3.1.74 \(\int x^{-1+n} (b+2 c x^n) (a+b x^n+c x^{2 n})^{13} \, dx\)
Optimal. Leaf size=23 \[ \frac {\left (a+b x^n+c x^{2 n}\right )^{14}}{14 n} \]
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Rubi [A] time = 0.06, antiderivative size = 23, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used =
{1468, 629} \begin {gather*} \frac {\left (a+b x^n+c x^{2 n}\right )^{14}}{14 n} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[x^(-1 + n)*(b + 2*c*x^n)*(a + b*x^n + c*x^(2*n))^13,x]
[Out]
(a + b*x^n + c*x^(2*n))^14/(14*n)
Rule 629
Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d*(a + b*x + c*x^2)^(p +
1))/(b*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]
Rule 1468
Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :>
Dist[1/n, Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x]
&& EqQ[n2, 2*n] && EqQ[Simplify[m - n + 1], 0]
Rubi steps
\begin {align*} \int x^{-1+n} \left (b+2 c x^n\right ) \left (a+b x^n+c x^{2 n}\right )^{13} \, dx &=\frac {\operatorname {Subst}\left (\int (b+2 c x) \left (a+b x+c x^2\right )^{13} \, dx,x,x^n\right )}{n}\\ &=\frac {\left (a+b x^n+c x^{2 n}\right )^{14}}{14 n}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 22, normalized size = 0.96 \begin {gather*} \frac {\left (a+x^n \left (b+c x^n\right )\right )^{14}}{14 n} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[x^(-1 + n)*(b + 2*c*x^n)*(a + b*x^n + c*x^(2*n))^13,x]
[Out]
(a + x^n*(b + c*x^n))^14/(14*n)
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IntegrateAlgebraic [B] time = 0.35, size = 1485, normalized size = 64.57 \begin {gather*} \frac {x^n \left (c x^n+b\right ) \left (91 a^{12} b x^n+364 a^{11} b^2 x^{2 n}+91 a^{12} c x^{2 n}+1001 a^{10} b^3 x^{3 n}+728 a^{11} b c x^{3 n}+2002 a^9 b^4 x^{4 n}+364 a^{11} c^2 x^{4 n}+3003 a^{10} b^2 c x^{4 n}+3003 a^8 b^5 x^{5 n}+3003 a^{10} b c^2 x^{5 n}+8008 a^9 b^3 c x^{5 n}+3432 a^7 b^6 x^{6 n}+1001 a^{10} c^3 x^{6 n}+12012 a^9 b^2 c^2 x^{6 n}+15015 a^8 b^4 c x^{6 n}+3003 a^6 b^7 x^{7 n}+8008 a^9 b c^3 x^{7 n}+30030 a^8 b^3 c^2 x^{7 n}+20592 a^7 b^5 c x^{7 n}+2002 a^5 b^8 x^{8 n}+2002 a^9 c^4 x^{8 n}+30030 a^8 b^2 c^3 x^{8 n}+51480 a^7 b^4 c^2 x^{8 n}+21021 a^6 b^6 c x^{8 n}+1001 a^4 b^9 x^{9 n}+15015 a^8 b c^4 x^{9 n}+68640 a^7 b^3 c^3 x^{9 n}+63063 a^6 b^5 c^2 x^{9 n}+16016 a^5 b^7 c x^{9 n}+364 a^3 b^{10} x^{10 n}+3003 a^8 c^5 x^{10 n}+51480 a^7 b^2 c^4 x^{10 n}+105105 a^6 b^4 c^3 x^{10 n}+56056 a^5 b^6 c^2 x^{10 n}+9009 a^4 b^8 c x^{10 n}+91 a^2 b^{11} x^{11 n}+20592 a^7 b c^5 x^{11 n}+105105 a^6 b^3 c^4 x^{11 n}+112112 a^5 b^5 c^3 x^{11 n}+36036 a^4 b^7 c^2 x^{11 n}+3640 a^3 b^9 c x^{11 n}+14 a b^{12} x^{12 n}+3432 a^7 c^6 x^{12 n}+63063 a^6 b^2 c^5 x^{12 n}+140140 a^5 b^4 c^4 x^{12 n}+84084 a^4 b^6 c^3 x^{12 n}+16380 a^3 b^8 c^2 x^{12 n}+1001 a^2 b^{10} c x^{12 n}+b^{13} x^{13 n}+21021 a^6 b c^6 x^{13 n}+112112 a^5 