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

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

[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)

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Rubi [A]  time = 0.234766, antiderivative size = 98, 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, Rules used = {77} \[ -\frac{b (c+d x)^{19} (3 b c-2 a d)}{19 d^4}+\frac{(c+d x)^{18} (b c-a d) (3 b c-a d)}{18 d^4}-\frac{c (c+d x)^{17} (b c-a d)^2}{17 d^4}+\frac{b^2 (c+d x)^{20}}{20 d^4} \]

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 = 0.0866463, size = 583, normalized size = 5.95 \[ \frac{1}{18} d^{14} x^{18} \left (a^2 d^2+32 a b c d+120 b^2 c^2\right )+\frac{16}{17} c d^{13} x^{17} \left (a^2 d^2+15 a b c d+35 b^2 c^2\right )+\frac{5}{4} c^2 d^{12} x^{16} \left (6 a^2 d^2+56 a b c d+91 b^2 c^2\right )+\frac{56}{15} c^3 d^{11} x^{15} \left (10 a^2 d^2+65 a b c d+78 b^2 c^2\right )+26 c^4 d^{10} x^{14} \left (5 a^2 d^2+24 a b c d+22 b^2 c^2\right )+16 c^5 d^9 x^{13} \left (21 a^2 d^2+77 a b c d+55 b^2 c^2\right )+\frac{143}{6} c^6 d^8 x^{12} \left (28 a^2 d^2+80 a b c d+45 b^2 c^2\right )+260 c^7 d^7 x^{11} \left (4 a^2 d^2+9 a b c d+4 b^2 c^2\right )+\frac{143}{5} c^8 d^6 x^{10} \left (45 a^2 d^2+80 a b c d+28 b^2 c^2\right )+\frac{208}{9} c^9 d^5 x^9 \left (55 a^2 d^2+77 a b c d+21 b^2 c^2\right )+\frac{91}{2} c^{10} d^4 x^8 \left (22 a^2 d^2+24 a b c d+5 b^2 c^2\right )+8 c^{11} d^3 x^7 \left (78 a^2 d^2+65 a b c d+10 b^2 c^2\right )+\frac{10}{3} c^{12} d^2 x^6 \left (91 a^2 d^2+56 a b c d+6 b^2 c^2\right )+\frac{16}{5} c^{13} d x^5 \left (35 a^2 d^2+15 a b c d+b^2 c^2\right )+\frac{1}{4} c^{14} x^4 \left (120 a^2 d^2+32 a b c d+b^2 c^2\right )+\frac{1}{2} a^2 c^{16} x^2+\frac{2}{3} a c^{15} x^3 (8 a d+b c)+\frac{2}{19} b d^{15} x^{19} (a d+8 b c)+\frac{1}{20} b^2 d^{16} x^{20} \]

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

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Maple [B]  time = 0.002, size = 622, normalized size = 6.4 \begin{align*}{\frac{{b}^{2}{d}^{16}{x}^{20}}{20}}+{\frac{ \left ( 2\,ab{d}^{16}+16\,{b}^{2}c{d}^{15} \right ){x}^{19}}{19}}+{\frac{ \left ({a}^{2}{d}^{16}+32\,abc{d}^{15}+120\,{b}^{2}{c}^{2}{d}^{14} \right ){x}^{18}}{18}}+{\frac{ \left ( 16\,{a}^{2}c{d}^{15}+240\,ab{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\,ab{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\,ab{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\,ab{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\,ab{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\,ab{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\,ab{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\,ab{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\,ab{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\,ab{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\,ab{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\,ab{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\,ab{c}^{14}{d}^{2}+16\,{b}^{2}{c}^{15}d \right ){x}^{5}}{5}}+{\frac{ \left ( 120\,{a}^{2}{c}^{14}{d}^{2}+32\,ab{c}^{15}d+{b}^{2}{c}^{16} \right ){x}^{4}}{4}}+{\frac{ \left ( 16\,{a}^{2}{c}^{15}d+2\,ab{c}^{16} \right ){x}^{3}}{3}}+{\frac{{a}^{2}{c}^{16}{x}^{2}}{2}} \end{align*}

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

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Maxima [B]  time = 1.0682, size = 833, normalized size = 8.5 \begin{align*} \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{align*}

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

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Fricas [B]  time = 1.64238, size = 1639, normalized size = 16.72 \begin{align*} \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{align*}

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

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Sympy [B]  time = 0.212229, size = 682, normalized size = 6.96 \begin{align*} \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{align*}

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)

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Giac [B]  time = 1.19861, size = 902, normalized size = 9.2 \begin{align*} \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{align*}

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