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

Optimal. Leaf size=62 \[ -\frac {(c+d x)^{18} (2 b c-a d)}{18 d^3}+\frac {c (c+d x)^{17} (b c-a d)}{17 d^3}+\frac {b (c+d x)^{19}}{19 d^3} \]

[Out]

1/17*c*(-a*d+b*c)*(d*x+c)^17/d^3-1/18*(-a*d+2*b*c)*(d*x+c)^18/d^3+1/19*b*(d*x+c)^19/d^3

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Rubi [A]  time = 0.13, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {76} \[ -\frac {(c+d x)^{18} (2 b c-a d)}{18 d^3}+\frac {c (c+d x)^{17} (b c-a d)}{17 d^3}+\frac {b (c+d x)^{19}}{19 d^3} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x)*(c + d*x)^16,x]

[Out]

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

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

\begin {align*} \int x (a+b x) (c+d x)^{16} \, dx &=\int \left (\frac {c (b c-a d) (c+d x)^{16}}{d^2}+\frac {(-2 b c+a d) (c+d x)^{17}}{d^2}+\frac {b (c+d x)^{18}}{d^2}\right ) \, dx\\ &=\frac {c (b c-a d) (c+d x)^{17}}{17 d^3}-\frac {(2 b c-a d) (c+d x)^{18}}{18 d^3}+\frac {b (c+d x)^{19}}{19 d^3}\\ \end {align*}

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Mathematica [B]  time = 0.07, size = 347, normalized size = 5.60 \[ \frac {1}{3} c^{15} x^3 (16 a d+b c)+2 c^{14} d x^4 (15 a d+2 b c)+8 c^{13} d^2 x^5 (14 a d+3 b c)+\frac {70}{3} c^{12} d^3 x^6 (13 a d+4 b c)+52 c^{11} d^4 x^7 (12 a d+5 b c)+91 c^{10} d^5 x^8 (11 a d+6 b c)+\frac {1144}{9} c^9 d^6 x^9 (10 a d+7 b c)+143 c^8 d^7 x^{10} (9 a d+8 b c)+130 c^7 d^8 x^{11} (8 a d+9 b c)+\frac {286}{3} c^6 d^9 x^{12} (7 a d+10 b c)+56 c^5 d^{10} x^{13} (6 a d+11 b c)+26 c^4 d^{11} x^{14} (5 a d+12 b c)+\frac {28}{3} c^3 d^{12} x^{15} (4 a d+13 b c)+\frac {5}{2} c^2 d^{13} x^{16} (3 a d+14 b c)+\frac {1}{18} d^{15} x^{18} (a d+16 b c)+\frac {8}{17} c d^{14} x^{17} (2 a d+15 b c)+\frac {1}{2} a c^{16} x^2+\frac {1}{19} b d^{16} x^{19} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x)*(c + d*x)^16,x]

[Out]

(a*c^16*x^2)/2 + (c^15*(b*c + 16*a*d)*x^3)/3 + 2*c^14*d*(2*b*c + 15*a*d)*x^4 + 8*c^13*d^2*(3*b*c + 14*a*d)*x^5
+ (70*c^12*d^3*(4*b*c + 13*a*d)*x^6)/3 + 52*c^11*d^4*(5*b*c + 12*a*d)*x^7 + 91*c^10*d^5*(6*b*c + 11*a*d)*x^8
+ (1144*c^9*d^6*(7*b*c + 10*a*d)*x^9)/9 + 143*c^8*d^7*(8*b*c + 9*a*d)*x^10 + 130*c^7*d^8*(9*b*c + 8*a*d)*x^11
+ (286*c^6*d^9*(10*b*c + 7*a*d)*x^12)/3 + 56*c^5*d^10*(11*b*c + 6*a*d)*x^13 + 26*c^4*d^11*(12*b*c + 5*a*d)*x^1
4 + (28*c^3*d^12*(13*b*c + 4*a*d)*x^15)/3 + (5*c^2*d^13*(14*b*c + 3*a*d)*x^16)/2 + (8*c*d^14*(15*b*c + 2*a*d)*
x^17)/17 + (d^15*(16*b*c + a*d)*x^18)/18 + (b*d^16*x^19)/19

