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

Optimal. Leaf size=137 \[ \frac {(c+d x)^{19} \left (a^2 d^2-6 a b c d+6 b^2 c^2\right )}{19 d^5}+\frac {c^2 (c+d x)^{17} (b c-a d)^2}{17 d^5}-\frac {b (c+d x)^{20} (2 b c-a d)}{10 d^5}-\frac {c (c+d x)^{18} (b c-a d) (2 b c-a d)}{9 d^5}+\frac {b^2 (c+d x)^{21}}{21 d^5} \]

[Out]

1/17*c^2*(-a*d+b*c)^2*(d*x+c)^17/d^5-1/9*c*(-a*d+b*c)*(-a*d+2*b*c)*(d*x+c)^18/d^5+1/19*(a^2*d^2-6*a*b*c*d+6*b^
2*c^2)*(d*x+c)^19/d^5-1/10*b*(-a*d+2*b*c)*(d*x+c)^20/d^5+1/21*b^2*(d*x+c)^21/d^5

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Rubi [A]  time = 0.34, antiderivative size = 137, normalized size of antiderivative = 1.00, 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} \[ \frac {(c+d x)^{19} \left (a^2 d^2-6 a b c d+6 b^2 c^2\right )}{19 d^5}+\frac {c^2 (c+d x)^{17} (b c-a d)^2}{17 d^5}-\frac {b (c+d x)^{20} (2 b c-a d)}{10 d^5}-\frac {c (c+d x)^{18} (b c-a d) (2 b c-a d)}{9 d^5}+\frac {b^2 (c+d x)^{21}}{21 d^5} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x)^2*(c + d*x)^16,x]

[Out]

(c^2*(b*c - a*d)^2*(c + d*x)^17)/(17*d^5) - (c*(b*c - a*d)*(2*b*c - a*d)*(c + d*x)^18)/(9*d^5) + ((6*b^2*c^2 -
6*a*b*c*d + a^2*d^2)*(c + d*x)^19)/(19*d^5) - (b*(2*b*c - a*d)*(c + d*x)^20)/(10*d^5) + (b^2*(c + d*x)^21)/(2
1*d^5)

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^2 (a+b x)^2 (c+d x)^{16} \, dx &=\int \left (\frac {c^2 (b c-a d)^2 (c+d x)^{16}}{d^4}+\frac {2 c (b c-a d) (-2 b c+a d) (c+d x)^{17}}{d^4}+\frac {\left (6 b^2 c^2-6 a b c d+a^2 d^2\right ) (c+d x)^{18}}{d^4}-\frac {2 b (2 b c-a d) (c+d x)^{19}}{d^4}+\frac {b^2 (c+d x)^{20}}{d^4}\right ) \, dx\\ &=\frac {c^2 (b c-a d)^2 (c+d x)^{17}}{17 d^5}-\frac {c (b c-a d) (2 b c-a d) (c+d x)^{18}}{9 d^5}+\frac {\left (6 b^2 c^2-6 a b c d+a^2 d^2\right ) (c+d x)^{19}}{19 d^5}-\frac {b (2 b c-a d) (c+d x)^{20}}{10 d^5}+\frac {b^2 (c+d x)^{21}}{21 d^5}\\ \end {align*}

