3.14.3 \(\int (a+b x)^8 (c+d x)^{10} \, dx\) [1303]

Optimal. Leaf size=225 \[ \frac {(b c-a d)^8 (c+d x)^{11}}{11 d^9}-\frac {2 b (b c-a d)^7 (c+d x)^{12}}{3 d^9}+\frac {28 b^2 (b c-a d)^6 (c+d x)^{13}}{13 d^9}-\frac {4 b^3 (b c-a d)^5 (c+d x)^{14}}{d^9}+\frac {14 b^4 (b c-a d)^4 (c+d x)^{15}}{3 d^9}-\frac {7 b^5 (b c-a d)^3 (c+d x)^{16}}{2 d^9}+\frac {28 b^6 (b c-a d)^2 (c+d x)^{17}}{17 d^9}-\frac {4 b^7 (b c-a d) (c+d x)^{18}}{9 d^9}+\frac {b^8 (c+d x)^{19}}{19 d^9} \]

[Out]

1/11*(-a*d+b*c)^8*(d*x+c)^11/d^9-2/3*b*(-a*d+b*c)^7*(d*x+c)^12/d^9+28/13*b^2*(-a*d+b*c)^6*(d*x+c)^13/d^9-4*b^3
*(-a*d+b*c)^5*(d*x+c)^14/d^9+14/3*b^4*(-a*d+b*c)^4*(d*x+c)^15/d^9-7/2*b^5*(-a*d+b*c)^3*(d*x+c)^16/d^9+28/17*b^
6*(-a*d+b*c)^2*(d*x+c)^17/d^9-4/9*b^7*(-a*d+b*c)*(d*x+c)^18/d^9+1/19*b^8*(d*x+c)^19/d^9

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Rubi [A]
time = 0.62, antiderivative size = 225, normalized size of antiderivative = 1.00, 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 = {45} \begin {gather*} -\frac {4 b^7 (c+d x)^{18} (b c-a d)}{9 d^9}+\frac {28 b^6 (c+d x)^{17} (b c-a d)^2}{17 d^9}-\frac {7 b^5 (c+d x)^{16} (b c-a d)^3}{2 d^9}+\frac {14 b^4 (c+d x)^{15} (b c-a d)^4}{3 d^9}-\frac {4 b^3 (c+d x)^{14} (b c-a d)^5}{d^9}+\frac {28 b^2 (c+d x)^{13} (b c-a d)^6}{13 d^9}-\frac {2 b (c+d x)^{12} (b c-a d)^7}{3 d^9}+\frac {(c+d x)^{11} (b c-a d)^8}{11 d^9}+\frac {b^8 (c+d x)^{19}}{19 d^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^8*(c + d*x)^10,x]

[Out]

((b*c - a*d)^8*(c + d*x)^11)/(11*d^9) - (2*b*(b*c - a*d)^7*(c + d*x)^12)/(3*d^9) + (28*b^2*(b*c - a*d)^6*(c +
d*x)^13)/(13*d^9) - (4*b^3*(b*c - a*d)^5*(c + d*x)^14)/d^9 + (14*b^4*(b*c - a*d)^4*(c + d*x)^15)/(3*d^9) - (7*
b^5*(b*c - a*d)^3*(c + d*x)^16)/(2*d^9) + (28*b^6*(b*c - a*d)^2*(c + d*x)^17)/(17*d^9) - (4*b^7*(b*c - a*d)*(c
+ d*x)^18)/(9*d^9) + (b^8*(c + d*x)^19)/(19*d^9)

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*} \int (a+b x)^8 (c+d x)^{10} \, dx &=\int \left (\frac {(-b c+a d)^8 (c+d x)^{10}}{d^8}-\frac {8 b (b c-a d)^7 (c+d x)^{11}}{d^8}+\frac {28 b^2 (b c-a d)^6 (c+d x)^{12}}{d^8}-\frac {56 b^3 (b c-a d)^5 (c+d x)^{13}}{d^8}+\frac {70 b^4 (b c-a d)^4 (c+d x)^{14}}{d^8}-\frac {56 b^5 (b c-a d)^3 (c+d x)^{15}}{d^8}+\frac {28 b^6 (b c-a d)^2 (c+d x)^{16}}{d^8}-\frac {8 b^7 (b c-a d) (c+d x)^{17}}{d^8}+\frac {b^8 (c+d x)^{18}}{d^8}\right ) \, dx\\ &=\frac {(b c-a d)^8 (c+d x)^{11}}{11 d^9}-\frac {2 b (b c-a d)^7 (c+d x)^{12}}{3 d^9}+\frac {28 b^2 (b c-a d)^6 (c+d x)^{13}}{13 d^9}-\frac {4 b^3 (b c-a d)^5 (c+d x)^{14}}{d^9}+\frac {14 b^4 (b c-a d)^4 (c+d x)^{15}}{3 d^9}-\frac {7 b^5 (b c-a d)^3 (c+d x)^{16}}{2 d^9}+\frac {28 b^6 (b c-a d)^2 (c+d x)^{17}}{17 d^9}-\frac {4 b^7 (b c-a d) (c+d x)^{18}}{9 d^9}+\frac {b^8 (c+d x)^{19}}{19 d^9}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1241\) vs. \(2(225)=450\).
time = 0.11, size = 1241, normalized size = 5.52 \begin {gather*} a^8 c^{10} x+a^7 c^9 (4 b c+5 a d) x^2+\frac {1}{3} a^6 c^8 \left (28 b^2 c^2+80 a b c d+45 a^2 d^2\right ) x^3+2 a^5 c^7 \left (7 b^3 c^3+35 a b^2 c^2 d+45 a^2 b c d^2+15 a^3 d^3\right ) x^4+2 a^4 c^6 \left (7 b^4 c^4+56 a b^3 c^3 d+126 a^2 b^2 c^2 d^2+96 a^3 b c d^3+21 a^4 d^4\right ) x^5+\frac {14}{3} a^3 c^5 \left (2 b^5 c^5+25 a b^4 c^4 d+90 a^2 b^3 c^3 d^2+120 a^3 b^2 c^2 d^3+60 a^4 b c d^4+9 a^5 d^5\right ) x^6+2 a^2 c^4 \left (2 b^6 c^6+40 a b^5 c^5 d+225 a^2 b^4 c^4 d^2+480 a^3 b^3 c^3 d^3+420 a^4 b^2 c^2 d^4+144 a^5 b c d^5+15 a^6 d^6\right ) x^7+a c^3 \left (b^7 c^7+35 a b^6 c^6 d+315 a^2 b^5 c^5 d^2+1050 a^3 b^4 c^4 d^3+1470 a^4 b^3 c^3 d^4+882 a^5 b^2 c^2 d^5+210 a^6 b c d^6+15 a^7 d^7\right ) x^8+\frac {1}{9} c^2 \left (b^8 c^8+80 a b^7 c^7 d+1260 a^2 b^6 c^6 d^2+6720 a^3 b^5 c^5 d^3+14700 a^4 b^4 c^4 d^4+14112 a^5 b^3 c^3 d^5+5880 a^6 b^2 c^2 d^6+960 a^7 b c d^7+45 a^8 d^8\right ) x^9+c d \left (b^8 c^8+36 a b^7 c^7 d+336 a^2 b^6 c^6 d^2+1176 a^3 b^5 c^5 d^3+1764 a^4 b^4 c^4 d^4+1176 a^5 b^3 c^3 d^5+336 a^6 b^2 c^2 d^6+36 a^7 b c d^7+a^8 d^8\right ) x^{10}+\frac {1}{11} d^2 \left (45 b^8 c^8+960 a b^7 c^7 d+5880 a^2 b^6 c^6 d^2+14112 a^3 b^5 c^5 d^3+14700 a^4 b^4 c^4 d^4+6720 a^5 b^3 c^3 d^5+1260 a^6 b^2 c^2 d^6+80 a^7 b c d^7+a^8 d^8\right ) x^{11}+\frac {2}{3} b d^3 \left (15 b^7 c^7+210 a b^6 c^6 d+882 a^2 b^5 c^5 d^2+1470 a^3 b^4 c^4 d^3+1050 a^4 b^3 c^3 d^4+315 a^5 b^2 c^2 d^5+35 a^6 b c d^6+a^7 d^7\right ) x^{12}+\frac {14}{13} b^2 d^4 \left (15 b^6 c^6+144 a b^5 c^5 d+420 a^2 b^4 c^4 d^2+480 a^3 b^3 c^3 d^3+225 a^4 b^2 c^2 d^4+40 a^5 b c d^5+2 a^6 d^6\right ) x^{13}+2 b^3 d^5 \left (9 b^5 c^5+60 a b^4 c^4 d+120 a^2 b^3 c^3 d^2+90 a^3 b^2 c^2 d^3+25 a^4 b c d^4+2 a^5 d^5\right ) x^{14}+\frac {2}{3} b^4 d^6 \left (21 b^4 c^4+96 a b^3 c^3 d+126 a^2 b^2 c^2 d^2+56 a^3 b c d^3+7 a^4 d^4\right ) x^{15}+\frac {1}{2} b^5 d^7 \left (15 b^3 c^3+45 a b^2 c^2 d+35 a^2 b c d^2+7 a^3 d^3\right ) x^{16}+\frac {1}{17} b^6 d^8 \left (45 b^2 c^2+80 a b c d+28 a^2 d^2\right ) x^{17}+\frac {1}{9} b^7 d^9 (5 b c+4 a d) x^{18}+\frac {1}{19} b^8 d^{10} x^{19} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^8*(c + d*x)^10,x]

