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

Optimal. Leaf size=38 \[ \frac{b (c+d x)^{18}}{18 d^2}-\frac{(c+d x)^{17} (b c-a d)}{17 d^2} \]

[Out]

-((b*c - a*d)*(c + d*x)^17)/(17*d^2) + (b*(c + d*x)^18)/(18*d^2)

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Rubi [A]  time = 0.0603763, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{b (c+d x)^{18}}{18 d^2}-\frac{(c+d x)^{17} (b c-a d)}{17 d^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)*(c + d*x)^16,x]

[Out]

-((b*c - a*d)*(c + d*x)^17)/(17*d^2) + (b*(c + d*x)^18)/(18*d^2)

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Rubi in Sympy [A]  time = 56.7719, size = 31, normalized size = 0.82 \[ \frac{b \left (c + d x\right )^{18}}{18 d^{2}} + \frac{\left (c + d x\right )^{17} \left (a d - b c\right )}{17 d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*(c + d*x)**18/(18*d**2) + (c + d*x)**17*(a*d - b*c)/(17*d**2)

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Mathematica [B]  time = 0.098997, size = 342, normalized size = 9. \[ \frac{1}{2} c^{15} x^2 (16 a d+b c)+\frac{8}{3} c^{14} d x^3 (15 a d+2 b c)+10 c^{13} d^2 x^4 (14 a d+3 b c)+28 c^{12} d^3 x^5 (13 a d+4 b c)+\frac{182}{3} c^{11} d^4 x^6 (12 a d+5 b c)+104 c^{10} d^5 x^7 (11 a d+6 b c)+143 c^9 d^6 x^8 (10 a d+7 b c)+\frac{1430}{9} c^8 d^7 x^9 (9 a d+8 b c)+143 c^7 d^8 x^{10} (8 a d+9 b c)+104 c^6 d^9 x^{11} (7 a d+10 b c)+\frac{182}{3} c^5 d^{10} x^{12} (6 a d+11 b c)+28 c^4 d^{11} x^{13} (5 a d+12 b c)+10 c^3 d^{12} x^{14} (4 a d+13 b c)+\frac{8}{3} c^2 d^{13} x^{15} (3 a d+14 b c)+\frac{1}{17} d^{15} x^{17} (a d+16 b c)+\frac{1}{2} c d^{14} x^{16} (2 a d+15 b c)+a c^{16} x+\frac{1}{18} b d^{16} x^{18} \]

Antiderivative was successfully verified.

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

[Out]

a*c^16*x + (c^15*(b*c + 16*a*d)*x^2)/2 + (8*c^14*d*(2*b*c + 15*a*d)*x^3)/3 + 10*
c^13*d^2*(3*b*c + 14*a*d)*x^4 + 28*c^12*d^3*(4*b*c + 13*a*d)*x^5 + (182*c^11*d^4
*(5*b*c + 12*a*d)*x^6)/3 + 104*c^10*d^5*(6*b*c + 11*a*d)*x^7 + 143*c^9*d^6*(7*b*
c + 10*a*d)*x^8 + (1430*c^8*d^7*(8*b*c + 9*a*d)*x^9)/9 + 143*c^7*d^8*(9*b*c + 8*
a*d)*x^10 + 104*c^6*d^9*(10*b*c + 7*a*d)*x^11 + (182*c^5*d^10*(11*b*c + 6*a*d)*x
^12)/3 + 28*c^4*d^11*(12*b*c + 5*a*d)*x^13 + 10*c^3*d^12*(13*b*c + 4*a*d)*x^14 +
(8*c^2*d^13*(14*b*c + 3*a*d)*x^15)/3 + (c*d^14*(15*b*c + 2*a*d)*x^16)/2 + (d^15
*(16*b*c + a*d)*x^17)/17 + (b*d^16*x^18)/18