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Mathematica [B]  time = 0.11, size = 585, normalized size = 4.27 \[ \frac {1}{19} d^{14} x^{19} \left (a^2 d^2+32 a b c d+120 b^2 c^2\right )+\frac {8}{9} c d^{13} x^{18} \left (a^2 d^2+15 a b c d+35 b^2 c^2\right )+\frac {20}{17} c^2 d^{12} x^{17} \left (6 a^2 d^2+56 a b c d+91 b^2 c^2\right )+\frac {1}{5} c^{14} x^5 \left (120 a^2 d^2+32 a b c d+b^2 c^2\right )+\frac {8}{3} c^{13} d x^6 \left (35 a^2 d^2+15 a b c d+b^2 c^2\right )+\frac {20}{7} c^{12} d^2 x^7 \left (91 a^2 d^2+56 a b c d+6 b^2 c^2\right )+7 c^{11} d^3 x^8 \left (78 a^2 d^2+65 a b c d+10 b^2 c^2\right )+\frac {364}{9} c^{10} d^4 x^9 \left (22 a^2 d^2+24 a b c d+5 b^2 c^2\right )+\frac {104}{5} c^9 d^5 x^{10} \left (55 a^2 d^2+77 a b c d+21 b^2 c^2\right )+26 c^8 d^6 x^{11} \left (45 a^2 d^2+80 a b c d+28 b^2 c^2\right )+\frac {715}{3} c^7 d^7 x^{12} \left (4 a^2 d^2+9 a b c d+4 b^2 c^2\right )+22 c^6 d^8 x^{13} \left (28 a^2 d^2+80 a b c d+45 b^2 c^2\right )+\frac {104}{7} c^5 d^9 x^{14} \left (21 a^2 d^2+77 a b c d+55 b^2 c^2\right )+\frac {364}{15} c^4 d^{10} x^{15} \left (5 a^2 d^2+24 a b c d+22 b^2 c^2\right )+\frac {7}{2} c^3 d^{11} x^{16} \left (10 a^2 d^2+65 a b c d+78 b^2 c^2\right )+\frac {1}{3} a^2 c^{16} x^3+\frac {1}{2} a c^{15} x^4 (8 a d+b c)+\frac {1}{10} b d^{15} x^{20} (a d+8 b c)+\frac {1}{21} b^2 d^{16} x^{21} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*c^16*x^3)/3 + (a*c^15*(b*c + 8*a*d)*x^4)/2 + (c^14*(b^2*c^2 + 32*a*b*c*d + 120*a^2*d^2)*x^5)/5 + (8*c^13*
d*(b^2*c^2 + 15*a*b*c*d + 35*a^2*d^2)*x^6)/3 + (20*c^12*d^2*(6*b^2*c^2 + 56*a*b*c*d + 91*a^2*d^2)*x^7)/7 + 7*c
^11*d^3*(10*b^2*c^2 + 65*a*b*c*d + 78*a^2*d^2)*x^8 + (364*c^10*d^4*(5*b^2*c^2 + 24*a*b*c*d + 22*a^2*d^2)*x^9)/
9 + (104*c^9*d^5*(21*b^2*c^2 + 77*a*b*c*d + 55*a^2*d^2)*x^10)/5 + 26*c^8*d^6*(28*b^2*c^2 + 80*a*b*c*d + 45*a^2
*d^2)*x^11 + (715*c^7*d^7*(4*b^2*c^2 + 9*a*b*c*d + 4*a^2*d^2)*x^12)/3 + 22*c^6*d^8*(45*b^2*c^2 + 80*a*b*c*d +
28*a^2*d^2)*x^13 + (104*c^5*d^9*(55*b^2*c^2 + 77*a*b*c*d + 21*a^2*d^2)*x^14)/7 + (364*c^4*d^10*(22*b^2*c^2 + 2
4*a*b*c*d + 5*a^2*d^2)*x^15)/15 + (7*c^3*d^11*(78*b^2*c^2 + 65*a*b*c*d + 10*a^2*d^2)*x^16)/2 + (20*c^2*d^12*(9
1*b^2*c^2 + 56*a*b*c*d + 6*a^2*d^2)*x^17)/17 + (8*c*d^13*(35*b^2*c^2 + 15*a*b*c*d + a^2*d^2)*x^18)/9 + (d^14*(
120*b^2*c^2 + 32*a*b*c*d + a^2*d^2)*x^19)/19 + (b*d^15*(8*b*c + a*d)*x^20)/10 + (b^2*d^16*x^21)/21