[Out]

a^8*c^10*x + a^7*c^9*(4*b*c + 5*a*d)*x^2 + (a^6*c^8*(28*b^2*c^2 + 80*a*b*c*d + 45*a^2*d^2)*x^3)/3 + 2*a^5*c^7*
(7*b^3*c^3 + 35*a*b^2*c^2*d + 45*a^2*b*c*d^2 + 15*a^3*d^3)*x^4 + 2*a^4*c^6*(7*b^4*c^4 + 56*a*b^3*c^3*d + 126*a
^2*b^2*c^2*d^2 + 96*a^3*b*c*d^3 + 21*a^4*d^4)*x^5 + (14*a^3*c^5*(2*b^5*c^5 + 25*a*b^4*c^4*d + 90*a^2*b^3*c^3*d
^2 + 120*a^3*b^2*c^2*d^3 + 60*a^4*b*c*d^4 + 9*a^5*d^5)*x^6)/3 + 2*a^2*c^4*(2*b^6*c^6 + 40*a*b^5*c^5*d + 225*a^
2*b^4*c^4*d^2 + 480*a^3*b^3*c^3*d^3 + 420*a^4*b^2*c^2*d^4 + 144*a^5*b*c*d^5 + 15*a^6*d^6)*x^7 + a*c^3*(b^7*c^7
+ 35*a*b^6*c^6*d + 315*a^2*b^5*c^5*d^2 + 1050*a^3*b^4*c^4*d^3 + 1470*a^4*b^3*c^3*d^4 + 882*a^5*b^2*c^2*d^5 +
210*a^6*b*c*d^6 + 15*a^7*d^7)*x^8 + (c^2*(b^8*c^8 + 80*a*b^7*c^7*d + 1260*a^2*b^6*c^6*d^2 + 6720*a^3*b^5*c^5*d
^3 + 14700*a^4*b^4*c^4*d^4 + 14112*a^5*b^3*c^3*d^5 + 5880*a^6*b^2*c^2*d^6 + 960*a^7*b*c*d^7 + 45*a^8*d^8)*x^9)
/9 + c*d*(b^8*c^8 + 36*a*b^7*c^7*d + 336*a^2*b^6*c^6*d^2 + 1176*a^3*b^5*c^5*d^3 + 1764*a^4*b^4*c^4*d^4 + 1176*
a^5*b^3*c^3*d^5 + 336*a^6*b^2*c^2*d^6 + 36*a^7*b*c*d^7 + a^8*d^8)*x^10 + (d^2*(45*b^8*c^8 + 960*a*b^7*c^7*d +
5880*a^2*b^6*c^6*d^2 + 14112*a^3*b^5*c^5*d^3 + 14700*a^4*b^4*c^4*d^4 + 6720*a^5*b^3*c^3*d^5 + 1260*a^6*b^2*c^2
*d^6 + 80*a^7*b*c*d^7 + a^8*d^8)*x^11)/11 + (2*b*d^3*(15*b^7*c^7 + 210*a*b^6*c^6*d + 882*a^2*b^5*c^5*d^2 + 147
0*a^3*b^4*c^4*d^3 + 1050*a^4*b^3*c^3*d^4 + 315*a^5*b^2*c^2*d^5 + 35*a^6*b*c*d^6 + a^7*d^7)*x^12)/3 + (14*b^2*d
^4*(15*b^6*c^6 + 144*a*b^5*c^5*d + 420*a^2*b^4*c^4*d^2 + 480*a^3*b^3*c^3*d^3 + 225*a^4*b^2*c^2*d^4 + 40*a^5*b*
c*d^5 + 2*a^6*d^6)*x^13)/13 + 2*b^3*d^5*(9*b^5*c^5 + 60*a*b^4*c^4*d + 120*a^2*b^3*c^3*d^2 + 90*a^3*b^2*c^2*d^3
+ 25*a^4*b*c*d^4 + 2*a^5*d^5)*x^14 + (2*b^4*d^6*(21*b^4*c^4 + 96*a*b^3*c^3*d + 126*a^2*b^2*c^2*d^2 + 56*a^3*b
*c*d^3 + 7*a^4*d^4)*x^15)/3 + (b^5*d^7*(15*b^3*c^3 + 45*a*b^2*c^2*d + 35*a^2*b*c*d^2 + 7*a^3*d^3)*x^16)/2 + (b
^6*d^8*(45*b^2*c^2 + 80*a*b*c*d + 28*a^2*d^2)*x^17)/17 + (b^7*d^9*(5*b*c + 4*a*d)*x^18)/9 + (b^8*d^10*x^19)/19

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1290\) vs. \(2(209)=418\).
time = 0.15, size = 1291, normalized size = 5.74 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^8*(d*x+c)^10,x,method=_RETURNVERBOSE)

[Out]

1/19*b^8*d^10*x^19+1/18*(8*a*b^7*d^10+10*b^8*c*d^9)*x^18+1/17*(28*a^2*b^6*d^10+80*a*b^7*c*d^9+45*b^8*c^2*d^8)*
x^17+1/16*(56*a^3*b^5*d^10+280*a^2*b^6*c*d^9+360*a*b^7*c^2*d^8+120*b^8*c^3*d^7)*x^16+1/15*(70*a^4*b^4*d^10+560
*a^3*b^5*c*d^9+1260*a^2*b^6*c^2*d^8+960*a*b^7*c^3*d^7+210*b^8*c^4*d^6)*x^15+1/14*(56*a^5*b^3*d^10+700*a^4*b^4*
c*d^9+2520*a^3*b^5*c^2*d^8+3360*a^2*b^6*c^3*d^7+1680*a*b^7*c^4*d^6+252*b^8*c^5*d^5)*x^14+1/13*(28*a^6*b^2*d^10
+560*a^5*b^3*c*d^9+3150*a^4*b^4*c^2*d^8+6720*a^3*b^5*c^3*d^7+5880*a^2*b^6*c^4*d^6+2016*a*b^7*c^5*d^5+210*b^8*c
^6*d^4)*x^13+1/12*(8*a^7*b*d^10+280*a^6*b^2*c*d^9+2520*a^5*b^3*c^2*d^8+8400*a^4*b^4*c^3*d^7+11760*a^3*b^5*c^4*
d^6+7056*a^2*b^6*c^5*d^5+1680*a*b^7*c^6*d^4+120*b^8*c^7*d^3)*x^12+1/11*(a^8*d^10+80*a^7*b*c*d^9+1260*a^6*b^2*c
^2*d^8+6720*a^5*b^3*c^3*d^7+14700*a^4*b^4*c^4*d^6+14112*a^3*b^5*c^5*d^5+5880*a^2*b^6*c^6*d^4+960*a*b^7*c^7*d^3
+45*b^8*c^8*d^2)*x^11+1/10*(10*a^8*c*d^9+360*a^7*b*c^2*d^8+3360*a^6*b^2*c^3*d^7+11760*a^5*b^3*c^4*d^6+17640*a^
4*b^4*c^5*d^5+11760*a^3*b^5*c^6*d^4+3360*a^2*b^6*c^7*d^3+360*a*b^7*c^8*d^2+10*b^8*c^9*d)*x^10+1/9*(45*a^8*c^2*
d^8+960*a^7*b*c^3*d^7+5880*a^6*b^2*c^4*d^6+14112*a^5*b^3*c^5*d^5+14700*a^4*b^4*c^6*d^4+6720*a^3*b^5*c^7*d^3+12
60*a^2*b^6*c^8*d^2+80*a*b^7*c^9*d+b^8*c^10)*x^9+1/8*(120*a^8*c^3*d^7+1680*a^7*b*c^4*d^6+7056*a^6*b^2*c^5*d^5+1
1760*a^5*b^3*c^6*d^4+8400*a^4*b^4*c^7*d^3+2520*a^3*b^5*c^8*d^2+280*a^2*b^6*c^9*d+8*a*b^7*c^10)*x^8+1/7*(210*a^
8*c^4*d^6+2016*a^7*b*c^5*d^5+5880*a^6*b^2*c^6*d^4+6720*a^5*b^3*c^7*d^3+3150*a^4*b^4*c^8*d^2+560*a^3*b^5*c^9*d+
28*a^2*b^6*c^10)*x^7+1/6*(252*a^8*c^5*d^5+1680*a^7*b*c^6*d^4+3360*a^6*b^2*c^7*d^3+2520*a^5*b^3*c^8*d^2+700*a^4
*b^4*c^9*d+56*a^3*b^5*c^10)*x^6+1/5*(210*a^8*c^6*d^4+960*a^7*b*c^7*d^3+1260*a^6*b^2*c^8*d^2+560*a^5*b^3*c^9*d+
70*a^4*b^4*c^10)*x^5+1/4*(120*a^8*c^7*d^3+360*a^7*b*c^8*d^2+280*a^6*b^2*c^9*d+56*a^5*b^3*c^10)*x^4+1/3*(45*a^8
*c^8*d^2+80*a^7*b*c^9*d+28*a^6*b^2*c^10)*x^3+1/2*(10*a^8*c^9*d+8*a^7*b*c^10)*x^2+a^8*c^10*x

