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

Optimal. Leaf size=98 \[ -\frac {b (c+d x)^{19} (3 b c-2 a d)}{19 d^4}+\frac {(c+d x)^{18} (b c-a d) (3 b c-a d)}{18 d^4}-\frac {c (c+d x)^{17} (b c-a d)^2}{17 d^4}+\frac {b^2 (c+d x)^{20}}{20 d^4} \]

________________________________________________________________________________________

Rubi [A]  time = 0.23, antiderivative size = 98, 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 = {77} \begin {gather*} -\frac {b (c+d x)^{19} (3 b c-2 a d)}{19 d^4}+\frac {(c+d x)^{18} (b c-a d) (3 b c-a d)}{18 d^4}-\frac {c (c+d x)^{17} (b c-a d)^2}{17 d^4}+\frac {b^2 (c+d x)^{20}}{20 d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x)^2*(c + d*x)^16,x]

[Out]

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

________________________________________________________________________________________

Mathematica [B]  time = 0.12, size = 583, normalized size = 5.95 \begin {gather*} \frac {1}{18} d^{14} x^{18} \left (a^2 d^2+32 a b c d+120 b^2 c^2\right )+\frac {16}{17} c d^{13} x^{17} \left (a^2 d^2+15 a b c d+35 b^2 c^2\right )+\frac {5}{4} c^2 d^{12} x^{16} \left (6 a^2 d^2+56 a b c d+91 b^2 c^2\right )+\frac {1}{4} c^{14} x^4 \left (120 a^2 d^2+32 a b c d+b^2 c^2\right )+\frac {16}{5} c^{13} d x^5 \left (35 a^2 d^2+15 a b c d+b^2 c^2\right )+\frac {10}{3} c^{12} d^2 x^6 \left (91 a^2 d^2+56 a b c d+6 b^2 c^2\right )+8 c^{11} d^3 x^7 \left (78 a^2 d^2+65 a b c d+10 b^2 c^2\right )+\frac {91}{2} c^{10} d^4 x^8 \left (22 a^2 d^2+24 a b c d+5 b^2 c^2\right )+\frac {208}{9} c^9 d^5 x^9 \left (55 a^2 d^2+77 a b c d+21 b^2 c^2\right )+\frac {143}{5} c^8 d^6 x^{10} \left (45 a^2 d^2+80 a b c d+28 b^2 c^2\right )+260 c^7 d^7 x^{11} \left (4 a^2 d^2+9 a b c d+4 b^2 c^2\right )+\frac {143}{6} c^6 d^8 x^{12} \left (28 a^2 d^2+80 a b c d+45 b^2 c^2\right )+16 c^5 d^9 x^{13} \left (21 a^2 d^2+77 a b c d+55 b^2 c^2\right )+26 c^4 d^{10} x^{14} \left (5 a^2 d^2+24 a b c d+22 b^2 c^2\right )+\frac {56}{15} c^3 d^{11} x^{15} \left (10 a^2 d^2+65 a b c d+78 b^2 c^2\right )+\frac {1}{2} a^2 c^{16} x^2+\frac {2}{3} a c^{15} x^3 (8 a d+b c)+\frac {2}{19} b d^{15} x^{19} (a d+8 b c)+\frac {1}{20} b^2 d^{16} x^{20} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*c^16*x^2)/2 + (2*a*c^15*(b*c + 8*a*d)*x^3)/3 + (c^14*(b^2*c^2 + 32*a*b*c*d + 120*a^2*d^2)*x^4)/4 + (16*c^
13*d*(b^2*c^2 + 15*a*b*c*d + 35*a^2*d^2)*x^5)/5 + (10*c^12*d^2*(6*b^2*c^2 + 56*a*b*c*d + 91*a^2*d^2)*x^6)/3 +
8*c^11*d^3*(10*b^2*c^2 + 65*a*b*c*d + 78*a^2*d^2)*x^7 + (91*c^10*d^4*(5*b^2*c^2 + 24*a*b*c*d + 22*a^2*d^2)*x^8
)/2 + (208*c^9*d^5*(21*b^2*c^2 + 77*a*b*c*d + 55*a^2*d^2)*x^9)/9 + (143*c^8*d^6*(28*b^2*c^2 + 80*a*b*c*d + 45*
a^2*d^2)*x^10)/5 + 260*c^7*d^7*(4*b^2*c^2 + 9*a*b*c*d + 4*a^2*d^2)*x^11 + (143*c^6*d^8*(45*b^2*c^2 + 80*a*b*c*
d + 28*a^2*d^2)*x^12)/6 + 16*c^5*d^9*(55*b^2*c^2 + 77*a*b*c*d + 21*a^2*d^2)*x^13 + 26*c^4*d^10*(22*b^2*c^2 + 2
4*a*b*c*d + 5*a^2*d^2)*x^14 + (56*c^3*d^11*(78*b^2*c^2 + 65*a*b*c*d + 10*a^2*d^2)*x^15)/15 + (5*c^2*d^12*(91*b
^2*c^2 + 56*a*b*c*d + 6*a^2*d^2)*x^16)/4 + (16*c*d^13*(35*b^2*c^2 + 15*a*b*c*d + a^2*d^2)*x^17)/17 + (d^14*(12
0*b^2*c^2 + 32*a*b*c*d + a^2*d^2)*x^18)/18 + (2*b*d^15*(8*b*c + a*d)*x^19)/19 + (b^2*d^16*x^20)/20

