3.1302 \(\int (a+b x)^9 (c+d x)^{10} \, dx\)

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

[Out]

-((b*c - a*d)^9*(c + d*x)^11)/(11*d^10) + (3*b*(b*c - a*d)^8*(c + d*x)^12)/(4*d^
10) - (36*b^2*(b*c - a*d)^7*(c + d*x)^13)/(13*d^10) + (6*b^3*(b*c - a*d)^6*(c +
d*x)^14)/d^10 - (42*b^4*(b*c - a*d)^5*(c + d*x)^15)/(5*d^10) + (63*b^5*(b*c - a*
d)^4*(c + d*x)^16)/(8*d^10) - (84*b^6*(b*c - a*d)^3*(c + d*x)^17)/(17*d^10) + (2
*b^7*(b*c - a*d)^2*(c + d*x)^18)/d^10 - (9*b^8*(b*c - a*d)*(c + d*x)^19)/(19*d^1
0) + (b^9*(c + d*x)^20)/(20*d^10)

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Rubi [A]  time = 2.10657, antiderivative size = 250, 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 \[ -\frac{9 b^8 (c+d x)^{19} (b c-a d)}{19 d^{10}}+\frac{2 b^7 (c+d x)^{18} (b c-a d)^2}{d^{10}}-\frac{84 b^6 (c+d x)^{17} (b c-a d)^3}{17 d^{10}}+\frac{63 b^5 (c+d x)^{16} (b c-a d)^4}{8 d^{10}}-\frac{42 b^4 (c+d x)^{15} (b c-a d)^5}{5 d^{10}}+\frac{6 b^3 (c+d x)^{14} (b c-a d)^6}{d^{10}}-\frac{36 b^2 (c+d x)^{13} (b c-a d)^7}{13 d^{10}}+\frac{3 b (c+d x)^{12} (b c-a d)^8}{4 d^{10}}-\frac{(c+d x)^{11} (b c-a d)^9}{11 d^{10}}+\frac{b^9 (c+d x)^{20}}{20 d^{10}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^9*(c + d*x)^10,x]

[Out]

-((b*c - a*d)^9*(c + d*x)^11)/(11*d^10) + (3*b*(b*c - a*d)^8*(c + d*x)^12)/(4*d^
10) - (36*b^2*(b*c - a*d)^7*(c + d*x)^13)/(13*d^10) + (6*b^3*(b*c - a*d)^6*(c +
d*x)^14)/d^10 - (42*b^4*(b*c - a*d)^5*(c + d*x)^15)/(5*d^10) + (63*b^5*(b*c - a*
d)^4*(c + d*x)^16)/(8*d^10) - (84*b^6*(b*c - a*d)^3*(c + d*x)^17)/(17*d^10) + (2
*b^7*(b*c - a*d)^2*(c + d*x)^18)/d^10 - (9*b^8*(b*c - a*d)*(c + d*x)^19)/(19*d^1
0) + (b^9*(c + d*x)^20)/(20*d^10)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**9*(d*x+c)**10,x)

[Out]

