3.202 \(\int x^3 (a+b x)^2 (c+d x)^{16} \, dx\)

Optimal. Leaf size=177 \[ \frac{(c+d x)^{20} \left (a^2 d^2-8 a b c d+10 b^2 c^2\right )}{20 d^6}-\frac{c (c+d x)^{19} \left (3 a^2 d^2-12 a b c d+10 b^2 c^2\right )}{19 d^6}+\frac{c^2 (c+d x)^{18} (5 b c-3 a d) (b c-a d)}{18 d^6}-\frac{c^3 (c+d x)^{17} (b c-a d)^2}{17 d^6}-\frac{b (c+d x)^{21} (5 b c-2 a d)}{21 d^6}+\frac{b^2 (c+d x)^{22}}{22 d^6} \]

[Out]

-(c^3*(b*c - a*d)^2*(c + d*x)^17)/(17*d^6) + (c^2*(5*b*c - 3*a*d)*(b*c - a*d)*(c + d*x)^18)/(18*d^6) - (c*(10*
b^2*c^2 - 12*a*b*c*d + 3*a^2*d^2)*(c + d*x)^19)/(19*d^6) + ((10*b^2*c^2 - 8*a*b*c*d + a^2*d^2)*(c + d*x)^20)/(
20*d^6) - (b*(5*b*c - 2*a*d)*(c + d*x)^21)/(21*d^6) + (b^2*(c + d*x)^22)/(22*d^6)

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Rubi [A]  time = 0.577246, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {88} \[ \frac{(c+d x)^{20} \left (a^2 d^2-8 a b c d+10 b^2 c^2\right )}{20 d^6}-\frac{c (c+d x)^{19} \left (3 a^2 d^2-12 a b c d+10 b^2 c^2\right )}{19 d^6}+\frac{c^2 (c+d x)^{18} (5 b c-3 a d) (b c-a d)}{18 d^6}-\frac{c^3 (c+d x)^{17} (b c-a d)^2}{17 d^6}-\frac{b (c+d x)^{21} (5 b c-2 a d)}{21 d^6}+\frac{b^2 (c+d x)^{22}}{22 d^6} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x)^2*(c + d*x)^16,x]

[Out]

-(c^3*(b*c - a*d)^2*(c + d*x)^17)/(17*d^6) + (c^2*(5*b*c - 3*a*d)*(b*c - a*d)*(c + d*x)^18)/(18*d^6) - (c*(10*
b^2*c^2 - 12*a*b*c*d + 3*a^2*d^2)*(c + d*x)^19)/(19*d^6) + ((10*b^2*c^2 - 8*a*b*c*d + a^2*d^2)*(c + d*x)^20)/(
20*d^6) - (b*(5*b*c - 2*a*d)*(c + d*x)^21)/(21*d^6) + (b^2*(c + d*x)^22)/(22*d^6)

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int x^3 (a+b x)^2 (c+d x)^{16} \, dx &=\int \left (-\frac{c^3 (b c-a d)^2 (c+d x)^{16}}{d^5}+\frac{c^2 (5 b c-3 a d) (b c-a d) (c+d x)^{17}}{d^5}-\frac{c \left (10 b^2 c^2-12 a b c d+3 a^2 d^2\right ) (c+d x)^{18}}{d^5}+\frac{\left (10 b^2 c^2-8 a b c d+a^2 d^2\right ) (c+d x)^{19}}{d^5}-\frac{b (5 b c-2 a d) (c+d x)^{20}}{d^5}+\frac{b^2 (c+d x)^{21}}{d^5}\right ) \, dx\\ &=-\frac{c^3 (b c-a d)^2 (c+d x)^{17}}{17 d^6}+\frac{c^2 (5 b c-3 a d) (b c-a d) (c+d x)^{18}}{18 d^6}-\frac{c \left (10 b^2 c^2-12 a b c d+3 a^2 d^2\right ) (c+d x)^{19}}{19 d^6}+\frac{\left (10 b^2 c^2-8 a b c d+a^2 d^2\right ) (c+d x)^{20}}{20 d^6}-\frac{b (5 b c-2 a d) (c+d x)^{21}}{21 d^6}+\frac{b^2 (c+d x)^{22}}{22 d^6}\\ \end{align*}