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x (a+b x)^2 (c+d x)^{16} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 1.13, size = 668, normalized size = 6.82 \begin {gather*} \frac {1}{20} x^{20} d^{16} b^{2} + \frac {16}{19} x^{19} d^{15} c b^{2} + \frac {2}{19} x^{19} d^{16} b a + \frac {20}{3} x^{18} d^{14} c^{2} b^{2} + \frac {16}{9} x^{18} d^{15} c b a + \frac {1}{18} x^{18} d^{16} a^{2} + \frac {560}{17} x^{17} d^{13} c^{3} b^{2} + \frac {240}{17} x^{17} d^{14} c^{2} b a + \frac {16}{17} x^{17} d^{15} c a^{2} + \frac {455}{4} x^{16} d^{12} c^{4} b^{2} + 70 x^{16} d^{13} c^{3} b a + \frac {15}{2} x^{16} d^{14} c^{2} a^{2} + \frac {1456}{5} x^{15} d^{11} c^{5} b^{2} + \frac {728}{3} x^{15} d^{12} c^{4} b a + \frac {112}{3} x^{15} d^{13} c^{3} a^{2} + 572 x^{14} d^{10} c^{6} b^{2} + 624 x^{14} d^{11} c^{5} b a + 130 x^{14} d^{12} c^{4} a^{2} + 880 x^{13} d^{9} c^{7} b^{2} + 1232 x^{13} d^{10} c^{6} b a + 336 x^{13} d^{11} c^{5} a^{2} + \frac {2145}{2} x^{12} d^{8} c^{8} b^{2} + \frac {5720}{3} x^{12} d^{9} c^{7} b a + \frac {2002}{3} x^{12} d^{10} c^{6} a^{2} + 1040 x^{11} d^{7} c^{9} b^{2} + 2340 x^{11} d^{8} c^{8} b a + 1040 x^{11} d^{9} c^{7} a^{2} + \frac {4004}{5} x^{10} d^{6} c^{10} b^{2} + 2288 x^{10} d^{7} c^{9} b a + 1287 x^{10} d^{8} c^{8} a^{2} + \frac {1456}{3} x^{9} d^{5} c^{11} b^{2} + \frac {16016}{9} x^{9} d^{6} c^{10} b a + \frac {11440}{9} x^{9} d^{7} c^{9} a^{2} + \frac {455}{2} x^{8} d^{4} c^{12} b^{2} + 1092 x^{8} d^{5} c^{11} b a + 1001 x^{8} d^{6} c^{10} a^{2} + 80 x^{7} d^{3} c^{13} b^{2} + 520 x^{7} d^{4} c^{12} b a + 624 x^{7} d^{5} c^{11} a^{2} + 20 x^{6} d^{2} c^{14} b^{2} + \frac {560}{3} x^{6} d^{3} c^{13} b a + \frac {910}{3} x^{6} d^{4} c^{12} a^{2} + \frac {16}{5} x^{5} d c^{15} b^{2} + 48 x^{5} d^{2} c^{14} b a + 112 x^{5} d^{3} c^{13} a^{2} + \frac {1}{4} x^{4} c^{16} b^{2} + 8 x^{4} d c^{15} b a + 30 x^{4} d^{2} c^{14} a^{2} + \frac {2}{3} x^{3} c^{16} b a + \frac {16}{3} x^{3} d c^{15} a^{2} + \frac {1}{2} x^{2} c^{16} a^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*x^20*d^16*b^2 + 16/19*x^19*d^15*c*b^2 + 2/19*x^19*d^16*b*a + 20/3*x^18*d^14*c^2*b^2 + 16/9*x^18*d^15*c*b*
a + 1/18*x^18*d^16*a^2 + 560/17*x^17*d^13*c^3*b^2 + 240/17*x^17*d^14*c^2*b*a + 16/17*x^17*d^15*c*a^2 + 455/4*x
^16*d^12*c^4*b^2 + 70*x^16*d^13*c^3*b*a + 15/2*x^16*d^14*c^2*a^2 + 1456/5*x^15*d^11*c^5*b^2 + 728/3*x^15*d^12*
c^4*b*a + 112/3*x^15*d^13*c^3*a^2 + 572*x^14*d^10*c^6*b^2 + 624*x^14*d^11*c^5*b*a + 130*x^14*d^12*c^4*a^2 + 88
0*x^13*d^9*c^7*b^2 + 1232*x^13*d^10*c^6*b*a + 336*x^13*d^11*c^5*a^2 + 2145/2*x^12*d^8*c^8*b^2 + 5720/3*x^12*d^
9*c^7*b*a + 2002/3*x^12*d^10*c^6*a^2 + 1040*x^11*d^7*c^9*b^2 + 2340*x^11*d^8*c^8*b*a + 1040*x^11*d^9*c^7*a^2 +
4004/5*x^10*d^6*c^10*b^2 + 2288*x^10*d^7*c^9*b*a + 1287*x^10*d^8*c^8*a^2 + 1456/3*x^9*d^5*c^11*b^2 + 16016/9*
x^9*d^6*c^10*b*a + 11440/9*x^9*d^7*c^9*a^2 + 455/2*x^8*d^4*c^12*b^2 + 1092*x^8*d^5*c^11*b*a + 1001*x^8*d^6*c^1
0*a^2 + 80*x^7*d^3*c^13*b^2 + 520*x^7*d^4*c^12*b*a + 624*x^7*d^5*c^11*a^2 + 20*x^6*d^2*c^14*b^2 + 560/3*x^6*d^
3*c^13*b*a + 910/3*x^6*d^4*c^12*a^2 + 16/5*x^5*d*c^15*b^2 + 48*x^5*d^2*c^14*b*a + 112*x^5*d^3*c^13*a^2 + 1/4*x
^4*c^16*b^2 + 8*x^4*d*c^15*b*a + 30*x^4*d^2*c^14*a^2 + 2/3*x^3*c^16*b*a + 16/3*x^3*d*c^15*a^2 + 1/2*x^2*c^16*a
^2

