3.139 \(\int x^2 (a+b x) (c+d x)^{16} \, dx\)

Optimal. Leaf size=88 \[ -\frac{c^2 (c+d x)^{17} (b c-a d)}{17 d^4}-\frac{(c+d x)^{19} (3 b c-a d)}{19 d^4}+\frac{c (c+d x)^{18} (3 b c-2 a d)}{18 d^4}+\frac{b (c+d x)^{20}}{20 d^4} \]

[Out]

-(c^2*(b*c - a*d)*(c + d*x)^17)/(17*d^4) + (c*(3*b*c - 2*a*d)*(c + d*x)^18)/(18*
d^4) - ((3*b*c - a*d)*(c + d*x)^19)/(19*d^4) + (b*(c + d*x)^20)/(20*d^4)

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Rubi [A]  time = 0.722085, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ -\frac{c^2 (c+d x)^{17} (b c-a d)}{17 d^4}-\frac{(c+d x)^{19} (3 b c-a d)}{19 d^4}+\frac{c (c+d x)^{18} (3 b c-2 a d)}{18 d^4}+\frac{b (c+d x)^{20}}{20 d^4} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(a + b*x)*(c + d*x)^16,x]

[Out]

-(c^2*(b*c - a*d)*(c + d*x)^17)/(17*d^4) + (c*(3*b*c - 2*a*d)*(c + d*x)^18)/(18*
d^4) - ((3*b*c - a*d)*(c + d*x)^19)/(19*d^4) + (b*(c + d*x)^20)/(20*d^4)

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Rubi in Sympy [A]  time = 75.1066, size = 78, normalized size = 0.89 \[ \frac{b \left (c + d x\right )^{20}}{20 d^{4}} + \frac{c^{2} \left (c + d x\right )^{17} \left (a d - b c\right )}{17 d^{4}} - \frac{c \left (c + d x\right )^{18} \left (2 a d - 3 b c\right )}{18 d^{4}} + \frac{\left (c + d x\right )^{19} \left (a d - 3 b c\right )}{19 d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(b*x+a)*(d*x+c)**16,x)

[Out]

b*(c + d*x)**20/(20*d**4) + c**2*(c + d*x)**17*(a*d - b*c)/(17*d**4) - c*(c + d*
x)**18*(2*a*d - 3*b*c)/(18*d**4) + (c + d*x)**19*(a*d - 3*b*c)/(19*d**4)

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Mathematica [B]  time = 0.115993, size = 355, normalized size = 4.03 \[ \frac{1}{4} c^{15} x^4 (16 a d+b c)+\frac{8}{5} c^{14} d x^5 (15 a d+2 b c)+\frac{20}{3} c^{13} d^2 x^6 (14 a d+3 b c)+20 c^{12} d^3 x^7 (13 a d+4 b c)+\frac{91}{2} c^{11} d^4 x^8 (12 a d+5 b c)+\frac{728}{9} c^{10} d^5 x^9 (11 a d+6 b c)+\frac{572}{5} c^9 d^6 x^{10} (10 a d+7 b c)+130 c^8 d^7 x^{11} (9 a d+8 b c)+\frac{715}{6} c^7 d^8 x^{12} (8 a d+9 b c)+88 c^6 d^9 x^{13} (7 a d+10 b c)+52 c^5 d^{10} x^{14} (6 a d+11 b c)+\frac{364}{15} c^4 d^{11} x^{15} (5 a d+12 b c)+\frac{35}{4} c^3 d^{12} x^{16} (4 a d+13 b c)+\frac{40}{17} c^2 d^{13} x^{17} (3 a d+14 b c)+\frac{1}{19} d^{15} x^{19} (a d+16 b c)+\frac{4}{9} c d^{14} x^{18} (2 a d+15 b c)+\frac{1}{3} a c^{16} x^3+\frac{1}{20} b d^{16} x^{20} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*(a + b*x)*(c + d*x)^16,x]

[Out]