Mathematica [B]  time = 0.116656, size = 589, normalized size = 3.33 \[ \frac{1}{20} d^{14} x^{20} \left (a^2 d^2+32 a b c d+120 b^2 c^2\right )+\frac{16}{19} c d^{13} x^{19} \left (a^2 d^2+15 a b c d+35 b^2 c^2\right )+\frac{10}{9} c^2 d^{12} x^{18} \left (6 a^2 d^2+56 a b c d+91 b^2 c^2\right )+\frac{56}{17} c^3 d^{11} x^{17} \left (10 a^2 d^2+65 a b c d+78 b^2 c^2\right )+\frac{91}{4} c^4 d^{10} x^{16} \left (5 a^2 d^2+24 a b c d+22 b^2 c^2\right )+\frac{208}{15} c^5 d^9 x^{15} \left (21 a^2 d^2+77 a b c d+55 b^2 c^2\right )+\frac{143}{7} c^6 d^8 x^{14} \left (28 a^2 d^2+80 a b c d+45 b^2 c^2\right )+220 c^7 d^7 x^{13} \left (4 a^2 d^2+9 a b c d+4 b^2 c^2\right )+\frac{143}{6} c^8 d^6 x^{12} \left (45 a^2 d^2+80 a b c d+28 b^2 c^2\right )+\frac{208}{11} c^9 d^5 x^{11} \left (55 a^2 d^2+77 a b c d+21 b^2 c^2\right )+\frac{182}{5} c^{10} d^4 x^{10} \left (22 a^2 d^2+24 a b c d+5 b^2 c^2\right )+\frac{56}{9} c^{11} d^3 x^9 \left (78 a^2 d^2+65 a b c d+10 b^2 c^2\right )+\frac{5}{2} c^{12} d^2 x^8 \left (91 a^2 d^2+56 a b c d+6 b^2 c^2\right )+\frac{16}{7} c^{13} d x^7 \left (35 a^2 d^2+15 a b c d+b^2 c^2\right )+\frac{1}{6} c^{14} x^6 \left (120 a^2 d^2+32 a b c d+b^2 c^2\right )+\frac{1}{4} a^2 c^{16} x^4+\frac{2}{5} a c^{15} x^5 (8 a d+b c)+\frac{2}{21} b d^{15} x^{21} (a d+8 b c)+\frac{1}{22} b^2 d^{16} x^{22} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x)^2*(c + d*x)^16,x]

[Out]

(a^2*c^16*x^4)/4 + (2*a*c^15*(b*c + 8*a*d)*x^5)/5 + (c^14*(b^2*c^2 + 32*a*b*c*d + 120*a^2*d^2)*x^6)/6 + (16*c^
13*d*(b^2*c^2 + 15*a*b*c*d + 35*a^2*d^2)*x^7)/7 + (5*c^12*d^2*(6*b^2*c^2 + 56*a*b*c*d + 91*a^2*d^2)*x^8)/2 + (
56*c^11*d^3*(10*b^2*c^2 + 65*a*b*c*d + 78*a^2*d^2)*x^9)/9 + (182*c^10*d^4*(5*b^2*c^2 + 24*a*b*c*d + 22*a^2*d^2
)*x^10)/5 + (208*c^9*d^5*(21*b^2*c^2 + 77*a*b*c*d + 55*a^2*d^2)*x^11)/11 + (143*c^8*d^6*(28*b^2*c^2 + 80*a*b*c
*d + 45*a^2*d^2)*x^12)/6 + 220*c^7*d^7*(4*b^2*c^2 + 9*a*b*c*d + 4*a^2*d^2)*x^13 + (143*c^6*d^8*(45*b^2*c^2 + 8
0*a*b*c*d + 28*a^2*d^2)*x^14)/7 + (208*c^5*d^9*(55*b^2*c^2 + 77*a*b*c*d + 21*a^2*d^2)*x^15)/15 + (91*c^4*d^10*
(22*b^2*c^2 + 24*a*b*c*d + 5*a^2*d^2)*x^16)/4 + (56*c^3*d^11*(78*b^2*c^2 + 65*a*b*c*d + 10*a^2*d^2)*x^17)/17 +
(10*c^2*d^12*(91*b^2*c^2 + 56*a*b*c*d + 6*a^2*d^2)*x^18)/9 + (16*c*d^13*(35*b^2*c^2 + 15*a*b*c*d + a^2*d^2)*x
^19)/19 + (d^14*(120*b^2*c^2 + 32*a*b*c*d + a^2*d^2)*x^20)/20 + (2*b*d^15*(8*b*c + a*d)*x^21)/21 + (b^2*d^16*x
^22)/22