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1283 vs. \(2 (209) = 418\).
time = 0.28, size = 1283, normalized size = 5.70 \begin {gather*} \frac {1}{19} \, b^{8} d^{10} x^{19} + a^{8} c^{10} x + \frac {1}{9} \, {\left (5 \, b^{8} c d^{9} + 4 \, a b^{7} d^{10}\right )} x^{18} + \frac {1}{17} \, {\left (45 \, b^{8} c^{2} d^{8} + 80 \, a b^{7} c d^{9} + 28 \, a^{2} b^{6} d^{10}\right )} x^{17} + \frac {1}{2} \, {\left (15 \, b^{8} c^{3} d^{7} + 45 \, a b^{7} c^{2} d^{8} + 35 \, a^{2} b^{6} c d^{9} + 7 \, a^{3} b^{5} d^{10}\right )} x^{16} + \frac {2}{3} \, {\left (21 \, b^{8} c^{4} d^{6} + 96 \, a b^{7} c^{3} d^{7} + 126 \, a^{2} b^{6} c^{2} d^{8} + 56 \, a^{3} b^{5} c d^{9} + 7 \, a^{4} b^{4} d^{10}\right )} x^{15} + 2 \, {\left (9 \, b^{8} c^{5} d^{5} + 60 \, a b^{7} c^{4} d^{6} + 120 \, a^{2} b^{6} c^{3} d^{7} + 90 \, a^{3} b^{5} c^{2} d^{8} + 25 \, a^{4} b^{4} c d^{9} + 2 \, a^{5} b^{3} d^{10}\right )} x^{14} + \frac {14}{13} \, {\left (15 \, b^{8} c^{6} d^{4} + 144 \, a b^{7} c^{5} d^{5} + 420 \, a^{2} b^{6} c^{4} d^{6} + 480 \, a^{3} b^{5} c^{3} d^{7} + 225 \, a^{4} b^{4} c^{2} d^{8} + 40 \, a^{5} b^{3} c d^{9} + 2 \, a^{6} b^{2} d^{10}\right )} x^{13} + \frac {2}{3} \, {\left (15 \, b^{8} c^{7} d^{3} + 210 \, a b^{7} c^{6} d^{4} + 882 \, a^{2} b^{6} c^{5} d^{5} + 1470 \, a^{3} b^{5} c^{4} d^{6} + 1050 \, a^{4} b^{4} c^{3} d^{7} + 315 \, a^{5} b^{3} c^{2} d^{8} + 35 \, a^{6} b^{2} c d^{9} + a^{7} b d^{10}\right )} x^{12} + \frac {1}{11} \, {\left (45 \, b^{8} c^{8} d^{2} + 960 \, a b^{7} c^{7} d^{3} + 5880 \, a^{2} b^{6} c^{6} d^{4} + 14112 \, a^{3} b^{5} c^{5} d^{5} + 14700 \, a^{4} b^{4} c^{4} d^{6} + 6720 \, a^{5} b^{3} c^{3} d^{7} + 1260 \, a^{6} b^{2} c^{2} d^{8} + 80 \, a^{7} b c d^{9} + a^{8} d^{10}\right )} x^{11} + {\left (b^{8} c^{9} d + 36 \, a b^{7} c^{8} d^{2} + 336 \, a^{2} b^{6} c^{7} d^{3} + 1176 \, a^{3} b^{5} c^{6} d^{4} + 1764 \, a^{4} b^{4} c^{5} d^{5} + 1176 \, a^{5} b^{3} c^{4} d^{6} + 336 \, a^{6} b^{2} c^{3} d^{7} + 36 \, a^{7} b c^{2} d^{8} + a^{8} c d^{9}\right )} x^{10} + \frac {1}{9} \, {\left (b^{8} c^{10} + 80 \, a b^{7} c^{9} d + 1260 \, a^{2} b^{6} c^{8} d^{2} + 6720 \, a^{3} b^{5} c^{7} d^{3} + 14700 \, a^{4} b^{4} c^{6} d^{4} + 14112 \, a^{5} b^{3} c^{5} d^{5} + 5880 \, a^{6} b^{2} c^{4} d^{6} + 960 \, a^{7} b c^{3} d^{7} + 45 \, a^{8} c^{2} d^{8}\right )} x^{9} + {\left (a b^{7} c^{10} + 35 \, a^{2} b^{6} c^{9} d + 315 \, a^{3} b^{5} c^{8} d^{2} + 1050 \, a^{4} b^{4} c^{7} d^{3} + 1470 \, a^{5} b^{3} c^{6} d^{4} + 882 \, a^{6} b^{2} c^{5} d^{5} + 210 \, a^{7} b c^{4} d^{6} + 15 \, a^{8} c^{3} d^{7}\right )} x^{8} + 2 \, {\left (2 \, a^{2} b^{6} c^{10} + 40 \, a^{3} b^{5} c^{9} d + 225 \, a^{4} b^{4} c^{8} d^{2} + 480 \, a^{5} b^{3} c^{7} d^{3} + 420 \, a^{6} b^{2} c^{6} d^{4} + 144 \, a^{7} b c^{5} d^{5} + 15 \, a^{8} c^{4} d^{6}\right )} x^{7} + \frac {14}{3} \, {\left (2 \, a^{3} b^{5} c^{10} + 25 \, a^{4} b^{4} c^{9} d + 90 \, a^{5} b^{3} c^{8} d^{2} + 120 \, a^{6} b^{2} c^{7} d^{3} + 60 \, a^{7} b c^{6} d^{4} + 9 \, a^{8} c^{5} d^{5}\right )} x^{6} + 2 \, {\left (7 \, a^{4} b^{4} c^{10} + 56 \, a^{5} b^{3} c^{9} d + 126 \, a^{6} b^{2} c^{8} d^{2} + 96 \, a^{7} b c^{7} d^{3} + 21 \, a^{8} c^{6} d^{4}\right )} x^{5} + 2 \, {\left (7 \, a^{5} b^{3} c^{10} + 35 \, a^{6} b^{2} c^{9} d + 45 \, a^{7} b c^{8} d^{2} + 15 \, a^{8} c^{7} d^{3}\right )} x^{4} + \frac {1}{3} \, {\left (28 \, a^{6} b^{2} c^{10} + 80 \, a^{7} b c^{9} d + 45 \, a^{8} c^{8} d^{2}\right )} x^{3} + {\left (4 \, a^{7} b c^{10} + 5 \, a^{8} c^{9} d\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^8*(d*x+c)^10,x, algorithm="maxima")

[Out]

1/19*b^8*d^10*x^19 + a^8*c^10*x + 1/9*(5*b^8*c*d^9 + 4*a*b^7*d^10)*x^18 + 1/17*(45*b^8*c^2*d^8 + 80*a*b^7*c*d^
9 + 28*a^2*b^6*d^10)*x^17 + 1/2*(15*b^8*c^3*d^7 + 45*a*b^7*c^2*d^8 + 35*a^2*b^6*c*d^9 + 7*a^3*b^5*d^10)*x^16 +
2/3*(21*b^8*c^4*d^6 + 96*a*b^7*c^3*d^7 + 126*a^2*b^6*c^2*d^8 + 56*a^3*b^5*c*d^9 + 7*a^4*b^4*d^10)*x^15 + 2*(9
*b^8*c^5*d^5 + 60*a*b^7*c^4*d^6 + 120*a^2*b^6*c^3*d^7 + 90*a^3*b^5*c^2*d^8 + 25*a^4*b^4*c*d^9 + 2*a^5*b^3*d^10
)*x^14 + 14/13*(15*b^8*c^6*d^4 + 144*a*b^7*c^5*d^5 + 420*a^2*b^6*c^4*d^6 + 480*a^3*b^5*c^3*d^7 + 225*a^4*b^4*c
^2*d^8 + 40*a^5*b^3*c*d^9 + 2*a^6*b^2*d^10)*x^13 + 2/3*(15*b^8*c^7*d^3 + 210*a*b^7*c^6*d^4 + 882*a^2*b^6*c^5*d
^5 + 1470*a^3*b^5*c^4*d^6 + 1050*a^4*b^4*c^3*d^7 + 315*a^5*b^3*c^2*d^8 + 35*a^6*b^2*c*d^9 + a^7*b*d^10)*x^12 +
1/11*(45*b^8*c^8*d^2 + 960*a*b^7*c^7*d^3 + 5880*a^2*b^6*c^6*d^4 + 14112*a^3*b^5*c^5*d^5 + 14700*a^4*b^4*c^4*d
^6 + 6720*a^5*b^3*c^3*d^7 + 1260*a^6*b^2*c^2*d^8 + 80*a^7*b*c*d^9 + a^8*d^10)*x^11 + (b^8*c^9*d + 36*a*b^7*c^8
*d^2 + 336*a^2*b^6*c^7*d^3 + 1176*a^3*b^5*c^6*d^4 + 1764*a^4*b^4*c^5*d^5 + 1176*a^5*b^3*c^4*d^6 + 336*a^6*b^2*
c^3*d^7 + 36*a^7*b*c^2*d^8 + a^8*c*d^9)*x^10 + 1/9*(b^8*c^10 + 80*a*b^7*c^9*d + 1260*a^2*b^6*c^8*d^2 + 6720*a^
3*b^5*c^7*d^3 + 14700*a^4*b^4*c^6*d^4 + 14112*a^5*b^3*c^5*d^5 + 5880*a^6*b^2*c^4*d^6 + 960*a^7*b*c^3*d^7 + 45*
a^8*c^2*d^8)*x^9 + (a*b^7*c^10 + 35*a^2*b^6*c^9*d + 315*a^3*b^5*c^8*d^2 + 1050*a^4*b^4*c^7*d^3 + 1470*a^5*b^3*
c^6*d^4 + 882*a^6*b^2*c^5*d^5 + 210*a^7*b*c^4*d^6 + 15*a^8*c^3*d^7)*x^8 + 2*(2*a^2*b^6*c^10 + 40*a^3*b^5*c^9*d
+ 225*a^4*b^4*c^8*d^2 + 480*a^5*b^3*c^7*d^3 + 420*a^6*b^2*c^6*d^4 + 144*a^7*b*c^5*d^5 + 15*a^8*c^4*d^6)*x^7 +
14/3*(2*a^3*b^5*c^10 + 25*a^4*b^4*c^9*d + 90*a^5*b^3*c^8*d^2 + 120*a^6*b^2*c^7*d^3 + 60*a^7*b*c^6*d^4 + 9*a^8
*c^5*d^5)*x^6 + 2*(7*a^4*b^4*c^10 + 56*a^5*b^3*c^9*d + 126*a^6*b^2*c^8*d^2 + 96*a^7*b*c^7*d^3 + 21*a^8*c^6*d^4
)*x^5 + 2*(7*a^5*b^3*c^10 + 35*a^6*b^2*c^9*d + 45*a^7*b*c^8*d^2 + 15*a^8*c^7*d^3)*x^4 + 1/3*(28*a^6*b^2*c^10 +
80*a^7*b*c^9*d + 45*a^8*c^8*d^2)*x^3 + (4*a^7*b*c^10 + 5*a^8*c^9*d)*x^2