b^3 c^5 x^{13 n}+126126 a^4 b^5 c^4 x^{13 n}+43680 a^3 b^7 c^3 x^{13 n}+5005 a^2 b^9 c^2 x^{13 n}+168 a b^{11} c x^{13 n}+3003 a^6 c^7 x^{14 n}+56056 a^5 b^2 c^6 x^{14 n}+126126 a^4 b^4 c^5 x^{14 n}+76440 a^3 b^6 c^4 x^{14 n}+15015 a^2 b^8 c^3 x^{14 n}+924 a b^{10} c^2 x^{14 n}+13 b^{12} c x^{14 n}+16016 a^5 b c^7 x^{15 n}+84084 a^4 b^3 c^6 x^{15 n}+91728 a^3 b^5 c^5 x^{15 n}+30030 a^2 b^7 c^4 x^{15 n}+3080 a b^9 c^3 x^{15 n}+78 b^{11} c^2 x^{15 n}+2002 a^5 c^8 x^{16 n}+36036 a^4 b^2 c^7 x^{16 n}+76440 a^3 b^4 c^6 x^{16 n}+42042 a^2 b^6 c^5 x^{16 n}+6930 a b^8 c^4 x^{16 n}+286 b^{10} c^3 x^{16 n}+9009 a^4 b c^8 x^{17 n}+43680 a^3 b^3 c^7 x^{17 n}+42042 a^2 b^5 c^6 x^{17 n}+11088 a b^7 c^5 x^{17 n}+715 b^9 c^4 x^{17 n}+1001 a^4 c^9 x^{18 n}+16380 a^3 b^2 c^8 x^{18 n}+30030 a^2 b^4 c^7 x^{18 n}+12936 a b^6 c^6 x^{18 n}+1287 b^8 c^5 x^{18 n}+3640 a^3 b c^9 x^{19 n}+15015 a^2 b^3 c^8 x^{19 n}+11088 a b^5 c^7 x^{19 n}+1716 b^7 c^6 x^{19 n}+364 a^3 c^{10} x^{20 n}+5005 a^2 b^2 c^9 x^{20 n}+6930 a b^4 c^8 x^{20 n}+1716 b^6 c^7 x^{20 n}+1001 a^2 b c^{10} x^{21 n}+3080 a b^3 c^9 x^{21 n}+1287 b^5 c^8 x^{21 n}+91 a^2 c^{11} x^{22 n}+924 a b^2 c^{10} x^{22 n}+715 b^4 c^9 x^{22 n}+168 a b c^{11} x^{23 n}+286 b^3 c^{10} x^{23 n}+14 a c^{12} x^{24 n}+78 b^2 c^{11} x^{24 n}+13 b c^{12} x^{25 n}+c^{13} x^{26 n}+14 a^{13}\right )}{14 n} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[x^(-1 + n)*(b + 2*c*x^n)*(a + b*x^n + c*x^(2*n))^13,x]
[Out]
(x^n*(b + c*x^n)*(14*a^13 + 91*a^12*b*x^n + 364*a^11*b^2*x^(2*n) + 91*a^12*c*x^(2*n) + 1001*a^10*b^3*x^(3*n) +
728*a^11*b*c*x^(3*n) + 2002*a^9*b^4*x^(4*n) + 3003*a^10*b^2*c*x^(4*n) + 364*a^11*c^2*x^(4*n) + 3003*a^8*b^5*x
^(5*n) + 8008*a^9*b^3*c*x^(5*n) + 3003*a^10*b*c^2*x^(5*n) + 3432*a^7*b^6*x^(6*n) + 15015*a^8*b^4*c*x^(6*n) + 1
2012*a^9*b^2*c^2*x^(6*n) + 1001*a^10*c^3*x^(6*n) + 3003*a^6*b^7*x^(7*n) + 20592*a^7*b^5*c*x^(7*n) + 30030*a^8*
b^3*c^2*x^(7*n) + 8008*a^9*b*c^3*x^(7*n) + 2002*a^5*b^8*x^(8*n) + 21021*a^6*b^6*c*x^(8*n) + 51480*a^7*b^4*c^2*
x^(8*n) + 30030*a^8*b^2*c^3*x^(8*n) + 2002*a^9*c^4*x^(8*n) + 1001*a^4*b^9*x^(9*n) + 16016*a^5*b^7*c*x^(9*n) +
63063*a^6*b^5*c^2*x^(9*n) + 68640*a^7*b^3*c^3*x^(9*n) + 15015*a^8*b*c^4*x^(9*n) + 364*a^3*b^10*x^(10*n) + 9009
*a^4*b^8*c*x^(10*n) + 56056*a^5*b^6*c^2*x^(10*n) + 105105*a^6*b^4*c^3*x^(10*n) + 51480*a^7*b^2*c^4*x^(10*n) +
3003*a^8*c^5*x^(10*n) + 91*a^2*b^11*x^(11*n) + 3640*a^3*b^9*c*x^(11*n) + 36036*a^4*b^7*c^2*x^(11*n) + 112112*a
^5*b^5*c^3*x^(11*n) + 105105*a^6*b^3*c^4*x^(11*n) + 20592*a^7*b*c^5*x^(11*n) + 14*a*b^12*x^(12*n) + 1001*a^2*b
^10*c*x^(12*n) + 16380*a^3*b^8*c^2*x^(12*n) + 84084*a^4*b^6*c^3*x^(12*n) + 140140*a^5*b^4*c^4*x^(12*n) + 63063
*a^6*b^2*c^5*x^(12*n) + 3432*a^7*c^6*x^(12*n) + b^13*x^(13*n) + 168*a*b^11*c*x^(13*n) + 5005*a^2*b^9*c^2*x^(13
*n) + 43680*a^3*b^7*c^3*x^(13*n) + 126126*a^4*b^5*c^4*x^(13*n) + 112112*a^5*b^3*c^5*x^(13*n) + 21021*a^6*b*c^6