Timed out

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Mathematica [B]  time = 0.339223, size = 1397, normalized size = 5.59 \[ \frac{1}{20} b^9 d^{10} x^{20}+\frac{1}{19} b^8 d^9 (10 b c+9 a d) x^{19}+\frac{1}{2} b^7 d^8 \left (5 b^2 c^2+10 a b d c+4 a^2 d^2\right ) x^{18}+\frac{3}{17} b^6 d^7 \left (40 b^3 c^3+135 a b^2 d c^2+120 a^2 b d^2 c+28 a^3 d^3\right ) x^{17}+\frac{3}{8} b^5 d^6 \left (35 b^4 c^4+180 a b^3 d c^3+270 a^2 b^2 d^2 c^2+140 a^3 b d^3 c+21 a^4 d^4\right ) x^{16}+\frac{6}{5} b^4 d^5 \left (14 b^5 c^5+105 a b^4 d c^4+240 a^2 b^3 d^2 c^3+210 a^3 b^2 d^3 c^2+70 a^4 b d^4 c+7 a^5 d^5\right ) x^{15}+3 b^3 d^4 \left (5 b^6 c^6+54 a b^5 d c^5+180 a^2 b^4 d^2 c^4+240 a^3 b^3 d^3 c^3+135 a^4 b^2 d^4 c^2+30 a^5 b d^5 c+2 a^6 d^6\right ) x^{14}+\frac{6}{13} b^2 d^3 \left (20 b^7 c^7+315 a b^6 d c^6+1512 a^2 b^5 d^2 c^5+2940 a^3 b^4 d^3 c^4+2520 a^4 b^3 d^4 c^3+945 a^5 b^2 d^5 c^2+140 a^6 b d^6 c+6 a^7 d^7\right ) x^{13}+\frac{3}{4} b d^2 \left (5 b^8 c^8+120 a b^7 d c^7+840 a^2 b^6 d^2 c^6+2352 a^3 b^5 d^3 c^5+2940 a^4 b^4 d^4 c^4+1680 a^5 b^3 d^5 c^3+420 a^6 b^2 d^6 c^2+40 a^7 b d^7 c+a^8 d^8\right ) x^{12}+\frac{1}{11} d \left (10 b^9 c^9+405 a b^8 d c^8+4320 a^2 b^7 d^2 c^7+17640 a^3 b^6 d^3 c^6+31752 a^4 b^5 d^4 c^5+26460 a^5 b^4 d^5 c^4+10080 a^6 b^3 d^6 c^3+1620 a^7 b^2 d^7 c^2+90 a^8 b d^8 c+a^9 d^9\right ) x^{11}+\frac{1}{10} c \left (b^9 c^9+90 a b^8 d c^8+1620 a^2 b^7 d^2 c^7+10080 a^3 b^6 d^3 c^6+26460 a^4 b^5 d^4 c^5+31752 a^5 b^4 d^5 c^4+17640 a^6 b^3 d^6 c^3+4320 a^7 b^2 d^7 c^2+405 a^8 b d^8 c+10 a^9 d^9\right ) x^{10}+a c^2 \left (b^8 c^8+40 a b^7 d c^7+420 a^2 b^6 d^2 c^6+1680 a^3 b^5 d^3 c^5+2940 a^4 b^4 d^4 c^4+2352 a^5 b^3 d^5 c^3+840 a^6 b^2 d^6 c^2+120 a^7 b d^7 c+5 a^8 d^8\right ) x^9+\frac{3}{4} a^2 c^3 \left (6 b^7 c^7+140 a b^6 d c^6+945 a^2 b^5 d^2 c^5+2520 a^3 b^4 d^3 c^4+2940 a^4 b^3 d^4 c^3+1512 a^5 b^2 d^5 c^2+315 a^6 b d^6 c+20 a^7 d^7\right ) x^8+6 a^3 c^4 \left (2 b^6 c^6+30 a b^5 d c^5+135 a^2 b^4 d^2 c^4+240 a^3 b^3 d^3 c^3+180 a^4 b^2 d^4 c^2+54 a^5 b d^5 c+5 a^6 d^6\right ) x^7+3 a^4 c^5 \left (7 b^5 c^5+70 a b^4 d c^4+210 a^2 b^3 d^2 c^3+240 a^3 b^2 d^3 c^2+105 a^4 b d^4 c+14 a^5 d^5\right ) x^6+\frac{6}{5} a^5 c^6 \left (21 b^4 c^4+140 a b^3 d c^3+270 a^2 b^2 d^2 c^2+180 a^3 b d^3 c+35 a^4 d^4\right ) x^5+\frac{3}{4} a^6 c^7 \left (28 b^3 c^3+120 a b^2 d c^2+135 a^2 b d^2 c+40 a^3 d^3\right ) x^4+3 a^7 c^8 \left (4 b^2 c^2+10 a b d c+5 a^2 d^2\right ) x^3+\frac{1}{2} a^8 c^9 (9 b c+10 a d) x^2+a^9 c^{10} x \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^9*(c + d*x)^10,x]

[Out]