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Maple [B]  time = 0.002, size = 622, normalized size = 3.5 \begin{align*}{\frac{{b}^{2}{d}^{16}{x}^{22}}{22}}+{\frac{ \left ( 2\,ab{d}^{16}+16\,{b}^{2}c{d}^{15} \right ){x}^{21}}{21}}+{\frac{ \left ({a}^{2}{d}^{16}+32\,abc{d}^{15}+120\,{b}^{2}{c}^{2}{d}^{14} \right ){x}^{20}}{20}}+{\frac{ \left ( 16\,{a}^{2}c{d}^{15}+240\,ab{c}^{2}{d}^{14}+560\,{b}^{2}{c}^{3}{d}^{13} \right ){x}^{19}}{19}}+{\frac{ \left ( 120\,{a}^{2}{c}^{2}{d}^{14}+1120\,ab{c}^{3}{d}^{13}+1820\,{b}^{2}{c}^{4}{d}^{12} \right ){x}^{18}}{18}}+{\frac{ \left ( 560\,{a}^{2}{c}^{3}{d}^{13}+3640\,ab{c}^{4}{d}^{12}+4368\,{b}^{2}{c}^{5}{d}^{11} \right ){x}^{17}}{17}}+{\frac{ \left ( 1820\,{a}^{2}{c}^{4}{d}^{12}+8736\,ab{c}^{5}{d}^{11}+8008\,{b}^{2}{c}^{6}{d}^{10} \right ){x}^{16}}{16}}+{\frac{ \left ( 4368\,{a}^{2}{c}^{5}{d}^{11}+16016\,ab{c}^{6}{d}^{10}+11440\,{b}^{2}{c}^{7}{d}^{9} \right ){x}^{15}}{15}}+{\frac{ \left ( 8008\,{a}^{2}{c}^{6}{d}^{10}+22880\,ab{c}^{7}{d}^{9}+12870\,{b}^{2}{c}^{8}{d}^{8} \right ){x}^{14}}{14}}+{\frac{ \left ( 11440\,{a}^{2}{c}^{7}{d}^{9}+25740\,ab{c}^{8}{d}^{8}+11440\,{b}^{2}{c}^{9}{d}^{7} \right ){x}^{13}}{13}}+{\frac{ \left ( 12870\,{a}^{2}{c}^{8}{d}^{8}+22880\,ab{c}^{9}{d}^{7}+8008\,{b}^{2}{c}^{10}{d}^{6} \right ){x}^{12}}{12}}+{\frac{ \left ( 11440\,{a}^{2}{c}^{9}{d}^{7}+16016\,ab{c}^{10}{d}^{6}+4368\,{b}^{2}{c}^{11}{d}^{5} \right ){x}^{11}}{11}}+{\frac{ \left ( 8008\,{a}^{2}{c}^{10}{d}^{6}+8736\,ab{c}^{11}{d}^{5}+1820\,{b}^{2}{c}^{12}{d}^{4} \right ){x}^{10}}{10}}+{\frac{ \left ( 4368\,{a}^{2}{c}^{11}{d}^{5}+3640\,ab{c}^{12}{d}^{4}+560\,{b}^{2}{c}^{13}{d}^{3} \right ){x}^{9}}{9}}+{\frac{ \left ( 1820\,{a}^{2}{c}^{12}{d}^{4}+1120\,ab{c}^{13}{d}^{3}+120\,{b}^{2}{c}^{14}{d}^{2} \right ){x}^{8}}{8}}+{\frac{ \left ( 560\,{a}^{2}{c}^{13}{d}^{3}+240\,ab{c}^{14}{d}^{2}+16\,{b}^{2}{c}^{15}d \right ){x}^{7}}{7}}+{\frac{ \left ( 120\,{a}^{2}{c}^{14}{d}^{2}+32\,ab{c}^{15}d+{b}^{2}{c}^{16} \right ){x}^{6}}{6}}+{\frac{ \left ( 16\,{a}^{2}{c}^{15}d+2\,ab{c}^{16} \right ){x}^{5}}{5}}+{\frac{{a}^{2}{c}^{16}{x}^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x+a)^2*(d*x+c)^16,x)

[Out]