(a*c^16*x^3)/3 + (c^15*(b*c + 16*a*d)*x^4)/4 + (8*c^14*d*(2*b*c + 15*a*d)*x^5)/5
+ (20*c^13*d^2*(3*b*c + 14*a*d)*x^6)/3 + 20*c^12*d^3*(4*b*c + 13*a*d)*x^7 + (91
*c^11*d^4*(5*b*c + 12*a*d)*x^8)/2 + (728*c^10*d^5*(6*b*c + 11*a*d)*x^9)/9 + (572
*c^9*d^6*(7*b*c + 10*a*d)*x^10)/5 + 130*c^8*d^7*(8*b*c + 9*a*d)*x^11 + (715*c^7*
d^8*(9*b*c + 8*a*d)*x^12)/6 + 88*c^6*d^9*(10*b*c + 7*a*d)*x^13 + 52*c^5*d^10*(11
*b*c + 6*a*d)*x^14 + (364*c^4*d^11*(12*b*c + 5*a*d)*x^15)/15 + (35*c^3*d^12*(13*
b*c + 4*a*d)*x^16)/4 + (40*c^2*d^13*(14*b*c + 3*a*d)*x^17)/17 + (4*c*d^14*(15*b*
c + 2*a*d)*x^18)/9 + (d^15*(16*b*c + a*d)*x^19)/19 + (b*d^16*x^20)/20

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Maple [B]  time = 0.003, size = 388, normalized size = 4.4 \[{\frac{b{d}^{16}{x}^{20}}{20}}+{\frac{ \left ( a{d}^{16}+16\,bc{d}^{15} \right ){x}^{19}}{19}}+{\frac{ \left ( 16\,ac{d}^{15}+120\,b{c}^{2}{d}^{14} \right ){x}^{18}}{18}}+{\frac{ \left ( 120\,a{c}^{2}{d}^{14}+560\,b{c}^{3}{d}^{13} \right ){x}^{17}}{17}}+{\frac{ \left ( 560\,a{c}^{3}{d}^{13}+1820\,b{c}^{4}{d}^{12} \right ){x}^{16}}{16}}+{\frac{ \left ( 1820\,a{c}^{4}{d}^{12}+4368\,b{c}^{5}{d}^{11} \right ){x}^{15}}{15}}+{\frac{ \left ( 4368\,a{c}^{5}{d}^{11}+8008\,b{c}^{6}{d}^{10} \right ){x}^{14}}{14}}+{\frac{ \left ( 8008\,a{c}^{6}{d}^{10}+11440\,b{c}^{7}{d}^{9} \right ){x}^{13}}{13}}+{\frac{ \left ( 11440\,a{c}^{7}{d}^{9}+12870\,b{c}^{8}{d}^{8} \right ){x}^{12}}{12}}+{\frac{ \left ( 12870\,a{c}^{8}{d}^{8}+11440\,b{c}^{9}{d}^{7} \right ){x}^{11}}{11}}+{\frac{ \left ( 11440\,a{c}^{9}{d}^{7}+8008\,b{c}^{10}{d}^{6} \right ){x}^{10}}{10}}+{\frac{ \left ( 8008\,a{c}^{10}{d}^{6}+4368\,b{c}^{11}{d}^{5} \right ){x}^{9}}{9}}+{\frac{ \left ( 4368\,a{c}^{11}{d}^{5}+1820\,b{c}^{12}{d}^{4} \right ){x}^{8}}{8}}+{\frac{ \left ( 1820\,a{c}^{12}{d}^{4}+560\,b{c}^{13}{d}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( 560\,a{c}^{13}{d}^{3}+120\,b{c}^{14}{d}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 120\,a{c}^{14}{d}^{2}+16\,b{c}^{15}d \right ){x}^{5}}{5}}+{\frac{ \left ( 16\,a{c}^{15}d+b{c}^{16} \right ){x}^{4}}{4}}+{\frac{a{c}^{16}{x}^{3}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x+a)*(d*x+c)^16,x)

[Out]