a^9*c^10*x + (a^8*c^9*(9*b*c + 10*a*d)*x^2)/2 + 3*a^7*c^8*(4*b^2*c^2 + 10*a*b*c*
d + 5*a^2*d^2)*x^3 + (3*a^6*c^7*(28*b^3*c^3 + 120*a*b^2*c^2*d + 135*a^2*b*c*d^2
+ 40*a^3*d^3)*x^4)/4 + (6*a^5*c^6*(21*b^4*c^4 + 140*a*b^3*c^3*d + 270*a^2*b^2*c^
2*d^2 + 180*a^3*b*c*d^3 + 35*a^4*d^4)*x^5)/5 + 3*a^4*c^5*(7*b^5*c^5 + 70*a*b^4*c
^4*d + 210*a^2*b^3*c^3*d^2 + 240*a^3*b^2*c^2*d^3 + 105*a^4*b*c*d^4 + 14*a^5*d^5)
*x^6 + 6*a^3*c^4*(2*b^6*c^6 + 30*a*b^5*c^5*d + 135*a^2*b^4*c^4*d^2 + 240*a^3*b^3
*c^3*d^3 + 180*a^4*b^2*c^2*d^4 + 54*a^5*b*c*d^5 + 5*a^6*d^6)*x^7 + (3*a^2*c^3*(6
*b^7*c^7 + 140*a*b^6*c^6*d + 945*a^2*b^5*c^5*d^2 + 2520*a^3*b^4*c^4*d^3 + 2940*a
^4*b^3*c^3*d^4 + 1512*a^5*b^2*c^2*d^5 + 315*a^6*b*c*d^6 + 20*a^7*d^7)*x^8)/4 + a
*c^2*(b^8*c^8 + 40*a*b^7*c^7*d + 420*a^2*b^6*c^6*d^2 + 1680*a^3*b^5*c^5*d^3 + 29
40*a^4*b^4*c^4*d^4 + 2352*a^5*b^3*c^3*d^5 + 840*a^6*b^2*c^2*d^6 + 120*a^7*b*c*d^
7 + 5*a^8*d^8)*x^9 + (c*(b^9*c^9 + 90*a*b^8*c^8*d + 1620*a^2*b^7*c^7*d^2 + 10080
*a^3*b^6*c^6*d^3 + 26460*a^4*b^5*c^5*d^4 + 31752*a^5*b^4*c^4*d^5 + 17640*a^6*b^3
*c^3*d^6 + 4320*a^7*b^2*c^2*d^7 + 405*a^8*b*c*d^8 + 10*a^9*d^9)*x^10)/10 + (d*(1
0*b^9*c^9 + 405*a*b^8*c^8*d + 4320*a^2*b^7*c^7*d^2 + 17640*a^3*b^6*c^6*d^3 + 317
52*a^4*b^5*c^5*d^4 + 26460*a^5*b^4*c^4*d^5 + 10080*a^6*b^3*c^3*d^6 + 1620*a^7*b^
2*c^2*d^7 + 90*a^8*b*c*d^8 + a^9*d^9)*x^11)/11 + (3*b*d^2*(5*b^8*c^8 + 120*a*b^7
*c^7*d + 840*a^2*b^6*c^6*d^2 + 2352*a^3*b^5*c^5*d^3 + 2940*a^4*b^4*c^4*d^4 + 168
0*a^5*b^3*c^3*d^5 + 420*a^6*b^2*c^2*d^6 + 40*a^7*b*c*d^7 + a^8*d^8)*x^12)/4 + (6
*b^2*d^3*(20*b^7*c^7 + 315*a*b^6*c^6*d + 1512*a^2*b^5*c^5*d^2 + 2940*a^3*b^4*c^4
*d^3 + 2520*a^4*b^3*c^3*d^4 + 945*a^5*b^2*c^2*d^5 + 140*a^6*b*c*d^6 + 6*a^7*d^7)
*x^13)/13 + 3*b^3*d^4*(5*b^6*c^6 + 54*a*b^5*c^5*d + 180*a^2*b^4*c^4*d^2 + 240*a^
3*b^3*c^3*d^3 + 135*a^4*b^2*c^2*d^4 + 30*a^5*b*c*d^5 + 2*a^6*d^6)*x^14 + (6*b^4*
d^5*(14*b^5*c^5 + 105*a*b^4*c^4*d + 240*a^2*b^3*c^3*d^2 + 210*a^3*b^2*c^2*d^3 +
70*a^4*b*c*d^4 + 7*a^5*d^5)*x^15)/5 + (3*b^5*d^6*(35*b^4*c^4 + 180*a*b^3*c^3*d +
270*a^2*b^2*c^2*d^2 + 140*a^3*b*c*d^3 + 21*a^4*d^4)*x^16)/8 + (3*b^6*d^7*(40*b^
3*c^3 + 135*a*b^2*c^2*d + 120*a^2*b*c*d^2 + 28*a^3*d^3)*x^17)/17 + (b^7*d^8*(5*b
^2*c^2 + 10*a*b*c*d + 4*a^2*d^2)*x^18)/2 + (b^8*d^9*(10*b*c + 9*a*d)*x^19)/19 +
(b^9*d^10*x^20)/20

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Maple [B]  time = 0.003, size = 1441, normalized size = 5.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^9*(d*x+c)^10,x)

[Out]

