3.2.55 \(\int x^3 (a+b x) (c+d x)^{16} \, dx\)

Optimal. Leaf size=114 \[ \frac {c^3 (c+d x)^{17} (b c-a d)}{17 d^5}-\frac {c^2 (c+d x)^{18} (4 b c-3 a d)}{18 d^5}-\frac {(c+d x)^{20} (4 b c-a d)}{20 d^5}+\frac {3 c (c+d x)^{19} (2 b c-a d)}{19 d^5}+\frac {b (c+d x)^{21}}{21 d^5} \]

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Rubi [A]  time = 0.36, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {76} \begin {gather*} -\frac {c^2 (c+d x)^{18} (4 b c-3 a d)}{18 d^5}+\frac {c^3 (c+d x)^{17} (b c-a d)}{17 d^5}-\frac {(c+d x)^{20} (4 b c-a d)}{20 d^5}+\frac {3 c (c+d x)^{19} (2 b c-a d)}{19 d^5}+\frac {b (c+d x)^{21}}{21 d^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x)*(c + d*x)^16,x]

[Out]

(c^3*(b*c - a*d)*(c + d*x)^17)/(17*d^5) - (c^2*(4*b*c - 3*a*d)*(c + d*x)^18)/(18*d^5) + (3*c*(2*b*c - a*d)*(c
+ d*x)^19)/(19*d^5) - ((4*b*c - a*d)*(c + d*x)^20)/(20*d^5) + (b*(c + d*x)^21)/(21*d^5)

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^3 (a+b x) (c+d x)^{16} \, dx &=\int \left (\frac {c^3 (b c-a d) (c+d x)^{16}}{d^4}-\frac {c^2 (4 b c-3 a d) (c+d x)^{17}}{d^4}+\frac {3 c (2 b c-a d) (c+d x)^{18}}{d^4}+\frac {(-4 b c+a d) (c+d x)^{19}}{d^4}+\frac {b (c+d x)^{20}}{d^4}\right ) \, dx\\ &=\frac {c^3 (b c-a d) (c+d x)^{17}}{17 d^5}-\frac {c^2 (4 b c-3 a d) (c+d x)^{18}}{18 d^5}+\frac {3 c (2 b c-a d) (c+d x)^{19}}{19 d^5}-\frac {(4 b c-a d) (c+d x)^{20}}{20 d^5}+\frac {b (c+d x)^{21}}{21 d^5}\\ \end {align*}

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Mathematica [B]  time = 0.10, size = 359, normalized size = 3.15 \begin {gather*} \frac {1}{5} c^{15} x^5 (16 a d+b c)+\frac {4}{3} c^{14} d x^6 (15 a d+2 b c)+\frac {40}{7} c^{13} d^2 x^7 (14 a d+3 b c)+\frac {35}{2} c^{12} d^3 x^8 (13 a d+4 b c)+\frac {364}{9} c^{11} d^4 x^9 (12 a d+5 b c)+\frac {364}{5} c^{10} d^5 x^{10} (11 a d+6 b c)+104 c^9 d^6 x^{11} (10 a d+7 b c)+\frac {715}{6} c^8 d^7 x^{12} (9 a d+8 b c)+110 c^7 d^8 x^{13} (8 a d+9 b c)+\frac {572}{7} c^6 d^9 x^{14} (7 a d+10 b c)+\frac {728}{15} c^5 d^{10} x^{15} (6 a d+11 b c)+\frac {91}{4} c^4 d^{11} x^{16} (5 a d+12 b c)+\frac {140}{17} c^3 d^{12} x^{17} (4 a d+13 b c)+\frac {20}{9} c^2 d^{13} x^{18} (3 a d+14 b c)+\frac {1}{20} d^{15} x^{20} (a d+16 b c)+\frac {8}{19} c d^{14} x^{19} (2 a d+15 b c)+\frac {1}{4} a c^{16} x^4+\frac {1}{21} b d^{16} x^{21} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x)*(c + d*x)^16,x]

[Out]