________________________________________________________________________________________

giac [B]  time = 1.10, size = 668, normalized size = 6.82 \begin {gather*} \frac {1}{20} \, b^{2} d^{16} x^{20} + \frac {16}{19} \, b^{2} c d^{15} x^{19} + \frac {2}{19} \, a b d^{16} x^{19} + \frac {20}{3} \, b^{2} c^{2} d^{14} x^{18} + \frac {16}{9} \, a b c d^{15} x^{18} + \frac {1}{18} \, a^{2} d^{16} x^{18} + \frac {560}{17} \, b^{2} c^{3} d^{13} x^{17} + \frac {240}{17} \, a b c^{2} d^{14} x^{17} + \frac {16}{17} \, a^{2} c d^{15} x^{17} + \frac {455}{4} \, b^{2} c^{4} d^{12} x^{16} + 70 \, a b c^{3} d^{13} x^{16} + \frac {15}{2} \, a^{2} c^{2} d^{14} x^{16} + \frac {1456}{5} \, b^{2} c^{5} d^{11} x^{15} + \frac {728}{3} \, a b c^{4} d^{12} x^{15} + \frac {112}{3} \, a^{2} c^{3} d^{13} x^{15} + 572 \, b^{2} c^{6} d^{10} x^{14} + 624 \, a b c^{5} d^{11} x^{14} + 130 \, a^{2} c^{4} d^{12} x^{14} + 880 \, b^{2} c^{7} d^{9} x^{13} + 1232 \, a b c^{6} d^{10} x^{13} + 336 \, a^{2} c^{5} d^{11} x^{13} + \frac {2145}{2} \, b^{2} c^{8} d^{8} x^{12} + \frac {5720}{3} \, a b c^{7} d^{9} x^{12} + \frac {2002}{3} \, a^{2} c^{6} d^{10} x^{12} + 1040 \, b^{2} c^{9} d^{7} x^{11} + 2340 \, a b c^{8} d^{8} x^{11} + 1040 \, a^{2} c^{7} d^{9} x^{11} + \frac {4004}{5} \, b^{2} c^{10} d^{6} x^{10} + 2288 \, a b c^{9} d^{7} x^{10} + 1287 \, a^{2} c^{8} d^{8} x^{10} + \frac {1456}{3} \, b^{2} c^{11} d^{5} x^{9} + \frac {16016}{9} \, a b c^{10} d^{6} x^{9} + \frac {11440}{9} \, a^{2} c^{9} d^{7} x^{9} + \frac {455}{2} \, b^{2} c^{12} d^{4} x^{8} + 1092 \, a b c^{11} d^{5} x^{8} + 1001 \, a^{2} c^{10} d^{6} x^{8} + 80 \, b^{2} c^{13} d^{3} x^{7} + 520 \, a b c^{12} d^{4} x^{7} + 624 \, a^{2} c^{11} d^{5} x^{7} + 20 \, b^{2} c^{14} d^{2} x^{6} + \frac {560}{3} \, a b c^{13} d^{3} x^{6} + \frac {910}{3} \, a^{2} c^{12} d^{4} x^{6} + \frac {16}{5} \, b^{2} c^{15} d x^{5} + 48 \, a b c^{14} d^{2} x^{5} + 112 \, a^{2} c^{13} d^{3} x^{5} + \frac {1}{4} \, b^{2} c^{16} x^{4} + 8 \, a b c^{15} d x^{4} + 30 \, a^{2} c^{14} d^{2} x^{4} + \frac {2}{3} \, a b c^{16} x^{3} + \frac {16}{3} \, a^{2} c^{15} d x^{3} + \frac {1}{2} \, a^{2} c^{16} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*b^2*d^16*x^20 + 16/19*b^2*c*d^15*x^19 + 2/19*a*b*d^16*x^19 + 20/3*b^2*c^2*d^14*x^18 + 16/9*a*b*c*d^15*x^1
8 + 1/18*a^2*d^16*x^18 + 560/17*b^2*c^3*d^13*x^17 + 240/17*a*b*c^2*d^14*x^17 + 16/17*a^2*c*d^15*x^17 + 455/4*b
^2*c^4*d^12*x^16 + 70*a*b*c^3*d^13*x^16 + 15/2*a^2*c^2*d^14*x^16 + 1456/5*b^2*c^5*d^11*x^15 + 728/3*a*b*c^4*d^
12*x^15 + 112/3*a^2*c^3*d^13*x^15 + 572*b^2*c^6*d^10*x^14 + 624*a*b*c^5*d^11*x^14 + 130*a^2*c^4*d^12*x^14 + 88
0*b^2*c^7*d^9*x^13 + 1232*a*b*c^6*d^10*x^13 + 336*a^2*c^5*d^11*x^13 + 2145/2*b^2*c^8*d^8*x^12 + 5720/3*a*b*c^7
*d^9*x^12 + 2002/3*a^2*c^6*d^10*x^12 + 1040*b^2*c^9*d^7*x^11 + 2340*a*b*c^8*d^8*x^11 + 1040*a^2*c^7*d^9*x^11 +
4004/5*b^2*c^10*d^6*x^10 + 2288*a*b*c^9*d^7*x^10 + 1287*a^2*c^8*d^8*x^10 + 1456/3*b^2*c^11*d^5*x^9 + 16016/9*
a*b*c^10*d^6*x^9 + 11440/9*a^2*c^9*d^7*x^9 + 455/2*b^2*c^12*d^4*x^8 + 1092*a*b*c^11*d^5*x^8 + 1001*a^2*c^10*d^
6*x^8 + 80*b^2*c^13*d^3*x^7 + 520*a*b*c^12*d^4*x^7 + 624*a^2*c^11*d^5*x^7 + 20*b^2*c^14*d^2*x^6 + 560/3*a*b*c^
13*d^3*x^6 + 910/3*a^2*c^12*d^4*x^6 + 16/5*b^2*c^15*d*x^5 + 48*a*b*c^14*d^2*x^5 + 112*a^2*c^13*d^3*x^5 + 1/4*b
^2*c^16*x^4 + 8*a*b*c^15*d*x^4 + 30*a^2*c^14*d^2*x^4 + 2/3*a*b*c^16*x^3 + 16/3*a^2*c^15*d*x^3 + 1/2*a^2*c^16*x
^2