1/22*b^2*d^16*x^22+1/21*(2*a*b*d^16+16*b^2*c*d^15)*x^21+1/20*(a^2*d^16+32*a*b*c*d^15+120*b^2*c^2*d^14)*x^20+1/
19*(16*a^2*c*d^15+240*a*b*c^2*d^14+560*b^2*c^3*d^13)*x^19+1/18*(120*a^2*c^2*d^14+1120*a*b*c^3*d^13+1820*b^2*c^
4*d^12)*x^18+1/17*(560*a^2*c^3*d^13+3640*a*b*c^4*d^12+4368*b^2*c^5*d^11)*x^17+1/16*(1820*a^2*c^4*d^12+8736*a*b
*c^5*d^11+8008*b^2*c^6*d^10)*x^16+1/15*(4368*a^2*c^5*d^11+16016*a*b*c^6*d^10+11440*b^2*c^7*d^9)*x^15+1/14*(800
8*a^2*c^6*d^10+22880*a*b*c^7*d^9+12870*b^2*c^8*d^8)*x^14+1/13*(11440*a^2*c^7*d^9+25740*a*b*c^8*d^8+11440*b^2*c
^9*d^7)*x^13+1/12*(12870*a^2*c^8*d^8+22880*a*b*c^9*d^7+8008*b^2*c^10*d^6)*x^12+1/11*(11440*a^2*c^9*d^7+16016*a
*b*c^10*d^6+4368*b^2*c^11*d^5)*x^11+1/10*(8008*a^2*c^10*d^6+8736*a*b*c^11*d^5+1820*b^2*c^12*d^4)*x^10+1/9*(436
8*a^2*c^11*d^5+3640*a*b*c^12*d^4+560*b^2*c^13*d^3)*x^9+1/8*(1820*a^2*c^12*d^4+1120*a*b*c^13*d^3+120*b^2*c^14*d
^2)*x^8+1/7*(560*a^2*c^13*d^3+240*a*b*c^14*d^2+16*b^2*c^15*d)*x^7+1/6*(120*a^2*c^14*d^2+32*a*b*c^15*d+b^2*c^16
)*x^6+1/5*(16*a^2*c^15*d+2*a*b*c^16)*x^5+1/4*a^2*c^16*x^4

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Maxima [B]  time = 1.21376, size = 833, normalized size = 4.71 \begin{align*} \frac{1}{22} \, b^{2} d^{16} x^{22} + \frac{1}{4} \, a^{2} c^{16} x^{4} + \frac{2}{21} \,{\left (8 \, b^{2} c d^{15} + a b d^{16}\right )} x^{21} + \frac{1}{20} \,{\left (120 \, b^{2} c^{2} d^{14} + 32 \, a b c d^{15} + a^{2} d^{16}\right )} x^{20} + \frac{16}{19} \,{\left (35 \, b^{2} c^{3} d^{13} + 15 \, a b c^{2} d^{14} + a^{2} c d^{15}\right )} x^{19} + \frac{10}{9} \,{\left (91 \, b^{2} c^{4} d^{12} + 56 \, a b c^{3} d^{13} + 6 \, a^{2} c^{2} d^{14}\right )} x^{18} + \frac{56}{17} \,{\left (78 \, b^{2} c^{5} d^{11} + 65 \, a b c^{4} d^{12} + 10 \, a^{2} c^{3} d^{13}\right )} x^{17} + \frac{91}{4} \,{\left (22 \, b^{2} c^{6} d^{10} + 24 \, a b c^{5} d^{11} + 5 \, a^{2} c^{4} d^{12}\right )} x^{16} + \frac{208}{15} \,{\left (55 \, b^{2} c^{7} d^{9} + 77 \, a b c^{6} d^{10} + 21 \, a^{2} c^{5} d^{11}\right )} x^{15} + \frac{143}{7} \,{\left (45 \, b^{2} c^{8} d^{8} + 80 \, a b c^{7} d^{9} + 28 \, a^{2} c^{6} d^{10}\right )} x^{14} + 220 \,{\left (4 \, b^{2} c^{9} d^{7} + 9 \, a b c^{8} d^{8} + 4 \, a^{2} c^{7} d^{9}\right )} x^{13} + \frac{143}{6} \,{\left (28 \, b^{2} c^{10} d^{6} + 80 \, a b c^{9} d^{7} + 45 \, a^{2} c^{8} d^{8}\right )} x^{12} + \frac{208}{11} \,{\left (21 \, b^{2} c^{11} d^{5} + 77 \, a b c^{10} d^{6} + 55 \, a^{2} c^{9} d^{7}\right )} x^{11} + \frac{182}{5} \,{\left (5 \, b^{2} c^{12} d^{4} + 24 \, a b c^{11} d^{5} + 22 \, a^{2} c^{10} d^{6}\right )} x^{10} + \frac{56}{9} \,{\left (10 \, b^{2} c^{13} d^{3} + 65 \, a b c^{12} d^{4} + 78 \, a^{2} c^{11} d^{5}\right )} x^{9} + \frac{5}{2} \,{\left (6 \, b^{2} c^{14} d^{2} + 56 \, a b c^{13} d^{3} + 91 \, a^{2} c^{12} d^{4}\right )} x^{8} + \frac{16}{7} \,{\left (b^{2} c^{15} d + 15 \, a b c^{14} d^{2} + 35 \, a^{2} c^{13} d^{3}\right )} x^{7} + \frac{1}{6} \,{\left (b^{2} c^{16} + 32 \, a b c^{15} d + 120 \, a^{2} c^{14} d^{2}\right )} x^{6} + \frac{2}{5} \,{\left (a b c^{16} + 8 \, a^{2} c^{15} d\right )} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^2*(d*x+c)^16,x, algorithm="maxima")