1/20*b*d^16*x^20+1/19*(a*d^16+16*b*c*d^15)*x^19+1/18*(16*a*c*d^15+120*b*c^2*d^14
)*x^18+1/17*(120*a*c^2*d^14+560*b*c^3*d^13)*x^17+1/16*(560*a*c^3*d^13+1820*b*c^4
*d^12)*x^16+1/15*(1820*a*c^4*d^12+4368*b*c^5*d^11)*x^15+1/14*(4368*a*c^5*d^11+80
08*b*c^6*d^10)*x^14+1/13*(8008*a*c^6*d^10+11440*b*c^7*d^9)*x^13+1/12*(11440*a*c^
7*d^9+12870*b*c^8*d^8)*x^12+1/11*(12870*a*c^8*d^8+11440*b*c^9*d^7)*x^11+1/10*(11
440*a*c^9*d^7+8008*b*c^10*d^6)*x^10+1/9*(8008*a*c^10*d^6+4368*b*c^11*d^5)*x^9+1/
8*(4368*a*c^11*d^5+1820*b*c^12*d^4)*x^8+1/7*(1820*a*c^12*d^4+560*b*c^13*d^3)*x^7
+1/6*(560*a*c^13*d^3+120*b*c^14*d^2)*x^6+1/5*(120*a*c^14*d^2+16*b*c^15*d)*x^5+1/
4*(16*a*c^15*d+b*c^16)*x^4+1/3*a*c^16*x^3

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Maxima [A]  time = 1.36182, size = 522, normalized size = 5.93 \[ \frac{1}{20} \, b d^{16} x^{20} + \frac{1}{3} \, a c^{16} x^{3} + \frac{1}{19} \,{\left (16 \, b c d^{15} + a d^{16}\right )} x^{19} + \frac{4}{9} \,{\left (15 \, b c^{2} d^{14} + 2 \, a c d^{15}\right )} x^{18} + \frac{40}{17} \,{\left (14 \, b c^{3} d^{13} + 3 \, a c^{2} d^{14}\right )} x^{17} + \frac{35}{4} \,{\left (13 \, b c^{4} d^{12} + 4 \, a c^{3} d^{13}\right )} x^{16} + \frac{364}{15} \,{\left (12 \, b c^{5} d^{11} + 5 \, a c^{4} d^{12}\right )} x^{15} + 52 \,{\left (11 \, b c^{6} d^{10} + 6 \, a c^{5} d^{11}\right )} x^{14} + 88 \,{\left (10 \, b c^{7} d^{9} + 7 \, a c^{6} d^{10}\right )} x^{13} + \frac{715}{6} \,{\left (9 \, b c^{8} d^{8} + 8 \, a c^{7} d^{9}\right )} x^{12} + 130 \,{\left (8 \, b c^{9} d^{7} + 9 \, a c^{8} d^{8}\right )} x^{11} + \frac{572}{5} \,{\left (7 \, b c^{10} d^{6} + 10 \, a c^{9} d^{7}\right )} x^{10} + \frac{728}{9} \,{\left (6 \, b c^{11} d^{5} + 11 \, a c^{10} d^{6}\right )} x^{9} + \frac{91}{2} \,{\left (5 \, b c^{12} d^{4} + 12 \, a c^{11} d^{5}\right )} x^{8} + 20 \,{\left (4 \, b c^{13} d^{3} + 13 \, a c^{12} d^{4}\right )} x^{7} + \frac{20}{3} \,{\left (3 \, b c^{14} d^{2} + 14 \, a c^{13} d^{3}\right )} x^{6} + \frac{8}{5} \,{\left (2 \, b c^{15} d + 15 \, a c^{14} d^{2}\right )} x^{5} + \frac{1}{4} \,{\left (b c^{16} + 16 \, a c^{15} d\right )} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(d*x + c)^16*x^2,x, algorithm="maxima")

[Out]