________________________________________________________________________________________

maple [B]  time = 0.00, size = 622, normalized size = 6.35 \begin {gather*} \frac {b^{2} d^{16} x^{20}}{20}+\frac {a^{2} c^{16} x^{2}}{2}+\frac {\left (2 a b \,d^{16}+16 b^{2} c \,d^{15}\right ) x^{19}}{19}+\frac {\left (a^{2} d^{16}+32 a b c \,d^{15}+120 b^{2} c^{2} d^{14}\right ) x^{18}}{18}+\frac {\left (16 a^{2} c \,d^{15}+240 a b \,c^{2} d^{14}+560 b^{2} c^{3} d^{13}\right ) x^{17}}{17}+\frac {\left (120 a^{2} c^{2} d^{14}+1120 a b \,c^{3} d^{13}+1820 b^{2} c^{4} d^{12}\right ) x^{16}}{16}+\frac {\left (560 a^{2} c^{3} d^{13}+3640 a b \,c^{4} d^{12}+4368 b^{2} c^{5} d^{11}\right ) x^{15}}{15}+\frac {\left (1820 a^{2} c^{4} d^{12}+8736 a b \,c^{5} d^{11}+8008 b^{2} c^{6} d^{10}\right ) x^{14}}{14}+\frac {\left (4368 a^{2} c^{5} d^{11}+16016 a b \,c^{6} d^{10}+11440 b^{2} c^{7} d^{9}\right ) x^{13}}{13}+\frac {\left (8008 a^{2} c^{6} d^{10}+22880 a b \,c^{7} d^{9}+12870 b^{2} c^{8} d^{8}\right ) x^{12}}{12}+\frac {\left (11440 a^{2} c^{7} d^{9}+25740 a b \,c^{8} d^{8}+11440 b^{2} c^{9} d^{7}\right ) x^{11}}{11}+\frac {\left (12870 a^{2} c^{8} d^{8}+22880 a b \,c^{9} d^{7}+8008 b^{2} c^{10} d^{6}\right ) x^{10}}{10}+\frac {\left (11440 a^{2} c^{9} d^{7}+16016 a b \,c^{10} d^{6}+4368 b^{2} c^{11} d^{5}\right ) x^{9}}{9}+\frac {\left (8008 a^{2} c^{10} d^{6}+8736 a b \,c^{11} d^{5}+1820 b^{2} c^{12} d^{4}\right ) x^{8}}{8}+\frac {\left (4368 a^{2} c^{11} d^{5}+3640 a b \,c^{12} d^{4}+560 b^{2} c^{13} d^{3}\right ) x^{7}}{7}+\frac {\left (1820 a^{2} c^{12} d^{4}+1120 a b \,c^{13} d^{3}+120 b^{2} c^{14} d^{2}\right ) x^{6}}{6}+\frac {\left (560 a^{2} c^{13} d^{3}+240 a b \,c^{14} d^{2}+16 b^{2} c^{15} d \right ) x^{5}}{5}+\frac {\left (120 a^{2} c^{14} d^{2}+32 a b \,c^{15} d +b^{2} c^{16}\right ) x^{4}}{4}+\frac {\left (16 a^{2} c^{15} d +2 a b \,c^{16}\right ) x^{3}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b*x+a)^2*(d*x+c)^16,x)