[Out]

1/22*b^2*d^16*x^22 + 1/4*a^2*c^16*x^4 + 2/21*(8*b^2*c*d^15 + a*b*d^16)*x^21 + 1/20*(120*b^2*c^2*d^14 + 32*a*b*
c*d^15 + a^2*d^16)*x^20 + 16/19*(35*b^2*c^3*d^13 + 15*a*b*c^2*d^14 + a^2*c*d^15)*x^19 + 10/9*(91*b^2*c^4*d^12
+ 56*a*b*c^3*d^13 + 6*a^2*c^2*d^14)*x^18 + 56/17*(78*b^2*c^5*d^11 + 65*a*b*c^4*d^12 + 10*a^2*c^3*d^13)*x^17 +
91/4*(22*b^2*c^6*d^10 + 24*a*b*c^5*d^11 + 5*a^2*c^4*d^12)*x^16 + 208/15*(55*b^2*c^7*d^9 + 77*a*b*c^6*d^10 + 21
*a^2*c^5*d^11)*x^15 + 143/7*(45*b^2*c^8*d^8 + 80*a*b*c^7*d^9 + 28*a^2*c^6*d^10)*x^14 + 220*(4*b^2*c^9*d^7 + 9*
a*b*c^8*d^8 + 4*a^2*c^7*d^9)*x^13 + 143/6*(28*b^2*c^10*d^6 + 80*a*b*c^9*d^7 + 45*a^2*c^8*d^8)*x^12 + 208/11*(2
1*b^2*c^11*d^5 + 77*a*b*c^10*d^6 + 55*a^2*c^9*d^7)*x^11 + 182/5*(5*b^2*c^12*d^4 + 24*a*b*c^11*d^5 + 22*a^2*c^1
0*d^6)*x^10 + 56/9*(10*b^2*c^13*d^3 + 65*a*b*c^12*d^4 + 78*a^2*c^11*d^5)*x^9 + 5/2*(6*b^2*c^14*d^2 + 56*a*b*c^
13*d^3 + 91*a^2*c^12*d^4)*x^8 + 16/7*(b^2*c^15*d + 15*a*b*c^14*d^2 + 35*a^2*c^13*d^3)*x^7 + 1/6*(b^2*c^16 + 32
*a*b*c^15*d + 120*a^2*c^14*d^2)*x^6 + 2/5*(a*b*c^16 + 8*a^2*c^15*d)*x^5

