3.1303 \(\int (a+b x)^8 (c+d x)^{10} \, dx\)
Optimal. Leaf size=225 \[ -\frac{4 b^7 (c+d x)^{18} (b c-a d)}{9 d^9}+\frac{28 b^6 (c+d x)^{17} (b c-a d)^2}{17 d^9}-\frac{7 b^5 (c+d x)^{16} (b c-a d)^3}{2 d^9}+\frac{14 b^4 (c+d x)^{15} (b c-a d)^4}{3 d^9}-\frac{4 b^3 (c+d x)^{14} (b c-a d)^5}{d^9}+\frac{28 b^2 (c+d x)^{13} (b c-a d)^6}{13 d^9}-\frac{2 b (c+d x)^{12} (b c-a d)^7}{3 d^9}+\frac{(c+d x)^{11} (b c-a d)^8}{11 d^9}+\frac{b^8 (c+d x)^{19}}{19 d^9} \]
[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)
________________________________________________________________________________________
Rubi [A] time = 0.899381, antiderivative size = 225, 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} \[ -\frac{4 b^7 (c+d x)^{18} (b c-a d)}{9 d^9}+\frac{28 b^6 (c+d x)^{17} (b c-a d)^2}{17 d^9}-\frac{7 b^5 (c+d x)^{16} (b c-a d)^3}{2 d^9}+\frac{14 b^4 (c+d x)^{15} (b c-a d)^4}{3 d^9}-\frac{4 b^3 (c+d x)^{14} (b c-a d)^5}{d^9}+\frac{28 b^2 (c+d x)^{13} (b c-a d)^6}{13 d^9}-\frac{2 b (c+d x)^{12} (b c-a d)^7}{3 d^9}+\frac{(c+d x)^{11} (b c-a d)^8}{11 d^9}+\frac{b^8 (c+d x)^{19}}{19 d^9} \]
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 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)^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*}
Mathematica [B] time = 0.153915, size = 1241, normalized size = 5.52 \[ \frac{1}{19} b^8 d^{10} x^{19}+\frac{1}{9} b^7 d^9 (5 b c+4 a d) x^{18}+\frac{1}{17} b^6 d^8 \left (45 b^2 c^2+80 a b d c+28 a^2 d^2\right ) x^{17}+\frac{1}{2} b^5 d^7 \left (15 b^3 c^3+45 a b^2 d c^2+35 a^2 b d^2 c+7 a^3 d^3\right ) x^{16}+\frac{2}{3} b^4 d^6 \left (21 b^4 c^4+96 a b^3 d c^3+126 a^2 b^2 d^2 c^2+56 a^3 b d^3 c+7 a^4 d^4\right ) x^{15}+2 b^3 d^5 \left (9 b^5 c^5+60 a b^4 d c^4+120 a^2 b^3 d^2 c^3+90 a^3 b^2 d^3 c^2+25 a^4 b d^4 c+2 a^5 d^5\right ) x^{14}+\frac{14}{13} b^2 d^4 \left (15 b^6 c^6+144 a b^5 d c^5+420 a^2 b^4 d^2 c^4+480 a^3 b^3 d^3 c^3+225 a^4 b^2 d^4 c^2+40 a^5 b d^5 c+2 a^6 d^6\right ) x^{13}+\frac{2}{3} b d^3 \left (15 b^7 c^7+210 a b^6 d c^6+882 a^2 b^5 d^2 c^5+1470 a^3 b^4 d^3 c^4+1050 a^4 b^3 d^4 c^3+315 a^5 b^2 d^5 c^2+35 a^6 b d^6 c+a^7 d^7\right ) x^{12}+\frac{1}{11} d^2 \left (45 b^8 c^8+960 a b^7 d c^7+5880 a^2 b^6 d^2 c^6+14112 a^3 b^5 d^3 c^5+14700 a^4 b^4 d^4 c^4+6720 a^5 b^3 d^5 c^3+1260 a^6 b^2 d^6 c^2+80 a^7 b d^7 c+a^8 d^8\right ) x^{11}+c d \left (b^8 c^8+36 a b^7 d c^7+336 a^2 b^6 