(a*c^16*x^4)/4 + (c^15*(b*c + 16*a*d)*x^5)/5 + (4*c^14*d*(2*b*c + 15*a*d)*x^6)/3 + (40*c^13*d^2*(3*b*c + 14*a*
d)*x^7)/7 + (35*c^12*d^3*(4*b*c + 13*a*d)*x^8)/2 + (364*c^11*d^4*(5*b*c + 12*a*d)*x^9)/9 + (364*c^10*d^5*(6*b*
c + 11*a*d)*x^10)/5 + 104*c^9*d^6*(7*b*c + 10*a*d)*x^11 + (715*c^8*d^7*(8*b*c + 9*a*d)*x^12)/6 + 110*c^7*d^8*(
9*b*c + 8*a*d)*x^13 + (572*c^6*d^9*(10*b*c + 7*a*d)*x^14)/7 + (728*c^5*d^10*(11*b*c + 6*a*d)*x^15)/15 + (91*c^
4*d^11*(12*b*c + 5*a*d)*x^16)/4 + (140*c^3*d^12*(13*b*c + 4*a*d)*x^17)/17 + (20*c^2*d^13*(14*b*c + 3*a*d)*x^18
)/9 + (8*c*d^14*(15*b*c + 2*a*d)*x^19)/19 + (d^15*(16*b*c + a*d)*x^20)/20 + (b*d^16*x^21)/21

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^3 (a+b x) (c+d x)^{16} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^3*(a + b*x)*(c + d*x)^16,x]

[Out]

IntegrateAlgebraic[x^3*(a + b*x)*(c + d*x)^16, x]

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fricas [B]  time = 0.91, size = 389, normalized size = 3.41 \begin {gather*} \frac {1}{21} x^{21} d^{16} b + \frac {4}{5} x^{20} d^{15} c b + \frac {1}{20} x^{20} d^{16} a + \frac {120}{19} x^{19} d^{14} c^{2} b + \frac {16}{19} x^{19} d^{15} c a + \frac {280}{9} x^{18} d^{13} c^{3} b + \frac {20}{3} x^{18} d^{14} c^{2} a + \frac {1820}{17} x^{17} d^{12} c^{4} b + \frac {560}{17} x^{17} d^{13} c^{3} a + 273 x^{16} d^{11} c^{5} b + \frac {455}{4} x^{16} d^{12} c^{4} a + \frac {8008}{15} x^{15} d^{10} c^{6} b + \frac {1456}{5} x^{15} d^{11} c^{5} a + \frac {5720}{7} x^{14} d^{9} c^{7} b + 572 x^{14} d^{10} c^{6} a + 990 x^{13} d^{8} c^{8} b + 880 x^{13} d^{9} c^{7} a + \frac {2860}{3} x^{12} d^{7} c^{9} b + \frac {2145}{2} x^{12} d^{8} c^{8} a + 728 x^{11} d^{6} c^{10} b + 1040 x^{11} d^{7} c^{9} a + \frac {2184}{5} x^{10} d^{5} c^{11} b + \frac {4004}{5} x^{10} d^{6} c^{10} a + \frac {1820}{9} x^{9} d^{4} c^{12} b + \frac {1456}{3} x^{9} d^{5} c^{11} a + 70 x^{8} d^{3} c^{13} b + \frac {455}{2} x^{8} d^{4} c^{12} a + \frac {120}{7} x^{7} d^{2} c^{14} b + 80 x^{7} d^{3} c^{13} a + \frac {8}{3} x^{6} d c^{15} b + 20 x^{6} d^{2} c^{14} a + \frac {1}{5} x^{5} c^{16} b + \frac {16}{5} x^{5} d c^{15} a + \frac {1}{4} x^{4} c^{16} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)*(d*x+c)^16,x, algorithm="fricas")

[Out]