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Fricas [B]  time = 1.66193, size = 1683, normalized size = 9.51 \begin{align*} \frac{1}{22} x^{22} d^{16} b^{2} + \frac{16}{21} x^{21} d^{15} c b^{2} + \frac{2}{21} x^{21} d^{16} b a + 6 x^{20} d^{14} c^{2} b^{2} + \frac{8}{5} x^{20} d^{15} c b a + \frac{1}{20} x^{20} d^{16} a^{2} + \frac{560}{19} x^{19} d^{13} c^{3} b^{2} + \frac{240}{19} x^{19} d^{14} c^{2} b a + \frac{16}{19} x^{19} d^{15} c a^{2} + \frac{910}{9} x^{18} d^{12} c^{4} b^{2} + \frac{560}{9} x^{18} d^{13} c^{3} b a + \frac{20}{3} x^{18} d^{14} c^{2} a^{2} + \frac{4368}{17} x^{17} d^{11} c^{5} b^{2} + \frac{3640}{17} x^{17} d^{12} c^{4} b a + \frac{560}{17} x^{17} d^{13} c^{3} a^{2} + \frac{1001}{2} x^{16} d^{10} c^{6} b^{2} + 546 x^{16} d^{11} c^{5} b a + \frac{455}{4} x^{16} d^{12} c^{4} a^{2} + \frac{2288}{3} x^{15} d^{9} c^{7} b^{2} + \frac{16016}{15} x^{15} d^{10} c^{6} b a + \frac{1456}{5} x^{15} d^{11} c^{5} a^{2} + \frac{6435}{7} x^{14} d^{8} c^{8} b^{2} + \frac{11440}{7} x^{14} d^{9} c^{7} b a + 572 x^{14} d^{10} c^{6} a^{2} + 880 x^{13} d^{7} c^{9} b^{2} + 1980 x^{13} d^{8} c^{8} b a + 880 x^{13} d^{9} c^{7} a^{2} + \frac{2002}{3} x^{12} d^{6} c^{10} b^{2} + \frac{5720}{3} x^{12} d^{7} c^{9} b a + \frac{2145}{2} x^{12} d^{8} c^{8} a^{2} + \frac{4368}{11} x^{11} d^{5} c^{11} b^{2} + 1456 x^{11} d^{6} c^{10} b a + 1040 x^{11} d^{7} c^{9} a^{2} + 182 x^{10} d^{4} c^{12} b^{2} + \frac{4368}{5} x^{10} d^{5} c^{11} b a + \frac{4004}{5} x^{10} d^{6} c^{10} a^{2} + \frac{560}{9} x^{9} d^{3} c^{13} b^{2} + \frac{3640}{9} x^{9} d^{4} c^{12} b a + \frac{1456}{3} x^{9} d^{5} c^{11} a^{2} + 15 x^{8} d^{2} c^{14} b^{2} + 140 x^{8} d^{3} c^{13} b a + \frac{455}{2} x^{8} d^{4} c^{12} a^{2} + \frac{16}{7} x^{7} d c^{15} b^{2} + \frac{240}{7} x^{7} d^{2} c^{14} b a + 80 x^{7} d^{3} c^{13} a^{2} + \frac{1}{6} x^{6} c^{16} b^{2} + \frac{16}{3} x^{6} d c^{15} b a + 20 x^{6} d^{2} c^{14} a^{2} + \frac{2}{5} x^{5} c^{16} b a + \frac{16}{5} x^{5} d c^{15} a^{2} + \frac{1}{4} x^{4} c^{16} a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^2*(d*x+c)^16,x, algorithm="fricas")

[Out]

1/22*x^22*d^16*b^2 + 16/21*x^21*d^15*c*b^2 + 2/21*x^21*d^16*b*a + 6*x^20*d^14*c^2*b^2 + 8/5*x^20*d^15*c*b*a +
1/20*x^20*d^16*a^2 + 560/19*x^19*d^13*c^3*b^2 + 240/19*x^19*d^14*c^2*b*a + 16/19*x^19*d^15*c*a^2 + 910/9*x^18*
d^12*c^4*b^2 + 560/9*x^18*d^13*c^3*b*a + 20/3*x^18*d^14*c^2*a^2 + 4368/17*x^17*d^11*c^5*b^2 + 3640/17*x^17*d^1
2*c^4*b*a + 560/17*x^17*d^13*c^3*a^2 + 1001/2*x^16*d^10*c^6*b^2 + 546*x^16*d^11*c^5*b*a + 455/4*x^16*d^12*c^4*
a^2 + 2288/3*x^15*d^9*c^7*b^2 + 16016/15*x^15*d^10*c^6*b*a + 1456/5*x^15*d^11*c^5*a^2 + 6435/7*x^14*d^8*c^8*b^
2 + 11440/7*x^14*d^9*c^7*b*a + 572*x^14*d^10*c^6*a^2 + 880*x^13*d^7*c^9*b^2 + 1980*x^13*d^8*c^8*b*a + 880*x^13
*d^9*c^7*a^2 + 2002/3*x^12*d^6*c^10*b^2 + 5720/3*x^12*d^7*c^9*b*a + 2145/2*x^12*d^8*c^8*a^2 + 4368/11*x^11*d^5
*c^11*b^2 + 1456*x^11*d^6*c^10*b*a + 1040*x^11*d^7*c^9*a^2 + 182*x^10*d^4*c^12*b^2 + 4368/5*x^10*d^5*c^11*b*a
+ 4004/5*x^10*d^6*c^10*a^2 + 560/9*x^9*d^3*c^13*b^2 + 3640/9*x^9*d^4*c^12*b*a + 1456/3*x^9*d^5*c^11*a^2 + 15*x
^8*d^2*c^14*b^2 + 140*x^8*d^3*c^13*b*a + 455/2*x^8*d^4*c^12*a^2 + 16/7*x^7*d*c^15*b^2 + 240/7*x^7*d^2*c^14*b*a
+ 80*x^7*d^3*c^13*a^2 + 1/6*x^6*c^16*b^2 + 16/3*x^6*d*c^15*b*a + 20*x^6*d^2*c^14*a^2 + 2/5*x^5*c^16*b*a + 16/
5*x^5*d*c^15*a^2 + 1/4*x^4*c^16*a^2