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fricas [B]  time = 0.84, size = 389, normalized size = 6.27 \[ \frac {1}{19} x^{19} d^{16} b + \frac {8}{9} x^{18} d^{15} c b + \frac {1}{18} x^{18} d^{16} a + \frac {120}{17} x^{17} d^{14} c^{2} b + \frac {16}{17} x^{17} d^{15} c a + 35 x^{16} d^{13} c^{3} b + \frac {15}{2} x^{16} d^{14} c^{2} a + \frac {364}{3} x^{15} d^{12} c^{4} b + \frac {112}{3} x^{15} d^{13} c^{3} a + 312 x^{14} d^{11} c^{5} b + 130 x^{14} d^{12} c^{4} a + 616 x^{13} d^{10} c^{6} b + 336 x^{13} d^{11} c^{5} a + \frac {2860}{3} x^{12} d^{9} c^{7} b + \frac {2002}{3} x^{12} d^{10} c^{6} a + 1170 x^{11} d^{8} c^{8} b + 1040 x^{11} d^{9} c^{7} a + 1144 x^{10} d^{7} c^{9} b + 1287 x^{10} d^{8} c^{8} a + \frac {8008}{9} x^{9} d^{6} c^{10} b + \frac {11440}{9} x^{9} d^{7} c^{9} a + 546 x^{8} d^{5} c^{11} b + 1001 x^{8} d^{6} c^{10} a + 260 x^{7} d^{4} c^{12} b + 624 x^{7} d^{5} c^{11} a + \frac {280}{3} x^{6} d^{3} c^{13} b + \frac {910}{3} x^{6} d^{4} c^{12} a + 24 x^{5} d^{2} c^{14} b + 112 x^{5} d^{3} c^{13} a + 4 x^{4} d c^{15} b + 30 x^{4} d^{2} c^{14} a + \frac {1}{3} x^{3} c^{16} b + \frac {16}{3} x^{3} d c^{15} a + \frac {1}{2} x^{2} c^{16} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/19*x^19*d^16*b + 8/9*x^18*d^15*c*b + 1/18*x^18*d^16*a + 120/17*x^17*d^14*c^2*b + 16/17*x^17*d^15*c*a + 35*x^
16*d^13*c^3*b + 15/2*x^16*d^14*c^2*a + 364/3*x^15*d^12*c^4*b + 112/3*x^15*d^13*c^3*a + 312*x^14*d^11*c^5*b + 1
30*x^14*d^12*c^4*a + 616*x^13*d^10*c^6*b + 336*x^13*d^11*c^5*a + 2860/3*x^12*d^9*c^7*b + 2002/3*x^12*d^10*c^6*
a + 1170*x^11*d^8*c^8*b + 1040*x^11*d^9*c^7*a + 1144*x^10*d^7*c^9*b + 1287*x^10*d^8*c^8*a + 8008/9*x^9*d^6*c^1
0*b + 11440/9*x^9*d^7*c^9*a + 546*x^8*d^5*c^11*b + 1001*x^8*d^6*c^10*a + 260*x^7*d^4*c^12*b + 624*x^7*d^5*c^11
*a + 280/3*x^6*d^3*c^13*b + 910/3*x^6*d^4*c^12*a + 24*x^5*d^2*c^14*b + 112*x^5*d^3*c^13*a + 4*x^4*d*c^15*b + 3
0*x^4*d^2*c^14*a + 1/3*x^3*c^16*b + 16/3*x^3*d*c^15*a + 1/2*x^2*c^16*a