1/21*x^21*d^16*b + 4/5*x^20*d^15*c*b + 1/20*x^20*d^16*a + 120/19*x^19*d^14*c^2*b + 16/19*x^19*d^15*c*a + 280/9
*x^18*d^13*c^3*b + 20/3*x^18*d^14*c^2*a + 1820/17*x^17*d^12*c^4*b + 560/17*x^17*d^13*c^3*a + 273*x^16*d^11*c^5
*b + 455/4*x^16*d^12*c^4*a + 8008/15*x^15*d^10*c^6*b + 1456/5*x^15*d^11*c^5*a + 5720/7*x^14*d^9*c^7*b + 572*x^
14*d^10*c^6*a + 990*x^13*d^8*c^8*b + 880*x^13*d^9*c^7*a + 2860/3*x^12*d^7*c^9*b + 2145/2*x^12*d^8*c^8*a + 728*
x^11*d^6*c^10*b + 1040*x^11*d^7*c^9*a + 2184/5*x^10*d^5*c^11*b + 4004/5*x^10*d^6*c^10*a + 1820/9*x^9*d^4*c^12*
b + 1456/3*x^9*d^5*c^11*a + 70*x^8*d^3*c^13*b + 455/2*x^8*d^4*c^12*a + 120/7*x^7*d^2*c^14*b + 80*x^7*d^3*c^13*
a + 8/3*x^6*d*c^15*b + 20*x^6*d^2*c^14*a + 1/5*x^5*c^16*b + 16/5*x^5*d*c^15*a + 1/4*x^4*c^16*a

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giac [B]  time = 0.99, size = 389, normalized size = 3.41 \begin {gather*} \frac {1}{21} \, b d^{16} x^{21} + \frac {4}{5} \, b c d^{15} x^{20} + \frac {1}{20} \, a d^{16} x^{20} + \frac {120}{19} \, b c^{2} d^{14} x^{19} + \frac {16}{19} \, a c d^{15} x^{19} + \frac {280}{9} \, b c^{3} d^{13} x^{18} + \frac {20}{3} \, a c^{2} d^{14} x^{18} + \frac {1820}{17} \, b c^{4} d^{12} x^{17} + \frac {560}{17} \, a c^{3} d^{13} x^{17} + 273 \, b c^{5} d^{11} x^{16} + \frac {455}{4} \, a c^{4} d^{12} x^{16} + \frac {8008}{15} \, b c^{6} d^{10} x^{15} + \frac {1456}{5} \, a c^{5} d^{11} x^{15} + \frac {5720}{7} \, b c^{7} d^{9} x^{14} + 572 \, a c^{6} d^{10} x^{14} + 990 \, b c^{8} d^{8} x^{13} + 880 \, a c^{7} d^{9} x^{13} + \frac {2860}{3} \, b c^{9} d^{7} x^{12} + \frac {2145}{2} \, a c^{8} d^{8} x^{12} + 728 \, b c^{10} d^{6} x^{11} + 1040 \, a c^{9} d^{7} x^{11} + \frac {2184}{5} \, b c^{11} d^{5} x^{10} + \frac {4004}{5} \, a c^{10} d^{6} x^{10} + \frac {1820}{9} \, b c^{12} d^{4} x^{9} + \frac {1456}{3} \, a c^{11} d^{5} x^{9} + 70 \, b c^{13} d^{3} x^{8} + \frac {455}{2} \, a c^{12} d^{4} x^{8} + \frac {120}{7} \, b c^{14} d^{2} x^{7} + 80 \, a c^{13} d^{3} x^{7} + \frac {8}{3} \, b c^{15} d x^{6} + 20 \, a c^{14} d^{2} x^{6} + \frac {1}{5} \, b c^{16} x^{5} + \frac {16}{5} \, a c^{15} d x^{5} + \frac {1}{4} \, a c^{16} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)*(d*x+c)^16,x, algorithm="giac")

[Out]