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1283 vs. \(2 (209) = 418\).
time = 0.59, size = 1283, normalized size = 5.70 \begin {gather*} \frac {1}{19} \, b^{8} d^{10} x^{19} + a^{8} c^{10} x + \frac {1}{9} \, {\left (5 \, b^{8} c d^{9} + 4 \, a b^{7} d^{10}\right )} x^{18} + \frac {1}{17} \, {\left (45 \, b^{8} c^{2} d^{8} + 80 \, a b^{7} c d^{9} + 28 \, a^{2} b^{6} d^{10}\right )} x^{17} + \frac {1}{2} \, {\left (15 \, b^{8} c^{3} d^{7} + 45 \, a b^{7} c^{2} d^{8} + 35 \, a^{2} b^{6} c d^{9} + 7 \, a^{3} b^{5} d^{10}\right )} x^{16} + \frac {2}{3} \, {\left (21 \, b^{8} c^{4} d^{6} + 96 \, a b^{7} c^{3} d^{7} + 126 \, a^{2} b^{6} c^{2} d^{8} + 56 \, a^{3} b^{5} c d^{9} + 7 \, a^{4} b^{4} d^{10}\right )} x^{15} + 2 \, {\left (9 \, b^{8} c^{5} d^{5} + 60 \, a b^{7} c^{4} d^{6} + 120 \, a^{2} b^{6} c^{3} d^{7} + 90 \, a^{3} b^{5} c^{2} d^{8} + 25 \, a^{4} b^{4} c d^{9} + 2 \, a^{5} b^{3} d^{10}\right )} x^{14} + \frac {14}{13} \, {\left (15 \, b^{8} c^{6} d^{4} + 144 \, a b^{7} c^{5} d^{5} + 420 \, a^{2} b^{6} c^{4} d^{6} + 480 \, a^{3} b^{5} c^{3} d^{7} + 225 \, a^{4} b^{4} c^{2} d^{8} + 40 \, a^{5} b^{3} c d^{9} + 2 \, a^{6} b^{2} d^{10}\right )} x^{13} + \frac {2}{3} \, {\left (15 \, b^{8} c^{7} d^{3} + 210 \, a b^{7} c^{6} d^{4} + 882 \, a^{2} b^{6} c^{5} d^{5} + 1470 \, a^{3} b^{5} c^{4} d^{6} + 1050 \, a^{4} b^{4} c^{3} d^{7} + 315 \, a^{5} b^{3} c^{2} d^{8} + 35 \, a^{6} b^{2} c d^{9} + a^{7} b d^{10}\right )} x^{12} + \frac {1}{11} \, {\left (45 \, b^{8} c^{8} d^{2} + 960 \, a b^{7} c^{7} d^{3} + 5880 \, a^{2} b^{6} c^{6} d^{4} + 14112 \, a^{3} b^{5} c^{5} d^{5} + 14700 \, a^{4} b^{4} c^{4} d^{6} + 6720 \, a^{5} b^{3} c^{3} d^{7} + 1260 \, a^{6} b^{2} c^{2} d^{8} + 80 \, a^{7} b c d^{9} + a^{8} d^{10}\right )} x^{11} + {\left (b^{8} c^{9} d + 36 \, a b^{7} c^{8} d^{2} + 336 \, a^{2} b^{6} c^{7} d^{3} + 1176 \, a^{3} b^{5} c^{6} d^{4} + 1764 \, a^{4} b^{4} c^{5} d^{5} + 1176 \, a^{5} b^{3} c^{4} d^{6} + 336 \, a^{6} b^{2} c^{3} d^{7} + 36 \, a^{7} b c^{2} d^{8} + a^{8} c d^{9}\right )} x^{10} + \frac {1}{9} \, {\left (b^{8} c^{10} + 80 \, a b^{7} c^{9} d + 1260 \, a^{2} b^{6} c^{8} d^{2} + 6720 \, a^{3} b^{5} c^{7} d^{3} + 14700 \, a^{4} b^{4} c^{6} d^{4} + 14112 \, a^{5} b^{3} c^{5} d^{5} + 5880 \, a^{6} b^{2} c^{4} d^{6} + 960 \, a^{7} b c^{3} d^{7} + 45 \, a^{8} c^{2} d^{8}\right )} x^{9} + {\left (a b^{7} c^{10} + 35 \, a^{2} b^{6} c^{9} d + 315 \, a^{3} b^{5} c^{8} d^{2} + 1050 \, a^{4} b^{4} c^{7} d^{3} + 1470 \, a^{5} b^{3} c^{6} d^{4} + 882 \, a^{6} b^{2} c^{5} d^{5} + 210 \, a^{7} b c^{4} d^{6} + 15 \, a^{8} c^{3} d^{7}\right )} x^{8} + 2 \, {\left (2 \, a^{2} b^{6} c^{10} + 40 \, a^{3} b^{5} c^{9} d + 225 \, a^{4} b^{4} c^{8} d^{2} + 480 \, a^{5} b^{3} c^{7} d^{3} + 420 \, a^{6} b^{2} c^{6} d^{4} + 144 \, a^{7} b c^{5} d^{5} + 15 \, a^{8} c^{4} d^{6}\right )} x^{7} + \frac {14}{3} \, {\left (2 \, a^{3} b^{5} c^{10} + 25 \, a^{4} b^{4} c^{9} d + 90 \, a^{5} b^{3} c^{8} d^{2} + 120 \, a^{6} b^{2} c^{7} d^{3} + 60 \, a^{7} b c^{6} d^{4} + 9 \, a^{8} c^{5} d^{5}\right )} x^{6} + 2 \, {\left (7 \, a^{4} b^{4} c^{10} + 56 \, a^{5} b^{3} c^{9} d + 126 \, a^{6} b^{2} c^{8} d^{2} + 96 \, a^{7} b c^{7} d^{3} + 21 \, a^{8} c^{6} d^{4}\right )} x^{5} + 2 \, {\left (7 \, a^{5} b^{3} c^{10} + 35 \, a^{6} b^{2} c^{9} d + 45 \, a^{7} b c^{8} d^{2} + 15 \, a^{8} c^{7} d^{3}\right )} x^{4} + \frac {1}{3} \, {\left (28 \, a^{6} b^{2} c^{10} + 80 \, a^{7} b c^{9} d + 45 \, a^{8} c^{8} d^{2}\right )} x^{3} + {\left (4 \, a^{7} b c^{10} + 5 \, a^{8} c^{9} d\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^8*(d*x+c)^10,x, algorithm="fricas")

[Out]

