3.163 \(\int x^{14 (-1+n)} (b+2 c x^n) (b x+c x^{1+n})^{13} \, dx\)

Optimal. Leaf size=21 \[ \frac {x^{14 n} \left (b+c x^n\right )^{14}}{14 n} \]

[Out]

1/14*x^(14*n)*(b+c*x^n)^14/n

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Rubi [A]  time = 0.03, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {1584, 446, 74} \[ \frac {x^{14 n} \left (b+c x^n\right )^{14}}{14 n} \]

Antiderivative was successfully verified.

[In]

Int[x^(14*(-1 + n))*(b + 2*c*x^n)*(b*x + c*x^(1 + n))^13,x]

[Out]

(x^(14*n)*(b + c*x^n)^14)/(14*n)

Rule 74

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps

\begin {align*} \int x^{14 (-1+n)} \left (b+2 c x^n\right ) \left (b x+c x^{1+n}\right )^{13} \, dx &=\int x^{13+14 (-1+n)} \left (b+c x^n\right )^{13} \left (b+2 c x^n\right ) \, dx\\ &=\frac {\operatorname {Subst}\left (\int x^{13} (b+c x)^{13} (b+2 c x) \, dx,x,x^n\right )}{n}\\ &=\frac {x^{14 n} \left (b+c x^n\right )^{14}}{14 n}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 21, normalized size = 1.00 \[ \frac {x^{14 n} \left (b+c x^n\right )^{14}}{14 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(14*(-1 + n))*(b + 2*c*x^n)*(b*x + c*x^(1 + n))^13,x]

[Out]

(x^(14*n)*(b + c*x^n)^14)/(14*n)

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fricas [B]  time = 1.04, size = 262, normalized size = 12.48 \[ \frac {b^{14} x^{14} x^{14 \, n + 14} + 14 \, b^{13} c x^{13} x^{15 \, n + 15} + 91 \, b^{12} c^{2} x^{12} x^{16 \, n + 16} + 364 \, b^{11} c^{3} x^{11} x^{17 \, n + 17} + 1001 \, b^{10} c^{4} x^{10} x^{18 \, n + 18} + 2002 \, b^{9} c^{5} x^{9} x^{19 \, n + 19} + 3003 \, b^{8} c^{6} x^{8} x^{20 \, n + 20} + 3432 \, b^{7} c^{7} x^{7} x^{21 \, n + 21} + 3003 \, b^{6} c^{8} x^{6} x^{22 \, n + 22} + 2002 \, b^{5} c^{9} x^{5} x^{23 \, n + 23} + 1001 \, b^{4} c^{10} x^{4} x^{24 \, n + 24} + 364 \, b^{3} c^{11} x^{3} x^{25 \, n + 25} + 91 \, b^{2} c^{12} x^{2} x^{26 \, n + 26} + 14 \, b c^{13} x x^{27 \, n + 27} + c^{14} x^{28 \, n + 28}}{14 \, n x^{28}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-14+14*n)*(b+2*c*x^n)*(b*x+c*x^(1+n))^13,x, algorithm="fricas")

[Out]

1/14*(b^14*x^14*x^(14*n + 14) + 14*b^13*c*x^13*x^(15*n + 15) + 91*b^12*c^2*x^12*x^(16*n + 16) + 364*b^11*c^3*x
^11*x^(17*n + 17) + 1001*b^10*c^4*x^10*x^(18*n + 18) + 2002*b^9*c^5*x^9*x^(19*n + 19) + 3003*b^8*c^6*x^8*x^(20
*n + 20) + 3432*b^7*c^7*x^7*x^(21*n + 21) + 3003*b^6*c^8*x^6*x^(22*n + 22) + 2002*b^5*c^9*x^5*x^(23*n + 23) +
1001*b^4*c^10*x^4*x^(24*n + 24) + 364*b^3*c^11*x^3*x^(25*n + 25) + 91*b^2*c^12*x^2*x^(26*n + 26) + 14*b*c^13*x
*x^(27*n + 27) + c^14*x^(28*n + 28))/(n*x^28)