1/20*b*d^16*x^20 + 1/3*a*c^16*x^3 + 1/19*(16*b*c*d^15 + a*d^16)*x^19 + 4/9*(15*b
*c^2*d^14 + 2*a*c*d^15)*x^18 + 40/17*(14*b*c^3*d^13 + 3*a*c^2*d^14)*x^17 + 35/4*
(13*b*c^4*d^12 + 4*a*c^3*d^13)*x^16 + 364/15*(12*b*c^5*d^11 + 5*a*c^4*d^12)*x^15
+ 52*(11*b*c^6*d^10 + 6*a*c^5*d^11)*x^14 + 88*(10*b*c^7*d^9 + 7*a*c^6*d^10)*x^1
3 + 715/6*(9*b*c^8*d^8 + 8*a*c^7*d^9)*x^12 + 130*(8*b*c^9*d^7 + 9*a*c^8*d^8)*x^1
1 + 572/5*(7*b*c^10*d^6 + 10*a*c^9*d^7)*x^10 + 728/9*(6*b*c^11*d^5 + 11*a*c^10*d
^6)*x^9 + 91/2*(5*b*c^12*d^4 + 12*a*c^11*d^5)*x^8 + 20*(4*b*c^13*d^3 + 13*a*c^12
*d^4)*x^7 + 20/3*(3*b*c^14*d^2 + 14*a*c^13*d^3)*x^6 + 8/5*(2*b*c^15*d + 15*a*c^1
4*d^2)*x^5 + 1/4*(b*c^16 + 16*a*c^15*d)*x^4

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Fricas [A]  time = 0.179046, size = 1, normalized size = 0.01 \[ \frac{1}{20} x^{20} d^{16} b + \frac{16}{19} x^{19} d^{15} c b + \frac{1}{19} x^{19} d^{16} a + \frac{20}{3} x^{18} d^{14} c^{2} b + \frac{8}{9} x^{18} d^{15} c a + \frac{560}{17} x^{17} d^{13} c^{3} b + \frac{120}{17} x^{17} d^{14} c^{2} a + \frac{455}{4} x^{16} d^{12} c^{4} b + 35 x^{16} d^{13} c^{3} a + \frac{1456}{5} x^{15} d^{11} c^{5} b + \frac{364}{3} x^{15} d^{12} c^{4} a + 572 x^{14} d^{10} c^{6} b + 312 x^{14} d^{11} c^{5} a + 880 x^{13} d^{9} c^{7} b + 616 x^{13} d^{10} c^{6} a + \frac{2145}{2} x^{12} d^{8} c^{8} b + \frac{2860}{3} x^{12} d^{9} c^{7} a + 1040 x^{11} d^{7} c^{9} b + 1170 x^{11} d^{8} c^{8} a + \frac{4004}{5} x^{10} d^{6} c^{10} b + 1144 x^{10} d^{7} c^{9} a + \frac{1456}{3} x^{9} d^{5} c^{11} b + \frac{8008}{9} x^{9} d^{6} c^{10} a + \frac{455}{2} x^{8} d^{4} c^{12} b + 546 x^{8} d^{5} c^{11} a + 80 x^{7} d^{3} c^{13} b + 260 x^{7} d^{4} c^{12} a + 20 x^{6} d^{2} c^{14} b + \frac{280}{3} x^{6} d^{3} c^{13} a + \frac{16}{5} x^{5} d c^{15} b + 24 x^{5} d^{2} c^{14} a + \frac{1}{4} x^{4} c^{16} b + 4 x^{4} d c^{15} a + \frac{1}{3} x^{3} c^{16} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(d*x + c)^16*x^2,x, algorithm="fricas")

[Out]

1/20*x^20*d^16*b + 16/19*x^19*d^15*c*b + 1/19*x^19*d^16*a + 20/3*x^18*d^14*c^2*b
+ 8/9*x^18*d^15*c*a + 560/17*x^17*d^13*c^3*b + 120/17*x^17*d^14*c^2*a + 455/4*x
^16*d^12*c^4*b + 35*x^16*d^13*c^3*a + 1456/5*x^15*d^11*c^5*b + 364/3*x^15*d^12*c
^4*a + 572*x^14*d^10*c^6*b + 312*x^14*d^11*c^5*a + 880*x^13*d^9*c^7*b + 616*x^13
*d^10*c^6*a + 2145/2*x^12*d^8*c^8*b + 2860/3*x^12*d^9*c^7*a + 1040*x^11*d^7*c^9*
b + 1170*x^11*d^8*c^8*a + 4004/5*x^10*d^6*c^10*b + 1144*x^10*d^7*c^9*a + 1456/3*
x^9*d^5*c^11*b + 8008/9*x^9*d^6*c^10*a + 455/2*x^8*d^4*c^12*b + 546*x^8*d^5*c^11
*a + 80*x^7*d^3*c^13*b + 260*x^7*d^4*c^12*a + 20*x^6*d^2*c^14*b + 280/3*x^6*d^3*
c^13*a + 16/5*x^5*d*c^15*b + 24*x^5*d^2*c^14*a + 1/4*x^4*c^16*b + 4*x^4*d*c^15*a
+ 1/3*x^3*c^16*a