1/19*b^8*d^10*x^19 + a^8*c^10*x + 1/9*(5*b^8*c*d^9 + 4*a*b^7*d^10)*x^18 + 1/17*(45*b^8*c^2*d^8 + 80*a*b^7*c*d^
9 + 28*a^2*b^6*d^10)*x^17 + 1/2*(15*b^8*c^3*d^7 + 45*a*b^7*c^2*d^8 + 35*a^2*b^6*c*d^9 + 7*a^3*b^5*d^10)*x^16 +
2/3*(21*b^8*c^4*d^6 + 96*a*b^7*c^3*d^7 + 126*a^2*b^6*c^2*d^8 + 56*a^3*b^5*c*d^9 + 7*a^4*b^4*d^10)*x^15 + 2*(9
*b^8*c^5*d^5 + 60*a*b^7*c^4*d^6 + 120*a^2*b^6*c^3*d^7 + 90*a^3*b^5*c^2*d^8 + 25*a^4*b^4*c*d^9 + 2*a^5*b^3*d^10
)*x^14 + 14/13*(15*b^8*c^6*d^4 + 144*a*b^7*c^5*d^5 + 420*a^2*b^6*c^4*d^6 + 480*a^3*b^5*c^3*d^7 + 225*a^4*b^4*c
^2*d^8 + 40*a^5*b^3*c*d^9 + 2*a^6*b^2*d^10)*x^13 + 2/3*(15*b^8*c^7*d^3 + 210*a*b^7*c^6*d^4 + 882*a^2*b^6*c^5*d
^5 + 1470*a^3*b^5*c^4*d^6 + 1050*a^4*b^4*c^3*d^7 + 315*a^5*b^3*c^2*d^8 + 35*a^6*b^2*c*d^9 + a^7*b*d^10)*x^12 +
1/11*(45*b^8*c^8*d^2 + 960*a*b^7*c^7*d^3 + 5880*a^2*b^6*c^6*d^4 + 14112*a^3*b^5*c^5*d^5 + 14700*a^4*b^4*c^4*d
^6 + 6720*a^5*b^3*c^3*d^7 + 1260*a^6*b^2*c^2*d^8 + 80*a^7*b*c*d^9 + a^8*d^10)*x^11 + (b^8*c^9*d + 36*a*b^7*c^8
*d^2 + 336*a^2*b^6*c^7*d^3 + 1176*a^3*b^5*c^6*d^4 + 1764*a^4*b^4*c^5*d^5 + 1176*a^5*b^3*c^4*d^6 + 336*a^6*b^2*
c^3*d^7 + 36*a^7*b*c^2*d^8 + a^8*c*d^9)*x^10 + 1/9*(b^8*c^10 + 80*a*b^7*c^9*d + 1260*a^2*b^6*c^8*d^2 + 6720*a^
3*b^5*c^7*d^3 + 14700*a^4*b^4*c^6*d^4 + 14112*a^5*b^3*c^5*d^5 + 5880*a^6*b^2*c^4*d^6 + 960*a^7*b*c^3*d^7 + 45*
a^8*c^2*d^8)*x^9 + (a*b^7*c^10 + 35*a^2*b^6*c^9*d + 315*a^3*b^5*c^8*d^2 + 1050*a^4*b^4*c^7*d^3 + 1470*a^5*b^3*
c^6*d^4 + 882*a^6*b^2*c^5*d^5 + 210*a^7*b*c^4*d^6 + 15*a^8*c^3*d^7)*x^8 + 2*(2*a^2*b^6*c^10 + 40*a^3*b^5*c^9*d
+ 225*a^4*b^4*c^8*d^2 + 480*a^5*b^3*c^7*d^3 + 420*a^6*b^2*c^6*d^4 + 144*a^7*b*c^5*d^5 + 15*a^8*c^4*d^6)*x^7 +
14/3*(2*a^3*b^5*c^10 + 25*a^4*b^4*c^9*d + 90*a^5*b^3*c^8*d^2 + 120*a^6*b^2*c^7*d^3 + 60*a^7*b*c^6*d^4 + 9*a^8
*c^5*d^5)*x^6 + 2*(7*a^4*b^4*c^10 + 56*a^5*b^3*c^9*d + 126*a^6*b^2*c^8*d^2 + 96*a^7*b*c^7*d^3 + 21*a^8*c^6*d^4
)*x^5 + 2*(7*a^5*b^3*c^10 + 35*a^6*b^2*c^9*d + 45*a^7*b*c^8*d^2 + 15*a^8*c^7*d^3)*x^4 + 1/3*(28*a^6*b^2*c^10 +
80*a^7*b*c^9*d + 45*a^8*c^8*d^2)*x^3 + (4*a^7*b*c^10 + 5*a^8*c^9*d)*x^2

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1428 vs. \(2 (207) = 414\).
time = 0.10, size = 1428, normalized size = 6.35 \begin {gather*} a^{8} c^{10} x + \frac {b^{8} d^{10} x^{19}}{19} + x^{18} \cdot \left (\frac {4 a b^{7} d^{10}}{9} + \frac {5 b^{8} c d^{9}}{9}\right ) + x^{17} \cdot \left (\frac {28 a^{2} b^{6} d^{10}}{17} + \frac {80 a b^{7} c d^{9}}{17} + \frac {45 b^{8} c^{2} d^{8}}{17}\right ) + x^{16} \cdot \left (\frac {7 a^{3} b^{5} d^{10}}{2} + \frac {35 a^{2} b^{6} c d^{9}}{2} + \frac {45 a b^{7} c^{2} d^{8}}{2} + \frac {15 b^{8} c^{3} d^{7}}{2}\right ) + x^{15} \cdot \left (\frac {14 a^{4} b^{4} d^{10}}{3} + \frac {112 a^{3} b^{5} c d^{9}}{3} + 84 a^{2} b^{6} c^{2} d^{8} + 64 a b^{7} c^{3} d^{7} + 14 b^{8} c^{4} d^{6}\right ) + x^{14} \cdot \left (4 a^{5} b^{3} d^{10} + 50 a^{4} b^{4} c d^{9} + 180 a^{3} b^{5} c^{2} d^{8} + 240 a^{2} b^{6} c^{3} d^{7} + 120 a b^{7} c^{4} d^{6} + 18 b^{8} c^{5} d^{5}\right ) + x^{13} \cdot \left (\frac {28 a^{6} b^{2} d^{10}}{13} + \frac {560 a^{5} b^{3} c d^{9}}{13} + \frac {3150 a^{4} b^{4} c^{2} d^{8}}{13} + \frac {6720 a^{3} b^{5} c^{3} d^{7}}{13} + \frac {5880 a^{2} b^{6} c^{4} d^{6}}{13} + \frac {2016 a b^{7} c^{5} d^{5}}{13} + \frac {210 b^{8} c^{6} d^{4}}{13}\right ) + x^{12} \cdot \left (\frac {2 a^{7} b d^{10}}{3} + \frac {70 a^{6} b^{2} c d^{9}}{3} + 210 a^{5} b^{3} c^{2} d^{8} + 700 a^{4} b^{4} c^{3} d^{7} + 980 a^{3} b^{5} c^{4} d^{6} + 588 a^{2} b^{6} c^{5} d^{5} + 140 a b^{7} c^{6} d^{4} + 10 b^{8} c^{7} d^{3}\right ) + x^{11} \left (\frac {a^{8} d^{10}}{11} + \frac {80 a^{7} b c d^{9}}{11} + \frac {1260 a^{6} b^{2} c^{2} d^{8}}{11} + \frac {6720 a^{5} b^{3} c^{3} d^{7}}{11} + \frac {14700 a^{4} b^{4} c^{4} d^{6}}{11} + \frac {14112 a^{3} b^{5} c^{5} d^{5}}{11} + \frac {5880 a^{2} b^{6} c^{6} d^{4}}{11} + \frac {960 a b^{7} c^{7} d^{3}}{11} + \frac {45 b^{8} c^{8} d^{2}}{11}\right ) + x^{10} \left (a^{8} c d^{9} + 36 a^{7} b c^{2} d^{8} + 336 a^{6} b^{2} c^{3} d^{7} + 1176 a^{5} b^{3} c^{4} d^{6} + 1764 a^{4} b^{4} c^{5} d^{5} + 1176 a^{3} b^{5} c^{6} d^{4} + 336 a^{2} b^{6} c^{7} d^{3} + 36 a b^{7} c^{8} d^{2} + b^{8} c^{9} d\right ) + x^{9} \cdot \left (5 a^{8} c^{2} d^{8} + \frac {320 a^{7} b c^{3} d^{7}}{3} + \frac {1960 a^{6} b^{2} c^{4} d^{6}}{3} + 1568 a^{5} b^{3} c^{5} d^{5} + \frac {4900 a^{4} b^{4} c^{6} d^{4}}{3} + \frac {2240 a^{3} b^{5} c^{7} d^{3}}{3} + 140 a^{2} b^{6} c^{8} d^{2} + \frac {80 a b^{7} c^{9} d}{9} + \frac {b^{8} c^{10}}{9}\right ) + x^{8} \cdot \left (15 a^{8} c^{3} d^{7} + 210 a^{7} b c^{4} d^{6} + 882 a^{6} b^{2} c^{5} d^{5} + 1470 a^{5} b^{3} c^{6} d^{4} + 1050 a^{4} b^{4} c^{7} d^{3} + 315 a^{3} b^{5} c^{8} d^{2} + 35 a^{2} b^{6} c^{9} d + a b^{7} c^{10}\right ) + x^{7} \cdot \left (30 a^{8} c^{4} d^{6} + 288 a^{7} b c^{5} d^{5} + 840 a^{6} b^{2} c^{6} d^{4} + 960 a^{5} b^{3} c^{7} d^{3} + 450 a^{4} b^{4} c^{8} d^{2} + 80 a^{3} b^{5} c^{9} d + 4 a^{2} b^{6} c^{10}\right ) + x^{6} \cdot \left (42 a^{8} c^{5} d^{5} + 280 a^{7} b c^{6} d^{4} + 560 a^{6} b^{2} c^{7} d^{3} + 420 a^{5} b^{3} c^{8} d^{2} + \frac {350 a^{4} b^{4} c^{9} d}{3} + \frac {28 a^{3} b^{5} c^{10}}{3}\right ) + x^{5} \cdot \left (42 a^{8} c^{6} d^{4} + 192 a^{7} b c^{7} d^{3} + 252 a^{6} b^{2} c^{8} d^{2} + 112 a^{5} b^{3} c^{9} d + 14 a^{4} b^{4} c^{10}\right ) + x^{4} \cdot \left (30 a^{8} c^{7} d^{3} + 90 a^{7} b c^{8} d^{2} + 70 a^{6} b^{2} c^{9} d + 14 a^{5} b^{3} c^{10}\right ) + x^{3} \cdot \left (15 a^{8} c^{8} d^{2} + \frac {80 a^{7} b c^{9} d}{3} + \frac {28 a^{6} b^{2} c^{10}}{3}\right ) + x^{2} \cdot \left (5 a^{8} c^{9} d + 4 a^{7} b c^{10}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**8*(d*x+c)**10,x)