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giac [B]  time = 0.91, size = 389, normalized size = 6.27 \[ \frac {1}{19} \, b d^{16} x^{19} + \frac {8}{9} \, b c d^{15} x^{18} + \frac {1}{18} \, a d^{16} x^{18} + \frac {120}{17} \, b c^{2} d^{14} x^{17} + \frac {16}{17} \, a c d^{15} x^{17} + 35 \, b c^{3} d^{13} x^{16} + \frac {15}{2} \, a c^{2} d^{14} x^{16} + \frac {364}{3} \, b c^{4} d^{12} x^{15} + \frac {112}{3} \, a c^{3} d^{13} x^{15} + 312 \, b c^{5} d^{11} x^{14} + 130 \, a c^{4} d^{12} x^{14} + 616 \, b c^{6} d^{10} x^{13} + 336 \, a c^{5} d^{11} x^{13} + \frac {2860}{3} \, b c^{7} d^{9} x^{12} + \frac {2002}{3} \, a c^{6} d^{10} x^{12} + 1170 \, b c^{8} d^{8} x^{11} + 1040 \, a c^{7} d^{9} x^{11} + 1144 \, b c^{9} d^{7} x^{10} + 1287 \, a c^{8} d^{8} x^{10} + \frac {8008}{9} \, b c^{10} d^{6} x^{9} + \frac {11440}{9} \, a c^{9} d^{7} x^{9} + 546 \, b c^{11} d^{5} x^{8} + 1001 \, a c^{10} d^{6} x^{8} + 260 \, b c^{12} d^{4} x^{7} + 624 \, a c^{11} d^{5} x^{7} + \frac {280}{3} \, b c^{13} d^{3} x^{6} + \frac {910}{3} \, a c^{12} d^{4} x^{6} + 24 \, b c^{14} d^{2} x^{5} + 112 \, a c^{13} d^{3} x^{5} + 4 \, b c^{15} d x^{4} + 30 \, a c^{14} d^{2} x^{4} + \frac {1}{3} \, b c^{16} x^{3} + \frac {16}{3} \, a c^{15} d x^{3} + \frac {1}{2} \, a c^{16} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/19*b*d^16*x^19 + 8/9*b*c*d^15*x^18 + 1/18*a*d^16*x^18 + 120/17*b*c^2*d^14*x^17 + 16/17*a*c*d^15*x^17 + 35*b*
c^3*d^13*x^16 + 15/2*a*c^2*d^14*x^16 + 364/3*b*c^4*d^12*x^15 + 112/3*a*c^3*d^13*x^15 + 312*b*c^5*d^11*x^14 + 1
30*a*c^4*d^12*x^14 + 616*b*c^6*d^10*x^13 + 336*a*c^5*d^11*x^13 + 2860/3*b*c^7*d^9*x^12 + 2002/3*a*c^6*d^10*x^1
2 + 1170*b*c^8*d^8*x^11 + 1040*a*c^7*d^9*x^11 + 1144*b*c^9*d^7*x^10 + 1287*a*c^8*d^8*x^10 + 8008/9*b*c^10*d^6*
x^9 + 11440/9*a*c^9*d^7*x^9 + 546*b*c^11*d^5*x^8 + 1001*a*c^10*d^6*x^8 + 260*b*c^12*d^4*x^7 + 624*a*c^11*d^5*x
^7 + 280/3*b*c^13*d^3*x^6 + 910/3*a*c^12*d^4*x^6 + 24*b*c^14*d^2*x^5 + 112*a*c^13*d^3*x^5 + 4*b*c^15*d*x^4 + 3
0*a*c^14*d^2*x^4 + 1/3*b*c^16*x^3 + 16/3*a*c^15*d*x^3 + 1/2*a*c^16*x^2

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maple [B]  time = 0.00, size = 388, normalized size = 6.26 \[ \frac {b \,d^{16} x^{19}}{19}+\frac {a \,c^{16} x^{2}}{2}+\frac {\left (a \,d^{16}+16 b c \,d^{15}\right ) x^{18}}{18}+\frac {\left (16 a c \,d^{15}+120 b \,c^{2} d^{14}\right ) x^{17}}{17}+\frac {\left (120 a \,c^{2} d^{14}+560 b \,c^{3} d^{13}\right ) x^{16}}{16}+\frac {\left (560 a \,c^{3} d^{13}+1820 b \,c^{4} d^{12}\right ) x^{15}}{15}+\frac {\left (1820 a \,c^{4} d^{12}+4368 b \,c^{5} d^{11}\right ) x^{14}}{14}+\frac {\left (4368 a \,c^{5} d^{11}+8008 b \,c^{6} d^{10}\right ) x^{13}}{13}+\frac {\left (8008 a \,c^{6} d^{10}+11440 b \,c^{7} d^{9}\right ) x^{12}}{12}+\frac {\left (11440 a \,c^{7} d^{9}+12870 b \,c^{8} d^{8}\right ) x^{11}}{11}+\frac {\left (12870 a \,c^{8} d^{8}+11440 b \,c^{9} d^{7}\right ) x^{10}}{10}+\frac {\left (11440 a \,c^{9} d^{7}+8008 b \,c^{10} d^{6}\right ) x^{9}}{9}+\frac {\left (8008 a \,c^{10} d^{6}+4368 b \,c^{11} d^{5}\right ) x^{8}}{8}+\frac {\left (4368 a \,c^{11} d^{5}+1820 b \,c^{12} d^{4}\right ) x^{7}}{7}+\frac {\left (1820 a \,c^{12} d^{4}+560 b \,c^{13} d^{3}\right ) x^{6}}{6}+\frac {\left (560 a \,c^{13} d^{3}+120 b \,c^{14} d^{2}\right ) x^{5}}{5}+\frac {\left (120 a \,c^{14} d^{2}+16 b \,c^{15} d \right ) x^{4}}{4}+\frac {\left (16 a \,c^{15} d +b \,c^{16}\right ) x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b*x+a)*(d*x+c)^16,x)