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Maple [B]  time = 0.003, size = 385, normalized size = 10.1 \[{\frac{b{d}^{16}{x}^{18}}{18}}+{\frac{ \left ( a{d}^{16}+16\,bc{d}^{15} \right ){x}^{17}}{17}}+{\frac{ \left ( 16\,ac{d}^{15}+120\,b{c}^{2}{d}^{14} \right ){x}^{16}}{16}}+{\frac{ \left ( 120\,a{c}^{2}{d}^{14}+560\,b{c}^{3}{d}^{13} \right ){x}^{15}}{15}}+{\frac{ \left ( 560\,a{c}^{3}{d}^{13}+1820\,b{c}^{4}{d}^{12} \right ){x}^{14}}{14}}+{\frac{ \left ( 1820\,a{c}^{4}{d}^{12}+4368\,b{c}^{5}{d}^{11} \right ){x}^{13}}{13}}+{\frac{ \left ( 4368\,a{c}^{5}{d}^{11}+8008\,b{c}^{6}{d}^{10} \right ){x}^{12}}{12}}+{\frac{ \left ( 8008\,a{c}^{6}{d}^{10}+11440\,b{c}^{7}{d}^{9} \right ){x}^{11}}{11}}+{\frac{ \left ( 11440\,a{c}^{7}{d}^{9}+12870\,b{c}^{8}{d}^{8} \right ){x}^{10}}{10}}+{\frac{ \left ( 12870\,a{c}^{8}{d}^{8}+11440\,b{c}^{9}{d}^{7} \right ){x}^{9}}{9}}+{\frac{ \left ( 11440\,a{c}^{9}{d}^{7}+8008\,b{c}^{10}{d}^{6} \right ){x}^{8}}{8}}+{\frac{ \left ( 8008\,a{c}^{10}{d}^{6}+4368\,b{c}^{11}{d}^{5} \right ){x}^{7}}{7}}+{\frac{ \left ( 4368\,a{c}^{11}{d}^{5}+1820\,b{c}^{12}{d}^{4} \right ){x}^{6}}{6}}+{\frac{ \left ( 1820\,a{c}^{12}{d}^{4}+560\,b{c}^{13}{d}^{3} \right ){x}^{5}}{5}}+{\frac{ \left ( 560\,a{c}^{13}{d}^{3}+120\,b{c}^{14}{d}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( 120\,a{c}^{14}{d}^{2}+16\,b{c}^{15}d \right ){x}^{3}}{3}}+{\frac{ \left ( 16\,a{c}^{15}d+b{c}^{16} \right ){x}^{2}}{2}}+a{c}^{16}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(d*x+c)^16,x)

[Out]