1/20*b^9*d^10*x^20+1/19*(9*a*b^8*d^10+10*b^9*c*d^9)*x^19+1/18*(36*a^2*b^7*d^10+9
0*a*b^8*c*d^9+45*b^9*c^2*d^8)*x^18+1/17*(84*a^3*b^6*d^10+360*a^2*b^7*c*d^9+405*a
*b^8*c^2*d^8+120*b^9*c^3*d^7)*x^17+1/16*(126*a^4*b^5*d^10+840*a^3*b^6*c*d^9+1620
*a^2*b^7*c^2*d^8+1080*a*b^8*c^3*d^7+210*b^9*c^4*d^6)*x^16+1/15*(126*a^5*b^4*d^10
+1260*a^4*b^5*c*d^9+3780*a^3*b^6*c^2*d^8+4320*a^2*b^7*c^3*d^7+1890*a*b^8*c^4*d^6
+252*b^9*c^5*d^5)*x^15+1/14*(84*a^6*b^3*d^10+1260*a^5*b^4*c*d^9+5670*a^4*b^5*c^2
*d^8+10080*a^3*b^6*c^3*d^7+7560*a^2*b^7*c^4*d^6+2268*a*b^8*c^5*d^5+210*b^9*c^6*d
^4)*x^14+1/13*(36*a^7*b^2*d^10+840*a^6*b^3*c*d^9+5670*a^5*b^4*c^2*d^8+15120*a^4*
b^5*c^3*d^7+17640*a^3*b^6*c^4*d^6+9072*a^2*b^7*c^5*d^5+1890*a*b^8*c^6*d^4+120*b^
9*c^7*d^3)*x^13+1/12*(9*a^8*b*d^10+360*a^7*b^2*c*d^9+3780*a^6*b^3*c^2*d^8+15120*
a^5*b^4*c^3*d^7+26460*a^4*b^5*c^4*d^6+21168*a^3*b^6*c^5*d^5+7560*a^2*b^7*c^6*d^4
+1080*a*b^8*c^7*d^3+45*b^9*c^8*d^2)*x^12+1/11*(a^9*d^10+90*a^8*b*c*d^9+1620*a^7*
b^2*c^2*d^8+10080*a^6*b^3*c^3*d^7+26460*a^5*b^4*c^4*d^6+31752*a^4*b^5*c^5*d^5+17
640*a^3*b^6*c^6*d^4+4320*a^2*b^7*c^7*d^3+405*a*b^8*c^8*d^2+10*b^9*c^9*d)*x^11+1/
10*(10*a^9*c*d^9+405*a^8*b*c^2*d^8+4320*a^7*b^2*c^3*d^7+17640*a^6*b^3*c^4*d^6+31
752*a^5*b^4*c^5*d^5+26460*a^4*b^5*c^6*d^4+10080*a^3*b^6*c^7*d^3+1620*a^2*b^7*c^8
*d^2+90*a*b^8*c^9*d+b^9*c^10)*x^10+1/9*(45*a^9*c^2*d^8+1080*a^8*b*c^3*d^7+7560*a
^7*b^2*c^4*d^6+21168*a^6*b^3*c^5*d^5+26460*a^5*b^4*c^6*d^4+15120*a^4*b^5*c^7*d^3
+3780*a^3*b^6*c^8*d^2+360*a^2*b^7*c^9*d+9*a*b^8*c^10)*x^9+1/8*(120*a^9*c^3*d^7+1
890*a^8*b*c^4*d^6+9072*a^7*b^2*c^5*d^5+17640*a^6*b^3*c^6*d^4+15120*a^5*b^4*c^7*d
^3+5670*a^4*b^5*c^8*d^2+840*a^3*b^6*c^9*d+36*a^2*b^7*c^10)*x^8+1/7*(210*a^9*c^4*
d^6+2268*a^8*b*c^5*d^5+7560*a^7*b^2*c^6*d^4+10080*a^6*b^3*c^7*d^3+5670*a^5*b^4*c
^8*d^2+1260*a^4*b^5*c^9*d+84*a^3*b^6*c^10)*x^7+1/6*(252*a^9*c^5*d^5+1890*a^8*b*c
^6*d^4+4320*a^7*b^2*c^7*d^3+3780*a^6*b^3*c^8*d^2+1260*a^5*b^4*c^9*d+126*a^4*b^5*
c^10)*x^6+1/5*(210*a^9*c^6*d^4+1080*a^8*b*c^7*d^3+1620*a^7*b^2*c^8*d^2+840*a^6*b
^3*c^9*d+126*a^5*b^4*c^10)*x^5+1/4*(120*a^9*c^7*d^3+405*a^8*b*c^8*d^2+360*a^7*b^
2*c^9*d+84*a^6*b^3*c^10)*x^4+1/3*(45*a^9*c^8*d^2+90*a^8*b*c^9*d+36*a^7*b^2*c^10)
*x^3+1/2*(10*a^9*c^9*d+9*a^8*b*c^10)*x^2+a^9*c^10*x