*x^(13*n) + 13*b^12*c*x^(14*n) + 924*a*b^10*c^2*x^(14*n) + 15015*a^2*b^8*c^3*x^(14*n) + 76440*a^3*b^6*c^4*x^(1
4*n) + 126126*a^4*b^4*c^5*x^(14*n) + 56056*a^5*b^2*c^6*x^(14*n) + 3003*a^6*c^7*x^(14*n) + 78*b^11*c^2*x^(15*n)
+ 3080*a*b^9*c^3*x^(15*n) + 30030*a^2*b^7*c^4*x^(15*n) + 91728*a^3*b^5*c^5*x^(15*n) + 84084*a^4*b^3*c^6*x^(15
*n) + 16016*a^5*b*c^7*x^(15*n) + 286*b^10*c^3*x^(16*n) + 6930*a*b^8*c^4*x^(16*n) + 42042*a^2*b^6*c^5*x^(16*n)
+ 76440*a^3*b^4*c^6*x^(16*n) + 36036*a^4*b^2*c^7*x^(16*n) + 2002*a^5*c^8*x^(16*n) + 715*b^9*c^4*x^(17*n) + 110
88*a*b^7*c^5*x^(17*n) + 42042*a^2*b^5*c^6*x^(17*n) + 43680*a^3*b^3*c^7*x^(17*n) + 9009*a^4*b*c^8*x^(17*n) + 12
87*b^8*c^5*x^(18*n) + 12936*a*b^6*c^6*x^(18*n) + 30030*a^2*b^4*c^7*x^(18*n) + 16380*a^3*b^2*c^8*x^(18*n) + 100
1*a^4*c^9*x^(18*n) + 1716*b^7*c^6*x^(19*n) + 11088*a*b^5*c^7*x^(19*n) + 15015*a^2*b^3*c^8*x^(19*n) + 3640*a^3*
b*c^9*x^(19*n) + 1716*b^6*c^7*x^(20*n) + 6930*a*b^4*c^8*x^(20*n) + 5005*a^2*b^2*c^9*x^(20*n) + 364*a^3*c^10*x^
(20*n) + 1287*b^5*c^8*x^(21*n) + 3080*a*b^3*c^9*x^(21*n) + 1001*a^2*b*c^10*x^(21*n) + 715*b^4*c^9*x^(22*n) + 9
24*a*b^2*c^10*x^(22*n) + 91*a^2*c^11*x^(22*n) + 286*b^3*c^10*x^(23*n) + 168*a*b*c^11*x^(23*n) + 78*b^2*c^11*x^
(24*n) + 14*a*c^12*x^(24*n) + 13*b*c^12*x^(25*n) + c^13*x^(26*n)))/(14*n)
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fricas [B] time = 1.22, size = 1297, normalized size = 56.39
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^(-1+n)*(b+2*c*x^n)*(a+b*x^n+c*x^(2*n))^13,x, algorithm="fricas")
[Out]
1/14*(c^14*x^(28*n) + 14*b*c^13*x^(27*n) + 14*a^13*b*x^n + 7*(13*b^2*c^12 + 2*a*c^13)*x^(26*n) + 182*(2*b^3*c^
11 + a*b*c^12)*x^(25*n) + 91*(11*b^4*c^10 + 12*a*b^2*c^11 + a^2*c^12)*x^(24*n) + 182*(11*b^5*c^9 + 22*a*b^3*c^
10 + 6*a^2*b*c^11)*x^(23*n) + 91*(33*b^6*c^8 + 110*a*b^4*c^9 + 66*a^2*b^2*c^10 + 4*a^3*c^11)*x^(22*n) + 286*(1
2*b^7*c^7 + 63*a*b^5*c^8 + 70*a^2*b^3*c^9 + 14*a^3*b*c^10)*x^(21*n) + 1001*(3*b^8*c^6 + 24*a*b^6*c^7 + 45*a^2*
b^4*c^8 + 20*a^3*b^2*c^9 + a^4*c^10)*x^(20*n) + 2002*(b^9*c^5 + 12*a*b^7*c^6 + 36*a^2*b^5*c^7 + 30*a^3*b^3*c^8
+ 5*a^4*b*c^9)*x^(19*n) + 1001*(b^10*c^4 + 18*a*b^8*c^5 + 84*a^2*b^6*c^6 + 120*a^3*b^4*c^7 + 45*a^4*b^2*c^8 +
2*a^5*c^9)*x^(18*n) + 182*(2*b^11*c^3 + 55*a*b^9*c^4 + 396*a^2*b^7*c^5 + 924*a^3*b^5*c^6 + 660*a^4*b^3*c^7 +
99*a^5*b*c^8)*x^(17*n) + 91*(b^12*c^2 + 44*a*b^10*c^3 + 495*a^2*b^8*c^4 + 1848*a^3*b^6*c^5 + 2310*a^4*b^4*c^6
+ 792*a^5*b^2*c^7 + 33*a^6*c^8)*x^(16*n) + 14*(b^13*c + 78*a*b^11*c^2 + 1430*a^2*b^9*c^3 + 8580*a^3*b^7*c^4 +
18018*a^4*b^5*c^5 + 12012*a^5*b^3*c^6 + 1716*a^6*b*c^7)*x^(15*n) + (b^14 + 182*a*b^12*c + 6006*a^2*b^10*c^2 +
60060*a^3*b^8*c^3 + 210210*a^4*b^6*c^4 + 252252*a^5*b^4*c^5 + 84084*a^6*b^2*c^6 + 3432*a^7*c^7)*x^(14*n) + 14*