[Out]

1/19*b*d^16*x^19+1/18*(a*d^16+16*b*c*d^15)*x^18+1/17*(16*a*c*d^15+120*b*c^2*d^14)*x^17+1/16*(120*a*c^2*d^14+56
0*b*c^3*d^13)*x^16+1/15*(560*a*c^3*d^13+1820*b*c^4*d^12)*x^15+1/14*(1820*a*c^4*d^12+4368*b*c^5*d^11)*x^14+1/13
*(4368*a*c^5*d^11+8008*b*c^6*d^10)*x^13+1/12*(8008*a*c^6*d^10+11440*b*c^7*d^9)*x^12+1/11*(11440*a*c^7*d^9+1287
0*b*c^8*d^8)*x^11+1/10*(12870*a*c^8*d^8+11440*b*c^9*d^7)*x^10+1/9*(11440*a*c^9*d^7+8008*b*c^10*d^6)*x^9+1/8*(8
008*a*c^10*d^6+4368*b*c^11*d^5)*x^8+1/7*(4368*a*c^11*d^5+1820*b*c^12*d^4)*x^7+1/6*(1820*a*c^12*d^4+560*b*c^13*
d^3)*x^6+1/5*(560*a*c^13*d^3+120*b*c^14*d^2)*x^5+1/4*(120*a*c^14*d^2+16*b*c^15*d)*x^4+1/3*(16*a*c^15*d+b*c^16)
*x^3+1/2*a*c^16*x^2

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maxima [B]  time = 1.10, size = 387, normalized size = 6.24 \[ \frac {1}{19} \, b d^{16} x^{19} + \frac {1}{2} \, a c^{16} x^{2} + \frac {1}{18} \, {\left (16 \, b c d^{15} + a d^{16}\right )} x^{18} + \frac {8}{17} \, {\left (15 \, b c^{2} d^{14} + 2 \, a c d^{15}\right )} x^{17} + \frac {5}{2} \, {\left (14 \, b c^{3} d^{13} + 3 \, a c^{2} d^{14}\right )} x^{16} + \frac {28}{3} \, {\left (13 \, b c^{4} d^{12} + 4 \, a c^{3} d^{13}\right )} x^{15} + 26 \, {\left (12 \, b c^{5} d^{11} + 5 \, a c^{4} d^{12}\right )} x^{14} + 56 \, {\left (11 \, b c^{6} d^{10} + 6 \, a c^{5} d^{11}\right )} x^{13} + \frac {286}{3} \, {\left (10 \, b c^{7} d^{9} + 7 \, a c^{6} d^{10}\right )} x^{12} + 130 \, {\left (9 \, b c^{8} d^{8} + 8 \, a c^{7} d^{9}\right )} x^{11} + 143 \, {\left (8 \, b c^{9} d^{7} + 9 \, a c^{8} d^{8}\right )} x^{10} + \frac {1144}{9} \, {\left (7 \, b c^{10} d^{6} + 10 \, a c^{9} d^{7}\right )} x^{9} + 91 \, {\left (6 \, b c^{11} d^{5} + 11 \, a c^{10} d^{6}\right )} x^{8} + 52 \, {\left (5 \, b c^{12} d^{4} + 12 \, a c^{11} d^{5}\right )} x^{7} + \frac {70}{3} \, {\left (4 \, b c^{13} d^{3} + 13 \, a c^{12} d^{4}\right )} x^{6} + 8 \, {\left (3 \, b c^{14} d^{2} + 14 \, a c^{13} d^{3}\right )} x^{5} + 2 \, {\left (2 \, b c^{15} d + 15 \, a c^{14} d^{2}\right )} x^{4} + \frac {1}{3} \, {\left (b c^{16} + 16 \, a c^{15} d\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/19*b*d^16*x^19 + 1/2*a*c^16*x^2 + 1/18*(16*b*c*d^15 + a*d^16)*x^18 + 8/17*(15*b*c^2*d^14 + 2*a*c*d^15)*x^17
+ 5/2*(14*b*c^3*d^13 + 3*a*c^2*d^14)*x^16 + 28/3*(13*b*c^4*d^12 + 4*a*c^3*d^13)*x^15 + 26*(12*b*c^5*d^11 + 5*a
*c^4*d^12)*x^14 + 56*(11*b*c^6*d^10 + 6*a*c^5*d^11)*x^13 + 286/3*(10*b*c^7*d^9 + 7*a*c^6*d^10)*x^12 + 130*(9*b
*c^8*d^8 + 8*a*c^7*d^9)*x^11 + 143*(8*b*c^9*d^7 + 9*a*c^8*d^8)*x^10 + 1144/9*(7*b*c^10*d^6 + 10*a*c^9*d^7)*x^9
+ 91*(6*b*c^11*d^5 + 11*a*c^10*d^6)*x^8 + 52*(5*b*c^12*d^4 + 12*a*c^11*d^5)*x^7 + 70/3*(4*b*c^13*d^3 + 13*a*c
^12*d^4)*x^6 + 8*(3*b*c^14*d^2 + 14*a*c^13*d^3)*x^5 + 2*(2*b*c^15*d + 15*a*c^14*d^2)*x^4 + 1/3*(b*c^16 + 16*a*
c^15*d)*x^3