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

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Maxima [A]  time = 1.36046, size = 518, normalized size = 13.63 \[ \frac{1}{18} \, b d^{16} x^{18} + a c^{16} x + \frac{1}{17} \,{\left (16 \, b c d^{15} + a d^{16}\right )} x^{17} + \frac{1}{2} \,{\left (15 \, b c^{2} d^{14} + 2 \, a c d^{15}\right )} x^{16} + \frac{8}{3} \,{\left (14 \, b c^{3} d^{13} + 3 \, a c^{2} d^{14}\right )} x^{15} + 10 \,{\left (13 \, b c^{4} d^{12} + 4 \, a c^{3} d^{13}\right )} x^{14} + 28 \,{\left (12 \, b c^{5} d^{11} + 5 \, a c^{4} d^{12}\right )} x^{13} + \frac{182}{3} \,{\left (11 \, b c^{6} d^{10} + 6 \, a c^{5} d^{11}\right )} x^{12} + 104 \,{\left (10 \, b c^{7} d^{9} + 7 \, a c^{6} d^{10}\right )} x^{11} + 143 \,{\left (9 \, b c^{8} d^{8} + 8 \, a c^{7} d^{9}\right )} x^{10} + \frac{1430}{9} \,{\left (8 \, b c^{9} d^{7} + 9 \, a c^{8} d^{8}\right )} x^{9} + 143 \,{\left (7 \, b c^{10} d^{6} + 10 \, a c^{9} d^{7}\right )} x^{8} + 104 \,{\left (6 \, b c^{11} d^{5} + 11 \, a c^{10} d^{6}\right )} x^{7} + \frac{182}{3} \,{\left (5 \, b c^{12} d^{4} + 12 \, a c^{11} d^{5}\right )} x^{6} + 28 \,{\left (4 \, b c^{13} d^{3} + 13 \, a c^{12} d^{4}\right )} x^{5} + 10 \,{\left (3 \, b c^{14} d^{2} + 14 \, a c^{13} d^{3}\right )} x^{4} + \frac{8}{3} \,{\left (2 \, b c^{15} d + 15 \, a c^{14} d^{2}\right )} x^{3} + \frac{1}{2} \,{\left (b c^{16} + 16 \, a c^{15} d\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18*b*d^16*x^18 + a*c^16*x + 1/17*(16*b*c*d^15 + a*d^16)*x^17 + 1/2*(15*b*c^2*d
^14 + 2*a*c*d^15)*x^16 + 8/3*(14*b*c^3*d^13 + 3*a*c^2*d^14)*x^15 + 10*(13*b*c^4*
d^12 + 4*a*c^3*d^13)*x^14 + 28*(12*b*c^5*d^11 + 5*a*c^4*d^12)*x^13 + 182/3*(11*b
*c^6*d^10 + 6*a*c^5*d^11)*x^12 + 104*(10*b*c^7*d^9 + 7*a*c^6*d^10)*x^11 + 143*(9
*b*c^8*d^8 + 8*a*c^7*d^9)*x^10 + 1430/9*(8*b*c^9*d^7 + 9*a*c^8*d^8)*x^9 + 143*(7
*b*c^10*d^6 + 10*a*c^9*d^7)*x^8 + 104*(6*b*c^11*d^5 + 11*a*c^10*d^6)*x^7 + 182/3
*(5*b*c^12*d^4 + 12*a*c^11*d^5)*x^6 + 28*(4*b*c^13*d^3 + 13*a*c^12*d^4)*x^5 + 10
*(3*b*c^14*d^2 + 14*a*c^13*d^3)*x^4 + 8/3*(2*b*c^15*d + 15*a*c^14*d^2)*x^3 + 1/2
*(b*c^16 + 16*a*c^15*d)*x^2

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Fricas [A]  time = 0.180005, size = 1, normalized size = 0.03 \[ \frac{1}{18} x^{18} d^{16} b + \frac{16}{17} x^{17} d^{15} c b + \frac{1}{17} x^{17} d^{16} a + \frac{15}{2} x^{16} d^{14} c^{2} b + x^{16} d^{15} c a + \frac{112}{3} x^{15} d^{13} c^{3} b + 8 x^{15} d^{14} c^{2} a + 130 x^{14} d^{12} c^{4} b + 40 x^{14} d^{13} c^{3} a + 336 x^{13} d^{11} c^{5} b + 140 x^{13} d^{12} c^{4} a + \frac{2002}{3} x^{12} d^{10} c^{6} b + 364 x^{12} d^{11} c^{5} a + 1040 x^{11} d^{9} c^{7} b + 728 x^{11} d^{10} c^{6} a + 1287 x^{10} d^{8} c^{8} b + 1144 x^{10} d^{9} c^{7} a + \frac{11440}{9} x^{9} d^{7} c^{9} b + 1430 x^{9} d^{8} c^{8} a + 1001 x^{8} d^{6} c^{10} b + 1430 x^{8} d^{7} c^{9} a + 624 x^{7} d^{5} c^{11} b + 1144 x^{7} d^{6} c^{10} a + \frac{910}{3} x^{6} d^{4} c^{12} b + 728 x^{6} d^{5} c^{11} a + 112 x^{5} d^{3} c^{13} b + 364 x^{5} d^{4} c^{12} a + 30 x^{4} d^{2} c^{14} b + 140 x^{4} d^{3} c^{13} a + \frac{16}{3} x^{3} d c^{15} b + 40 x^{3} d^{2} c^{14} a + \frac{1}{2} x^{2} c^{16} b + 8 x^{2} d c^{15} a + x c^{16} a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18*x^18*d^16*b + 16/17*x^17*d^15*c*b + 1/17*x^17*d^16*a + 15/2*x^16*d^14*c^2*b
+ x^16*d^15*c*a + 112/3*x^15*d^13*c^3*b + 8*x^15*d^14*c^2*a + 130*x^14*d^12*c^4
*b + 40*x^14*d^13*c^3*a + 336*x^13*d^11*c^5*b + 140*x^13*d^12*c^4*a + 2002/3*x^1
2*d^10*c^6*b + 364*x^12*d^11*c^5*a + 1040*x^11*d^9*c^7*b + 728*x^11*d^10*c^6*a +
1287*x^10*d^8*c^8*b + 1144*x^10*d^9*c^7*a + 11440/9*x^9*d^7*c^9*b + 1430*x^9*d^
8*c^8*a + 1001*x^8*d^6*c^10*b + 1430*x^8*d^7*c^9*a + 624*x^7*d^5*c^11*b + 1144*x
^7*d^6*c^10*a + 910/3*x^6*d^4*c^12*b + 728*x^6*d^5*c^11*a + 112*x^5*d^3*c^13*b +
364*x^5*d^4*c^12*a + 30*x^4*d^2*c^14*b + 140*x^4*d^3*c^13*a + 16/3*x^3*d*c^15*b
+ 40*x^3*d^2*c^14*a + 1/2*x^2*c^16*b + 8*x^2*d*c^15*a + x*c^16*a

