 3960*a**2*b**10*c**7
*d**3/7 + 270*a*b**11*c**8*d**2/7 + 5*b**12*c**9*d/7) + x**13*(66*a**10*b**2*d**10/13 + 2200*a**9*b**3*c*d**9/
13 + 22275*a**8*b**4*c**2*d**8/13 + 95040*a**7*b**5*c**3*d**7/13 + 194040*a**6*b**6*c**4*d**6/13 + 199584*a**5
*b**7*c**5*d**5/13 + 103950*a**4*b**8*c**6*d**4/13 + 26400*a**3*b**9*c**7*d**3/13 + 2970*a**2*b**10*c**8*d**2/
13 + 120*a*b**11*c**9*d/13 + b**12*c**10/13) + x**12*(a**11*b*d**10 + 55*a**10*b**2*c*d**9 + 825*a**9*b**3*c**
2*d**8 + 4950*a**8*b**4*c**3*d**7 + 13860*a**7*b**5*c**4*d**6 + 19404*a**6*b**6*c**5*d**5 + 13860*a**5*b**7*c*
*6*d**4 + 4950*a**4*b**8*c**7*d**3 + 825*a**3*b**9*c**8*d**2 + 55*a**2*b**10*c**9*d + a*b**11*c**10) + x**11*(
a**12*d**10/11 + 120*a**11*b*c*d**9/11 + 270*a**10*b**2*c**2*d**8 + 2400*a**9*b**3*c**3*d**7 + 9450*a**8*b**4*
c**4*d**6 + 18144*a**7*b**5*c**5*d**5 + 17640*a**6*b**6*c**6*d**4 + 8640*a**5*b**7*c**7*d**3 + 2025*a**4*b**8*
c**8*d**2 + 200*a**3*b**9*c**9*d + 6*a**2*b**10*c**10) + x**10*(a**12*c*d**9 + 54*a**11*b*c**2*d**8 + 792*a**1
0*b**2*c**3*d**7 + 4620*a**9*b**3*c**4*d**6 + 12474*a**8*b**4*c**5*d**5 + 16632*a**7*b**5*c**6*d**4 + 11088*a*
*6*b**6*c**7*d**3 + 3564*a**5*b**7*c**8*d**2 + 495*a**4*b**8*c**9*d + 22*a**3*b**9*c**10) + x**9*(5*a**12*c**2
*d**8 + 160*a**11*b*c**3*d**7 + 1540*a**10*b**2*c**4*d**6 + 6160*a**9*b**3*c**5*d**5 + 11550*a**8*b**4*c**6*d*
*4 + 10560*a**7*b**5*c**7*d**3 + 4620*a**6*b**6*c**8*d**2 + 880*a**5*b**7*c**9*d + 55*a**4*b**8*c**10) + x**8*
(15*a**12*c**3*d**7 + 315*a**11*b*c**4*d**6 + 2079*a**10*b**2*c**5*d**5 + 5775*a**9*b**3*c**6*d**4 + 7425*a**8
*b**4*c**7*d**3 + 4455*a**7*b**5*c**8*d**2 + 1155*a**6*b**6*c**9*d + 99*a**5*b**7*c**10) + x**7*(30*a**12*c**4
*d**6 + 432*a**11*b*c**5*d**5 + 1980*a**10*b**2*c**6*d**4 + 26400*a**9*b**3*c**7*d**3/7 + 22275*a**8*b**4*c**8
*d**2/7 + 7920*a**7*b**5*c**9*d/7 + 132*a**6*b**6*c**10) + x**6*(42*a**12*c**5*d**5 + 420*a**11*b*c**6*d**4 +
1320*a**10*b**2*c**7*d**3 + 1650*a**9*b**3*c**8*d**2 + 825*a**8*b**4*c**9*d + 132*a**7*b**5*c**10) + x**5*(42*
a**12*c**6*d**4 + 288*a**11*b*c**7*d**3 + 594*a**10*b**2*c**8*d**2 + 440*a**9*b**3*c**9*d + 99*a**8*b**4*c**10