[Out]

a**8*c**10*x + b**8*d**10*x**19/19 + x**18*(4*a*b**7*d**10/9 + 5*b**8*c*d**9/9) + x**17*(28*a**2*b**6*d**10/17
+ 80*a*b**7*c*d**9/17 + 45*b**8*c**2*d**8/17) + x**16*(7*a**3*b**5*d**10/2 + 35*a**2*b**6*c*d**9/2 + 45*a*b**
7*c**2*d**8/2 + 15*b**8*c**3*d**7/2) + x**15*(14*a**4*b**4*d**10/3 + 112*a**3*b**5*c*d**9/3 + 84*a**2*b**6*c**
2*d**8 + 64*a*b**7*c**3*d**7 + 14*b**8*c**4*d**6) + x**14*(4*a**5*b**3*d**10 + 50*a**4*b**4*c*d**9 + 180*a**3*
b**5*c**2*d**8 + 240*a**2*b**6*c**3*d**7 + 120*a*b**7*c**4*d**6 + 18*b**8*c**5*d**5) + x**13*(28*a**6*b**2*d**
10/13 + 560*a**5*b**3*c*d**9/13 + 3150*a**4*b**4*c**2*d**8/13 + 6720*a**3*b**5*c**3*d**7/13 + 5880*a**2*b**6*c
**4*d**6/13 + 2016*a*b**7*c**5*d**5/13 + 210*b**8*c**6*d**4/13) + x**12*(2*a**7*b*d**10/3 + 70*a**6*b**2*c*d**
9/3 + 210*a**5*b**3*c**2*d**8 + 700*a**4*b**4*c**3*d**7 + 980*a**3*b**5*c**4*d**6 + 588*a**2*b**6*c**5*d**5 +
140*a*b**7*c**6*d**4 + 10*b**8*c**7*d**3) + x**11*(a**8*d**10/11 + 80*a**7*b*c*d**9/11 + 1260*a**6*b**2*c**2*d
**8/11 + 6720*a**5*b**3*c**3*d**7/11 + 14700*a**4*b**4*c**4*d**6/11 + 14112*a**3*b**5*c**5*d**5/11 + 5880*a**2
*b**6*c**6*d**4/11 + 960*a*b**7*c**7*d**3/11 + 45*b**8*c**8*d**2/11) + x**10*(a**8*c*d**9 + 36*a**7*b*c**2*d**
8 + 336*a**6*b**2*c**3*d**7 + 1176*a**5*b**3*c**4*d**6 + 1764*a**4*b**4*c**5*d**5 + 1176*a**3*b**5*c**6*d**4 +
336*a**2*b**6*c**7*d**3 + 36*a*b**7*c**8*d**2 + b**8*c**9*d) + x**9*(5*a**8*c**2*d**8 + 320*a**7*b*c**3*d**7/
3 + 1960*a**6*b**2*c**4*d**6/3 + 1568*a**5*b**3*c**5*d**5 + 4900*a**4*b**4*c**6*d**4/3 + 2240*a**3*b**5*c**7*d
**3/3 + 140*a**2*b**6*c**8*d**2 + 80*a*b**7*c**9*d/9 + b**8*c**10/9) + x**8*(15*a**8*c**3*d**7 + 210*a**7*b*c*
*4*d**6 + 882*a**6*b**2*c**5*d**5 + 1470*a**5*b**3*c**6*d**4 + 1050*a**4*b**4*c**7*d**3 + 315*a**3*b**5*c**8*d
**2 + 35*a**2*b**6*c**9*d + a*b**7*c**10) + x**7*(30*a**8*c**4*d**6 + 288*a**7*b*c**5*d**5 + 840*a**6*b**2*c**
6*d**4 + 960*a**5*b**3*c**7*d**3 + 450*a**4*b**4*c**8*d**2 + 80*a**3*b**5*c**9*d + 4*a**2*b**6*c**10) + x**6*(
42*a**8*c**5*d**5 + 280*a**7*b*c**6*d**4 + 560*a**6*b**2*c**7*d**3 + 420*a**5*b**3*c**8*d**2 + 350*a**4*b**4*c
**9*d/3 + 28*a**3*b**5*c**10/3) + x**5*(42*a**8*c**6*d**4 + 192*a**7*b*c**7*d**3 + 252*a**6*b**2*c**8*d**2 + 1
12*a**5*b**3*c**9*d + 14*a**4*b**4*c**10) + x**4*(30*a**8*c**7*d**3 + 90*a**7*b*c**8*d**2 + 70*a**6*b**2*c**9*
d + 14*a**5*b**3*c**10) + x**3*(15*a**8*c**8*d**2 + 80*a**7*b*c**9*d/3 + 28*a**6*b**2*c**10/3) + x**2*(5*a**8*
c**9*d + 4*a**7*b*c**10)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1478 vs. \(2 (209) = 418\).
time = 1.00, size = 1478, normalized size = 6.57 \begin {gather*} \frac {1}{19} \, b^{8} d^{10} x^{19} + \frac {5}{9} \, b^{8} c d^{9} x^{18} + \frac {4}{9} \, a b^{7} d^{10} x^{18} + \frac {45}{17} \, b^{8} c^{2} d^{8} x^{17} + \frac {80}{17} \, a b^{7} c d^{9} x^{17} + \frac {28}{17} \, a^{2} b^{6} d^{10} x^{17} + \frac {15}{2} \, b^{8} c^{3} d^{7} x^{16} + \frac {45}{2} \, a b^{7} c^{2} d^{8} x^{16} + \frac {35}{2} \, a^{2} b^{6} c d^{9} x^{16} + \frac {7}{2} \, a^{3} b^{5} d^{10} x^{16} + 14 \, b^{8} c^{4} d^{6} x^{15} + 64 \, a b^{7} c^{3} d^{7} x^{15} + 84 \, a^{2} b^{6} c^{2} d^{8} x^{15} + \frac {112}{3} \, a^{3} b^{5} c d^{9} x^{15} + \frac {14}{3} \, a^{4} b^{4} d^{10} x^{15} + 18 \, b^{8} c^{5} d^{5} x^{14} + 120 \, a b^{7} c^{4} d^{6} x^{14} + 240 \, a^{2} b^{6} c^{3} d^{7} x^{14} + 180 \, a^{3} b^{5} c^{2} d^{8} x^{14} + 50 \, a^{4} b^{4} c d^{9} x^{14} + 4 \, a^{5} b^{3} d^{10} x^{14} + \frac {210}{13} \, b^{8} c^{6} d^{4} x^{13} + \frac {2016}{13} \, a b^{7} c^{5} d^{5} x^{13} + \frac {5880}{13} \, a^{2} b^{6} c^{4} d^{6} x^{13} + \frac {6720}{13} \, a^{3} b^{5} c^{3} d^{7} x^{13} + \frac {3150}{13} \, a^{4} b^{4} c^{2} d^{8} x^{13} + \frac {560}{13} \, a^{5} b^{3} c d^{9} x^{13} + \frac {28}{13} \, a^{6} b^{2} d^{10} x^{13} + 10 \, b^{8} c^{7} d^{3} x^{12} + 140 \, a b^{7} c^{6} d^{4} x^{12} + 588 \, a^{2} b^{6} c^{5} d^{5} x^{12} + 980 \, a^{3} b^{5} c^{4} d^{6} x^{12} + 700 \, a^{4} b^{4} c^{3} d^{7} x^{12} + 210 \, a^{5} b^{3} c^{2} d^{8} x^{12} + \frac {70}{3} \, a^{6} b^{2} c d^{9} x^{12} + \frac {2}{3} \, a^{7} b d^{10} x^{12} + \frac {45}{11} \, b^{8} c^{8} d^{2} x^{11} + \frac {960}{11} \, a b^{7} c^{7} d^{3} x^{11} + \frac {5880}{11} \, a^{2} b^{6} c^{6} d^{4} x^{11} + \frac {14112}{11} \, a^{3} b^{5} c^{5} d^{5} x^{11} + \frac {14700}{11} \, a^{4} b^{4} c^{4} d^{6} x^{11} + \frac {6720}{11} \, a^{5} b^{3} c^{3} d^{7} x^{11} + \frac {1260}{11} \, a^{6} b^{2} c^{2} d^{8} x^{11} + \frac {80}{11} \, a^{7} b c d^{9} x^{11} + \frac {1}{11} \, a^{8} d^{10} x^{11} + b^{8} c^{9} d x^{10} + 36 \, a b^{7} c^{8} d^{2} x^{10} + 336 \, a^{2} b^{6} c^{7} d^{3} x^{10} + 1176 \, a^{3} b^{5} c^{6} d^{4} x^{10} + 1764 \, a^{4} b^{4} c^{5} d^{5} x^{10} + 1176 \, a^{5} b^{3} c^{4} d^{6} x^{10} + 336 \, a^{6} b^{2} c^{3} d^{7} x^{10} + 36 \, a^{7} b c^{2} d^{8} x^{10} + a^{8} c d^{9} x^{10} + \frac {1}{9} \, b^{8} c^{10} x^{9} + \frac {80}{9} \, a b^{7} c^{9} d x^{9} + 140 \, a^{2} b^{6} c^{8} d^{2} x^{9} + \frac {2240}{3} \, a^{3} b^{5} c^{7} d^{3} x^{9} + \frac {4900}{3} \, a^{4} b^{4} c^{6} d^{4} x^{9} + 1568 \, a^{5} b^{3} c^{5} d^{5} x^{9} + \frac {1960}{3} \, a^{6} b^{2} c^{4} d^{6} x^{9} + \frac {320}{3} \, a^{7} b c^{3} d^{7} x^{9} + 5 \, a^{8} c^{2} d^{8} x^{9} + a b^{7} c^{10} x^{8} + 35 \, a^{2} b^{6} c^{9} d x^{8} + 315 \, a^{3} b^{5} c^{8} d^{2} x^{8} + 1050 \, a^{4} b^{4} c^{7} d^{3} x^{8} + 1470 \, a^{5} b^{3} c^{6} d^{4} x^{8} + 882 \, a^{6} b^{2} c^{5} d^{5} x^{8} + 210 \, a^{7} b c^{4} d^{6} x^{8} + 15 \, a^{8} c^{3} d^{7} x^{8} + 4 \, a^{2} b^{6} c^{10} x^{7} + 80 \, a^{3} b^{5} c^{9} d x^{7} + 450 \, a^{4} b^{4} c^{8} d^{2} x^{7} + 960 \, a^{5} b^{3} c^{7} d^{3} x^{7} + 840 \, a^{6} b^{2} c^{6} d^{4} x^{7} + 288 \, a^{7} b c^{5} d^{5} x^{7} + 30 \, a^{8} c^{4} d^{6} x^{7} + \frac {28}{3} \, a^{3} b^{5} c^{10} x^{6} + \frac {350}{3} \, a^{4} b^{4} c^{9} d x^{6} + 420 \, a^{5} b^{3} c^{8} d^{2} x^{6} + 560 \, a^{6} b^{2} c^{7} d^{3} x^{6} + 280 \, a^{7} b c^{6} d^{4} x^{6} + 42 \, a^{8} c^{5} d^{5} x^{6} + 14 \, a^{4} b^{4} c^{10} x^{5} + 112 \, a^{5} b^{3} c^{9} d x^{5} + 252 \, a^{6} b^{2} c^{8} d^{2} x^{5} + 192 \, a^{7} b c^{7} d^{3} x^{5} + 42 \, a^{8} c^{6} d^{4} x^{5} + 14 \, a^{5} b^{3} c^{10} x^{4} + 70 \, a^{6} b^{2} c^{9} d x^{4} + 90 \, a^{7} b c^{8} d^{2} x^{4} + 30 \, a^{8} c^{7} d^{3} x^{4} + \frac {28}{3} \, a^{6} b^{2} c^{10} x^{3} + \frac {80}{3} \, a^{7} b c^{9} d x^{3} + 15 \, a^{8} c^{8} d^{2} x^{3} + 4 \, a^{7} b c^{10} x^{2} + 5 \, a^{8} c^{9} d x^{2} + a^{8} c^{10} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^8*(d*x+c)^10,x, algorithm="giac")