d^2 c^6+1176 a^3 b^5 d^3 c^5+1764 a^4 b^4 d^4 c^4+1176 a^5 b^3 d^5 c^3+336 a^6 b^2 d^6 c^2+36 a^7 b d^7 c+a^8 d^8\right ) x^{10}+\frac{1}{9} c^2 \left (b^8 c^8+80 a b^7 d c^7+1260 a^2 b^6 d^2 c^6+6720 a^3 b^5 d^3 c^5+14700 a^4 b^4 d^4 c^4+14112 a^5 b^3 d^5 c^3+5880 a^6 b^2 d^6 c^2+960 a^7 b d^7 c+45 a^8 d^8\right ) x^9+a c^3 \left (b^7 c^7+35 a b^6 d c^6+315 a^2 b^5 d^2 c^5+1050 a^3 b^4 d^3 c^4+1470 a^4 b^3 d^4 c^3+882 a^5 b^2 d^5 c^2+210 a^6 b d^6 c+15 a^7 d^7\right ) x^8+2 a^2 c^4 \left (2 b^6 c^6+40 a b^5 d c^5+225 a^2 b^4 d^2 c^4+480 a^3 b^3 d^3 c^3+420 a^4 b^2 d^4 c^2+144 a^5 b d^5 c+15 a^6 d^6\right ) x^7+\frac{14}{3} a^3 c^5 \left (2 b^5 c^5+25 a b^4 d c^4+90 a^2 b^3 d^2 c^3+120 a^3 b^2 d^3 c^2+60 a^4 b d^4 c+9 a^5 d^5\right ) x^6+2 a^4 c^6 \left (7 b^4 c^4+56 a b^3 d c^3+126 a^2 b^2 d^2 c^2+96 a^3 b d^3 c+21 a^4 d^4\right ) x^5+2 a^5 c^7 \left (7 b^3 c^3+35 a b^2 d c^2+45 a^2 b d^2 c+15 a^3 d^3\right ) x^4+\frac{1}{3} a^6 c^8 \left (28 b^2 c^2+80 a b d c+45 a^2 d^2\right ) x^3+a^7 c^9 (4 b c+5 a d) x^2+a^8 c^{10} x \]
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
________________________________________________________________________________________
Maple [B] time = 0.002, size = 1291, normalized size = 5.7 \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)^8*(d*x+c)^10,x)
[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
________________________________________________________________________________________
Maxima [B] time = 0.997121, size = 1732, normalized size = 7.7 \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)^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
________________________________________________________________________________________
Fricas [B] time = 1.6589, size = 3305, normalized size = 14.69 \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)^8*(d*x+c)^10,x, algorithm="fricas")
[Out]
1/19*x^19*d^10*b^8 + 5/9*x^18*d^9*c*b^8 + 4/9*x^18*d^10*b^7*a + 45/17*x^17*d^8*c^2*b^8 + 80/17*x^17*d^9*c*b^7*
a + 28/17*x^17*d^10*b^6*a^2 + 15/2*x^16*d^7*c^3*b^8 + 45/2*x^16*d^8*c^2*b^7*a + 35/2*x^16*d^9*c*b^6*a^2 + 7/2*
x^16*d^10*b^5*a^3 + 14*x^15*d^6*c^4*b^8 + 64*x^15*d^7*c^3*b^7*a + 84*x^15*d^8*c^2*b^6*a^2 + 112/3*x^15*d^9*c*b
^5*a^3 + 14/3*x^15*d^10*b^4*a^4 + 18*x^14*d^5*c^5*b^8 + 120*x^14*d^6*c^4*b^7*a + 240*x^14*d^7*c^3*b^6*a^2 + 18
0*x^14*d^8*c^2*b^5*a^3 + 50*x^14*d^9*c*b^4*a^4 + 4*x^14*d^10*b^3*a^5 + 210/13*x^13*d^4*c^6*b^8 + 2016/13*x^13*