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Maxima [A]  time = 1.36834, size = 1940, normalized size = 7.76 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/20*b^9*d^10*x^20 + a^9*c^10*x + 1/19*(10*b^9*c*d^9 + 9*a*b^8*d^10)*x^19 + 1/2*
(5*b^9*c^2*d^8 + 10*a*b^8*c*d^9 + 4*a^2*b^7*d^10)*x^18 + 3/17*(40*b^9*c^3*d^7 +
135*a*b^8*c^2*d^8 + 120*a^2*b^7*c*d^9 + 28*a^3*b^6*d^10)*x^17 + 3/8*(35*b^9*c^4*
d^6 + 180*a*b^8*c^3*d^7 + 270*a^2*b^7*c^2*d^8 + 140*a^3*b^6*c*d^9 + 21*a^4*b^5*d
^10)*x^16 + 6/5*(14*b^9*c^5*d^5 + 105*a*b^8*c^4*d^6 + 240*a^2*b^7*c^3*d^7 + 210*
a^3*b^6*c^2*d^8 + 70*a^4*b^5*c*d^9 + 7*a^5*b^4*d^10)*x^15 + 3*(5*b^9*c^6*d^4 + 5
4*a*b^8*c^5*d^5 + 180*a^2*b^7*c^4*d^6 + 240*a^3*b^6*c^3*d^7 + 135*a^4*b^5*c^2*d^
8 + 30*a^5*b^4*c*d^9 + 2*a^6*b^3*d^10)*x^14 + 6/13*(20*b^9*c^7*d^3 + 315*a*b^8*c
^6*d^4 + 1512*a^2*b^7*c^5*d^5 + 2940*a^3*b^6*c^4*d^6 + 2520*a^4*b^5*c^3*d^7 + 94
5*a^5*b^4*c^2*d^8 + 140*a^6*b^3*c*d^9 + 6*a^7*b^2*d^10)*x^13 + 3/4*(5*b^9*c^8*d^
2 + 120*a*b^8*c^7*d^3 + 840*a^2*b^7*c^6*d^4 + 2352*a^3*b^6*c^5*d^5 + 2940*a^4*b^
5*c^4*d^6 + 1680*a^5*b^4*c^3*d^7 + 420*a^6*b^3*c^2*d^8 + 40*a^7*b^2*c*d^9 + a^8*
b*d^10)*x^12 + 1/11*(10*b^9*c^9*d + 405*a*b^8*c^8*d^2 + 4320*a^2*b^7*c^7*d^3 + 1
7640*a^3*b^6*c^6*d^4 + 31752*a^4*b^5*c^5*d^5 + 26460*a^5*b^4*c^4*d^6 + 10080*a^6
*b^3*c^3*d^7 + 1620*a^7*b^2*c^2*d^8 + 90*a^8*b*c*d^9 + a^9*d^10)*x^11 + 1/10*(b^
9*c^10 + 90*a*b^8*c^9*d + 1620*a^2*b^7*c^8*d^2 + 10080*a^3*b^6*c^7*d^3 + 26460*a
^4*b^5*c^6*d^4 + 31752*a^5*b^4*c^5*d^5 + 17640*a^6*b^3*c^4*d^6 + 4320*a^7*b^2*c^
3*d^7 + 405*a^8*b*c^2*d^8 + 10*a^9*c*d^9)*x^10 + (a*b^8*c^10 + 40*a^2*b^7*c^9*d
+ 420*a^3*b^6*c^8*d^2 + 1680*a^4*b^5*c^7*d^3 + 2940*a^5*b^4*c^6*d^4 + 2352*a^6*b
^3*c^5*d^5 + 840*a^7*b^2*c^4*d^6 + 120*a^8*b*c^3*d^7 + 5*a^9*c^2*d^8)*x^9 + 3/4*
(6*a^2*b^7*c^10 + 140*a^3*b^6*c^9*d + 945*a^4*b^5*c^8*d^2 + 2520*a^5*b^4*c^7*d^3
+ 2940*a^6*b^3*c^6*d^4 + 1512*a^7*b^2*c^5*d^5 + 315*a^8*b*c^4*d^6 + 20*a^9*c^3*
d^7)*x^8 + 6*(2*a^3*b^6*c^10 + 30*a^4*b^5*c^9*d + 135*a^5*b^4*c^8*d^2 + 240*a^6*
b^3*c^7*d^3 + 180*a^7*b^2*c^6*d^4 + 54*a^8*b*c^5*d^5 + 5*a^9*c^4*d^6)*x^7 + 3*(7
*a^4*b^5*c^10 + 70*a^5*b^4*c^9*d + 210*a^6*b^3*c^8*d^2 + 240*a^7*b^2*c^7*d^3 + 1
05*a^8*b*c^6*d^4 + 14*a^9*c^5*d^5)*x^6 + 6/5*(21*a^5*b^4*c^10 + 140*a^6*b^3*c^9*
d + 270*a^7*b^2*c^8*d^2 + 180*a^8*b*c^7*d^3 + 35*a^9*c^6*d^4)*x^5 + 3/4*(28*a^6*
b^3*c^10 + 120*a^7*b^2*c^9*d + 135*a^8*b*c^8*d^2 + 40*a^9*c^7*d^3)*x^4 + 3*(4*a^
7*b^2*c^10 + 10*a^8*b*c^9*d + 5*a^9*c^8*d^2)*x^3 + 1/2*(9*a^8*b*c^10 + 10*a^9*c^
9*d)*x^2