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giac [B]  time = 1.86, size = 189, normalized size = 9.00 \[ \frac {c^{14} x^{28 \, n} + 14 \, b c^{13} x^{27 \, n} + 91 \, b^{2} c^{12} x^{26 \, n} + 364 \, b^{3} c^{11} x^{25 \, n} + 1001 \, b^{4} c^{10} x^{24 \, n} + 2002 \, b^{5} c^{9} x^{23 \, n} + 3003 \, b^{6} c^{8} x^{22 \, n} + 3432 \, b^{7} c^{7} x^{21 \, n} + 3003 \, b^{8} c^{6} x^{20 \, n} + 2002 \, b^{9} c^{5} x^{19 \, n} + 1001 \, b^{10} c^{4} x^{18 \, n} + 364 \, b^{11} c^{3} x^{17 \, n} + 91 \, b^{12} c^{2} x^{16 \, n} + 14 \, b^{13} c x^{15 \, n} + b^{14} x^{14 \, n}}{14 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-14+14*n)*(b+2*c*x^n)*(b*x+c*x^(1+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) + 364*b^3*c^11*x^(25*n) + 1001*b^4*c^10*x^(24*
n) + 2002*b^5*c^9*x^(23*n) + 3003*b^6*c^8*x^(22*n) + 3432*b^7*c^7*x^(21*n) + 3003*b^8*c^6*x^(20*n) + 2002*b^9*
c^5*x^(19*n) + 1001*b^10*c^4*x^(18*n) + 364*b^11*c^3*x^(17*n) + 91*b^12*c^2*x^(16*n) + 14*b^13*c*x^(15*n) + b^
14*x^(14*n))/n

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maple [B]  time = 0.05, size = 230, normalized size = 10.95 \[ \frac {b^{14} x^{14 n}}{14 n}+\frac {b^{13} c \,x^{15 n}}{n}+\frac {13 b^{12} c^{2} x^{16 n}}{2 n}+\frac {26 b^{11} c^{3} x^{17 n}}{n}+\frac {143 b^{10} c^{4} x^{18 n}}{2 n}+\frac {143 b^{9} c^{5} x^{19 n}}{n}+\frac {429 b^{8} c^{6} x^{20 n}}{2 n}+\frac {1716 b^{7} c^{7} x^{21 n}}{7 n}+\frac {429 b^{6} c^{8} x^{22 n}}{2 n}+\frac {143 b^{5} c^{9} x^{23 n}}{n}+\frac {143 b^{4} c^{10} x^{24 n}}{2 n}+\frac {26 b^{3} c^{11} x^{25 n}}{n}+\frac {13 b^{2} c^{12} x^{26 n}}{2 n}+\frac {b \,c^{13} x^{27 n}}{n}+\frac {c^{14} x^{28 n}}{14 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-14+14*n)*(b+2*c*x^n)*(b*x+c*x^(1+n))^13,x)

[Out]

1/14*c^14/n*(x^n)^28+b*c^13/n*(x^n)^27+13/2*b^2*c^12/n*(x^n)^26+26*b^3*c^11/n*(x^n)^25+143/2*b^4*c^10/n*(x^n)^
24+143*b^5*c^9/n*(x^n)^23+429/2*b^6*c^8/n*(x^n)^22+1716/7*b^7*c^7/n*(x^n)^21+429/2*b^8*c^6/n*(x^n)^20+143*b^9*
c^5/n*(x^n)^19+143/2*b^10*c^4/n*(x^n)^18+26*b^11*c^3/n*(x^n)^17+13/2*b^12*c^2/n*(x^n)^16+b^13*c/n*(x^n)^15+1/1
4*b^14/n*(x^n)^14

