### 3.100 $$\int x^{-1+n} (b+2 c x^n) (-a+b x^n+c x^{2 n})^{13} \, dx$$

Optimal. Leaf size=25 $\frac{\left (a-b x^n-c x^{2 n}\right )^{14}}{14 n}$

[Out]

(a - b*x^n - c*x^(2*n))^14/(14*n)

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Rubi [A]  time = 0.05999, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 32, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.062, Rules used = {1468, 629} $\frac{\left (a-b x^n-c x^{2 n}\right )^{14}}{14 n}$

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 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]

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]

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*}

Mathematica [A]  time = 0.0555686, size = 24, normalized size = 0.96 $\frac{\left (x^n \left (b+c x^n\right )-a\right )^{14}}{14 n}$

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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Maple [B]  time = 0.06, size = 2046, normalized size = 81.8 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1+n)*(b+2*c*x^n)*(-a+b*x^n+c*x^(2*n))^13,x)

[Out]

1716/7*b^7*c^7/n*(x^n)^21+143*b^5*c^9/n*(x^n)^23+26*b^3*c^11/n*(x^n)^25-26*a^11*b^3/n*(x^n)^3-143*b^5*a^9/n*(x
^n)^5-1716/7*b^7*a^7/n*(x^n)^7-143*a^5*b^9/n*(x^n)^9-26*b^11*a^3/n*(x^n)^11-a*b^13/n*(x^n)^13+b^13*c/n*(x^n)^1
5+26*b^11*c^3/n*(x^n)^17+143*b^9*c^5/n*(x^n)^19-a^13/n*(x^n)^2*c+13/2*a^12/n*(x^n)^2*b^2-1716/7/n*(x^n)^14*a^7
*c^7-c^13/n*(x^n)^26*a+13/2*c^12/n*(x^n)^26*b^2+143/2*c^10/n*(x^n)^24*b^4+429/2*c^6/n*(x^n)^20*b^8-26*c^11/n*(
x^n)^22*a^3+429/2*c^8/n*(x^n)^22*b^6+13/2*c^12/n*(x^n)^24*a^2-143*a^9/n*(x^n)^10*c^5+143/2*a^4/n*(x^n)^10*b^10
+429/2*a^8/n*(x^n)^12*c^6+13/2*a^2/n*(x^n)^12*b^12-143*c^9/n*(x^n)^18*a^5+143/2*c^4/n*(x^n)^18*b^10+429/2*c^8/
n*(x^n)^16*a^6+13/2*c^2/n*(x^n)^16*b^12+143/2*c^10/n*(x^n)^20*a^4+13/2*a^12/n*(x^n)^4*c^2+143/2*a^10/n*(x^n)^4
*b^4+143/2*a^10/n*(x^n)^8*c^4+429/2*a^6/n*(x^n)^8*b^8-26*a^11/n*(x^n)^6*c^3+429/2*a^8/n*(x^n)^6*b^6+b*c^13/n*(
x^n)^27-b*a^13/n*x^n+1/14/n*(x^n)^14*b^14-8580*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-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+6435/2*a^8/n*(x^n)^8*b^4*c^2-1716*a^7/n*(x^n)^8*b^6*c+429*a^10/n*(x^n)^6*b^2*c^2-715*a^9/n
*(x^n)^6*b^4*c+6435/2*a^8/n*(x^n)^10*b^2*c^4+78*b*c^11/n*(x^n)^23*a^2-286*b^3*c^10/n*(x^n)^23*a-13*b*c^12/n*(x
^n)^25*a+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-286*b*c^10/n*(x^n)^21*a^3+1430*b^3*c^9/n*(x^n)^21*a^2-1287*b^5*c^8/n*(x^n)^21*a-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+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+1716*b*c^7/n*(x^n)^15*a^6-1
2012*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-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+1287*b*a^8/n*(x^n)^11*c^5-8580*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*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^4-78*b*a^11/n*(x^n)^5*c^2+286*b^3*a^10/n*(x^n)^5*c+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+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+13*a^12*b/n*(x^n)^
3*c-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^4-78*c^11/n*(x^n)^24*a*b^2-286*a^3/n*(x^n)^12*b^10*c+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^8-5148*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+1/14*c^14/n*(x^n)^28-78*a^11/n*(x^n)^4*b^2*c-1430*a^9/n*(x^n)^8*b^2*c^3

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [B]  time = 1.35823, size = 2969, normalized size = 118.76 \begin{align*} \text{result too large to display} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Giac [B]  time = 1.2868, size = 2286, normalized size = 91.44 \begin{align*} \text{result too large to display} \end{align*}

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