[Out]

1/20*b^2*d^16*x^20+1/19*(2*a*b*d^16+16*b^2*c*d^15)*x^19+1/18*(a^2*d^16+32*a*b*c*d^15+120*b^2*c^2*d^14)*x^18+1/
17*(16*a^2*c*d^15+240*a*b*c^2*d^14+560*b^2*c^3*d^13)*x^17+1/16*(120*a^2*c^2*d^14+1120*a*b*c^3*d^13+1820*b^2*c^
4*d^12)*x^16+1/15*(560*a^2*c^3*d^13+3640*a*b*c^4*d^12+4368*b^2*c^5*d^11)*x^15+1/14*(1820*a^2*c^4*d^12+8736*a*b
*c^5*d^11+8008*b^2*c^6*d^10)*x^14+1/13*(4368*a^2*c^5*d^11+16016*a*b*c^6*d^10+11440*b^2*c^7*d^9)*x^13+1/12*(800
8*a^2*c^6*d^10+22880*a*b*c^7*d^9+12870*b^2*c^8*d^8)*x^12+1/11*(11440*a^2*c^7*d^9+25740*a*b*c^8*d^8+11440*b^2*c
^9*d^7)*x^11+1/10*(12870*a^2*c^8*d^8+22880*a*b*c^9*d^7+8008*b^2*c^10*d^6)*x^10+1/9*(11440*a^2*c^9*d^7+16016*a*
b*c^10*d^6+4368*b^2*c^11*d^5)*x^9+1/8*(8008*a^2*c^10*d^6+8736*a*b*c^11*d^5+1820*b^2*c^12*d^4)*x^8+1/7*(4368*a^
2*c^11*d^5+3640*a*b*c^12*d^4+560*b^2*c^13*d^3)*x^7+1/6*(1820*a^2*c^12*d^4+1120*a*b*c^13*d^3+120*b^2*c^14*d^2)*
x^6+1/5*(560*a^2*c^13*d^3+240*a*b*c^14*d^2+16*b^2*c^15*d)*x^5+1/4*(120*a^2*c^14*d^2+32*a*b*c^15*d+b^2*c^16)*x^
4+1/3*(16*a^2*c^15*d+2*a*b*c^16)*x^3+1/2*a^2*c^16*x^2