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mupad [B]  time = 0.28, size = 331, normalized size = 5.34 \[ x^3\,\left (\frac {b\,c^{16}}{3}+\frac {16\,a\,d\,c^{15}}{3}\right )+x^{18}\,\left (\frac {a\,d^{16}}{18}+\frac {8\,b\,c\,d^{15}}{9}\right )+\frac {a\,c^{16}\,x^2}{2}+\frac {b\,d^{16}\,x^{19}}{19}+2\,c^{14}\,d\,x^4\,\left (15\,a\,d+2\,b\,c\right )+\frac {8\,c\,d^{14}\,x^{17}\,\left (2\,a\,d+15\,b\,c\right )}{17}+8\,c^{13}\,d^2\,x^5\,\left (14\,a\,d+3\,b\,c\right )+\frac {70\,c^{12}\,d^3\,x^6\,\left (13\,a\,d+4\,b\,c\right )}{3}+52\,c^{11}\,d^4\,x^7\,\left (12\,a\,d+5\,b\,c\right )+91\,c^{10}\,d^5\,x^8\,\left (11\,a\,d+6\,b\,c\right )+\frac {1144\,c^9\,d^6\,x^9\,\left (10\,a\,d+7\,b\,c\right )}{9}+143\,c^8\,d^7\,x^{10}\,\left (9\,a\,d+8\,b\,c\right )+130\,c^7\,d^8\,x^{11}\,\left (8\,a\,d+9\,b\,c\right )+\frac {286\,c^6\,d^9\,x^{12}\,\left (7\,a\,d+10\,b\,c\right )}{3}+56\,c^5\,d^{10}\,x^{13}\,\left (6\,a\,d+11\,b\,c\right )+26\,c^4\,d^{11}\,x^{14}\,\left (5\,a\,d+12\,b\,c\right )+\frac {28\,c^3\,d^{12}\,x^{15}\,\left (4\,a\,d+13\,b\,c\right )}{3}+\frac {5\,c^2\,d^{13}\,x^{16}\,\left (3\,a\,d+14\,b\,c\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*x)*(c + d*x)^16,x)