) + x**4*(30*a**12*c**7*d**3 + 135*a**11*b*c**8*d**2 + 165*a**10*b**2*c**9*d + 55*a**9*b**3*c**10) + x**3*(15*
a**12*c**8*d**2 + 40*a**11*b*c**9*d + 22*a**10*b**2*c**10) + x**2*(5*a**12*c**9*d + 6*a**11*b*c**10)

________________________________________________________________________________________

Giac [B]  time = 1.07181, size = 2951, normalized size = 10.73 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^12*(d*x+c)^10,x, algorithm="giac")

[Out]

1/23*b^12*d^10*x^23 + 5/11*b^12*c*d^9*x^22 + 6/11*a*b^11*d^10*x^22 + 15/7*b^12*c^2*d^8*x^21 + 40/7*a*b^11*c*d^
9*x^21 + 22/7*a^2*b^10*d^10*x^21 + 6*b^12*c^3*d^7*x^20 + 27*a*b^11*c^2*d^8*x^20 + 33*a^2*b^10*c*d^9*x^20 + 11*
a^3*b^9*d^10*x^20 + 210/19*b^12*c^4*d^6*x^19 + 1440/19*a*b^11*c^3*d^7*x^19 + 2970/19*a^2*b^10*c^2*d^8*x^19 + 2
200/19*a^3*b^9*c*d^9*x^19 + 495/19*a^4*b^8*d^10*x^19 + 14*b^12*c^5*d^5*x^18 + 140*a*b^11*c^4*d^6*x^18 + 440*a^
2*b^10*c^3*d^7*x^18 + 550*a^3*b^9*c^2*d^8*x^18 + 275*a^4*b^8*c*d^9*x^18 + 44*a^5*b^7*d^10*x^18 + 210/17*b^12*c
^6*d^4*x^17 + 3024/17*a*b^11*c^5*d^5*x^17 + 13860/17*a^2*b^10*c^4*d^6*x^17 + 26400/17*a^3*b^9*c^3*d^7*x^17 + 2
2275/17*a^4*b^8*c^2*d^8*x^17 + 7920/17*a^5*b^7*c*d^9*x^17 + 924/17*a^6*b^6*d^10*x^17 + 15/2*b^12*c^7*d^3*x^16
+ 315/2*a*b^11*c^6*d^4*x^16 + 2079/2*a^2*b^10*c^5*d^5*x^16 + 5775/2*a^3*b^9*c^4*d^6*x^16 + 7425/2*a^4*b^8*c^3*
d^7*x^16 + 4455/2*a^5*b^7*c^2*d^8*x^16 + 1155/2*a^6*b^6*c*d^9*x^16 + 99/2*a^7*b^5*d^10*x^16 + 3*b^12*c^8*d^2*x
^15 + 96*a*b^11*c^7*d^3*x^15 + 924*a^2*b^10*c^6*d^4*x^15 + 3696*a^3*b^9*c^5*d^5*x^15 + 6930*a^4*b^8*c^4*d^6*x^
15 + 6336*a^5*b^7*c^3*d^7*x^15 + 2772*a^6*b^6*c^2*d^8*x^15 + 528*a^7*b^5*c*d^9*x^15 + 33*a^8*b^4*d^10*x^15 + 5
/7*b^12*c^9*d*x^14 + 270/7*a*b^11*c^8*d^2*x^14 + 3960/7*a^2*b^10*c^7*d^3*x^14 + 3300*a^3*b^9*c^6*d^4*x^14 + 89
10*a^4*b^8*c^5*d^5*x^14 + 11880*a^5*b^7*c^4*d^6*x^14 + 7920*a^6*b^6*c^3*d^7*x^14 + 17820/7*a^7*b^5*c^2*d^8*x^1
4 + 2475/7*a^8*b^4*c*d^9*x^14 + 110/7*a^9*b^3*d^10*x^14 + 1/13*b^12*c^10*x^13 + 120/13*a*b^11*c^9*d*x^13 + 297
0/13*a^2*b^10*c^8*d^2*x^13 + 26400/13*a^3*b^9*c^7*d^3*x^13 + 103950/13*a^4*b^8*c^6*d^4*x^13 + 199584/13*a^5*b^