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fricas [B]  time = 0.72, size = 668, normalized size = 4.88 \[ \frac {1}{21} x^{21} d^{16} b^{2} + \frac {4}{5} x^{20} d^{15} c b^{2} + \frac {1}{10} x^{20} d^{16} b a + \frac {120}{19} x^{19} d^{14} c^{2} b^{2} + \frac {32}{19} x^{19} d^{15} c b a + \frac {1}{19} x^{19} d^{16} a^{2} + \frac {280}{9} x^{18} d^{13} c^{3} b^{2} + \frac {40}{3} x^{18} d^{14} c^{2} b a + \frac {8}{9} x^{18} d^{15} c a^{2} + \frac {1820}{17} x^{17} d^{12} c^{4} b^{2} + \frac {1120}{17} x^{17} d^{13} c^{3} b a + \frac {120}{17} x^{17} d^{14} c^{2} a^{2} + 273 x^{16} d^{11} c^{5} b^{2} + \frac {455}{2} x^{16} d^{12} c^{4} b a + 35 x^{16} d^{13} c^{3} a^{2} + \frac {8008}{15} x^{15} d^{10} c^{6} b^{2} + \frac {2912}{5} x^{15} d^{11} c^{5} b a + \frac {364}{3} x^{15} d^{12} c^{4} a^{2} + \frac {5720}{7} x^{14} d^{9} c^{7} b^{2} + 1144 x^{14} d^{10} c^{6} b a + 312 x^{14} d^{11} c^{5} a^{2} + 990 x^{13} d^{8} c^{8} b^{2} + 1760 x^{13} d^{9} c^{7} b a + 616 x^{13} d^{10} c^{6} a^{2} + \frac {2860}{3} x^{12} d^{7} c^{9} b^{2} + 2145 x^{12} d^{8} c^{8} b a + \frac {2860}{3} x^{12} d^{9} c^{7} a^{2} + 728 x^{11} d^{6} c^{10} b^{2} + 2080 x^{11} d^{7} c^{9} b a + 1170 x^{11} d^{8} c^{8} a^{2} + \frac {2184}{5} x^{10} d^{5} c^{11} b^{2} + \frac {8008}{5} x^{10} d^{6} c^{10} b a + 1144 x^{10} d^{7} c^{9} a^{2} + \frac {1820}{9} x^{9} d^{4} c^{12} b^{2} + \frac {2912}{3} x^{9} d^{5} c^{11} b a + \frac {8008}{9} x^{9} d^{6} c^{10} a^{2} + 70 x^{8} d^{3} c^{13} b^{2} + 455 x^{8} d^{4} c^{12} b a + 546 x^{8} d^{5} c^{11} a^{2} + \frac {120}{7} x^{7} d^{2} c^{14} b^{2} + 160 x^{7} d^{3} c^{13} b a + 260 x^{7} d^{4} c^{12} a^{2} + \frac {8}{3} x^{6} d c^{15} b^{2} + 40 x^{6} d^{2} c^{14} b a + \frac {280}{3} x^{6} d^{3} c^{13} a^{2} + \frac {1}{5} x^{5} c^{16} b^{2} + \frac {32}{5} x^{5} d c^{15} b a + 24 x^{5} d^{2} c^{14} a^{2} + \frac {1}{2} x^{4} c^{16} b a + 4 x^{4} d c^{15} a^{2} + \frac {1}{3} x^{3} c^{16} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21*x^21*d^16*b^2 + 4/5*x^20*d^15*c*b^2 + 1/10*x^20*d^16*b*a + 120/19*x^19*d^14*c^2*b^2 + 32/19*x^19*d^15*c*b
*a + 1/19*x^19*d^16*a^2 + 280/9*x^18*d^13*c^3*b^2 + 40/3*x^18*d^14*c^2*b*a + 8/9*x^18*d^15*c*a^2 + 1820/17*x^1
7*d^12*c^4*b^2 + 1120/17*x^17*d^13*c^3*b*a + 120/17*x^17*d^14*c^2*a^2 + 273*x^16*d^11*c^5*b^2 + 455/2*x^16*d^1
2*c^4*b*a + 35*x^16*d^13*c^3*a^2 + 8008/15*x^15*d^10*c^6*b^2 + 2912/5*x^15*d^11*c^5*b*a + 364/3*x^15*d^12*c^4*
a^2 + 5720/7*x^14*d^9*c^7*b^2 + 1144*x^14*d^10*c^6*b*a + 312*x^14*d^11*c^5*a^2 + 990*x^13*d^8*c^8*b^2 + 1760*x
^13*d^9*c^7*b*a + 616*x^13*d^10*c^6*a^2 + 2860/3*x^12*d^7*c^9*b^2 + 2145*x^12*d^8*c^8*b*a + 2860/3*x^12*d^9*c^
7*a^2 + 728*x^11*d^6*c^10*b^2 + 2080*x^11*d^7*c^9*b*a + 1170*x^11*d^8*c^8*a^2 + 2184/5*x^10*d^5*c^11*b^2 + 800
8/5*x^10*d^6*c^10*b*a + 1144*x^10*d^7*c^9*a^2 + 1820/9*x^9*d^4*c^12*b^2 + 2912/3*x^9*d^5*c^11*b*a + 8008/9*x^9
*d^6*c^10*a^2 + 70*x^8*d^3*c^13*b^2 + 455*x^8*d^4*c^12*b*a + 546*x^8*d^5*c^11*a^2 + 120/7*x^7*d^2*c^14*b^2 + 1
60*x^7*d^3*c^13*b*a + 260*x^7*d^4*c^12*a^2 + 8/3*x^6*d*c^15*b^2 + 40*x^6*d^2*c^14*b*a + 280/3*x^6*d^3*c^13*a^2
+ 1/5*x^5*c^16*b^2 + 32/5*x^5*d*c^15*b*a + 24*x^5*d^2*c^14*a^2 + 1/2*x^4*c^16*b*a + 4*x^4*d*c^15*a^2 + 1/3*x^
3*c^16*a^2