(a*b^13 + 78*a^2*b^11*c + 1430*a^3*b^9*c^2 + 8580*a^4*b^7*c^3 + 18018*a^5*b^5*c^4 + 12012*a^6*b^3*c^5 + 1716*a
^7*b*c^6)*x^(13*n) + 91*(a^2*b^12 + 44*a^3*b^10*c + 495*a^4*b^8*c^2 + 1848*a^5*b^6*c^3 + 2310*a^6*b^4*c^4 + 79
2*a^7*b^2*c^5 + 33*a^8*c^6)*x^(12*n) + 182*(2*a^3*b^11 + 55*a^4*b^9*c + 396*a^5*b^7*c^2 + 924*a^6*b^5*c^3 + 66
0*a^7*b^3*c^4 + 99*a^8*b*c^5)*x^(11*n) + 1001*(a^4*b^10 + 18*a^5*b^8*c + 84*a^6*b^6*c^2 + 120*a^7*b^4*c^3 + 45
*a^8*b^2*c^4 + 2*a^9*c^5)*x^(10*n) + 2002*(a^5*b^9 + 12*a^6*b^7*c + 36*a^7*b^5*c^2 + 30*a^8*b^3*c^3 + 5*a^9*b*
c^4)*x^(9*n) + 1001*(3*a^6*b^8 + 24*a^7*b^6*c + 45*a^8*b^4*c^2 + 20*a^9*b^2*c^3 + a^10*c^4)*x^(8*n) + 286*(12*
a^7*b^7 + 63*a^8*b^5*c + 70*a^9*b^3*c^2 + 14*a^10*b*c^3)*x^(7*n) + 91*(33*a^8*b^6 + 110*a^9*b^4*c + 66*a^10*b^
2*c^2 + 4*a^11*c^3)*x^(6*n) + 182*(11*a^9*b^5 + 22*a^10*b^3*c + 6*a^11*b*c^2)*x^(5*n) + 91*(11*a^10*b^4 + 12*a
^11*b^2*c + a^12*c^2)*x^(4*n) + 182*(2*a^11*b^3 + a^12*b*c)*x^(3*n) + 7*(13*a^12*b^2 + 2*a^13*c)*x^(2*n))/n
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giac [B] time = 1.00, size = 1693, normalized size = 73.61
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^(-1+n)*(b+2*c*x^n)*(a+b*x^n+c*x^(2*n))^13,x, algorithm="giac")
[Out]
1/14*(c^14*x^(28*n) + 14*b*c^13*x^(27*n) + 91*b^2*c^12*x^(26*n) + 14*a*c^13*x^(26*n) + 364*b^3*c^11*x^(25*n) +
182*a*b*c^12*x^(25*n) + 1001*b^4*c^10*x^(24*n) + 1092*a*b^2*c^11*x^(24*n) + 91*a^2*c^12*x^(24*n) + 2002*b^5*c
^9*x^(23*n) + 4004*a*b^3*c^10*x^(23*n) + 1092*a^2*b*c^11*x^(23*n) + 3003*b^6*c^8*x^(22*n) + 10010*a*b^4*c^9*x^
(22*n) + 6006*a^2*b^2*c^10*x^(22*n) + 364*a^3*c^11*x^(22*n) + 3432*b^7*c^7*x^(21*n) + 18018*a*b^5*c^8*x^(21*n)
+ 20020*a^2*b^3*c^9*x^(21*n) + 4004*a^3*b*c^10*x^(21*n) + 3003*b^8*c^6*x^(20*n) + 24024*a*b^6*c^7*x^(20*n) +
45045*a^2*b^4*c^8*x^(20*n) + 20020*a^3*b^2*c^9*x^(20*n) + 1001*a^4*c^10*x^(20*n) + 2002*b^9*c^5*x^(19*n) + 240
24*a*b^7*c^6*x^(19*n) + 72072*a^2*b^5*c^7*x^(19*n) + 60060*a^3*b^3*c^8*x^(19*n) + 10010*a^4*b*c^9*x^(19*n) + 1
001*b^10*c^4*x^(18*n) + 18018*a*b^8*c^5*x^(18*n) + 84084*a^2*b^6*c^6*x^(18*n) + 120120*a^3*b^4*c^7*x^(18*n) +
45045*a^4*b^2*c^8*x^(18*n) + 2002*a^5*c^9*x^(18*n) + 364*b^11*c^3*x^(17*n) + 10010*a*b^9*c^4*x^(17*n) + 72072*
a^2*b^7*c^5*x^(17*n) + 168168*a^3*b^5*c^6*x^(17*n) + 120120*a^4*b^3*c^7*x^(17*n) + 18018*a^5*b*c^8*x^(17*n) +
91*b^12*c^2*x^(16*n) + 4004*a*b^10*c^3*x^(16*n) + 45045*a^2*b^8*c^4*x^(16*n) + 168168*a^3*b^6*c^5*x^(16*n) + 2
10210*a^4*b^4*c^6*x^(16*n) + 72072*a^5*b^2*c^7*x^(16*n) + 3003*a^6*c^8*x^(16*n) + 14*b^13*c*x^(15*n) + 1092*a*
b^11*c^2*x^(15*n) + 20020*a^2*b^9*c^3*x^(15*n) + 120120*a^3*b^7*c^4*x^(15*n) + 252252*a^4*b^5*c^5*x^(15*n) + 1
68168*a^5*b^3*c^6*x^(15*n) + 24024*a^6*b*c^7*x^(15*n) + b^14*x^(14*n) + 182*a*b^12*c*x^(14*n) + 6006*a^2*b^10*
c^2*x^(14*n) + 60060*a^3*b^8*c^3*x^(14*n) + 210210*a^4*b^6*c^4*x^(14*n) + 252252*a^5*b^4*c^5*x^(14*n) + 84084*