[Out]

x^3*((b*c^16)/3 + (16*a*c^15*d)/3) + x^18*((a*d^16)/18 + (8*b*c*d^15)/9) + (a*c^16*x^2)/2 + (b*d^16*x^19)/19 +
2*c^14*d*x^4*(15*a*d + 2*b*c) + (8*c*d^14*x^17*(2*a*d + 15*b*c))/17 + 8*c^13*d^2*x^5*(14*a*d + 3*b*c) + (70*c
^12*d^3*x^6*(13*a*d + 4*b*c))/3 + 52*c^11*d^4*x^7*(12*a*d + 5*b*c) + 91*c^10*d^5*x^8*(11*a*d + 6*b*c) + (1144*
c^9*d^6*x^9*(10*a*d + 7*b*c))/9 + 143*c^8*d^7*x^10*(9*a*d + 8*b*c) + 130*c^7*d^8*x^11*(8*a*d + 9*b*c) + (286*c
^6*d^9*x^12*(7*a*d + 10*b*c))/3 + 56*c^5*d^10*x^13*(6*a*d + 11*b*c) + 26*c^4*d^11*x^14*(5*a*d + 12*b*c) + (28*
c^3*d^12*x^15*(4*a*d + 13*b*c))/3 + (5*c^2*d^13*x^16*(3*a*d + 14*b*c))/2

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sympy [B]  time = 0.17, size = 408, normalized size = 6.58 \[ \frac {a c^{16} x^{2}}{2} + \frac {b d^{16} x^{19}}{19} + x^{18} \left (\frac {a d^{16}}{18} + \frac {8 b c d^{15}}{9}\right ) + x^{17} \left (\frac {16 a c d^{15}}{17} + \frac {120 b c^{2} d^{14}}{17}\right ) + x^{16} \left (\frac {15 a c^{2} d^{14}}{2} + 35 b c^{3} d^{13}\right ) + x^{15} \left (\frac {112 a c^{3} d^{13}}{3} + \frac {364 b c^{4} d^{12}}{3}\right ) + x^{14} \left (130 a c^{4} d^{12} + 312 b c^{5} d^{11}\right ) + x^{13} \left (336 a c^{5} d^{11} + 616 b c^{6} d^{10}\right ) + x^{12} \left (\frac {2002 a c^{6} d^{10}}{3} + \frac {2860 b c^{7} d^{9}}{3}\right ) + x^{11} \left (1040 a c^{7} d^{9} + 1170 b c^{8} d^{8}\right ) + x^{10} \left (1287 a c^{8} d^{8} + 1144 b c^{9} d^{7}\right ) + x^{9} \left (\frac {11440 a c^{9} d^{7}}{9} + \frac {8008 b c^{10} d^{6}}{9}\right ) + x^{8} \left (1001 a c^{10} d^{6} + 546 b c^{11} d^{5}\right ) + x^{7} \left (624 a c^{11} d^{5} + 260 b c^{12} d^{4}\right ) + x^{6} \left (\frac {910 a c^{12} d^{4}}{3} + \frac {280 b c^{13} d^{3}}{3}\right ) + x^{5} \left (112 a c^{13} d^{3} + 24 b c^{14} d^{2}\right ) + x^{4} \left (30 a c^{14} d^{2} + 4 b c^{15} d\right ) + x^{3} \left (\frac {16 a c^{15} d}{3} + \frac {b c^{16}}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*c**16*x**2/2 + b*d**16*x**19/19 + x**18*(a*d**16/18 + 8*b*c*d**15/9) + x**17*(16*a*c*d**15/17 + 120*b*c**2*d
**14/17) + x**16*(15*a*c**2*d**14/2 + 35*b*c**3*d**13) + x**15*(112*a*c**3*d**13/3 + 364*b*c**4*d**12/3) + x**
14*(130*a*c**4*d**12 + 312*b*c**5*d**11) + x**13*(336*a*c**5*d**11 + 616*b*c**6*d**10) + x**12*(2002*a*c**6*d*
*10/3 + 2860*b*c**7*d**9/3) + x**11*(1040*a*c**7*d**9 + 1170*b*c**8*d**8) + x**10*(1287*a*c**8*d**8 + 1144*b*c
**9*d**7) + x**9*(11440*a*c**9*d**7/9 + 8008*b*c**10*d**6/9) + x**8*(1001*a*c**10*d**6 + 546*b*c**11*d**5) + x
**7*(624*a*c**11*d**5 + 260*b*c**12*d**4) + x**6*(910*a*c**12*d**4/3 + 280*b*c**13*d**3/3) + x**5*(112*a*c**13
*d**3 + 24*b*c**14*d**2) + x**4*(30*a*c**14*d**2 + 4*b*c**15*d) + x**3*(16*a*c**15*d/3 + b*c**16/3)

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