1/21*b*d^16*x^21 + 4/5*b*c*d^15*x^20 + 1/20*a*d^16*x^20 + 120/19*b*c^2*d^14*x^19 + 16/19*a*c*d^15*x^19 + 280/9
*b*c^3*d^13*x^18 + 20/3*a*c^2*d^14*x^18 + 1820/17*b*c^4*d^12*x^17 + 560/17*a*c^3*d^13*x^17 + 273*b*c^5*d^11*x^
16 + 455/4*a*c^4*d^12*x^16 + 8008/15*b*c^6*d^10*x^15 + 1456/5*a*c^5*d^11*x^15 + 5720/7*b*c^7*d^9*x^14 + 572*a*
c^6*d^10*x^14 + 990*b*c^8*d^8*x^13 + 880*a*c^7*d^9*x^13 + 2860/3*b*c^9*d^7*x^12 + 2145/2*a*c^8*d^8*x^12 + 728*
b*c^10*d^6*x^11 + 1040*a*c^9*d^7*x^11 + 2184/5*b*c^11*d^5*x^10 + 4004/5*a*c^10*d^6*x^10 + 1820/9*b*c^12*d^4*x^
9 + 1456/3*a*c^11*d^5*x^9 + 70*b*c^13*d^3*x^8 + 455/2*a*c^12*d^4*x^8 + 120/7*b*c^14*d^2*x^7 + 80*a*c^13*d^3*x^
7 + 8/3*b*c^15*d*x^6 + 20*a*c^14*d^2*x^6 + 1/5*b*c^16*x^5 + 16/5*a*c^15*d*x^5 + 1/4*a*c^16*x^4

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maple [B]  time = 0.00, size = 388, normalized size = 3.40 \begin {gather*} \frac {b \,d^{16} x^{21}}{21}+\frac {a \,c^{16} x^{4}}{4}+\frac {\left (a \,d^{16}+16 b c \,d^{15}\right ) x^{20}}{20}+\frac {\left (16 a c \,d^{15}+120 b \,c^{2} d^{14}\right ) x^{19}}{19}+\frac {\left (120 a \,c^{2} d^{14}+560 b \,c^{3} d^{13}\right ) x^{18}}{18}+\frac {\left (560 a \,c^{3} d^{13}+1820 b \,c^{4} d^{12}\right ) x^{17}}{17}+\frac {\left (1820 a \,c^{4} d^{12}+4368 b \,c^{5} d^{11}\right ) x^{16}}{16}+\frac {\left (4368 a \,c^{5} d^{11}+8008 b \,c^{6} d^{10}\right ) x^{15}}{15}+\frac {\left (8008 a \,c^{6} d^{10}+11440 b \,c^{7} d^{9}\right ) x^{14}}{14}+\frac {\left (11440 a \,c^{7} d^{9}+12870 b \,c^{8} d^{8}\right ) x^{13}}{13}+\frac {\left (12870 a \,c^{8} d^{8}+11440 b \,c^{9} d^{7}\right ) x^{12}}{12}+\frac {\left (11440 a \,c^{9} d^{7}+8008 b \,c^{10} d^{6}\right ) x^{11}}{11}+\frac {\left (8008 a \,c^{10} d^{6}+4368 b \,c^{11} d^{5}\right ) x^{10}}{10}+\frac {\left (4368 a \,c^{11} d^{5}+1820 b \,c^{12} d^{4}\right ) x^{9}}{9}+\frac {\left (1820 a \,c^{12} d^{4}+560 b \,c^{13} d^{3}\right ) x^{8}}{8}+\frac {\left (560 a \,c^{13} d^{3}+120 b \,c^{14} d^{2}\right ) x^{7}}{7}+\frac {\left (120 a \,c^{14} d^{2}+16 b \,c^{15} d \right ) x^{6}}{6}+\frac {\left (16 a \,c^{15} d +b \,c^{16}\right ) x^{5}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x+a)*(d*x+c)^16,x)

[Out]