_______________________________________________________________________________________

Fricas [A]  time = 0.187822, 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)^9*(d*x + c)^10,x, algorithm="fricas")

[Out]

1/20*x^20*d^10*b^9 + 10/19*x^19*d^9*c*b^9 + 9/19*x^19*d^10*b^8*a + 5/2*x^18*d^8*
c^2*b^9 + 5*x^18*d^9*c*b^8*a + 2*x^18*d^10*b^7*a^2 + 120/17*x^17*d^7*c^3*b^9 + 4
05/17*x^17*d^8*c^2*b^8*a + 360/17*x^17*d^9*c*b^7*a^2 + 84/17*x^17*d^10*b^6*a^3 +
105/8*x^16*d^6*c^4*b^9 + 135/2*x^16*d^7*c^3*b^8*a + 405/4*x^16*d^8*c^2*b^7*a^2
+ 105/2*x^16*d^9*c*b^6*a^3 + 63/8*x^16*d^10*b^5*a^4 + 84/5*x^15*d^5*c^5*b^9 + 12
6*x^15*d^6*c^4*b^8*a + 288*x^15*d^7*c^3*b^7*a^2 + 252*x^15*d^8*c^2*b^6*a^3 + 84*
x^15*d^9*c*b^5*a^4 + 42/5*x^15*d^10*b^4*a^5 + 15*x^14*d^4*c^6*b^9 + 162*x^14*d^5
*c^5*b^8*a + 540*x^14*d^6*c^4*b^7*a^2 + 720*x^14*d^7*c^3*b^6*a^3 + 405*x^14*d^8*
c^2*b^5*a^4 + 90*x^14*d^9*c*b^4*a^5 + 6*x^14*d^10*b^3*a^6 + 120/13*x^13*d^3*c^7*
b^9 + 1890/13*x^13*d^4*c^6*b^8*a + 9072/13*x^13*d^5*c^5*b^7*a^2 + 17640/13*x^13*
d^6*c^4*b^6*a^3 + 15120/13*x^13*d^7*c^3*b^5*a^4 + 5670/13*x^13*d^8*c^2*b^4*a^5 +
840/13*x^13*d^9*c*b^3*a^6 + 36/13*x^13*d^10*b^2*a^7 + 15/4*x^12*d^2*c^8*b^9 + 9
0*x^12*d^3*c^7*b^8*a + 630*x^12*d^4*c^6*b^7*a^2 + 1764*x^12*d^5*c^5*b^6*a^3 + 22
05*x^12*d^6*c^4*b^5*a^4 + 1260*x^12*d^7*c^3*b^4*a^5 + 315*x^12*d^8*c^2*b^3*a^6 +
30*x^12*d^9*c*b^2*a^7 + 3/4*x^12*d^10*b*a^8 + 10/11*x^11*d*c^9*b^9 + 405/11*x^1
1*d^2*c^8*b^8*a + 4320/11*x^11*d^3*c^7*b^7*a^2 + 17640/11*x^11*d^4*c^6*b^6*a^3 +
