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)
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Rubi [A] time = 1.85141, 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
\[ -\frac{4 b^7 (c+d x)^{18} (b c-a d)}{9 d^9}+\frac{28 b^6 (c+d x)^{17} (b c-a d)^2}{17 d^9}-\frac{7 b^5 (c+d x)^{16} (b c-a d)^3}{2 d^9}+\frac{14 b^4 (c+d x)^{15} (b c-a d)^4}{3 d^9}-\frac{4 b^3 (c+d x)^{14} (b c-a d)^5}{d^9}+\frac{28 b^2 (c+d x)^{13} (b c-a d)^6}{13 d^9}-\frac{2 b (c+d x)^{12} (b c-a d)^7}{3 d^9}+\frac{(c+d x)^{11} (b c-a d)^8}{11 d^9}+\frac{b^8 (c+d x)^{19}}{19 d^9} \]
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)
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Rubi in Sympy [A] time = 178.323, size = 207, normalized size = 0.92 \[ \frac{b^{8} \left (c + d x\right )^{19}}{19 d^{9}} + \frac{4 b^{7} \left (c + d x\right )^{18} \left (a d - b c\right )}{9 d^{9}} + \frac{28 b^{6} \left (c + d x\right )^{17} \left (a d - b c\right )^{2}}{17 d^{9}} + \frac{7 b^{5} \left (c + d x\right )^{16} \left (a d - b c\right )^{3}}{2 d^{9}} + \frac{14 b^{4} \left (c + d x\right )^{15} \left (a d - b c\right )^{4}}{3 d^{9}} + \frac{4 b^{3} \left (c + d x\right )^{14} \left (a d - b c\right )^{5}}{d^{9}} + \frac{28 b^{2} \left (c + d x\right )^{13} \left (a d - b c\right )^{6}}{13 d^{9}} + \frac{2 b \left (c + d x\right )^{12} \left (a d - b c\right )^{7}}{3 d^{9}} + \frac{\left (c + d x\right )^{11} \left (a d - b c\right )^{8}}{11 d^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**8*(d*x+c)**10,x)
[Out]
b**8*(c + d*x)**19/(19*d**9) + 4*b**7*(c + d*x)**18*(a*d - b*c)/(9*d**9) + 28*b*
*6*(c + d*x)**17*(a*d - b*c)**2/(17*d**9) + 7*b**5*(c + d*x)**16*(a*d - b*c)**3/
(2*d**9) + 14*b**4*(c + d*x)**15*(a*d - b*c)**4/(3*d**9) + 4*b**3*(c + d*x)**14*
(a*d - b*c)**5/d**9 + 28*b**2*(c + d*x)**13*(a*d - b*c)**6/(13*d**9) + 2*b*(c +
d*x)**12*(a*d - b*c)**7/(3*d**9) + (c + d*x)**11*(a*d - b*c)**8/(11*d**9)
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Mathematica [B] time = 0.297295, 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 + 4
5*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 + 42
0*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 + 88
2*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 + 1
4112*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 + 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)*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] time = 0.004, size = 1291, normalized size = 5.7 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^8*(d*x+c)^10,x)
[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+8
0*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+70
0*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+14
700*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+1260*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+1
680*a^7*b*c^4*d^6+7056*a^6*b^2*c^5*d^5+11760*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 [A] time = 1.3713, size = 1732, normalized size = 7.7 \[ \text{result too large to display} \]
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 + 4
0*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^1
0 + 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 [A] time = 0.200447, size = 1, normalized size = 0. \[ \text{result too large to display} \]
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 + 180*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 + 560/13*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 + 15
68*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 [A] time = 0.674398, size = 1428, normalized size = 6.35 \[ \text{result too large to display} \]
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 + 56
0*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**7/1
3 + 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 + 2
240*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 + 112*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**1
0/3) + x**2*(5*a**8*c**9*d + 4*a**7*b*c**10)
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.214816, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^8*(d*x + c)^10,x, algorithm="giac")
[Out]
Done