a^6*b^2*c^6*x^(14*n) + 3432*a^7*c^7*x^(14*n) + 14*a*b^13*x^(13*n) + 1092*a^2*b^11*c*x^(13*n) + 20020*a^3*b^9*c
^2*x^(13*n) + 120120*a^4*b^7*c^3*x^(13*n) + 252252*a^5*b^5*c^4*x^(13*n) + 168168*a^6*b^3*c^5*x^(13*n) + 24024*
a^7*b*c^6*x^(13*n) + 91*a^2*b^12*x^(12*n) + 4004*a^3*b^10*c*x^(12*n) + 45045*a^4*b^8*c^2*x^(12*n) + 168168*a^5
*b^6*c^3*x^(12*n) + 210210*a^6*b^4*c^4*x^(12*n) + 72072*a^7*b^2*c^5*x^(12*n) + 3003*a^8*c^6*x^(12*n) + 364*a^3
*b^11*x^(11*n) + 10010*a^4*b^9*c*x^(11*n) + 72072*a^5*b^7*c^2*x^(11*n) + 168168*a^6*b^5*c^3*x^(11*n) + 120120*
a^7*b^3*c^4*x^(11*n) + 18018*a^8*b*c^5*x^(11*n) + 1001*a^4*b^10*x^(10*n) + 18018*a^5*b^8*c*x^(10*n) + 84084*a^
6*b^6*c^2*x^(10*n) + 120120*a^7*b^4*c^3*x^(10*n) + 45045*a^8*b^2*c^4*x^(10*n) + 2002*a^9*c^5*x^(10*n) + 2002*a
^5*b^9*x^(9*n) + 24024*a^6*b^7*c*x^(9*n) + 72072*a^7*b^5*c^2*x^(9*n) + 60060*a^8*b^3*c^3*x^(9*n) + 10010*a^9*b
*c^4*x^(9*n) + 3003*a^6*b^8*x^(8*n) + 24024*a^7*b^6*c*x^(8*n) + 45045*a^8*b^4*c^2*x^(8*n) + 20020*a^9*b^2*c^3*
x^(8*n) + 1001*a^10*c^4*x^(8*n) + 3432*a^7*b^7*x^(7*n) + 18018*a^8*b^5*c*x^(7*n) + 20020*a^9*b^3*c^2*x^(7*n) +
4004*a^10*b*c^3*x^(7*n) + 3003*a^8*b^6*x^(6*n) + 10010*a^9*b^4*c*x^(6*n) + 6006*a^10*b^2*c^2*x^(6*n) + 364*a^
11*c^3*x^(6*n) + 2002*a^9*b^5*x^(5*n) + 4004*a^10*b^3*c*x^(5*n) + 1092*a^11*b*c^2*x^(5*n) + 1001*a^10*b^4*x^(4
*n) + 1092*a^11*b^2*c*x^(4*n) + 91*a^12*c^2*x^(4*n) + 364*a^11*b^3*x^(3*n) + 182*a^12*b*c*x^(3*n) + 91*a^12*b^
2*x^(2*n) + 14*a^13*c*x^(2*n) + 14*a^13*b*x^n)/n
________________________________________________________________________________________
maple [B] time = 0.06, size = 2042, normalized size = 88.78
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^(n-1)*(b+2*c*x^n)*(b*x^n+c*x^(2*n)+a)^13,x)
[Out]
26*b^11*c^3/n*(x^n)^17+1716/7/n*(x^n)^14*a^7*c^7+1716/7*b^7*a^7/n*(x^n)^7+143*b^9*c^5/n*(x^n)^19+26*b^3*c^11/n
*(x^n)^25+a*b^13/n*(x^n)^13+143*a^5*b^9/n*(x^n)^9+1716/7*b^7*c^7/n*(x^n)^21+143*b^5*c^9/n*(x^n)^23+143/2*a^10/
n*(x^n)^8*c^4+429/2*a^6/n*(x^n)^8*b^8+143*b^5*a^9/n*(x^n)^5+26*b^11*a^3/n*(x^n)^11+b^13*c/n*(x^n)^15+13/2*a^12
/n*(x^n)^4*c^2+143/2*a^10/n*(x^n)^4*b^4+26*a^11/n*(x^n)^6*c^3+429/2*a^8/n*(x^n)^6*b^6+143*a^9/n*(x^n)^10*c^5+1
43/2*a^4/n*(x^n)^10*b^10+429/2*c^8/n*(x^n)^22*b^6+c^13/n*(x^n)^26*a+13/2*c^12/n*(x^n)^26*b^2+429/2*c^8/n*(x^n)
^16*a^6+13/2*c^2/n*(x^n)^16*b^12+143*c^9/n*(x^n)^18*a^5+143/2*c^4/n*(x^n)^18*b^10+143/2*c^10/n*(x^n)^20*a^4+42
9/2*c^6/n*(x^n)^20*b^8+26*c^11/n*(x^n)^22*a^3+429/2*a^8/n*(x^n)^12*c^6+13/2*a^2/n*(x^n)^12*b^12+26*a^11*b^3/n*
(x^n)^3+13/2*c^12/n*(x^n)^24*a^2+143/2*c^10/n*(x^n)^24*b^4+a^13/n*(x^n)^2*c+13/2*a^12/n*(x^n)^2*b^2+b*a^13/n*x
^n+b*c^13/n*(x^n)^27+1287*b^5*c^8/n*(x^n)^21*a+78*b*c^11/n*(x^n)^23*a^2+286*b^3*c^10/n*(x^n)^23*a+286*b*a^10/n
*(x^n)^7*c^3+1430*b^3*a^9/n*(x^n)^7*c^2+1287*b^5*a^8/n*(x^n)^7*c+715*b*c^9/n*(x^n)^19*a^4+4290*b^3*c^8/n*(x^n)