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Sympy [A]  time = 0.559934, size = 413, normalized size = 4.69 \[ \frac{a c^{16} x^{3}}{3} + \frac{b d^{16} x^{20}}{20} + x^{19} \left (\frac{a d^{16}}{19} + \frac{16 b c d^{15}}{19}\right ) + x^{18} \left (\frac{8 a c d^{15}}{9} + \frac{20 b c^{2} d^{14}}{3}\right ) + x^{17} \left (\frac{120 a c^{2} d^{14}}{17} + \frac{560 b c^{3} d^{13}}{17}\right ) + x^{16} \left (35 a c^{3} d^{13} + \frac{455 b c^{4} d^{12}}{4}\right ) + x^{15} \left (\frac{364 a c^{4} d^{12}}{3} + \frac{1456 b c^{5} d^{11}}{5}\right ) + x^{14} \left (312 a c^{5} d^{11} + 572 b c^{6} d^{10}\right ) + x^{13} \left (616 a c^{6} d^{10} + 880 b c^{7} d^{9}\right ) + x^{12} \left (\frac{2860 a c^{7} d^{9}}{3} + \frac{2145 b c^{8} d^{8}}{2}\right ) + x^{11} \left (1170 a c^{8} d^{8} + 1040 b c^{9} d^{7}\right ) + x^{10} \left (1144 a c^{9} d^{7} + \frac{4004 b c^{10} d^{6}}{5}\right ) + x^{9} \left (\frac{8008 a c^{10} d^{6}}{9} + \frac{1456 b c^{11} d^{5}}{3}\right ) + x^{8} \left (546 a c^{11} d^{5} + \frac{455 b c^{12} d^{4}}{2}\right ) + x^{7} \left (260 a c^{12} d^{4} + 80 b c^{13} d^{3}\right ) + x^{6} \left (\frac{280 a c^{13} d^{3}}{3} + 20 b c^{14} d^{2}\right ) + x^{5} \left (24 a c^{14} d^{2} + \frac{16 b c^{15} d}{5}\right ) + x^{4} \left (4 a c^{15} d + \frac{b c^{16}}{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(b*x+a)*(d*x+c)**16,x)

[Out]

a*c**16*x**3/3 + b*d**16*x**20/20 + x**19*(a*d**16/19 + 16*b*c*d**15/19) + x**18
*(8*a*c*d**15/9 + 20*b*c**2*d**14/3) + x**17*(120*a*c**2*d**14/17 + 560*b*c**3*d
**13/17) + x**16*(35*a*c**3*d**13 + 455*b*c**4*d**12/4) + x**15*(364*a*c**4*d**1
2/3 + 1456*b*c**5*d**11/5) + x**14*(312*a*c**5*d**11 + 572*b*c**6*d**10) + x**13
*(616*a*c**6*d**10 + 880*b*c**7*d**9) + x**12*(2860*a*c**7*d**9/3 + 2145*b*c**8*
d**8/2) + x**11*(1170*a*c**8*d**8 + 1040*b*c**9*d**7) + x**10*(1144*a*c**9*d**7
+ 4004*b*c**10*d**6/5) + x**9*(8008*a*c**10*d**6/9 + 1456*b*c**11*d**5/3) + x**8
*(546*a*c**11*d**5 + 455*b*c**12*d**4/2) + x**7*(260*a*c**12*d**4 + 80*b*c**13*d
**3) + x**6*(280*a*c**13*d**3/3 + 20*b*c**14*d**2) + x**5*(24*a*c**14*d**2 + 16*
b*c**15*d/5) + x**4*(4*a*c**15*d + b*c**16/4)