1/21*b*d^16*x^21+1/20*(a*d^16+16*b*c*d^15)*x^20+1/19*(16*a*c*d^15+120*b*c^2*d^14)*x^19+1/18*(120*a*c^2*d^14+56
0*b*c^3*d^13)*x^18+1/17*(560*a*c^3*d^13+1820*b*c^4*d^12)*x^17+1/16*(1820*a*c^4*d^12+4368*b*c^5*d^11)*x^16+1/15
*(4368*a*c^5*d^11+8008*b*c^6*d^10)*x^15+1/14*(8008*a*c^6*d^10+11440*b*c^7*d^9)*x^14+1/13*(11440*a*c^7*d^9+1287
0*b*c^8*d^8)*x^13+1/12*(12870*a*c^8*d^8+11440*b*c^9*d^7)*x^12+1/11*(11440*a*c^9*d^7+8008*b*c^10*d^6)*x^11+1/10
*(8008*a*c^10*d^6+4368*b*c^11*d^5)*x^10+1/9*(4368*a*c^11*d^5+1820*b*c^12*d^4)*x^9+1/8*(1820*a*c^12*d^4+560*b*c
^13*d^3)*x^8+1/7*(560*a*c^13*d^3+120*b*c^14*d^2)*x^7+1/6*(120*a*c^14*d^2+16*b*c^15*d)*x^6+1/5*(16*a*c^15*d+b*c
^16)*x^5+1/4*a*c^16*x^4

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maxima [B]  time = 1.04, size = 387, normalized size = 3.39 \begin {gather*} \frac {1}{21} \, b d^{16} x^{21} + \frac {1}{4} \, a c^{16} x^{4} + \frac {1}{20} \, {\left (16 \, b c d^{15} + a d^{16}\right )} x^{20} + \frac {8}{19} \, {\left (15 \, b c^{2} d^{14} + 2 \, a c d^{15}\right )} x^{19} + \frac {20}{9} \, {\left (14 \, b c^{3} d^{13} + 3 \, a c^{2} d^{14}\right )} x^{18} + \frac {140}{17} \, {\left (13 \, b c^{4} d^{12} + 4 \, a c^{3} d^{13}\right )} x^{17} + \frac {91}{4} \, {\left (12 \, b c^{5} d^{11} + 5 \, a c^{4} d^{12}\right )} x^{16} + \frac {728}{15} \, {\left (11 \, b c^{6} d^{10} + 6 \, a c^{5} d^{11}\right )} x^{15} + \frac {572}{7} \, {\left (10 \, b c^{7} d^{9} + 7 \, a c^{6} d^{10}\right )} x^{14} + 110 \, {\left (9 \, b c^{8} d^{8} + 8 \, a c^{7} d^{9}\right )} x^{13} + \frac {715}{6} \, {\left (8 \, b c^{9} d^{7} + 9 \, a c^{8} d^{8}\right )} x^{12} + 104 \, {\left (7 \, b c^{10} d^{6} + 10 \, a c^{9} d^{7}\right )} x^{11} + \frac {364}{5} \, {\left (6 \, b c^{11} d^{5} + 11 \, a c^{10} d^{6}\right )} x^{10} + \frac {364}{9} \, {\left (5 \, b c^{12} d^{4} + 12 \, a c^{11} d^{5}\right )} x^{9} + \frac {35}{2} \, {\left (4 \, b c^{13} d^{3} + 13 \, a c^{12} d^{4}\right )} x^{8} + \frac {40}{7} \, {\left (3 \, b c^{14} d^{2} + 14 \, a c^{13} d^{3}\right )} x^{7} + \frac {4}{3} \, {\left (2 \, b c^{15} d + 15 \, a c^{14} d^{2}\right )} x^{6} + \frac {1}{5} \, {\left (b c^{16} + 16 \, a c^{15} d\right )} x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)*(d*x+c)^16,x, algorithm="maxima")

[Out]

1/21*b*d^16*x^21 + 1/4*a*c^16*x^4 + 1/20*(16*b*c*d^15 + a*d^16)*x^20 + 8/19*(15*b*c^2*d^14 + 2*a*c*d^15)*x^19
+ 20/9*(14*b*c^3*d^13 + 3*a*c^2*d^14)*x^18 + 140/17*(13*b*c^4*d^12 + 4*a*c^3*d^13)*x^17 + 91/4*(12*b*c^5*d^11
+ 5*a*c^4*d^12)*x^16 + 728/15*(11*b*c^6*d^10 + 6*a*c^5*d^11)*x^15 + 572/7*(10*b*c^7*d^9 + 7*a*c^6*d^10)*x^14 +
110*(9*b*c^8*d^8 + 8*a*c^7*d^9)*x^13 + 715/6*(8*b*c^9*d^7 + 9*a*c^8*d^8)*x^12 + 104*(7*b*c^10*d^6 + 10*a*c^9*
d^7)*x^11 + 364/5*(6*b*c^11*d^5 + 11*a*c^10*d^6)*x^10 + 364/9*(5*b*c^12*d^4 + 12*a*c^11*d^5)*x^9 + 35/2*(4*b*c
^13*d^3 + 13*a*c^12*d^4)*x^8 + 40/7*(3*b*c^14*d^2 + 14*a*c^13*d^3)*x^7 + 4/3*(2*b*c^15*d + 15*a*c^14*d^2)*x^6
+ 1/5*(b*c^16 + 16*a*c^15*d)*x^5