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Sympy [A]  time = 0.547865, size = 393, normalized size = 10.34 \[ a c^{16} x + \frac{b d^{16} x^{18}}{18} + x^{17} \left (\frac{a d^{16}}{17} + \frac{16 b c d^{15}}{17}\right ) + x^{16} \left (a c d^{15} + \frac{15 b c^{2} d^{14}}{2}\right ) + x^{15} \left (8 a c^{2} d^{14} + \frac{112 b c^{3} d^{13}}{3}\right ) + x^{14} \left (40 a c^{3} d^{13} + 130 b c^{4} d^{12}\right ) + x^{13} \left (140 a c^{4} d^{12} + 336 b c^{5} d^{11}\right ) + x^{12} \left (364 a c^{5} d^{11} + \frac{2002 b c^{6} d^{10}}{3}\right ) + x^{11} \left (728 a c^{6} d^{10} + 1040 b c^{7} d^{9}\right ) + x^{10} \left (1144 a c^{7} d^{9} + 1287 b c^{8} d^{8}\right ) + x^{9} \left (1430 a c^{8} d^{8} + \frac{11440 b c^{9} d^{7}}{9}\right ) + x^{8} \left (1430 a c^{9} d^{7} + 1001 b c^{10} d^{6}\right ) + x^{7} \left (1144 a c^{10} d^{6} + 624 b c^{11} d^{5}\right ) + x^{6} \left (728 a c^{11} d^{5} + \frac{910 b c^{12} d^{4}}{3}\right ) + x^{5} \left (364 a c^{12} d^{4} + 112 b c^{13} d^{3}\right ) + x^{4} \left (140 a c^{13} d^{3} + 30 b c^{14} d^{2}\right ) + x^{3} \left (40 a c^{14} d^{2} + \frac{16 b c^{15} d}{3}\right ) + x^{2} \left (8 a c^{15} d + \frac{b c^{16}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*c**16*x + b*d**16*x**18/18 + x**17*(a*d**16/17 + 16*b*c*d**15/17) + x**16*(a*c
*d**15 + 15*b*c**2*d**14/2) + x**15*(8*a*c**2*d**14 + 112*b*c**3*d**13/3) + x**1
4*(40*a*c**3*d**13 + 130*b*c**4*d**12) + x**13*(140*a*c**4*d**12 + 336*b*c**5*d*
*11) + x**12*(364*a*c**5*d**11 + 2002*b*c**6*d**10/3) + x**11*(728*a*c**6*d**10
+ 1040*b*c**7*d**9) + x**10*(1144*a*c**7*d**9 + 1287*b*c**8*d**8) + x**9*(1430*a
*c**8*d**8 + 11440*b*c**9*d**7/9) + x**8*(1430*a*c**9*d**7 + 1001*b*c**10*d**6)
+ x**7*(1144*a*c**10*d**6 + 624*b*c**11*d**5) + x**6*(728*a*c**11*d**5 + 910*b*c
**12*d**4/3) + x**5*(364*a*c**12*d**4 + 112*b*c**13*d**3) + x**4*(140*a*c**13*d*
*3 + 30*b*c**14*d**2) + x**3*(40*a*c**14*d**2 + 16*b*c**15*d/3) + x**2*(8*a*c**1
5*d + b*c**16/2)