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GIAC/XCAS [A]  time = 0.333622, size = 525, normalized size = 5.97 \[ \frac{1}{20} \, b d^{16} x^{20} + \frac{16}{19} \, b c d^{15} x^{19} + \frac{1}{19} \, a d^{16} x^{19} + \frac{20}{3} \, b c^{2} d^{14} x^{18} + \frac{8}{9} \, a c d^{15} x^{18} + \frac{560}{17} \, b c^{3} d^{13} x^{17} + \frac{120}{17} \, a c^{2} d^{14} x^{17} + \frac{455}{4} \, b c^{4} d^{12} x^{16} + 35 \, a c^{3} d^{13} x^{16} + \frac{1456}{5} \, b c^{5} d^{11} x^{15} + \frac{364}{3} \, a c^{4} d^{12} x^{15} + 572 \, b c^{6} d^{10} x^{14} + 312 \, a c^{5} d^{11} x^{14} + 880 \, b c^{7} d^{9} x^{13} + 616 \, a c^{6} d^{10} x^{13} + \frac{2145}{2} \, b c^{8} d^{8} x^{12} + \frac{2860}{3} \, a c^{7} d^{9} x^{12} + 1040 \, b c^{9} d^{7} x^{11} + 1170 \, a c^{8} d^{8} x^{11} + \frac{4004}{5} \, b c^{10} d^{6} x^{10} + 1144 \, a c^{9} d^{7} x^{10} + \frac{1456}{3} \, b c^{11} d^{5} x^{9} + \frac{8008}{9} \, a c^{10} d^{6} x^{9} + \frac{455}{2} \, b c^{12} d^{4} x^{8} + 546 \, a c^{11} d^{5} x^{8} + 80 \, b c^{13} d^{3} x^{7} + 260 \, a c^{12} d^{4} x^{7} + 20 \, b c^{14} d^{2} x^{6} + \frac{280}{3} \, a c^{13} d^{3} x^{6} + \frac{16}{5} \, b c^{15} d x^{5} + 24 \, a c^{14} d^{2} x^{5} + \frac{1}{4} \, b c^{16} x^{4} + 4 \, a c^{15} d x^{4} + \frac{1}{3} \, a c^{16} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(d*x + c)^16*x^2,x, algorithm="giac")

[Out]

1/20*b*d^16*x^20 + 16/19*b*c*d^15*x^19 + 1/19*a*d^16*x^19 + 20/3*b*c^2*d^14*x^18
+ 8/9*a*c*d^15*x^18 + 560/17*b*c^3*d^13*x^17 + 120/17*a*c^2*d^14*x^17 + 455/4*b
*c^4*d^12*x^16 + 35*a*c^3*d^13*x^16 + 1456/5*b*c^5*d^11*x^15 + 364/3*a*c^4*d^12*
x^15 + 572*b*c^6*d^10*x^14 + 312*a*c^5*d^11*x^14 + 880*b*c^7*d^9*x^13 + 616*a*c^
6*d^10*x^13 + 2145/2*b*c^8*d^8*x^12 + 2860/3*a*c^7*d^9*x^12 + 1040*b*c^9*d^7*x^1
1 + 1170*a*c^8*d^8*x^11 + 4004/5*b*c^10*d^6*x^10 + 1144*a*c^9*d^7*x^10 + 1456/3*
b*c^11*d^5*x^9 + 8008/9*a*c^10*d^6*x^9 + 455/2*b*c^12*d^4*x^8 + 546*a*c^11*d^5*x
^8 + 80*b*c^13*d^3*x^7 + 260*a*c^12*d^4*x^7 + 20*b*c^14*d^2*x^6 + 280/3*a*c^13*d
^3*x^6 + 16/5*b*c^15*d*x^5 + 24*a*c^14*d^2*x^5 + 1/4*b*c^16*x^4 + 4*a*c^15*d*x^4
+ 1/3*a*c^16*x^3