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mupad [B]  time = 0.33, size = 331, normalized size = 2.90 \begin {gather*} x^5\,\left (\frac {b\,c^{16}}{5}+\frac {16\,a\,d\,c^{15}}{5}\right )+x^{20}\,\left (\frac {a\,d^{16}}{20}+\frac {4\,b\,c\,d^{15}}{5}\right )+\frac {a\,c^{16}\,x^4}{4}+\frac {b\,d^{16}\,x^{21}}{21}+\frac {4\,c^{14}\,d\,x^6\,\left (15\,a\,d+2\,b\,c\right )}{3}+\frac {8\,c\,d^{14}\,x^{19}\,\left (2\,a\,d+15\,b\,c\right )}{19}+\frac {40\,c^{13}\,d^2\,x^7\,\left (14\,a\,d+3\,b\,c\right )}{7}+\frac {35\,c^{12}\,d^3\,x^8\,\left (13\,a\,d+4\,b\,c\right )}{2}+\frac {364\,c^{11}\,d^4\,x^9\,\left (12\,a\,d+5\,b\,c\right )}{9}+\frac {364\,c^{10}\,d^5\,x^{10}\,\left (11\,a\,d+6\,b\,c\right )}{5}+104\,c^9\,d^6\,x^{11}\,\left (10\,a\,d+7\,b\,c\right )+\frac {715\,c^8\,d^7\,x^{12}\,\left (9\,a\,d+8\,b\,c\right )}{6}+110\,c^7\,d^8\,x^{13}\,\left (8\,a\,d+9\,b\,c\right )+\frac {572\,c^6\,d^9\,x^{14}\,\left (7\,a\,d+10\,b\,c\right )}{7}+\frac {728\,c^5\,d^{10}\,x^{15}\,\left (6\,a\,d+11\,b\,c\right )}{15}+\frac {91\,c^4\,d^{11}\,x^{16}\,\left (5\,a\,d+12\,b\,c\right )}{4}+\frac {140\,c^3\,d^{12}\,x^{17}\,\left (4\,a\,d+13\,b\,c\right )}{17}+\frac {20\,c^2\,d^{13}\,x^{18}\,\left (3\,a\,d+14\,b\,c\right )}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*x)*(c + d*x)^16,x)

[Out]

x^5*((b*c^16)/5 + (16*a*c^15*d)/5) + x^20*((a*d^16)/20 + (4*b*c*d^15)/5) + (a*c^16*x^4)/4 + (b*d^16*x^21)/21 +
(4*c^14*d*x^6*(15*a*d + 2*b*c))/3 + (8*c*d^14*x^19*(2*a*d + 15*b*c))/19 + (40*c^13*d^2*x^7*(14*a*d + 3*b*c))/
7 + (35*c^12*d^3*x^8*(13*a*d + 4*b*c))/2 + (364*c^11*d^4*x^9*(12*a*d + 5*b*c))/9 + (364*c^10*d^5*x^10*(11*a*d
+ 6*b*c))/5 + 104*c^9*d^6*x^11*(10*a*d + 7*b*c) + (715*c^8*d^7*x^12*(9*a*d + 8*b*c))/6 + 110*c^7*d^8*x^13*(8*a
*d + 9*b*c) + (572*c^6*d^9*x^14*(7*a*d + 10*b*c))/7 + (728*c^5*d^10*x^15*(6*a*d + 11*b*c))/15 + (91*c^4*d^11*x
^16*(5*a*d + 12*b*c))/4 + (140*c^3*d^12*x^17*(4*a*d + 13*b*c))/17 + (20*c^2*d^13*x^18*(3*a*d + 14*b*c))/9