31752/11*x^11*d^5*c^5*b^5*a^4 + 26460/11*x^11*d^6*c^4*b^4*a^5 + 10080/11*x^11*d
^7*c^3*b^3*a^6 + 1620/11*x^11*d^8*c^2*b^2*a^7 + 90/11*x^11*d^9*c*b*a^8 + 1/11*x^
11*d^10*a^9 + 1/10*x^10*c^10*b^9 + 9*x^10*d*c^9*b^8*a + 162*x^10*d^2*c^8*b^7*a^2
+ 1008*x^10*d^3*c^7*b^6*a^3 + 2646*x^10*d^4*c^6*b^5*a^4 + 15876/5*x^10*d^5*c^5*
b^4*a^5 + 1764*x^10*d^6*c^4*b^3*a^6 + 432*x^10*d^7*c^3*b^2*a^7 + 81/2*x^10*d^8*c
^2*b*a^8 + x^10*d^9*c*a^9 + x^9*c^10*b^8*a + 40*x^9*d*c^9*b^7*a^2 + 420*x^9*d^2*
c^8*b^6*a^3 + 1680*x^9*d^3*c^7*b^5*a^4 + 2940*x^9*d^4*c^6*b^4*a^5 + 2352*x^9*d^5
*c^5*b^3*a^6 + 840*x^9*d^6*c^4*b^2*a^7 + 120*x^9*d^7*c^3*b*a^8 + 5*x^9*d^8*c^2*a
^9 + 9/2*x^8*c^10*b^7*a^2 + 105*x^8*d*c^9*b^6*a^3 + 2835/4*x^8*d^2*c^8*b^5*a^4 +
1890*x^8*d^3*c^7*b^4*a^5 + 2205*x^8*d^4*c^6*b^3*a^6 + 1134*x^8*d^5*c^5*b^2*a^7
+ 945/4*x^8*d^6*c^4*b*a^8 + 15*x^8*d^7*c^3*a^9 + 12*x^7*c^10*b^6*a^3 + 180*x^7*d
*c^9*b^5*a^4 + 810*x^7*d^2*c^8*b^4*a^5 + 1440*x^7*d^3*c^7*b^3*a^6 + 1080*x^7*d^4
*c^6*b^2*a^7 + 324*x^7*d^5*c^5*b*a^8 + 30*x^7*d^6*c^4*a^9 + 21*x^6*c^10*b^5*a^4
+ 210*x^6*d*c^9*b^4*a^5 + 630*x^6*d^2*c^8*b^3*a^6 + 720*x^6*d^3*c^7*b^2*a^7 + 31
5*x^6*d^4*c^6*b*a^8 + 42*x^6*d^5*c^5*a^9 + 126/5*x^5*c^10*b^4*a^5 + 168*x^5*d*c^
9*b^3*a^6 + 324*x^5*d^2*c^8*b^2*a^7 + 216*x^5*d^3*c^7*b*a^8 + 42*x^5*d^4*c^6*a^9
+ 21*x^4*c^10*b^3*a^6 + 90*x^4*d*c^9*b^2*a^7 + 405/4*x^4*d^2*c^8*b*a^8 + 30*x^4
*d^3*c^7*a^9 + 12*x^3*c^10*b^2*a^7 + 30*x^3*d*c^9*b*a^8 + 15*x^3*d^2*c^8*a^9 + 9
/2*x^2*c^10*b*a^8 + 5*x^2*d*c^9*a^9 + x*c^10*a^9