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giac [B]  time = 0.99, size = 668, normalized size = 4.88 \[ \frac {1}{21} \, b^{2} d^{16} x^{21} + \frac {4}{5} \, b^{2} c d^{15} x^{20} + \frac {1}{10} \, a b d^{16} x^{20} + \frac {120}{19} \, b^{2} c^{2} d^{14} x^{19} + \frac {32}{19} \, a b c d^{15} x^{19} + \frac {1}{19} \, a^{2} d^{16} x^{19} + \frac {280}{9} \, b^{2} c^{3} d^{13} x^{18} + \frac {40}{3} \, a b c^{2} d^{14} x^{18} + \frac {8}{9} \, a^{2} c d^{15} x^{18} + \frac {1820}{17} \, b^{2} c^{4} d^{12} x^{17} + \frac {1120}{17} \, a b c^{3} d^{13} x^{17} + \frac {120}{17} \, a^{2} c^{2} d^{14} x^{17} + 273 \, b^{2} c^{5} d^{11} x^{16} + \frac {455}{2} \, a b c^{4} d^{12} x^{16} + 35 \, a^{2} c^{3} d^{13} x^{16} + \frac {8008}{15} \, b^{2} c^{6} d^{10} x^{15} + \frac {2912}{5} \, a b c^{5} d^{11} x^{15} + \frac {364}{3} \, a^{2} c^{4} d^{12} x^{15} + \frac {5720}{7} \, b^{2} c^{7} d^{9} x^{14} + 1144 \, a b c^{6} d^{10} x^{14} + 312 \, a^{2} c^{5} d^{11} x^{14} + 990 \, b^{2} c^{8} d^{8} x^{13} + 1760 \, a b c^{7} d^{9} x^{13} + 616 \, a^{2} c^{6} d^{10} x^{13} + \frac {2860}{3} \, b^{2} c^{9} d^{7} x^{12} + 2145 \, a b c^{8} d^{8} x^{12} + \frac {2860}{3} \, a^{2} c^{7} d^{9} x^{12} + 728 \, b^{2} c^{10} d^{6} x^{11} + 2080 \, a b c^{9} d^{7} x^{11} + 1170 \, a^{2} c^{8} d^{8} x^{11} + \frac {2184}{5} \, b^{2} c^{11} d^{5} x^{10} + \frac {8008}{5} \, a b c^{10} d^{6} x^{10} + 1144 \, a^{2} c^{9} d^{7} x^{10} + \frac {1820}{9} \, b^{2} c^{12} d^{4} x^{9} + \frac {2912}{3} \, a b c^{11} d^{5} x^{9} + \frac {8008}{9} \, a^{2} c^{10} d^{6} x^{9} + 70 \, b^{2} c^{13} d^{3} x^{8} + 455 \, a b c^{12} d^{4} x^{8} + 546 \, a^{2} c^{11} d^{5} x^{8} + \frac {120}{7} \, b^{2} c^{14} d^{2} x^{7} + 160 \, a b c^{13} d^{3} x^{7} + 260 \, a^{2} c^{12} d^{4} x^{7} + \frac {8}{3} \, b^{2} c^{15} d x^{6} + 40 \, a b c^{14} d^{2} x^{6} + \frac {280}{3} \, a^{2} c^{13} d^{3} x^{6} + \frac {1}{5} \, b^{2} c^{16} x^{5} + \frac {32}{5} \, a b c^{15} d x^{5} + 24 \, a^{2} c^{14} d^{2} x^{5} + \frac {1}{2} \, a b c^{16} x^{4} + 4 \, a^{2} c^{15} d x^{4} + \frac {1}{3} \, a^{2} c^{16} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21*b^2*d^16*x^21 + 4/5*b^2*c*d^15*x^20 + 1/10*a*b*d^16*x^20 + 120/19*b^2*c^2*d^14*x^19 + 32/19*a*b*c*d^15*x^
19 + 1/19*a^2*d^16*x^19 + 280/9*b^2*c^3*d^13*x^18 + 40/3*a*b*c^2*d^14*x^18 + 8/9*a^2*c*d^15*x^18 + 1820/17*b^2
*c^4*d^12*x^17 + 1120/17*a*b*c^3*d^13*x^17 + 120/17*a^2*c^2*d^14*x^17 + 273*b^2*c^5*d^11*x^16 + 455/2*a*b*c^4*
d^12*x^16 + 35*a^2*c^3*d^13*x^16 + 8008/15*b^2*c^6*d^10*x^15 + 2912/5*a*b*c^5*d^11*x^15 + 364/3*a^2*c^4*d^12*x
^15 + 5720/7*b^2*c^7*d^9*x^14 + 1144*a*b*c^6*d^10*x^14 + 312*a^2*c^5*d^11*x^14 + 990*b^2*c^8*d^8*x^13 + 1760*a
*b*c^7*d^9*x^13 + 616*a^2*c^6*d^10*x^13 + 2860/3*b^2*c^9*d^7*x^12 + 2145*a*b*c^8*d^8*x^12 + 2860/3*a^2*c^7*d^9
*x^12 + 728*b^2*c^10*d^6*x^11 + 2080*a*b*c^9*d^7*x^11 + 1170*a^2*c^8*d^8*x^11 + 2184/5*b^2*c^11*d^5*x^10 + 800
8/5*a*b*c^10*d^6*x^10 + 1144*a^2*c^9*d^7*x^10 + 1820/9*b^2*c^12*d^4*x^9 + 2912/3*a*b*c^11*d^5*x^9 + 8008/9*a^2
*c^10*d^6*x^9 + 70*b^2*c^13*d^3*x^8 + 455*a*b*c^12*d^4*x^8 + 546*a^2*c^11*d^5*x^8 + 120/7*b^2*c^14*d^2*x^7 + 1
60*a*b*c^13*d^3*x^7 + 260*a^2*c^12*d^4*x^7 + 8/3*b^2*c^15*d*x^6 + 40*a*b*c^14*d^2*x^6 + 280/3*a^2*c^13*d^3*x^6
+ 1/5*b^2*c^16*x^5 + 32/5*a*b*c^15*d*x^5 + 24*a^2*c^14*d^2*x^5 + 1/2*a*b*c^16*x^4 + 4*a^2*c^15*d*x^4 + 1/3*a^
2*c^16*x^3