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sympy [B]  time = 0.18, size = 422, normalized size = 3.70 \begin {gather*} \frac {a c^{16} x^{4}}{4} + \frac {b d^{16} x^{21}}{21} + x^{20} \left (\frac {a d^{16}}{20} + \frac {4 b c d^{15}}{5}\right ) + x^{19} \left (\frac {16 a c d^{15}}{19} + \frac {120 b c^{2} d^{14}}{19}\right ) + x^{18} \left (\frac {20 a c^{2} d^{14}}{3} + \frac {280 b c^{3} d^{13}}{9}\right ) + x^{17} \left (\frac {560 a c^{3} d^{13}}{17} + \frac {1820 b c^{4} d^{12}}{17}\right ) + x^{16} \left (\frac {455 a c^{4} d^{12}}{4} + 273 b c^{5} d^{11}\right ) + x^{15} \left (\frac {1456 a c^{5} d^{11}}{5} + \frac {8008 b c^{6} d^{10}}{15}\right ) + x^{14} \left (572 a c^{6} d^{10} + \frac {5720 b c^{7} d^{9}}{7}\right ) + x^{13} \left (880 a c^{7} d^{9} + 990 b c^{8} d^{8}\right ) + x^{12} \left (\frac {2145 a c^{8} d^{8}}{2} + \frac {2860 b c^{9} d^{7}}{3}\right ) + x^{11} \left (1040 a c^{9} d^{7} + 728 b c^{10} d^{6}\right ) + x^{10} \left (\frac {4004 a c^{10} d^{6}}{5} + \frac {2184 b c^{11} d^{5}}{5}\right ) + x^{9} \left (\frac {1456 a c^{11} d^{5}}{3} + \frac {1820 b c^{12} d^{4}}{9}\right ) + x^{8} \left (\frac {455 a c^{12} d^{4}}{2} + 70 b c^{13} d^{3}\right ) + x^{7} \left (80 a c^{13} d^{3} + \frac {120 b c^{14} d^{2}}{7}\right ) + x^{6} \left (20 a c^{14} d^{2} + \frac {8 b c^{15} d}{3}\right ) + x^{5} \left (\frac {16 a c^{15} d}{5} + \frac {b c^{16}}{5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(b*x+a)*(d*x+c)**16,x)

[Out]

a*c**16*x**4/4 + b*d**16*x**21/21 + x**20*(a*d**16/20 + 4*b*c*d**15/5) + x**19*(16*a*c*d**15/19 + 120*b*c**2*d
**14/19) + x**18*(20*a*c**2*d**14/3 + 280*b*c**3*d**13/9) + x**17*(560*a*c**3*d**13/17 + 1820*b*c**4*d**12/17)
+ x**16*(455*a*c**4*d**12/4 + 273*b*c**5*d**11) + x**15*(1456*a*c**5*d**11/5 + 8008*b*c**6*d**10/15) + x**14*
(572*a*c**6*d**10 + 5720*b*c**7*d**9/7) + x**13*(880*a*c**7*d**9 + 990*b*c**8*d**8) + x**12*(2145*a*c**8*d**8/
2 + 2860*b*c**9*d**7/3) + x**11*(1040*a*c**9*d**7 + 728*b*c**10*d**6) + x**10*(4004*a*c**10*d**6/5 + 2184*b*c*
*11*d**5/5) + x**9*(1456*a*c**11*d**5/3 + 1820*b*c**12*d**4/9) + x**8*(455*a*c**12*d**4/2 + 70*b*c**13*d**3) +
x**7*(80*a*c**13*d**3 + 120*b*c**14*d**2/7) + x**6*(20*a*c**14*d**2 + 8*b*c**15*d/3) + x**5*(16*a*c**15*d/5 +
b*c**16/5)

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