_______________________________________________________________________________________

Sympy [A]  time = 0.720809, size = 1598, normalized size = 6.39 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**9*c**10*x + b**9*d**10*x**20/20 + x**19*(9*a*b**8*d**10/19 + 10*b**9*c*d**9/1
9) + x**18*(2*a**2*b**7*d**10 + 5*a*b**8*c*d**9 + 5*b**9*c**2*d**8/2) + x**17*(8
4*a**3*b**6*d**10/17 + 360*a**2*b**7*c*d**9/17 + 405*a*b**8*c**2*d**8/17 + 120*b
**9*c**3*d**7/17) + x**16*(63*a**4*b**5*d**10/8 + 105*a**3*b**6*c*d**9/2 + 405*a
**2*b**7*c**2*d**8/4 + 135*a*b**8*c**3*d**7/2 + 105*b**9*c**4*d**6/8) + x**15*(4
2*a**5*b**4*d**10/5 + 84*a**4*b**5*c*d**9 + 252*a**3*b**6*c**2*d**8 + 288*a**2*b
**7*c**3*d**7 + 126*a*b**8*c**4*d**6 + 84*b**9*c**5*d**5/5) + x**14*(6*a**6*b**3
*d**10 + 90*a**5*b**4*c*d**9 + 405*a**4*b**5*c**2*d**8 + 720*a**3*b**6*c**3*d**7
+ 540*a**2*b**7*c**4*d**6 + 162*a*b**8*c**5*d**5 + 15*b**9*c**6*d**4) + x**13*(
36*a**7*b**2*d**10/13 + 840*a**6*b**3*c*d**9/13 + 5670*a**5*b**4*c**2*d**8/13 +
15120*a**4*b**5*c**3*d**7/13 + 17640*a**3*b**6*c**4*d**6/13 + 9072*a**2*b**7*c**
5*d**5/13 + 1890*a*b**8*c**6*d**4/13 + 120*b**9*c**7*d**3/13) + x**12*(3*a**8*b*
d**10/4 + 30*a**7*b**2*c*d**9 + 315*a**6*b**3*c**2*d**8 + 1260*a**5*b**4*c**3*d*
*7 + 2205*a**4*b**5*c**4*d**6 + 1764*a**3*b**6*c**5*d**5 + 630*a**2*b**7*c**6*d*
*4 + 90*a*b**8*c**7*d**3 + 15*b**9*c**8*d**2/4) + x**11*(a**9*d**10/11 + 90*a**8
*b*c*d**9/11 + 1620*a**7*b**2*c**2*d**8/11 + 10080*a**6*b**3*c**3*d**7/11 + 2646
0*a**5*b**4*c**4*d**6/11 + 31752*a**4*b**5*c**5*d**5/11 + 17640*a**3*b**6*c**6*d
**4/11 + 4320*a**2*b**7*c**7*d**3/11 + 405*a*b**8*c**8*d**2/11 + 10*b**9*c**9*d/
11) + x**10*(a**9*c*d**9 + 81*a**8*b*c**2*d**8/2 + 432*a**7*b**2*c**3*d**7 + 176
4*a**6*b**3*c**4*d**6 + 15876*a**5*b**4*c**5*d**5/5 + 2646*a**4*b**5*c**6*d**4 +
1008*a**3*b**6*c**7*d**3 + 162*a**2*b**7*c**8*d**2 + 9*a*b**8*c**9*d + b**9*c**
10/10) + x**9*(5*a**9*c**2*d**8 + 120*a**8*b*c**3*d**7 + 840*a**7*b**2*c**4*d**6
+ 2352*a**6*b**3*c**5*d**5 + 2940*a**5*b**4*c**6*d**4 + 1680*a**4*b**5*c**7*d**
3 + 420*a**3*b**6*c**8*d**2 + 40*a**2*b**7*c**9*d + a*b**8*c**10) + x**8*(15*a**
9*c**3*d**7 + 945*a**8*b*c**4*d**6/4 + 1134*a**7*b**2*c**5*d**5 + 2205*a**6*b**3
*c**6*d**4 + 1890*a**5*b**4*c**7*d**3 + 2835*a**4*b**5*c**8*d**2/4 + 105*a**3*b*
*6*c**9*d + 9*a**2*b**7*c**10/2) + x**7*(30*a**9*c**4*d**6 + 324*a**8*b*c**5*d**
5 + 1080*a**7*b**2*c**6*d**4 + 1440*a**6*b**3*c**7*d**3 + 810*a**5*b**4*c**8*d**
2 + 180*a**4*b**5*c**9*d + 12*a**3*b**6*c**10) + x**6*(42*a**9*c**5*d**5 + 315*a
**8*b*c**6*d**4 + 720*a**7*b**2*c**7*d**3 + 630*a**6*b**3*c**8*d**2 + 210*a**5*b
**4*c**9*d + 21*a**4*b**5*c**10) + x**5*(42*a**9*c**6*d**4 + 216*a**8*b*c**7*d**
3 + 324*a**7*b**2*c**8*d**2 + 168*a**6*b**3*c**9*d + 126*a**5*b**4*c**10/5) + x*
*4*(30*a**9*c**7*d**3 + 405*a**8*b*c**8*d**2/4 + 90*a**7*b**2*c**9*d + 21*a**6*b
**3*c**10) + x**3*(15*a**9*c**8*d**2 + 30*a**8*b*c**9*d + 12*a**7*b**2*c**10) +
x**2*(5*a**9*c**9*d + 9*a**8*b*c**10/2)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.219954, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done