________________________________________________________________________________________

maxima [B]  time = 1.17, size = 617, normalized size = 6.30 \begin {gather*} \frac {1}{20} \, b^{2} d^{16} x^{20} + \frac {1}{2} \, a^{2} c^{16} x^{2} + \frac {2}{19} \, {\left (8 \, b^{2} c d^{15} + a b d^{16}\right )} x^{19} + \frac {1}{18} \, {\left (120 \, b^{2} c^{2} d^{14} + 32 \, a b c d^{15} + a^{2} d^{16}\right )} x^{18} + \frac {16}{17} \, {\left (35 \, b^{2} c^{3} d^{13} + 15 \, a b c^{2} d^{14} + a^{2} c d^{15}\right )} x^{17} + \frac {5}{4} \, {\left (91 \, b^{2} c^{4} d^{12} + 56 \, a b c^{3} d^{13} + 6 \, a^{2} c^{2} d^{14}\right )} x^{16} + \frac {56}{15} \, {\left (78 \, b^{2} c^{5} d^{11} + 65 \, a b c^{4} d^{12} + 10 \, a^{2} c^{3} d^{13}\right )} x^{15} + 26 \, {\left (22 \, b^{2} c^{6} d^{10} + 24 \, a b c^{5} d^{11} + 5 \, a^{2} c^{4} d^{12}\right )} x^{14} + 16 \, {\left (55 \, b^{2} c^{7} d^{9} + 77 \, a b c^{6} d^{10} + 21 \, a^{2} c^{5} d^{11}\right )} x^{13} + \frac {143}{6} \, {\left (45 \, b^{2} c^{8} d^{8} + 80 \, a b c^{7} d^{9} + 28 \, a^{2} c^{6} d^{10}\right )} x^{12} + 260 \, {\left (4 \, b^{2} c^{9} d^{7} + 9 \, a b c^{8} d^{8} + 4 \, a^{2} c^{7} d^{9}\right )} x^{11} + \frac {143}{5} \, {\left (28 \, b^{2} c^{10} d^{6} + 80 \, a b c^{9} d^{7} + 45 \, a^{2} c^{8} d^{8}\right )} x^{10} + \frac {208}{9} \, {\left (21 \, b^{2} c^{11} d^{5} + 77 \, a b c^{10} d^{6} + 55 \, a^{2} c^{9} d^{7}\right )} x^{9} + \frac {91}{2} \, {\left (5 \, b^{2} c^{12} d^{4} + 24 \, a b c^{11} d^{5} + 22 \, a^{2} c^{10} d^{6}\right )} x^{8} + 8 \, {\left (10 \, b^{2} c^{13} d^{3} + 65 \, a b c^{12} d^{4} + 78 \, a^{2} c^{11} d^{5}\right )} x^{7} + \frac {10}{3} \, {\left (6 \, b^{2} c^{14} d^{2} + 56 \, a b c^{13} d^{3} + 91 \, a^{2} c^{12} d^{4}\right )} x^{6} + \frac {16}{5} \, {\left (b^{2} c^{15} d + 15 \, a b c^{14} d^{2} + 35 \, a^{2} c^{13} d^{3}\right )} x^{5} + \frac {1}{4} \, {\left (b^{2} c^{16} + 32 \, a b c^{15} d + 120 \, a^{2} c^{14} d^{2}\right )} x^{4} + \frac {2}{3} \, {\left (a b c^{16} + 8 \, a^{2} c^{15} d\right )} x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*b^2*d^16*x^20 + 1/2*a^2*c^16*x^2 + 2/19*(8*b^2*c*d^15 + a*b*d^16)*x^19 + 1/18*(120*b^2*c^2*d^14 + 32*a*b*
c*d^15 + a^2*d^16)*x^18 + 16/17*(35*b^2*c^3*d^13 + 15*a*b*c^2*d^14 + a^2*c*d^15)*x^17 + 5/4*(91*b^2*c^4*d^12 +
56*a*b*c^3*d^13 + 6*a^2*c^2*d^14)*x^16 + 56/15*(78*b^2*c^5*d^11 + 65*a*b*c^4*d^12 + 10*a^2*c^3*d^13)*x^15 + 2
6*(22*b^2*c^6*d^10 + 24*a*b*c^5*d^11 + 5*a^2*c^4*d^12)*x^14 + 16*(55*b^2*c^7*d^9 + 77*a*b*c^6*d^10 + 21*a^2*c^
5*d^11)*x^13 + 143/6*(45*b^2*c^8*d^8 + 80*a*b*c^7*d^9 + 28*a^2*c^6*d^10)*x^12 + 260*(4*b^2*c^9*d^7 + 9*a*b*c^8
*d^8 + 4*a^2*c^7*d^9)*x^11 + 143/5*(28*b^2*c^10*d^6 + 80*a*b*c^9*d^7 + 45*a^2*c^8*d^8)*x^10 + 208/9*(21*b^2*c^
11*d^5 + 77*a*b*c^10*d^6 + 55*a^2*c^9*d^7)*x^9 + 91/2*(5*b^2*c^12*d^4 + 24*a*b*c^11*d^5 + 22*a^2*c^10*d^6)*x^8
+ 8*(10*b^2*c^13*d^3 + 65*a*b*c^12*d^4 + 78*a^2*c^11*d^5)*x^7 + 10/3*(6*b^2*c^14*d^2 + 56*a*b*c^13*d^3 + 91*a
^2*c^12*d^4)*x^6 + 16/5*(b^2*c^15*d + 15*a*b*c^14*d^2 + 35*a^2*c^13*d^3)*x^5 + 1/4*(b^2*c^16 + 32*a*b*c^15*d +
120*a^2*c^14*d^2)*x^4 + 2/3*(a*b*c^16 + 8*a^2*c^15*d)*x^3