^19*a^3+5148*b^5*c^7/n*(x^n)^19*a^2+1716*b^7*c^6/n*(x^n)^19*a+5148*a^7/n*(x^n)^12*b^2*c^5+15015*a^6/n*(x^n)^12
*b^4*c^4+12012*a^5/n*(x^n)^12*b^6*c^3+6435/2*a^4/n*(x^n)^12*b^8*c^2+1/14*c^14/n*(x^n)^28+715*a^9/n*(x^n)^6*b^4
*c+1/14/n*(x^n)^14*b^14+6435/2*a^8/n*(x^n)^10*b^2*c^4+8580*a^7/n*(x^n)^10*b^4*c^3+6006*a^6/n*(x^n)^10*b^6*c^2+
1287*a^5/n*(x^n)^10*b^8*c+1430*a^9/n*(x^n)^8*b^2*c^3+6435/2*a^8/n*(x^n)^8*b^4*c^2+1716*a^7/n*(x^n)^8*b^6*c+78*
b*a^11/n*(x^n)^5*c^2+286*b^3*a^10/n*(x^n)^5*c+1287*b*a^8/n*(x^n)^11*c^5+1716*a^7*b/n*(x^n)^13*c^6+12012*a^6*b^
3/n*(x^n)^13*c^5+18018*a^5*b^5/n*(x^n)^13*c^4+8580*a^4*b^7/n*(x^n)^13*c^3+1430*a^3*b^9/n*(x^n)^13*c^2+78*a^2*b
^11/n*(x^n)^13*c+715*a^9*b/n*(x^n)^9*c^4+4290*a^8*b^3/n*(x^n)^9*c^3+5148*a^7*b^5/n*(x^n)^9*c^2+1716*a^6*b^7/n*
(x^n)^9*c+286*b*c^10/n*(x^n)^21*a^3+1430*b^3*c^9/n*(x^n)^21*a^2+8580*b^3*a^7/n*(x^n)^11*c^4+12012*b^5*a^6/n*(x
^n)^11*c^3+5148*b^7*a^5/n*(x^n)^11*c^2+715*b^9*a^4/n*(x^n)^11*c+1716*b*c^7/n*(x^n)^15*a^6+12012*b^3*c^6/n*(x^n
)^15*a^5+18018*b^5*c^5/n*(x^n)^15*a^4+8580*b^7*c^4/n*(x^n)^15*a^3+1430*b^9*c^3/n*(x^n)^15*a^2+78*b^11*c^2/n*(x
^n)^15*a+13*b*c^12/n*(x^n)^25*a+1430*c^9/n*(x^n)^20*a^3*b^2+6435/2*c^8/n*(x^n)^20*a^2*b^4+1716*c^7/n*(x^n)^20*
a*b^6+429*c^10/n*(x^n)^22*a^2*b^2+715*c^9/n*(x^n)^22*a*b^4+13*a^12*b/n*(x^n)^3*c+78*c^11/n*(x^n)^24*a*b^2+78*a
^11/n*(x^n)^4*b^2*c+429*a^10/n*(x^n)^6*b^2*c^2+1287*b*c^8/n*(x^n)^17*a^5+8580*b^3*c^7/n*(x^n)^17*a^4+12012*b^5
*c^6/n*(x^n)^17*a^3+5148*b^7*c^5/n*(x^n)^17*a^2+715*b^9*c^4/n*(x^n)^17*a+6006/n*(x^n)^14*a^6*b^2*c^6+18018/n*(
x^n)^14*a^5*b^4*c^5+15015/n*(x^n)^14*a^4*b^6*c^4+4290/n*(x^n)^14*a^3*b^8*c^3+429/n*(x^n)^14*a^2*b^10*c^2+13/n*
(x^n)^14*a*b^12*c+286*a^3/n*(x^n)^12*b^10*c+5148*c^7/n*(x^n)^16*a^5*b^2+15015*c^6/n*(x^n)^16*a^4*b^4+12012*c^5
/n*(x^n)^16*a^3*b^6+6435/2*c^4/n*(x^n)^16*a^2*b^8+286*c^3/n*(x^n)^16*a*b^10+6435/2*c^8/n*(x^n)^18*a^4*b^2+8580
*c^7/n*(x^n)^18*a^3*b^4+6006*c^6/n*(x^n)^18*a^2*b^6+1287*c^5/n*(x^n)^18*a*b^8
________________________________________________________________________________________
maxima [B] time = 0.86, size = 2041, normalized size = 88.74
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^(-1+n)*(b+2*c*x^n)*(a+b*x^n+c*x^(2*n))^13,x, algorithm="maxima")
[Out]
1/14*c^14*x^(28*n)/n + b*c^13*x^(27*n)/n + 13/2*b^2*c^12*x^(26*n)/n + a*c^13*x^(26*n)/n + 26*b^3*c^11*x^(25*n)
/n + 13*a*b*c^12*x^(25*n)/n + 143/2*b^4*c^10*x^(24*n)/n + 78*a*b^2*c^11*x^(24*n)/n + 13/2*a^2*c^12*x^(24*n)/n
+ 143*b^5*c^9*x^(23*n)/n + 286*a*b^3*c^10*x^(23*n)/n + 78*a^2*b*c^11*x^(23*n)/n + 429/2*b^6*c^8*x^(22*n)/n + 7
15*a*b^4*c^9*x^(22*n)/n + 429*a^2*b^2*c^10*x^(22*n)/n + 26*a^3*c^11*x^(22*n)/n + 1716/7*b^7*c^7*x^(21*n)/n + 1
287*a*b^5*c^8*x^(21*n)/n + 1430*a^2*b^3*c^9*x^(21*n)/n + 286*a^3*b*c^10*x^(21*n)/n + 429/2*b^8*c^6*x^(20*n)/n