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maple [B]  time = 0.00, size = 622, normalized size = 4.54 \[ \frac {b^{2} d^{16} x^{21}}{21}+\frac {a^{2} c^{16} x^{3}}{3}+\frac {\left (2 a b \,d^{16}+16 b^{2} c \,d^{15}\right ) x^{20}}{20}+\frac {\left (a^{2} d^{16}+32 a b c \,d^{15}+120 b^{2} c^{2} d^{14}\right ) x^{19}}{19}+\frac {\left (16 a^{2} c \,d^{15}+240 a b \,c^{2} d^{14}+560 b^{2} c^{3} d^{13}\right ) x^{18}}{18}+\frac {\left (120 a^{2} c^{2} d^{14}+1120 a b \,c^{3} d^{13}+1820 b^{2} c^{4} d^{12}\right ) x^{17}}{17}+\frac {\left (560 a^{2} c^{3} d^{13}+3640 a b \,c^{4} d^{12}+4368 b^{2} c^{5} d^{11}\right ) x^{16}}{16}+\frac {\left (1820 a^{2} c^{4} d^{12}+8736 a b \,c^{5} d^{11}+8008 b^{2} c^{6} d^{10}\right ) x^{15}}{15}+\frac {\left (4368 a^{2} c^{5} d^{11}+16016 a b \,c^{6} d^{10}+11440 b^{2} c^{7} d^{9}\right ) x^{14}}{14}+\frac {\left (8008 a^{2} c^{6} d^{10}+22880 a b \,c^{7} d^{9}+12870 b^{2} c^{8} d^{8}\right ) x^{13}}{13}+\frac {\left (11440 a^{2} c^{7} d^{9}+25740 a b \,c^{8} d^{8}+11440 b^{2} c^{9} d^{7}\right ) x^{12}}{12}+\frac {\left (12870 a^{2} c^{8} d^{8}+22880 a b \,c^{9} d^{7}+8008 b^{2} c^{10} d^{6}\right ) x^{11}}{11}+\frac {\left (11440 a^{2} c^{9} d^{7}+16016 a b \,c^{10} d^{6}+4368 b^{2} c^{11} d^{5}\right ) x^{10}}{10}+\frac {\left (8008 a^{2} c^{10} d^{6}+8736 a b \,c^{11} d^{5}+1820 b^{2} c^{12} d^{4}\right ) x^{9}}{9}+\frac {\left (4368 a^{2} c^{11} d^{5}+3640 a b \,c^{12} d^{4}+560 b^{2} c^{13} d^{3}\right ) x^{8}}{8}+\frac {\left (1820 a^{2} c^{12} d^{4}+1120 a b \,c^{13} d^{3}+120 b^{2} c^{14} d^{2}\right ) x^{7}}{7}+\frac {\left (560 a^{2} c^{13} d^{3}+240 a b \,c^{14} d^{2}+16 b^{2} c^{15} d \right ) x^{6}}{6}+\frac {\left (120 a^{2} c^{14} d^{2}+32 a b \,c^{15} d +b^{2} c^{16}\right ) x^{5}}{5}+\frac {\left (16 a^{2} c^{15} d +2 a b \,c^{16}\right ) x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.05, size = 617, normalized size = 4.50 \[ \frac {1}{21} \, b^{2} d^{16} x^{21} + \frac {1}{3} \, a^{2} c^{16} x^{3} + \frac {1}{10} \, {\left (8 \, b^{2} c d^{15} + a b d^{16}\right )} x^{20} + \frac {1}{19} \, {\left (120 \, b^{2} c^{2} d^{14} + 32 \, a b c d^{15} + a^{2} d^{16}\right )} x^{19} + \frac {8}{9} \, {\left (35 \, b^{2} c^{3} d^{13} + 15 \, a b c^{2} d^{14} + a^{2} c d^{15}\right )} x^{18} + \frac {20}{17} \, {\left (91 \, b^{2} c^{4} d^{12} + 56 \, a b c^{3} d^{13} + 6 \, a^{2} c^{2} d^{14}\right )} x^{17} + \frac {7}{2} \, {\left (78 \, b^{2} c^{5} d^{11} + 65 \, a b c^{4} d^{12} + 10 \, a^{2} c^{3} d^{13}\right )} x^{16} + \frac {364}{15} \, {\left (22 \, b^{2} c^{6} d^{10} + 24 \, a b c^{5} d^{11} + 5 \, a^{2} c^{4} d^{12}\right )} x^{15} + \frac {104}{7} \, {\left (55 \, b^{2} c^{7} d^{9} + 77 \, a b c^{6} d^{10} + 21 \, a^{2} c^{5} d^{11}\right )} x^{14} + 22 \, {\left (45 \, b^{2} c^{8} d^{8} + 80 \, a b c^{7} d^{9} + 28 \, a^{2} c^{6} d^{10}\right )} x^{13} + \frac {715}{3} \, {\left (4 \, b^{2} c^{9} d^{7} + 9 \, a b c^{8} d^{8} + 4 \, a^{2} c^{7} d^{9}\right )} x^{12} + 26 \, {\left (28 \, b^{2} c^{10} d^{6} + 80 \, a b c^{9} d^{7} + 45 \, a^{2} c^{8} d^{8}\right )} x^{11} + \frac {104}{5} \, {\left (21 \, b^{2} c^{11} d^{5} + 77 \, a b c^{10} d^{6} + 55 \, a^{2} c^{9} d^{7}\right )} x^{10} + \frac {364}{9} \, {\left (5 \, b^{2} c^{12} d^{4} + 24 \, a b c^{11} d^{5} + 22 \, a^{2} c^{10} d^{6}\right )} x^{9} + 7 \, {\left (10 \, b^{2} c^{13} d^{3} + 65 \, a b c^{12} d^{4} + 78 \, a^{2} c^{11} d^{5}\right )} x^{8} + \frac {20}{7} \, {\left (6 \, b^{2} c^{14} d^{2} + 56 \, a b c^{13} d^{3} + 91 \, a^{2} c^{12} d^{4}\right )} x^{7} + \frac {8}{3} \, {\left (b^{2} c^{15} d + 15 \, a b c^{14} d^{2} + 35 \, a^{2} c^{13} d^{3}\right )} x^{6} + \frac {1}{5} \, {\left (b^{2} c^{16} + 32 \, a b c^{15} d + 120 \, a^{2} c^{14} d^{2}\right )} x^{5} + \frac {1}{2} \, {\left (a b c^{16} + 8 \, a^{2} c^{15} d\right )} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21*b^2*d^16*x^21 + 1/3*a^2*c^16*x^3 + 1/10*(8*b^2*c*d^15 + a*b*d^16)*x^20 + 1/19*(120*b^2*c^2*d^14 + 32*a*b*
c*d^15 + a^2*d^16)*x^19 + 8/9*(35*b^2*c^3*d^13 + 15*a*b*c^2*d^14 + a^2*c*d^15)*x^18 + 20/17*(91*b^2*c^4*d^12 +
56*a*b*c^3*d^13 + 6*a^2*c^2*d^14)*x^17 + 7/2*(78*b^2*c^5*d^11 + 65*a*b*c^4*d^12 + 10*a^2*c^3*d^13)*x^16 + 364
/15*(22*b^2*c^6*d^10 + 24*a*b*c^5*d^11 + 5*a^2*c^4*d^12)*x^15 + 104/7*(55*b^2*c^7*d^9 + 77*a*b*c^6*d^10 + 21*a
^2*c^5*d^11)*x^14 + 22*(45*b^2*c^8*d^8 + 80*a*b*c^7*d^9 + 28*a^2*c^6*d^10)*x^13 + 715/3*(4*b^2*c^9*d^7 + 9*a*b
*c^8*d^8 + 4*a^2*c^7*d^9)*x^12 + 26*(28*b^2*c^10*d^6 + 80*a*b*c^9*d^7 + 45*a^2*c^8*d^8)*x^11 + 104/5*(21*b^2*c
^11*d^5 + 77*a*b*c^10*d^6 + 55*a^2*c^9*d^7)*x^10 + 364/9*(5*b^2*c^12*d^4 + 24*a*b*c^11*d^5 + 22*a^2*c^10*d^6)*
x^9 + 7*(10*b^2*c^13*d^3 + 65*a*b*c^12*d^4 + 78*a^2*c^11*d^5)*x^8 + 20/7*(6*b^2*c^14*d^2 + 56*a*b*c^13*d^3 + 9
1*a^2*c^12*d^4)*x^7 + 8/3*(b^2*c^15*d + 15*a*b*c^14*d^2 + 35*a^2*c^13*d^3)*x^6 + 1/5*(b^2*c^16 + 32*a*b*c^15*d
+ 120*a^2*c^14*d^2)*x^5 + 1/2*(a*b*c^16 + 8*a^2*c^15*d)*x^4