[Out]

1/19*b^8*d^10*x^19 + 5/9*b^8*c*d^9*x^18 + 4/9*a*b^7*d^10*x^18 + 45/17*b^8*c^2*d^8*x^17 + 80/17*a*b^7*c*d^9*x^1
7 + 28/17*a^2*b^6*d^10*x^17 + 15/2*b^8*c^3*d^7*x^16 + 45/2*a*b^7*c^2*d^8*x^16 + 35/2*a^2*b^6*c*d^9*x^16 + 7/2*
a^3*b^5*d^10*x^16 + 14*b^8*c^4*d^6*x^15 + 64*a*b^7*c^3*d^7*x^15 + 84*a^2*b^6*c^2*d^8*x^15 + 112/3*a^3*b^5*c*d^
9*x^15 + 14/3*a^4*b^4*d^10*x^15 + 18*b^8*c^5*d^5*x^14 + 120*a*b^7*c^4*d^6*x^14 + 240*a^2*b^6*c^3*d^7*x^14 + 18
0*a^3*b^5*c^2*d^8*x^14 + 50*a^4*b^4*c*d^9*x^14 + 4*a^5*b^3*d^10*x^14 + 210/13*b^8*c^6*d^4*x^13 + 2016/13*a*b^7
*c^5*d^5*x^13 + 5880/13*a^2*b^6*c^4*d^6*x^13 + 6720/13*a^3*b^5*c^3*d^7*x^13 + 3150/13*a^4*b^4*c^2*d^8*x^13 + 5
60/13*a^5*b^3*c*d^9*x^13 + 28/13*a^6*b^2*d^10*x^13 + 10*b^8*c^7*d^3*x^12 + 140*a*b^7*c^6*d^4*x^12 + 588*a^2*b^
6*c^5*d^5*x^12 + 980*a^3*b^5*c^4*d^6*x^12 + 700*a^4*b^4*c^3*d^7*x^12 + 210*a^5*b^3*c^2*d^8*x^12 + 70/3*a^6*b^2
*c*d^9*x^12 + 2/3*a^7*b*d^10*x^12 + 45/11*b^8*c^8*d^2*x^11 + 960/11*a*b^7*c^7*d^3*x^11 + 5880/11*a^2*b^6*c^6*d
^4*x^11 + 14112/11*a^3*b^5*c^5*d^5*x^11 + 14700/11*a^4*b^4*c^4*d^6*x^11 + 6720/11*a^5*b^3*c^3*d^7*x^11 + 1260/
11*a^6*b^2*c^2*d^8*x^11 + 80/11*a^7*b*c*d^9*x^11 + 1/11*a^8*d^10*x^11 + b^8*c^9*d*x^10 + 36*a*b^7*c^8*d^2*x^10
+ 336*a^2*b^6*c^7*d^3*x^10 + 1176*a^3*b^5*c^6*d^4*x^10 + 1764*a^4*b^4*c^5*d^5*x^10 + 1176*a^5*b^3*c^4*d^6*x^1
0 + 336*a^6*b^2*c^3*d^7*x^10 + 36*a^7*b*c^2*d^8*x^10 + a^8*c*d^9*x^10 + 1/9*b^8*c^10*x^9 + 80/9*a*b^7*c^9*d*x^
9 + 140*a^2*b^6*c^8*d^2*x^9 + 2240/3*a^3*b^5*c^7*d^3*x^9 + 4900/3*a^4*b^4*c^6*d^4*x^9 + 1568*a^5*b^3*c^5*d^5*x
^9 + 1960/3*a^6*b^2*c^4*d^6*x^9 + 320/3*a^7*b*c^3*d^7*x^9 + 5*a^8*c^2*d^8*x^9 + a*b^7*c^10*x^8 + 35*a^2*b^6*c^
9*d*x^8 + 315*a^3*b^5*c^8*d^2*x^8 + 1050*a^4*b^4*c^7*d^3*x^8 + 1470*a^5*b^3*c^6*d^4*x^8 + 882*a^6*b^2*c^5*d^5*
x^8 + 210*a^7*b*c^4*d^6*x^8 + 15*a^8*c^3*d^7*x^8 + 4*a^2*b^6*c^10*x^7 + 80*a^3*b^5*c^9*d*x^7 + 450*a^4*b^4*c^8
*d^2*x^7 + 960*a^5*b^3*c^7*d^3*x^7 + 840*a^6*b^2*c^6*d^4*x^7 + 288*a^7*b*c^5*d^5*x^7 + 30*a^8*c^4*d^6*x^7 + 28
/3*a^3*b^5*c^10*x^6 + 350/3*a^4*b^4*c^9*d*x^6 + 420*a^5*b^3*c^8*d^2*x^6 + 560*a^6*b^2*c^7*d^3*x^6 + 280*a^7*b*
c^6*d^4*x^6 + 42*a^8*c^5*d^5*x^6 + 14*a^4*b^4*c^10*x^5 + 112*a^5*b^3*c^9*d*x^5 + 252*a^6*b^2*c^8*d^2*x^5 + 192
*a^7*b*c^7*d^3*x^5 + 42*a^8*c^6*d^4*x^5 + 14*a^5*b^3*c^10*x^4 + 70*a^6*b^2*c^9*d*x^4 + 90*a^7*b*c^8*d^2*x^4 +
30*a^8*c^7*d^3*x^4 + 28/3*a^6*b^2*c^10*x^3 + 80/3*a^7*b*c^9*d*x^3 + 15*a^8*c^8*d^2*x^3 + 4*a^7*b*c^10*x^2 + 5*
a^8*c^9*d*x^2 + a^8*c^10*x