d^5*c^5*b^7*a + 5880/13*x^13*d^6*c^4*b^6*a^2 + 6720/13*x^13*d^7*c^3*b^5*a^3 + 3150/13*x^13*d^8*c^2*b^4*a^4 + 5
60/13*x^13*d^9*c*b^3*a^5 + 28/13*x^13*d^10*b^2*a^6 + 10*x^12*d^3*c^7*b^8 + 140*x^12*d^4*c^6*b^7*a + 588*x^12*d
^5*c^5*b^6*a^2 + 980*x^12*d^6*c^4*b^5*a^3 + 700*x^12*d^7*c^3*b^4*a^4 + 210*x^12*d^8*c^2*b^3*a^5 + 70/3*x^12*d^
9*c*b^2*a^6 + 2/3*x^12*d^10*b*a^7 + 45/11*x^11*d^2*c^8*b^8 + 960/11*x^11*d^3*c^7*b^7*a + 5880/11*x^11*d^4*c^6*
b^6*a^2 + 14112/11*x^11*d^5*c^5*b^5*a^3 + 14700/11*x^11*d^6*c^4*b^4*a^4 + 6720/11*x^11*d^7*c^3*b^3*a^5 + 1260/
11*x^11*d^8*c^2*b^2*a^6 + 80/11*x^11*d^9*c*b*a^7 + 1/11*x^11*d^10*a^8 + x^10*d*c^9*b^8 + 36*x^10*d^2*c^8*b^7*a
+ 336*x^10*d^3*c^7*b^6*a^2 + 1176*x^10*d^4*c^6*b^5*a^3 + 1764*x^10*d^5*c^5*b^4*a^4 + 1176*x^10*d^6*c^4*b^3*a^
5 + 336*x^10*d^7*c^3*b^2*a^6 + 36*x^10*d^8*c^2*b*a^7 + x^10*d^9*c*a^8 + 1/9*x^9*c^10*b^8 + 80/9*x^9*d*c^9*b^7*
a + 140*x^9*d^2*c^8*b^6*a^2 + 2240/3*x^9*d^3*c^7*b^5*a^3 + 4900/3*x^9*d^4*c^6*b^4*a^4 + 1568*x^9*d^5*c^5*b^3*a
^5 + 1960/3*x^9*d^6*c^4*b^2*a^6 + 320/3*x^9*d^7*c^3*b*a^7 + 5*x^9*d^8*c^2*a^8 + x^8*c^10*b^7*a + 35*x^8*d*c^9*
b^6*a^2 + 315*x^8*d^2*c^8*b^5*a^3 + 1050*x^8*d^3*c^7*b^4*a^4 + 1470*x^8*d^4*c^6*b^3*a^5 + 882*x^8*d^5*c^5*b^2*
a^6 + 210*x^8*d^6*c^4*b*a^7 + 15*x^8*d^7*c^3*a^8 + 4*x^7*c^10*b^6*a^2 + 80*x^7*d*c^9*b^5*a^3 + 450*x^7*d^2*c^8
*b^4*a^4 + 960*x^7*d^3*c^7*b^3*a^5 + 840*x^7*d^4*c^6*b^2*a^6 + 288*x^7*d^5*c^5*b*a^7 + 30*x^7*d^6*c^4*a^8 + 28
/3*x^6*c^10*b^5*a^3 + 350/3*x^6*d*c^9*b^4*a^4 + 420*x^6*d^2*c^8*b^3*a^5 + 560*x^6*d^3*c^7*b^2*a^6 + 280*x^6*d^
4*c^6*b*a^7 + 42*x^6*d^5*c^5*a^8 + 14*x^5*c^10*b^4*a^4 + 112*x^5*d*c^9*b^3*a^5 + 252*x^5*d^2*c^8*b^2*a^6 + 192
*x^5*d^3*c^7*b*a^7 + 42*x^5*d^4*c^6*a^8 + 14*x^4*c^10*b^3*a^5 + 70*x^4*d*c^9*b^2*a^6 + 90*x^4*d^2*c^8*b*a^7 +
30*x^4*d^3*c^7*a^8 + 28/3*x^3*c^10*b^2*a^6 + 80/3*x^3*d*c^9*b*a^7 + 15*x^3*d^2*c^8*a^8 + 4*x^2*c^10*b*a^7 + 5*
x^2*d*c^9*a^8 + x*c^10*a^8
________________________________________________________________________________________
Sympy [B] time = 0.227105, size = 1428, normalized size = 6.35 \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)**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] time = 1.07491, size = 1995, normalized size = 8.87 \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)^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