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mupad [B]  time = 0.64, size = 557, normalized size = 4.07 \[ x^5\,\left (24\,a^2\,c^{14}\,d^2+\frac {32\,a\,b\,c^{15}\,d}{5}+\frac {b^2\,c^{16}}{5}\right )+x^{19}\,\left (\frac {a^2\,d^{16}}{19}+\frac {32\,a\,b\,c\,d^{15}}{19}+\frac {120\,b^2\,c^2\,d^{14}}{19}\right )+\frac {a^2\,c^{16}\,x^3}{3}+\frac {b^2\,d^{16}\,x^{21}}{21}+\frac {a\,c^{15}\,x^4\,\left (8\,a\,d+b\,c\right )}{2}+\frac {b\,d^{15}\,x^{20}\,\left (a\,d+8\,b\,c\right )}{10}+\frac {8\,c^{13}\,d\,x^6\,\left (35\,a^2\,d^2+15\,a\,b\,c\,d+b^2\,c^2\right )}{3}+\frac {8\,c\,d^{13}\,x^{18}\,\left (a^2\,d^2+15\,a\,b\,c\,d+35\,b^2\,c^2\right )}{9}+\frac {715\,c^7\,d^7\,x^{12}\,\left (4\,a^2\,d^2+9\,a\,b\,c\,d+4\,b^2\,c^2\right )}{3}+\frac {364\,c^{10}\,d^4\,x^9\,\left (22\,a^2\,d^2+24\,a\,b\,c\,d+5\,b^2\,c^2\right )}{9}+\frac {364\,c^4\,d^{10}\,x^{15}\,\left (5\,a^2\,d^2+24\,a\,b\,c\,d+22\,b^2\,c^2\right )}{15}+\frac {20\,c^{12}\,d^2\,x^7\,\left (91\,a^2\,d^2+56\,a\,b\,c\,d+6\,b^2\,c^2\right )}{7}+7\,c^{11}\,d^3\,x^8\,\left (78\,a^2\,d^2+65\,a\,b\,c\,d+10\,b^2\,c^2\right )+\frac {104\,c^9\,d^5\,x^{10}\,\left (55\,a^2\,d^2+77\,a\,b\,c\,d+21\,b^2\,c^2\right )}{5}+26\,c^8\,d^6\,x^{11}\,\left (45\,a^2\,d^2+80\,a\,b\,c\,d+28\,b^2\,c^2\right )+22\,c^6\,d^8\,x^{13}\,\left (28\,a^2\,d^2+80\,a\,b\,c\,d+45\,b^2\,c^2\right )+\frac {104\,c^5\,d^9\,x^{14}\,\left (21\,a^2\,d^2+77\,a\,b\,c\,d+55\,b^2\,c^2\right )}{7}+\frac {7\,c^3\,d^{11}\,x^{16}\,\left (10\,a^2\,d^2+65\,a\,b\,c\,d+78\,b^2\,c^2\right )}{2}+\frac {20\,c^2\,d^{12}\,x^{17}\,\left (6\,a^2\,d^2+56\,a\,b\,c\,d+91\,b^2\,c^2\right )}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*x)^2*(c + d*x)^16,x)