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Mupad [B]
time = 0.71, size = 1253, normalized size = 5.57 \begin {gather*} x^7\,\left (30\,a^8\,c^4\,d^6+288\,a^7\,b\,c^5\,d^5+840\,a^6\,b^2\,c^6\,d^4+960\,a^5\,b^3\,c^7\,d^3+450\,a^4\,b^4\,c^8\,d^2+80\,a^3\,b^5\,c^9\,d+4\,a^2\,b^6\,c^{10}\right )+x^{13}\,\left (\frac {28\,a^6\,b^2\,d^{10}}{13}+\frac {560\,a^5\,b^3\,c\,d^9}{13}+\frac {3150\,a^4\,b^4\,c^2\,d^8}{13}+\frac {6720\,a^3\,b^5\,c^3\,d^7}{13}+\frac {5880\,a^2\,b^6\,c^4\,d^6}{13}+\frac {2016\,a\,b^7\,c^5\,d^5}{13}+\frac {210\,b^8\,c^6\,d^4}{13}\right )+x^8\,\left (15\,a^8\,c^3\,d^7+210\,a^7\,b\,c^4\,d^6+882\,a^6\,b^2\,c^5\,d^5+1470\,a^5\,b^3\,c^6\,d^4+1050\,a^4\,b^4\,c^7\,d^3+315\,a^3\,b^5\,c^8\,d^2+35\,a^2\,b^6\,c^9\,d+a\,b^7\,c^{10}\right )+x^{12}\,\left (\frac {2\,a^7\,b\,d^{10}}{3}+\frac {70\,a^6\,b^2\,c\,d^9}{3}+210\,a^5\,b^3\,c^2\,d^8+700\,a^4\,b^4\,c^3\,d^7+980\,a^3\,b^5\,c^4\,d^6+588\,a^2\,b^6\,c^5\,d^5+140\,a\,b^7\,c^6\,d^4+10\,b^8\,c^7\,d^3\right )+x^{10}\,\left (a^8\,c\,d^9+36\,a^7\,b\,c^2\,d^8+336\,a^6\,b^2\,c^3\,d^7+1176\,a^5\,b^3\,c^4\,d^6+1764\,a^4\,b^4\,c^5\,d^5+1176\,a^3\,b^5\,c^6\,d^4+336\,a^2\,b^6\,c^7\,d^3+36\,a\,b^7\,c^8\,d^2+b^8\,c^9\,d\right )+x^5\,\left (42\,a^8\,c^6\,d^4+192\,a^7\,b\,c^7\,d^3+252\,a^6\,b^2\,c^8\,d^2+112\,a^5\,b^3\,c^9\,d+14\,a^4\,b^4\,c^{10}\right )+x^{15}\,\left (\frac {14\,a^4\,b^4\,d^{10}}{3}+\frac {112\,a^3\,b^5\,c\,d^9}{3}+84\,a^2\,b^6\,c^2\,d^8+64\,a\,b^7\,c^3\,d^7+14\,b^8\,c^4\,d^6\right )+x^6\,\left (42\,a^8\,c^5\,d^5+280\,a^7\,b\,c^6\,d^4+560\,a^6\,b^2\,c^7\,d^3+420\,a^5\,b^3\,c^8\,d^2+\frac {350\,a^4\,b^4\,c^9\,d}{3}+\frac {28\,a^3\,b^5\,c^{10}}{3}\right )+x^{14}\,\left (4\,a^5\,b^3\,d^{10}+50\,a^4\,b^4\,c\,d^9+180\,a^3\,b^5\,c^2\,d^8+240\,a^2\,b^6\,c^3\,d^7+120\,a\,b^7\,c^4\,d^6+18\,b^8\,c^5\,d^5\right )+x^9\,\left (5\,a^8\,c^2\,d^8+\frac {320\,a^7\,b\,c^3\,d^7}{3}+\frac {1960\,a^6\,b^2\,c^4\,d^6}{3}+1568\,a^5\,b^3\,c^5\,d^5+\frac {4900\,a^4\,b^4\,c^6\,d^4}{3}+\frac {2240\,a^3\,b^5\,c^7\,d^3}{3}+140\,a^2\,b^6\,c^8\,d^2+\frac {80\,a\,b^7\,c^9\,d}{9}+\frac {b^8\,c^{10}}{9}\right )+x^{11}\,\left (\frac {a^8\,d^{10}}{11}+\frac {80\,a^7\,b\,c\,d^9}{11}+\frac {1260\,a^6\,b^2\,c^2\,d^8}{11}+\frac {6720\,a^5\,b^3\,c^3\,d^7}{11}+\frac {14700\,a^4\,b^4\,c^4\,d^6}{11}+\frac {14112\,a^3\,b^5\,c^5\,d^5}{11}+\frac {5880\,a^2\,b^6\,c^6\,d^4}{11}+\frac {960\,a\,b^7\,c^7\,d^3}{11}+\frac {45\,b^8\,c^8\,d^2}{11}\right )+a^8\,c^{10}\,x+\frac {b^8\,d^{10}\,x^{19}}{19}+2\,a^5\,c^7\,x^4\,\left (15\,a^3\,d^3+45\,a^2\,b\,c\,d^2+35\,a\,b^2\,c^2\,d+7\,b^3\,c^3\right )+\frac {b^5\,d^7\,x^{16}\,\left (7\,a^3\,d^3+35\,a^2\,b\,c\,d^2+45\,a\,b^2\,c^2\,d+15\,b^3\,c^3\right )}{2}+a^7\,c^9\,x^2\,\left (5\,a\,d+4\,b\,c\right )+\frac {b^7\,d^9\,x^{18}\,\left (4\,a\,d+5\,b\,c\right )}{9}+\frac {a^6\,c^8\,x^3\,\left (45\,a^2\,d^2+80\,a\,b\,c\,d+28\,b^2\,c^2\right )}{3}+\frac {b^6\,d^8\,x^{17}\,\left (28\,a^2\,d^2+80\,a\,b\,c\,d+45\,b^2\,c^2\right )}{17} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^8*(c + d*x)^10,x)

[Out]

x^7*(4*a^2*b^6*c^10 + 30*a^8*c^4*d^6 + 80*a^3*b^5*c^9*d + 288*a^7*b*c^5*d^5 + 450*a^4*b^4*c^8*d^2 + 960*a^5*b^
3*c^7*d^3 + 840*a^6*b^2*c^6*d^4) + x^13*((28*a^6*b^2*d^10)/13 + (210*b^8*c^6*d^4)/13 + (2016*a*b^7*c^5*d^5)/13
+ (560*a^5*b^3*c*d^9)/13 + (5880*a^2*b^6*c^4*d^6)/13 + (6720*a^3*b^5*c^3*d^7)/13 + (3150*a^4*b^4*c^2*d^8)/13)
+ x^8*(a*b^7*c^10 + 15*a^8*c^3*d^7 + 35*a^2*b^6*c^9*d + 210*a^7*b*c^4*d^6 + 315*a^3*b^5*c^8*d^2 + 1050*a^4*b^
4*c^7*d^3 + 1470*a^5*b^3*c^6*d^4 + 882*a^6*b^2*c^5*d^5) + x^12*((2*a^7*b*d^10)/3 + 10*b^8*c^7*d^3 + 140*a*b^7*
c^6*d^4 + (70*a^6*b^2*c*d^9)/3 + 588*a^2*b^6*c^5*d^5 + 980*a^3*b^5*c^4*d^6 + 700*a^4*b^4*c^3*d^7 + 210*a^5*b^3
*c^2*d^8) + x^10*(a^8*c*d^9 + b^8*c^9*d + 36*a*b^7*c^8*d^2 + 36*a^7*b*c^2*d^8 + 336*a^2*b^6*c^7*d^3 + 1176*a^3
*b^5*c^6*d^4 + 1764*a^4*b^4*c^5*d^5 + 1176*a^5*b^3*c^4*d^6 + 336*a^6*b^2*c^3*d^7) + x^5*(14*a^4*b^4*c^10 + 42*
a^8*c^6*d^4 + 112*a^5*b^3*c^9*d + 192*a^7*b*c^7*d^3 + 252*a^6*b^2*c^8*d^2) + x^15*((14*a^4*b^4*d^10)/3 + 14*b^
8*c^4*d^6 + 64*a*b^7*c^3*d^7 + (112*a^3*b^5*c*d^9)/3 + 84*a^2*b^6*c^2*d^8) + x^6*((28*a^3*b^5*c^10)/3 + 42*a^8
*c^5*d^5 + (350*a^4*b^4*c^9*d)/3 + 280*a^7*b*c^6*d^4 + 420*a^5*b^3*c^8*d^2 + 560*a^6*b^2*c^7*d^3) + x^14*(4*a^
5*b^3*d^10 + 18*b^8*c^5*d^5 + 120*a*b^7*c^4*d^6 + 50*a^4*b^4*c*d^9 + 240*a^2*b^6*c^3*d^7 + 180*a^3*b^5*c^2*d^8
) + x^9*((b^8*c^10)/9 + 5*a^8*c^2*d^8 + (320*a^7*b*c^3*d^7)/3 + 140*a^2*b^6*c^8*d^2 + (2240*a^3*b^5*c^7*d^3)/3
+ (4900*a^4*b^4*c^6*d^4)/3 + 1568*a^5*b^3*c^5*d^5 + (1960*a^6*b^2*c^4*d^6)/3 + (80*a*b^7*c^9*d)/9) + x^11*((a
^8*d^10)/11 + (45*b^8*c^8*d^2)/11 + (960*a*b^7*c^7*d^3)/11 + (5880*a^2*b^6*c^6*d^4)/11 + (14112*a^3*b^5*c^5*d^
5)/11 + (14700*a^4*b^4*c^4*d^6)/11 + (6720*a^5*b^3*c^3*d^7)/11 + (1260*a^6*b^2*c^2*d^8)/11 + (80*a^7*b*c*d^9)/
11) + a^8*c^10*x + (b^8*d^10*x^19)/19 + 2*a^5*c^7*x^4*(15*a^3*d^3 + 7*b^3*c^3 + 35*a*b^2*c^2*d + 45*a^2*b*c*d^
2) + (b^5*d^7*x^16*(7*a^3*d^3 + 15*b^3*c^3 + 45*a*b^2*c^2*d + 35*a^2*b*c*d^2))/2 + a^7*c^9*x^2*(5*a*d + 4*b*c)
+ (b^7*d^9*x^18*(4*a*d + 5*b*c))/9 + (a^6*c^8*x^3*(45*a^2*d^2 + 28*b^2*c^2 + 80*a*b*c*d))/3 + (b^6*d^8*x^17*(
28*a^2*d^2 + 45*b^2*c^2 + 80*a*b*c*d))/17

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