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Sympy [B]  time = 0.212878, size = 697, normalized size = 3.94 \begin{align*} \frac{a^{2} c^{16} x^{4}}{4} + \frac{b^{2} d^{16} x^{22}}{22} + x^{21} \left (\frac{2 a b d^{16}}{21} + \frac{16 b^{2} c d^{15}}{21}\right ) + x^{20} \left (\frac{a^{2} d^{16}}{20} + \frac{8 a b c d^{15}}{5} + 6 b^{2} c^{2} d^{14}\right ) + x^{19} \left (\frac{16 a^{2} c d^{15}}{19} + \frac{240 a b c^{2} d^{14}}{19} + \frac{560 b^{2} c^{3} d^{13}}{19}\right ) + x^{18} \left (\frac{20 a^{2} c^{2} d^{14}}{3} + \frac{560 a b c^{3} d^{13}}{9} + \frac{910 b^{2} c^{4} d^{12}}{9}\right ) + x^{17} \left (\frac{560 a^{2} c^{3} d^{13}}{17} + \frac{3640 a b c^{4} d^{12}}{17} + \frac{4368 b^{2} c^{5} d^{11}}{17}\right ) + x^{16} \left (\frac{455 a^{2} c^{4} d^{12}}{4} + 546 a b c^{5} d^{11} + \frac{1001 b^{2} c^{6} d^{10}}{2}\right ) + x^{15} \left (\frac{1456 a^{2} c^{5} d^{11}}{5} + \frac{16016 a b c^{6} d^{10}}{15} + \frac{2288 b^{2} c^{7} d^{9}}{3}\right ) + x^{14} \left (572 a^{2} c^{6} d^{10} + \frac{11440 a b c^{7} d^{9}}{7} + \frac{6435 b^{2} c^{8} d^{8}}{7}\right ) + x^{13} \left (880 a^{2} c^{7} d^{9} + 1980 a b c^{8} d^{8} + 880 b^{2} c^{9} d^{7}\right ) + x^{12} \left (\frac{2145 a^{2} c^{8} d^{8}}{2} + \frac{5720 a b c^{9} d^{7}}{3} + \frac{2002 b^{2} c^{10} d^{6}}{3}\right ) + x^{11} \left (1040 a^{2} c^{9} d^{7} + 1456 a b c^{10} d^{6} + \frac{4368 b^{2} c^{11} d^{5}}{11}\right ) + x^{10} \left (\frac{4004 a^{2} c^{10} d^{6}}{5} + \frac{4368 a b c^{11} d^{5}}{5} + 182 b^{2} c^{12} d^{4}\right ) + x^{9} \left (\frac{1456 a^{2} c^{11} d^{5}}{3} + \frac{3640 a b c^{12} d^{4}}{9} + \frac{560 b^{2} c^{13} d^{3}}{9}\right ) + x^{8} \left (\frac{455 a^{2} c^{12} d^{4}}{2} + 140 a b c^{13} d^{3} + 15 b^{2} c^{14} d^{2}\right ) + x^{7} \left (80 a^{2} c^{13} d^{3} + \frac{240 a b c^{14} d^{2}}{7} + \frac{16 b^{2} c^{15} d}{7}\right ) + x^{6} \left (20 a^{2} c^{14} d^{2} + \frac{16 a b c^{15} d}{3} + \frac{b^{2} c^{16}}{6}\right ) + x^{5} \left (\frac{16 a^{2} c^{15} d}{5} + \frac{2 a b c^{16}}{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(b*x+a)**2*(d*x+c)**16,x)

[Out]