________________________________________________________________________________________

mupad [B]  time = 0.63, size = 557, normalized size = 5.68 \begin {gather*} x^4\,\left (30\,a^2\,c^{14}\,d^2+8\,a\,b\,c^{15}\,d+\frac {b^2\,c^{16}}{4}\right )+x^{18}\,\left (\frac {a^2\,d^{16}}{18}+\frac {16\,a\,b\,c\,d^{15}}{9}+\frac {20\,b^2\,c^2\,d^{14}}{3}\right )+\frac {a^2\,c^{16}\,x^2}{2}+\frac {b^2\,d^{16}\,x^{20}}{20}+\frac {2\,a\,c^{15}\,x^3\,\left (8\,a\,d+b\,c\right )}{3}+\frac {2\,b\,d^{15}\,x^{19}\,\left (a\,d+8\,b\,c\right )}{19}+\frac {16\,c^{13}\,d\,x^5\,\left (35\,a^2\,d^2+15\,a\,b\,c\,d+b^2\,c^2\right )}{5}+\frac {16\,c\,d^{13}\,x^{17}\,\left (a^2\,d^2+15\,a\,b\,c\,d+35\,b^2\,c^2\right )}{17}+260\,c^7\,d^7\,x^{11}\,\left (4\,a^2\,d^2+9\,a\,b\,c\,d+4\,b^2\,c^2\right )+\frac {91\,c^{10}\,d^4\,x^8\,\left (22\,a^2\,d^2+24\,a\,b\,c\,d+5\,b^2\,c^2\right )}{2}+26\,c^4\,d^{10}\,x^{14}\,\left (5\,a^2\,d^2+24\,a\,b\,c\,d+22\,b^2\,c^2\right )+\frac {10\,c^{12}\,d^2\,x^6\,\left (91\,a^2\,d^2+56\,a\,b\,c\,d+6\,b^2\,c^2\right )}{3}+8\,c^{11}\,d^3\,x^7\,\left (78\,a^2\,d^2+65\,a\,b\,c\,d+10\,b^2\,c^2\right )+\frac {208\,c^9\,d^5\,x^9\,\left (55\,a^2\,d^2+77\,a\,b\,c\,d+21\,b^2\,c^2\right )}{9}+\frac {143\,c^8\,d^6\,x^{10}\,\left (45\,a^2\,d^2+80\,a\,b\,c\,d+28\,b^2\,c^2\right )}{5}+\frac {143\,c^6\,d^8\,x^{12}\,\left (28\,a^2\,d^2+80\,a\,b\,c\,d+45\,b^2\,c^2\right )}{6}+16\,c^5\,d^9\,x^{13}\,\left (21\,a^2\,d^2+77\,a\,b\,c\,d+55\,b^2\,c^2\right )+\frac {56\,c^3\,d^{11}\,x^{15}\,\left (10\,a^2\,d^2+65\,a\,b\,c\,d+78\,b^2\,c^2\right )}{15}+\frac {5\,c^2\,d^{12}\,x^{16}\,\left (6\,a^2\,d^2+56\,a\,b\,c\,d+91\,b^2\,c^2\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*x)^2*(c + d*x)^16,x)

[Out]

x^4*((b^2*c^16)/4 + 30*a^2*c^14*d^2 + 8*a*b*c^15*d) + x^18*((a^2*d^16)/18 + (20*b^2*c^2*d^14)/3 + (16*a*b*c*d^
15)/9) + (a^2*c^16*x^2)/2 + (b^2*d^16*x^20)/20 + (2*a*c^15*x^3*(8*a*d + b*c))/3 + (2*b*d^15*x^19*(a*d + 8*b*c)
)/19 + (16*c^13*d*x^5*(35*a^2*d^2 + b^2*c^2 + 15*a*b*c*d))/5 + (16*c*d^13*x^17*(a^2*d^2 + 35*b^2*c^2 + 15*a*b*
c*d))/17 + 260*c^7*d^7*x^11*(4*a^2*d^2 + 4*b^2*c^2 + 9*a*b*c*d) + (91*c^10*d^4*x^8*(22*a^2*d^2 + 5*b^2*c^2 + 2
4*a*b*c*d))/2 + 26*c^4*d^10*x^14*(5*a^2*d^2 + 22*b^2*c^2 + 24*a*b*c*d) + (10*c^12*d^2*x^6*(91*a^2*d^2 + 6*b^2*
c^2 + 56*a*b*c*d))/3 + 8*c^11*d^3*x^7*(78*a^2*d^2 + 10*b^2*c^2 + 65*a*b*c*d) + (208*c^9*d^5*x^9*(55*a^2*d^2 +
21*b^2*c^2 + 77*a*b*c*d))/9 + (143*c^8*d^6*x^10*(45*a^2*d^2 + 28*b^2*c^2 + 80*a*b*c*d))/5 + (143*c^6*d^8*x^12*
(28*a^2*d^2 + 45*b^2*c^2 + 80*a*b*c*d))/6 + 16*c^5*d^9*x^13*(21*a^2*d^2 + 55*b^2*c^2 + 77*a*b*c*d) + (56*c^3*d
^11*x^15*(10*a^2*d^2 + 78*b^2*c^2 + 65*a*b*c*d))/15 + (5*c^2*d^12*x^16*(6*a^2*d^2 + 91*b^2*c^2 + 56*a*b*c*d))/
4