[Out]

x^5*((b^2*c^16)/5 + 24*a^2*c^14*d^2 + (32*a*b*c^15*d)/5) + x^19*((a^2*d^16)/19 + (120*b^2*c^2*d^14)/19 + (32*a
*b*c*d^15)/19) + (a^2*c^16*x^3)/3 + (b^2*d^16*x^21)/21 + (a*c^15*x^4*(8*a*d + b*c))/2 + (b*d^15*x^20*(a*d + 8*
b*c))/10 + (8*c^13*d*x^6*(35*a^2*d^2 + b^2*c^2 + 15*a*b*c*d))/3 + (8*c*d^13*x^18*(a^2*d^2 + 35*b^2*c^2 + 15*a*
b*c*d))/9 + (715*c^7*d^7*x^12*(4*a^2*d^2 + 4*b^2*c^2 + 9*a*b*c*d))/3 + (364*c^10*d^4*x^9*(22*a^2*d^2 + 5*b^2*c
^2 + 24*a*b*c*d))/9 + (364*c^4*d^10*x^15*(5*a^2*d^2 + 22*b^2*c^2 + 24*a*b*c*d))/15 + (20*c^12*d^2*x^7*(91*a^2*
d^2 + 6*b^2*c^2 + 56*a*b*c*d))/7 + 7*c^11*d^3*x^8*(78*a^2*d^2 + 10*b^2*c^2 + 65*a*b*c*d) + (104*c^9*d^5*x^10*(
55*a^2*d^2 + 21*b^2*c^2 + 77*a*b*c*d))/5 + 26*c^8*d^6*x^11*(45*a^2*d^2 + 28*b^2*c^2 + 80*a*b*c*d) + 22*c^6*d^8
*x^13*(28*a^2*d^2 + 45*b^2*c^2 + 80*a*b*c*d) + (104*c^5*d^9*x^14*(21*a^2*d^2 + 55*b^2*c^2 + 77*a*b*c*d))/7 + (
7*c^3*d^11*x^16*(10*a^2*d^2 + 78*b^2*c^2 + 65*a*b*c*d))/2 + (20*c^2*d^12*x^17*(6*a^2*d^2 + 91*b^2*c^2 + 56*a*b
*c*d))/17

