3.96 \(\int x^{-1+n} \left (b+2 c x^n\right ) \left (a+b x^n+c x^{2 n}\right )^{13} \, dx\)

Optimal. Leaf size=23 \[ \frac{\left (a+b x^n+c x^{2 n}\right )^{14}}{14 n} \]

[Out]

(a + b*x^n + c*x^(2*n))^14/(14*n)

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Rubi [A]  time = 0.0690415, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{\left (a+b x^n+c x^{2 n}\right )^{14}}{14 n} \]

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)

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Rubi in Sympy [A]  time = 14.2302, size = 17, normalized size = 0.74 \[ \frac{\left (a + b x^{n} + c x^{2 n}\right )^{14}}{14 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(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)

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Mathematica [B]  time = 0.726732, size = 260, normalized size = 11.3 \[ \frac{x^n \left (b+c x^n\right ) \left (14 a^{13}+91 a^{12} x^n \left (b+c x^n\right )+364 a^{11} x^{2 n} \left (b+c x^n\right )^2+1001 a^{10} x^{3 n} \left (b+c x^n\right )^3+2002 a^9 x^{4 n} \left (b+c x^n\right )^4+3003 a^8 x^{5 n} \left (b+c x^n\right )^5+3432 a^7 x^{6 n} \left (b+c x^n\right )^6+3003 a^6 x^{7 n} \left (b+c x^n\right )^7+2002 a^5 x^{8 n} \left (b+c x^n\right )^8+1001 a^4 x^{9 n} \left (b+c x^n\right )^9+364 a^3 x^{10 n} \left (b+c x^n\right )^{10}+91 a^2 x^{11 n} \left (b+c x^n\right )^{11}+14 a x^{12 n} \left (b+c x^n\right )^{12}+x^{13 n} \left (b+c x^n\right )^{13}\right )}{14 n} \]

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]

(x^n*(b + c*x^n)*(14*a^13 + 91*a^12*x^n*(b + c*x^n) + 364*a^11*x^(2*n)*(b + c*x^
n)^2 + 1001*a^10*x^(3*n)*(b + c*x^n)^3 + 2002*a^9*x^(4*n)*(b + c*x^n)^4 + 3003*a
^8*x^(5*n)*(b + c*x^n)^5 + 3432*a^7*x^(6*n)*(b + c*x^n)^6 + 3003*a^6*x^(7*n)*(b
+ c*x^n)^7 + 2002*a^5*x^(8*n)*(b + c*x^n)^8 + 1001*a^4*x^(9*n)*(b + c*x^n)^9 + 3
64*a^3*x^(10*n)*(b + c*x^n)^10 + 91*a^2*x^(11*n)*(b + c*x^n)^11 + 14*a*x^(12*n)*
(b + c*x^n)^12 + x^(13*n)*(b + c*x^n)^13))/(14*n)

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Maple [B]  time = 0.099, size = 2042, normalized size = 88.8 \[ \text{result too large to display} \]

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]

143*b^5*a^9/n*(x^n)^5+b^13*c/n*(x^n)^15+26*b^11*a^3/n*(x^n)^11+26*a^11*b^3/n*(x^
n)^3+143*a^5*b^9/n*(x^n)^9+143*b^5*c^9/n*(x^n)^23+26*c^11*b^3/n*(x^n)^25+a*b^13/
n*(x^n)^13+b*a^13/n*x^n+b*c^13/n*(x^n)^27+143*c^9/n*(x^n)^18*a^5+143/2*c^4/n*(x^
n)^18*b^10+26*c^11/n*(x^n)^22*a^3+429/2*c^8/n*(x^n)^22*b^6+26*b^11*c^3/n*(x^n)^1
7+c^13/n*(x^n)^26*a+13/2*c^12/n*(x^n)^26*b^2+1716/7*b^7*a^7/n*(x^n)^7+143*c^5*b^
9/n*(x^n)^19+1716/7*b^7*c^7/n*(x^n)^21+13/2*c^12/n*(x^n)^24*a^2+143/2*c^10/n*(x^
n)^24*b^4+143*a^9/n*(x^n)^10*c^5+143/2*a^4/n*(x^n)^10*b^10+a^13/n*(x^n)^2*c+13/2
*a^12/n*(x^n)^2*b^2+6006/n*(x^n)^14*a^6*b^2*c^6+18018/n*(x^n)^14*a^5*b^4*c^5+150
15/n*(x^n)^14*b^6*a^4*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+1/14*c^14/n*(x^n)^28+13*a^12*b/n*(x^n)^3*c+1287*a^5/n*(
x^n)^10*b^8*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+715*c^9*b/n*(x^n)^19*a^4+4290*c^8*b^3/n*(x^n)^19*a^3+5148*c^7*b^5/n*(
x^n)^19*a^2+1716*c^6*b^7/n*(x^n)^19*a+78*c^11/n*(x^n)^24*a*b^2+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+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+143/2*c^10/n*(x^n)^20*a^4+429/2*c^6/n*(x^n)^20*b^8+429/2*c^8/n*(x^n)^16*a^6+
13/2*c^2/n*(x^n)^16*b^12+429/2*a^8/n*(x^n)^12*c^6+13/2*a^2/n*(x^n)^12*b^12+13/2*
a^12/n*(x^n)^4*c^2+143/2*a^10/n*(x^n)^4*b^4+1716/7/n*(x^n)^14*a^7*c^7+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+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*b^
2*a^4+8580*c^7/n*(x^n)^18*b^4*a^3+6006*c^6/n*(x^n)^18*b^6*a^2+1287*c^5/n*(x^n)^1
8*a*b^8+429*c^10/n*(x^n)^22*a^2*b^2+715*c^9/n*(x^n)^22*a*b^4+1/14/n*(x^n)^14*b^1
4+13*c^12*b/n*(x^n)^25*a+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+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+429*a^10/n*(x^n)^6*b^2*c^2+715*a^9/n*(x^n)^6*b^4*
c+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)^1
7*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)^1
2*b^6*c^3+6435/2*a^4/n*(x^n)^12*b^8*c^2+286*a^3/n*(x^n)^12*b^10*c+78*a^11/n*(x^n
)^4*b^2*c+78*b*a^11/n*(x^n)^5*c^2+286*b^3*a^10/n*(x^n)^5*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+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+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+78*b*c^11/n*(x^n)
^23*a^2+286*b^3*c^10/n*(x^n)^23*a+1287*b*c^8/n*(x^n)^17*a^5+8580*b^3*c^7/n*(x^n)
^17*a^4+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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^(2*n) + b*x^n + a)^13*(2*c*x^n + b)*x^(n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.333561, size = 1751, normalized size = 76.13 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^(2*n) + b*x^n + a)^13*(2*c*x^n + b)*x^(n - 1),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*(12*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^(1
4*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 + 792*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 + 660*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^(1
0*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. \[ \text{Timed out} \]

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/XCAS [A]  time = 0.325759, size = 1, normalized size = 0.04 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^(2*n) + b*x^n + a)^13*(2*c*x^n + b)*x^(n - 1),x, algorithm="giac")

[Out]

Done