7*c^5*d^5*x^13 + 194040/13*a^6*b^6*c^4*d^6*x^13 + 95040/13*a^7*b^5*c^3*d^7*x^13 + 22275/13*a^8*b^4*c^2*d^8*x^1
3 + 2200/13*a^9*b^3*c*d^9*x^13 + 66/13*a^10*b^2*d^10*x^13 + a*b^11*c^10*x^12 + 55*a^2*b^10*c^9*d*x^12 + 825*a^
3*b^9*c^8*d^2*x^12 + 4950*a^4*b^8*c^7*d^3*x^12 + 13860*a^5*b^7*c^6*d^4*x^12 + 19404*a^6*b^6*c^5*d^5*x^12 + 138
60*a^7*b^5*c^4*d^6*x^12 + 4950*a^8*b^4*c^3*d^7*x^12 + 825*a^9*b^3*c^2*d^8*x^12 + 55*a^10*b^2*c*d^9*x^12 + a^11
*b*d^10*x^12 + 6*a^2*b^10*c^10*x^11 + 200*a^3*b^9*c^9*d*x^11 + 2025*a^4*b^8*c^8*d^2*x^11 + 8640*a^5*b^7*c^7*d^
3*x^11 + 17640*a^6*b^6*c^6*d^4*x^11 + 18144*a^7*b^5*c^5*d^5*x^11 + 9450*a^8*b^4*c^4*d^6*x^11 + 2400*a^9*b^3*c^
3*d^7*x^11 + 270*a^10*b^2*c^2*d^8*x^11 + 120/11*a^11*b*c*d^9*x^11 + 1/11*a^12*d^10*x^11 + 22*a^3*b^9*c^10*x^10
+ 495*a^4*b^8*c^9*d*x^10 + 3564*a^5*b^7*c^8*d^2*x^10 + 11088*a^6*b^6*c^7*d^3*x^10 + 16632*a^7*b^5*c^6*d^4*x^1
0 + 12474*a^8*b^4*c^5*d^5*x^10 + 4620*a^9*b^3*c^4*d^6*x^10 + 792*a^10*b^2*c^3*d^7*x^10 + 54*a^11*b*c^2*d^8*x^1
0 + a^12*c*d^9*x^10 + 55*a^4*b^8*c^10*x^9 + 880*a^5*b^7*c^9*d*x^9 + 4620*a^6*b^6*c^8*d^2*x^9 + 10560*a^7*b^5*c
^7*d^3*x^9 + 11550*a^8*b^4*c^6*d^4*x^9 + 6160*a^9*b^3*c^5*d^5*x^9 + 1540*a^10*b^2*c^4*d^6*x^9 + 160*a^11*b*c^3
*d^7*x^9 + 5*a^12*c^2*d^8*x^9 + 99*a^5*b^7*c^10*x^8 + 1155*a^6*b^6*c^9*d*x^8 + 4455*a^7*b^5*c^8*d^2*x^8 + 7425
*a^8*b^4*c^7*d^3*x^8 + 5775*a^9*b^3*c^6*d^4*x^8 + 2079*a^10*b^2*c^5*d^5*x^8 + 315*a^11*b*c^4*d^6*x^8 + 15*a^12
*c^3*d^7*x^8 + 132*a^6*b^6*c^10*x^7 + 7920/7*a^7*b^5*c^9*d*x^7 + 22275/7*a^8*b^4*c^8*d^2*x^7 + 26400/7*a^9*b^3
*c^7*d^3*x^7 + 1980*a^10*b^2*c^6*d^4*x^7 + 432*a^11*b*c^5*d^5*x^7 + 30*a^12*c^4*d^6*x^7 + 132*a^7*b^5*c^10*x^6
+ 825*a^8*b^4*c^9*d*x^6 + 1650*a^9*b^3*c^8*d^2*x^6 + 1320*a^10*b^2*c^7*d^3*x^6 + 420*a^11*b*c^6*d^4*x^6 + 42*
a^12*c^5*d^5*x^6 + 99*a^8*b^4*c^10*x^5 + 440*a^9*b^3*c^9*d*x^5 + 594*a^10*b^2*c^8*d^2*x^5 + 288*a^11*b*c^7*d^3
*x^5 + 42*a^12*c^6*d^4*x^5 + 55*a^9*b^3*c^10*x^4 + 165*a^10*b^2*c^9*d*x^4 + 135*a^11*b*c^8*d^2*x^4 + 30*a^12*c
^7*d^3*x^4 + 22*a^10*b^2*c^10*x^3 + 40*a^11*b*c^9*d*x^3 + 15*a^12*c^8*d^2*x^3 + 6*a^11*b*c^10*x^2 + 5*a^12*c^9
*d*x^2 + a^12*c^10*x