+ 1716*a*b^6*c^7*x^(20*n)/n + 6435/2*a^2*b^4*c^8*x^(20*n)/n + 1430*a^3*b^2*c^9*x^(20*n)/n + 143/2*a^4*c^10*x^(
20*n)/n + 143*b^9*c^5*x^(19*n)/n + 1716*a*b^7*c^6*x^(19*n)/n + 5148*a^2*b^5*c^7*x^(19*n)/n + 4290*a^3*b^3*c^8*
x^(19*n)/n + 715*a^4*b*c^9*x^(19*n)/n + 143/2*b^10*c^4*x^(18*n)/n + 1287*a*b^8*c^5*x^(18*n)/n + 6006*a^2*b^6*c
^6*x^(18*n)/n + 8580*a^3*b^4*c^7*x^(18*n)/n + 6435/2*a^4*b^2*c^8*x^(18*n)/n + 143*a^5*c^9*x^(18*n)/n + 26*b^11
*c^3*x^(17*n)/n + 715*a*b^9*c^4*x^(17*n)/n + 5148*a^2*b^7*c^5*x^(17*n)/n + 12012*a^3*b^5*c^6*x^(17*n)/n + 8580
*a^4*b^3*c^7*x^(17*n)/n + 1287*a^5*b*c^8*x^(17*n)/n + 13/2*b^12*c^2*x^(16*n)/n + 286*a*b^10*c^3*x^(16*n)/n + 6
435/2*a^2*b^8*c^4*x^(16*n)/n + 12012*a^3*b^6*c^5*x^(16*n)/n + 15015*a^4*b^4*c^6*x^(16*n)/n + 5148*a^5*b^2*c^7*
x^(16*n)/n + 429/2*a^6*c^8*x^(16*n)/n + b^13*c*x^(15*n)/n + 78*a*b^11*c^2*x^(15*n)/n + 1430*a^2*b^9*c^3*x^(15*
n)/n + 8580*a^3*b^7*c^4*x^(15*n)/n + 18018*a^4*b^5*c^5*x^(15*n)/n + 12012*a^5*b^3*c^6*x^(15*n)/n + 1716*a^6*b*
c^7*x^(15*n)/n + 1/14*b^14*x^(14*n)/n + 13*a*b^12*c*x^(14*n)/n + 429*a^2*b^10*c^2*x^(14*n)/n + 4290*a^3*b^8*c^
3*x^(14*n)/n + 15015*a^4*b^6*c^4*x^(14*n)/n + 18018*a^5*b^4*c^5*x^(14*n)/n + 6006*a^6*b^2*c^6*x^(14*n)/n + 171
6/7*a^7*c^7*x^(14*n)/n + a*b^13*x^(13*n)/n + 78*a^2*b^11*c*x^(13*n)/n + 1430*a^3*b^9*c^2*x^(13*n)/n + 8580*a^4
*b^7*c^3*x^(13*n)/n + 18018*a^5*b^5*c^4*x^(13*n)/n + 12012*a^6*b^3*c^5*x^(13*n)/n + 1716*a^7*b*c^6*x^(13*n)/n
+ 13/2*a^2*b^12*x^(12*n)/n + 286*a^3*b^10*c*x^(12*n)/n + 6435/2*a^4*b^8*c^2*x^(12*n)/n + 12012*a^5*b^6*c^3*x^(
12*n)/n + 15015*a^6*b^4*c^4*x^(12*n)/n + 5148*a^7*b^2*c^5*x^(12*n)/n + 429/2*a^8*c^6*x^(12*n)/n + 26*a^3*b^11*
x^(11*n)/n + 715*a^4*b^9*c*x^(11*n)/n + 5148*a^5*b^7*c^2*x^(11*n)/n + 12012*a^6*b^5*c^3*x^(11*n)/n + 8580*a^7*
b^3*c^4*x^(11*n)/n + 1287*a^8*b*c^5*x^(11*n)/n + 143/2*a^4*b^10*x^(10*n)/n + 1287*a^5*b^8*c*x^(10*n)/n + 6006*
a^6*b^6*c^2*x^(10*n)/n + 8580*a^7*b^4*c^3*x^(10*n)/n + 6435/2*a^8*b^2*c^4*x^(10*n)/n + 143*a^9*c^5*x^(10*n)/n
+ 143*a^5*b^9*x^(9*n)/n + 1716*a^6*b^7*c*x^(9*n)/n + 5148*a^7*b^5*c^2*x^(9*n)/n + 4290*a^8*b^3*c^3*x^(9*n)/n +
715*a^9*b*c^4*x^(9*n)/n + 429/2*a^6*b^8*x^(8*n)/n + 1716*a^7*b^6*c*x^(8*n)/n + 6435/2*a^8*b^4*c^2*x^(8*n)/n +
1430*a^9*b^2*c^3*x^(8*n)/n + 143/2*a^10*c^4*x^(8*n)/n + 1716/7*a^7*b^7*x^(7*n)/n + 1287*a^8*b^5*c*x^(7*n)/n +
1430*a^9*b^3*c^2*x^(7*n)/n + 286*a^10*b*c^3*x^(7*n)/n + 429/2*a^8*b^6*x^(6*n)/n + 715*a^9*b^4*c*x^(6*n)/n + 4
29*a^10*b^2*c^2*x^(6*n)/n + 26*a^11*c^3*x^(6*n)/n + 143*a^9*b^5*x^(5*n)/n + 286*a^10*b^3*c*x^(5*n)/n + 78*a^11
*b*c^2*x^(5*n)/n + 143/2*a^10*b^4*x^(4*n)/n + 78*a^11*b^2*c*x^(4*n)/n + 13/2*a^12*c^2*x^(4*n)/n + 26*a^11*b^3*
x^(3*n)/n + 13*a^12*b*c*x^(3*n)/n + 13/2*a^12*b^2*x^(2*n)/n + a^13*c*x^(2*n)/n + a^13*b*x^n/n
________________________________________________________________________________________
mupad [B] time = 5.78, size = 1395, normalized size = 60.65