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sympy [B]  time = 0.23, size = 682, normalized size = 4.98 \[ \frac {a^{2} c^{16} x^{3}}{3} + \frac {b^{2} d^{16} x^{21}}{21} + x^{20} \left (\frac {a b d^{16}}{10} + \frac {4 b^{2} c d^{15}}{5}\right ) + x^{19} \left (\frac {a^{2} d^{16}}{19} + \frac {32 a b c d^{15}}{19} + \frac {120 b^{2} c^{2} d^{14}}{19}\right ) + x^{18} \left (\frac {8 a^{2} c d^{15}}{9} + \frac {40 a b c^{2} d^{14}}{3} + \frac {280 b^{2} c^{3} d^{13}}{9}\right ) + x^{17} \left (\frac {120 a^{2} c^{2} d^{14}}{17} + \frac {1120 a b c^{3} d^{13}}{17} + \frac {1820 b^{2} c^{4} d^{12}}{17}\right ) + x^{16} \left (35 a^{2} c^{3} d^{13} + \frac {455 a b c^{4} d^{12}}{2} + 273 b^{2} c^{5} d^{11}\right ) + x^{15} \left (\frac {364 a^{2} c^{4} d^{12}}{3} + \frac {2912 a b c^{5} d^{11}}{5} + \frac {8008 b^{2} c^{6} d^{10}}{15}\right ) + x^{14} \left (312 a^{2} c^{5} d^{11} + 1144 a b c^{6} d^{10} + \frac {5720 b^{2} c^{7} d^{9}}{7}\right ) + x^{13} \left (616 a^{2} c^{6} d^{10} + 1760 a b c^{7} d^{9} + 990 b^{2} c^{8} d^{8}\right ) + x^{12} \left (\frac {2860 a^{2} c^{7} d^{9}}{3} + 2145 a b c^{8} d^{8} + \frac {2860 b^{2} c^{9} d^{7}}{3}\right ) + x^{11} \left (1170 a^{2} c^{8} d^{8} + 2080 a b c^{9} d^{7} + 728 b^{2} c^{10} d^{6}\right ) + x^{10} \left (1144 a^{2} c^{9} d^{7} + \frac {8008 a b c^{10} d^{6}}{5} + \frac {2184 b^{2} c^{11} d^{5}}{5}\right ) + x^{9} \left (\frac {8008 a^{2} c^{10} d^{6}}{9} + \frac {2912 a b c^{11} d^{5}}{3} + \frac {1820 b^{2} c^{12} d^{4}}{9}\right ) + x^{8} \left (546 a^{2} c^{11} d^{5} + 455 a b c^{12} d^{4} + 70 b^{2} c^{13} d^{3}\right ) + x^{7} \left (260 a^{2} c^{12} d^{4} + 160 a b c^{13} d^{3} + \frac {120 b^{2} c^{14} d^{2}}{7}\right ) + x^{6} \left (\frac {280 a^{2} c^{13} d^{3}}{3} + 40 a b c^{14} d^{2} + \frac {8 b^{2} c^{15} d}{3}\right ) + x^{5} \left (24 a^{2} c^{14} d^{2} + \frac {32 a b c^{15} d}{5} + \frac {b^{2} c^{16}}{5}\right ) + x^{4} \left (4 a^{2} c^{15} d + \frac {a b c^{16}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*c**16*x**3/3 + b**2*d**16*x**21/21 + x**20*(a*b*d**16/10 + 4*b**2*c*d**15/5) + x**19*(a**2*d**16/19 + 32*
a*b*c*d**15/19 + 120*b**2*c**2*d**14/19) + x**18*(8*a**2*c*d**15/9 + 40*a*b*c**2*d**14/3 + 280*b**2*c**3*d**13
/9) + x**17*(120*a**2*c**2*d**14/17 + 1120*a*b*c**3*d**13/17 + 1820*b**2*c**4*d**12/17) + x**16*(35*a**2*c**3*
d**13 + 455*a*b*c**4*d**12/2 + 273*b**2*c**5*d**11) + x**15*(364*a**2*c**4*d**12/3 + 2912*a*b*c**5*d**11/5 + 8
008*b**2*c**6*d**10/15) + x**14*(312*a**2*c**5*d**11 + 1144*a*b*c**6*d**10 + 5720*b**2*c**7*d**9/7) + x**13*(6
16*a**2*c**6*d**10 + 1760*a*b*c**7*d**9 + 990*b**2*c**8*d**8) + x**12*(2860*a**2*c**7*d**9/3 + 2145*a*b*c**8*d
**8 + 2860*b**2*c**9*d**7/3) + x**11*(1170*a**2*c**8*d**8 + 2080*a*b*c**9*d**7 + 728*b**2*c**10*d**6) + x**10*
(1144*a**2*c**9*d**7 + 8008*a*b*c**10*d**6/5 + 2184*b**2*c**11*d**5/5) + x**9*(8008*a**2*c**10*d**6/9 + 2912*a
*b*c**11*d**5/3 + 1820*b**2*c**12*d**4/9) + x**8*(546*a**2*c**11*d**5 + 455*a*b*c**12*d**4 + 70*b**2*c**13*d**
3) + x**7*(260*a**2*c**12*d**4 + 160*a*b*c**13*d**3 + 120*b**2*c**14*d**2/7) + x**6*(280*a**2*c**13*d**3/3 + 4
0*a*b*c**14*d**2 + 8*b**2*c**15*d/3) + x**5*(24*a**2*c**14*d**2 + 32*a*b*c**15*d/5 + b**2*c**16/5) + x**4*(4*a
**2*c**15*d + a*b*c**16/2)

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