________________________________________________________________________________________

sympy [B]  time = 0.63, size = 682, normalized size = 6.96 \begin {gather*} \frac {a^{2} c^{16} x^{2}}{2} + \frac {b^{2} d^{16} x^{20}}{20} + x^{19} \left (\frac {2 a b d^{16}}{19} + \frac {16 b^{2} c d^{15}}{19}\right ) + x^{18} \left (\frac {a^{2} d^{16}}{18} + \frac {16 a b c d^{15}}{9} + \frac {20 b^{2} c^{2} d^{14}}{3}\right ) + x^{17} \left (\frac {16 a^{2} c d^{15}}{17} + \frac {240 a b c^{2} d^{14}}{17} + \frac {560 b^{2} c^{3} d^{13}}{17}\right ) + x^{16} \left (\frac {15 a^{2} c^{2} d^{14}}{2} + 70 a b c^{3} d^{13} + \frac {455 b^{2} c^{4} d^{12}}{4}\right ) + x^{15} \left (\frac {112 a^{2} c^{3} d^{13}}{3} + \frac {728 a b c^{4} d^{12}}{3} + \frac {1456 b^{2} c^{5} d^{11}}{5}\right ) + x^{14} \left (130 a^{2} c^{4} d^{12} + 624 a b c^{5} d^{11} + 572 b^{2} c^{6} d^{10}\right ) + x^{13} \left (336 a^{2} c^{5} d^{11} + 1232 a b c^{6} d^{10} + 880 b^{2} c^{7} d^{9}\right ) + x^{12} \left (\frac {2002 a^{2} c^{6} d^{10}}{3} + \frac {5720 a b c^{7} d^{9}}{3} + \frac {2145 b^{2} c^{8} d^{8}}{2}\right ) + x^{11} \left (1040 a^{2} c^{7} d^{9} + 2340 a b c^{8} d^{8} + 1040 b^{2} c^{9} d^{7}\right ) + x^{10} \left (1287 a^{2} c^{8} d^{8} + 2288 a b c^{9} d^{7} + \frac {4004 b^{2} c^{10} d^{6}}{5}\right ) + x^{9} \left (\frac {11440 a^{2} c^{9} d^{7}}{9} + \frac {16016 a b c^{10} d^{6}}{9} + \frac {1456 b^{2} c^{11} d^{5}}{3}\right ) + x^{8} \left (1001 a^{2} c^{10} d^{6} + 1092 a b c^{11} d^{5} + \frac {455 b^{2} c^{12} d^{4}}{2}\right ) + x^{7} \left (624 a^{2} c^{11} d^{5} + 520 a b c^{12} d^{4} + 80 b^{2} c^{13} d^{3}\right ) + x^{6} \left (\frac {910 a^{2} c^{12} d^{4}}{3} + \frac {560 a b c^{13} d^{3}}{3} + 20 b^{2} c^{14} d^{2}\right ) + x^{5} \left (112 a^{2} c^{13} d^{3} + 48 a b c^{14} d^{2} + \frac {16 b^{2} c^{15} d}{5}\right ) + x^{4} \left (30 a^{2} c^{14} d^{2} + 8 a b c^{15} d + \frac {b^{2} c^{16}}{4}\right ) + x^{3} \left (\frac {16 a^{2} c^{15} d}{3} + \frac {2 a b c^{16}}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*c**16*x**2/2 + b**2*d**16*x**20/20 + x**19*(2*a*b*d**16/19 + 16*b**2*c*d**15/19) + x**18*(a**2*d**16/18 +
16*a*b*c*d**15/9 + 20*b**2*c**2*d**14/3) + x**17*(16*a**2*c*d**15/17 + 240*a*b*c**2*d**14/17 + 560*b**2*c**3*
d**13/17) + x**16*(15*a**2*c**2*d**14/2 + 70*a*b*c**3*d**13 + 455*b**2*c**4*d**12/4) + x**15*(112*a**2*c**3*d*
*13/3 + 728*a*b*c**4*d**12/3 + 1456*b**2*c**5*d**11/5) + x**14*(130*a**2*c**4*d**12 + 624*a*b*c**5*d**11 + 572
*b**2*c**6*d**10) + x**13*(336*a**2*c**5*d**11 + 1232*a*b*c**6*d**10 + 880*b**2*c**7*d**9) + x**12*(2002*a**2*
c**6*d**10/3 + 5720*a*b*c**7*d**9/3 + 2145*b**2*c**8*d**8/2) + x**11*(1040*a**2*c**7*d**9 + 2340*a*b*c**8*d**8
+ 1040*b**2*c**9*d**7) + x**10*(1287*a**2*c**8*d**8 + 2288*a*b*c**9*d**7 + 4004*b**2*c**10*d**6/5) + x**9*(11
440*a**2*c**9*d**7/9 + 16016*a*b*c**10*d**6/9 + 1456*b**2*c**11*d**5/3) + x**8*(1001*a**2*c**10*d**6 + 1092*a*
b*c**11*d**5 + 455*b**2*c**12*d**4/2) + x**7*(624*a**2*c**11*d**5 + 520*a*b*c**12*d**4 + 80*b**2*c**13*d**3) +
x**6*(910*a**2*c**12*d**4/3 + 560*a*b*c**13*d**3/3 + 20*b**2*c**14*d**2) + x**5*(112*a**2*c**13*d**3 + 48*a*b
*c**14*d**2 + 16*b**2*c**15*d/5) + x**4*(30*a**2*c**14*d**2 + 8*a*b*c**15*d + b**2*c**16/4) + x**3*(16*a**2*c*
*15*d/3 + 2*a*b*c**16/3)

________________________________________________________________________________________