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^(n - 1)*(b + 2*c*x^n)*(a + b*x^n + c*x^(2*n))^13,x)
[Out]
x^(n - 1)*((x^(11*n + 1)*((13*a^2*b^12)/2 + (429*a^8*c^6)/2 + 286*a^3*b^10*c + (6435*a^4*b^8*c^2)/2 + 12012*a^
5*b^6*c^3 + 15015*a^6*b^4*c^4 + 5148*a^7*b^2*c^5))/n + (x^(15*n + 1)*((429*a^6*c^8)/2 + (13*b^12*c^2)/2 + 286*
a*b^10*c^3 + (6435*a^2*b^8*c^4)/2 + 12012*a^3*b^6*c^5 + 15015*a^4*b^4*c^6 + 5148*a^5*b^2*c^7))/n + (x^(12*n +
1)*(a*b^13 + 78*a^2*b^11*c + 1716*a^7*b*c^6 + 1430*a^3*b^9*c^2 + 8580*a^4*b^7*c^3 + 18018*a^5*b^5*c^4 + 12012*
a^6*b^3*c^5))/n + (x^(14*n + 1)*(b^13*c + 78*a*b^11*c^2 + 1716*a^6*b*c^7 + 1430*a^2*b^9*c^3 + 8580*a^3*b^7*c^4
+ 18018*a^4*b^5*c^5 + 12012*a^5*b^3*c^6))/n + (x^(5*n + 1)*((429*a^8*b^6)/2 + 26*a^11*c^3 + 715*a^9*b^4*c + 4
29*a^10*b^2*c^2))/n + (x^(21*n + 1)*(26*a^3*c^11 + (429*b^6*c^8)/2 + 715*a*b^4*c^9 + 429*a^2*b^2*c^10))/n + (x
^(9*n + 1)*((143*a^4*b^10)/2 + 143*a^9*c^5 + 1287*a^5*b^8*c + 6006*a^6*b^6*c^2 + 8580*a^7*b^4*c^3 + (6435*a^8*
b^2*c^4)/2))/n + (x^(17*n + 1)*(143*a^5*c^9 + (143*b^10*c^4)/2 + 1287*a*b^8*c^5 + 6006*a^2*b^6*c^6 + 8580*a^3*
b^4*c^7 + (6435*a^4*b^2*c^8)/2))/n + (x^(13*n + 1)*(b^14/14 + (1716*a^7*c^7)/7 + 429*a^2*b^10*c^2 + 4290*a^3*b
^8*c^3 + 15015*a^4*b^6*c^4 + 18018*a^5*b^4*c^5 + 6006*a^6*b^2*c^6 + 13*a*b^12*c))/n + (x^(7*n + 1)*((429*a^6*b
^8)/2 + (143*a^10*c^4)/2 + 1716*a^7*b^6*c + (6435*a^8*b^4*c^2)/2 + 1430*a^9*b^2*c^3))/n + (x^(19*n + 1)*((143*
a^4*c^10)/2 + (429*b^8*c^6)/2 + 1716*a*b^6*c^7 + (6435*a^2*b^4*c^8)/2 + 1430*a^3*b^2*c^9))/n + (c^14*x^(27*n +
1))/(14*n) + (a^12*x^(n + 1)*(a*c + (13*b^2)/2))/n + (a^10*x^(3*n + 1)*((143*b^4)/2 + (13*a^2*c^2)/2 + 78*a*b
^2*c))/n + (c^10*x^(23*n + 1)*((143*b^4)/2 + (13*a^2*c^2)/2 + 78*a*b^2*c))/n + (b*c^13*x^(26*n + 1))/n + (c^12
*x^(25*n + 1)*(a*c + (13*b^2)/2))/n + (a^13*b*x)/n + (143*a^7*b*x^(6*n + 1)*(12*b^6 + 14*a^3*c^3 + 70*a^2*b^2*
c^2 + 63*a*b^4*c))/(7*n) + (143*b*c^7*x^(20*n + 1)*(12*b^6 + 14*a^3*c^3 + 70*a^2*b^2*c^2 + 63*a*b^4*c))/(7*n)
+ (143*a^5*b*x^(8*n + 1)*(b^8 + 5*a^4*c^4 + 36*a^2*b^4*c^2 + 30*a^3*b^2*c^3 + 12*a*b^6*c))/n + (143*b*c^5*x^(1
8*n + 1)*(b^8 + 5*a^4*c^4 + 36*a^2*b^4*c^2 + 30*a^3*b^2*c^3 + 12*a*b^6*c))/n + (13*a^3*b*x^(10*n + 1)*(2*b^10
+ 99*a^5*c^5 + 396*a^2*b^6*c^2 + 924*a^3*b^4*c^3 + 660*a^4*b^2*c^4 + 55*a*b^8*c))/n + (13*b*c^3*x^(16*n + 1)*(
2*b^10 + 99*a^5*c^5 + 396*a^2*b^6*c^2 + 924*a^3*b^4*c^3 + 660*a^4*b^2*c^4 + 55*a*b^8*c))/n + (13*a^9*b*x^(4*n
+ 1)*(11*b^4 + 6*a^2*c^2 + 22*a*b^2*c))/n + (13*b*c^9*x^(22*n + 1)*(11*b^4 + 6*a^2*c^2 + 22*a*b^2*c))/n + (13*
a^11*b*x^(2*n + 1)*(a*c + 2*b^2))/n + (13*b*c^11*x^(24*n + 1)*(a*c + 2*b^2))/n)
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**(-1+n)*(b+2*c*x**n)*(a+b*x**n+c*x**(2*n))**13,x)
[Out]
Timed out
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