a**2*c**16*x**4/4 + b**2*d**16*x**22/22 + x**21*(2*a*b*d**16/21 + 16*b**2*c*d**15/21) + x**20*(a**2*d**16/20 +
8*a*b*c*d**15/5 + 6*b**2*c**2*d**14) + x**19*(16*a**2*c*d**15/19 + 240*a*b*c**2*d**14/19 + 560*b**2*c**3*d**1
3/19) + x**18*(20*a**2*c**2*d**14/3 + 560*a*b*c**3*d**13/9 + 910*b**2*c**4*d**12/9) + x**17*(560*a**2*c**3*d**
13/17 + 3640*a*b*c**4*d**12/17 + 4368*b**2*c**5*d**11/17) + x**16*(455*a**2*c**4*d**12/4 + 546*a*b*c**5*d**11
+ 1001*b**2*c**6*d**10/2) + x**15*(1456*a**2*c**5*d**11/5 + 16016*a*b*c**6*d**10/15 + 2288*b**2*c**7*d**9/3) +
x**14*(572*a**2*c**6*d**10 + 11440*a*b*c**7*d**9/7 + 6435*b**2*c**8*d**8/7) + x**13*(880*a**2*c**7*d**9 + 198
0*a*b*c**8*d**8 + 880*b**2*c**9*d**7) + x**12*(2145*a**2*c**8*d**8/2 + 5720*a*b*c**9*d**7/3 + 2002*b**2*c**10*
d**6/3) + x**11*(1040*a**2*c**9*d**7 + 1456*a*b*c**10*d**6 + 4368*b**2*c**11*d**5/11) + x**10*(4004*a**2*c**10
*d**6/5 + 4368*a*b*c**11*d**5/5 + 182*b**2*c**12*d**4) + x**9*(1456*a**2*c**11*d**5/3 + 3640*a*b*c**12*d**4/9
+ 560*b**2*c**13*d**3/9) + x**8*(455*a**2*c**12*d**4/2 + 140*a*b*c**13*d**3 + 15*b**2*c**14*d**2) + x**7*(80*a
**2*c**13*d**3 + 240*a*b*c**14*d**2/7 + 16*b**2*c**15*d/7) + x**6*(20*a**2*c**14*d**2 + 16*a*b*c**15*d/3 + b**
2*c**16/6) + x**5*(16*a**2*c**15*d/5 + 2*a*b*c**16/5)

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Giac [B]  time = 1.21318, size = 902, normalized size = 5.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^2*(d*x+c)^16,x, algorithm="giac")

[Out]

1/22*b^2*d^16*x^22 + 16/21*b^2*c*d^15*x^21 + 2/21*a*b*d^16*x^21 + 6*b^2*c^2*d^14*x^20 + 8/5*a*b*c*d^15*x^20 +
1/20*a^2*d^16*x^20 + 560/19*b^2*c^3*d^13*x^19 + 240/19*a*b*c^2*d^14*x^19 + 16/19*a^2*c*d^15*x^19 + 910/9*b^2*c
^4*d^12*x^18 + 560/9*a*b*c^3*d^13*x^18 + 20/3*a^2*c^2*d^14*x^18 + 4368/17*b^2*c^5*d^11*x^17 + 3640/17*a*b*c^4*
d^12*x^17 + 560/17*a^2*c^3*d^13*x^17 + 1001/2*b^2*c^6*d^10*x^16 + 546*a*b*c^5*d^11*x^16 + 455/4*a^2*c^4*d^12*x
^16 + 2288/3*b^2*c^7*d^9*x^15 + 16016/15*a*b*c^6*d^10*x^15 + 1456/5*a^2*c^5*d^11*x^15 + 6435/7*b^2*c^8*d^8*x^1
4 + 11440/7*a*b*c^7*d^9*x^14 + 572*a^2*c^6*d^10*x^14 + 880*b^2*c^9*d^7*x^13 + 1980*a*b*c^8*d^8*x^13 + 880*a^2*
c^7*d^9*x^13 + 2002/3*b^2*c^10*d^6*x^12 + 5720/3*a*b*c^9*d^7*x^12 + 2145/2*a^2*c^8*d^8*x^12 + 4368/11*b^2*c^11
*d^5*x^11 + 1456*a*b*c^10*d^6*x^11 + 1040*a^2*c^9*d^7*x^11 + 182*b^2*c^12*d^4*x^10 + 4368/5*a*b*c^11*d^5*x^10
+ 4004/5*a^2*c^10*d^6*x^10 + 560/9*b^2*c^13*d^3*x^9 + 3640/9*a*b*c^12*d^4*x^9 + 1456/3*a^2*c^11*d^5*x^9 + 15*b
^2*c^14*d^2*x^8 + 140*a*b*c^13*d^3*x^8 + 455/2*a^2*c^12*d^4*x^8 + 16/7*b^2*c^15*d*x^7 + 240/7*a*b*c^14*d^2*x^7
+ 80*a^2*c^13*d^3*x^7 + 1/6*b^2*c^16*x^6 + 16/3*a*b*c^15*d*x^6 + 20*a^2*c^14*d^2*x^6 + 2/5*a*b*c^16*x^5 + 16/
5*a^2*c^15*d*x^5 + 1/4*a^2*c^16*x^4