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GIAC/XCAS [A]  time = 0.300092, size = 520, normalized size = 13.68 \[ \frac{1}{18} \, b d^{16} x^{18} + \frac{16}{17} \, b c d^{15} x^{17} + \frac{1}{17} \, a d^{16} x^{17} + \frac{15}{2} \, b c^{2} d^{14} x^{16} + a c d^{15} x^{16} + \frac{112}{3} \, b c^{3} d^{13} x^{15} + 8 \, a c^{2} d^{14} x^{15} + 130 \, b c^{4} d^{12} x^{14} + 40 \, a c^{3} d^{13} x^{14} + 336 \, b c^{5} d^{11} x^{13} + 140 \, a c^{4} d^{12} x^{13} + \frac{2002}{3} \, b c^{6} d^{10} x^{12} + 364 \, a c^{5} d^{11} x^{12} + 1040 \, b c^{7} d^{9} x^{11} + 728 \, a c^{6} d^{10} x^{11} + 1287 \, b c^{8} d^{8} x^{10} + 1144 \, a c^{7} d^{9} x^{10} + \frac{11440}{9} \, b c^{9} d^{7} x^{9} + 1430 \, a c^{8} d^{8} x^{9} + 1001 \, b c^{10} d^{6} x^{8} + 1430 \, a c^{9} d^{7} x^{8} + 624 \, b c^{11} d^{5} x^{7} + 1144 \, a c^{10} d^{6} x^{7} + \frac{910}{3} \, b c^{12} d^{4} x^{6} + 728 \, a c^{11} d^{5} x^{6} + 112 \, b c^{13} d^{3} x^{5} + 364 \, a c^{12} d^{4} x^{5} + 30 \, b c^{14} d^{2} x^{4} + 140 \, a c^{13} d^{3} x^{4} + \frac{16}{3} \, b c^{15} d x^{3} + 40 \, a c^{14} d^{2} x^{3} + \frac{1}{2} \, b c^{16} x^{2} + 8 \, a c^{15} d x^{2} + a c^{16} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18*b*d^16*x^18 + 16/17*b*c*d^15*x^17 + 1/17*a*d^16*x^17 + 15/2*b*c^2*d^14*x^16
+ a*c*d^15*x^16 + 112/3*b*c^3*d^13*x^15 + 8*a*c^2*d^14*x^15 + 130*b*c^4*d^12*x^
14 + 40*a*c^3*d^13*x^14 + 336*b*c^5*d^11*x^13 + 140*a*c^4*d^12*x^13 + 2002/3*b*c
^6*d^10*x^12 + 364*a*c^5*d^11*x^12 + 1040*b*c^7*d^9*x^11 + 728*a*c^6*d^10*x^11 +
1287*b*c^8*d^8*x^10 + 1144*a*c^7*d^9*x^10 + 11440/9*b*c^9*d^7*x^9 + 1430*a*c^8*
d^8*x^9 + 1001*b*c^10*d^6*x^8 + 1430*a*c^9*d^7*x^8 + 624*b*c^11*d^5*x^7 + 1144*a
*c^10*d^6*x^7 + 910/3*b*c^12*d^4*x^6 + 728*a*c^11*d^5*x^6 + 112*b*c^13*d^3*x^5 +
364*a*c^12*d^4*x^5 + 30*b*c^14*d^2*x^4 + 140*a*c^13*d^3*x^4 + 16/3*b*c^15*d*x^3
+ 40*a*c^14*d^2*x^3 + 1/2*b*c^16*x^2 + 8*a*c^15*d*x^2 + a*c^16*x