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maxima [B]  time = 0.65, size = 229, normalized size = 10.90 \[ \frac {c^{14} x^{28 \, n}}{14 \, n} + \frac {b c^{13} x^{27 \, n}}{n} + \frac {13 \, b^{2} c^{12} x^{26 \, n}}{2 \, n} + \frac {26 \, b^{3} c^{11} x^{25 \, n}}{n} + \frac {143 \, b^{4} c^{10} x^{24 \, n}}{2 \, n} + \frac {143 \, b^{5} c^{9} x^{23 \, n}}{n} + \frac {429 \, b^{6} c^{8} x^{22 \, n}}{2 \, n} + \frac {1716 \, b^{7} c^{7} x^{21 \, n}}{7 \, n} + \frac {429 \, b^{8} c^{6} x^{20 \, n}}{2 \, n} + \frac {143 \, b^{9} c^{5} x^{19 \, n}}{n} + \frac {143 \, b^{10} c^{4} x^{18 \, n}}{2 \, n} + \frac {26 \, b^{11} c^{3} x^{17 \, n}}{n} + \frac {13 \, b^{12} c^{2} x^{16 \, n}}{2 \, n} + \frac {b^{13} c x^{15 \, n}}{n} + \frac {b^{14} x^{14 \, n}}{14 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-14+14*n)*(b+2*c*x^n)*(b*x+c*x^(1+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 + 26*b^3*c^11*x^(25*n)/n + 143/2*b^4*c^10*
x^(24*n)/n + 143*b^5*c^9*x^(23*n)/n + 429/2*b^6*c^8*x^(22*n)/n + 1716/7*b^7*c^7*x^(21*n)/n + 429/2*b^8*c^6*x^(
20*n)/n + 143*b^9*c^5*x^(19*n)/n + 143/2*b^10*c^4*x^(18*n)/n + 26*b^11*c^3*x^(17*n)/n + 13/2*b^12*c^2*x^(16*n)
/n + b^13*c*x^(15*n)/n + 1/14*b^14*x^(14*n)/n

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mupad [B]  time = 4.02, size = 229, normalized size = 10.90 \[ \frac {b^{14}\,x^{14\,n}}{14\,n}+\frac {c^{14}\,x^{28\,n}}{14\,n}+\frac {13\,b^{12}\,c^2\,x^{16\,n}}{2\,n}+\frac {26\,b^{11}\,c^3\,x^{17\,n}}{n}+\frac {143\,b^{10}\,c^4\,x^{18\,n}}{2\,n}+\frac {143\,b^9\,c^5\,x^{19\,n}}{n}+\frac {429\,b^8\,c^6\,x^{20\,n}}{2\,n}+\frac {1716\,b^7\,c^7\,x^{21\,n}}{7\,n}+\frac {429\,b^6\,c^8\,x^{22\,n}}{2\,n}+\frac {143\,b^5\,c^9\,x^{23\,n}}{n}+\frac {143\,b^4\,c^{10}\,x^{24\,n}}{2\,n}+\frac {26\,b^3\,c^{11}\,x^{25\,n}}{n}+\frac {13\,b^2\,c^{12}\,x^{26\,n}}{2\,n}+\frac {b^{13}\,c\,x^{15\,n}}{n}+\frac {b\,c^{13}\,x^{27\,n}}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(14*n - 14)*(b*x + c*x^(n + 1))^13*(b + 2*c*x^n),x)

[Out]

(b^14*x^(14*n))/(14*n) + (c^14*x^(28*n))/(14*n) + (13*b^12*c^2*x^(16*n))/(2*n) + (26*b^11*c^3*x^(17*n))/n + (1
43*b^10*c^4*x^(18*n))/(2*n) + (143*b^9*c^5*x^(19*n))/n + (429*b^8*c^6*x^(20*n))/(2*n) + (1716*b^7*c^7*x^(21*n)
)/(7*n) + (429*b^6*c^8*x^(22*n))/(2*n) + (143*b^5*c^9*x^(23*n))/n + (143*b^4*c^10*x^(24*n))/(2*n) + (26*b^3*c^
11*x^(25*n))/n + (13*b^2*c^12*x^(26*n))/(2*n) + (b^13*c*x^(15*n))/n + (b*c^13*x^(27*n))/n

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-14+14*n)*(b+2*c*x**n)*(b*x+c*x**(1+n))**13,x)

[Out]

Timed out

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