3.2591 \(\int x^{-1+n} (a+b x^n)^{16} \, dx\)

Optimal. Leaf size=19 \[ \frac {\left (a+b x^n\right )^{17}}{17 b n} \]

[Out]

1/17*(a+b*x^n)^17/b/n

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Rubi [A]  time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {261} \[ \frac {\left (a+b x^n\right )^{17}}{17 b n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + n)*(a + b*x^n)^16,x]

[Out]

(a + b*x^n)^17/(17*b*n)

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin {align*} \int x^{-1+n} \left (a+b x^n\right )^{16} \, dx &=\frac {\left (a+b x^n\right )^{17}}{17 b n}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 19, normalized size = 1.00 \[ \frac {\left (a+b x^n\right )^{17}}{17 b n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 + n)*(a + b*x^n)^16,x]

[Out]

(a + b*x^n)^17/(17*b*n)

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fricas [B]  time = 0.70, size = 214, normalized size = 11.26 \[ \frac {b^{16} x^{17 \, n} + 17 \, a b^{15} x^{16 \, n} + 136 \, a^{2} b^{14} x^{15 \, n} + 680 \, a^{3} b^{13} x^{14 \, n} + 2380 \, a^{4} b^{12} x^{13 \, n} + 6188 \, a^{5} b^{11} x^{12 \, n} + 12376 \, a^{6} b^{10} x^{11 \, n} + 19448 \, a^{7} b^{9} x^{10 \, n} + 24310 \, a^{8} b^{8} x^{9 \, n} + 24310 \, a^{9} b^{7} x^{8 \, n} + 19448 \, a^{10} b^{6} x^{7 \, n} + 12376 \, a^{11} b^{5} x^{6 \, n} + 6188 \, a^{12} b^{4} x^{5 \, n} + 2380 \, a^{13} b^{3} x^{4 \, n} + 680 \, a^{14} b^{2} x^{3 \, n} + 136 \, a^{15} b x^{2 \, n} + 17 \, a^{16} x^{n}}{17 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)*(a+b*x^n)^16,x, algorithm="fricas")

[Out]

1/17*(b^16*x^(17*n) + 17*a*b^15*x^(16*n) + 136*a^2*b^14*x^(15*n) + 680*a^3*b^13*x^(14*n) + 2380*a^4*b^12*x^(13
*n) + 6188*a^5*b^11*x^(12*n) + 12376*a^6*b^10*x^(11*n) + 19448*a^7*b^9*x^(10*n) + 24310*a^8*b^8*x^(9*n) + 2431
0*a^9*b^7*x^(8*n) + 19448*a^10*b^6*x^(7*n) + 12376*a^11*b^5*x^(6*n) + 6188*a^12*b^4*x^(5*n) + 2380*a^13*b^3*x^
(4*n) + 680*a^14*b^2*x^(3*n) + 136*a^15*b*x^(2*n) + 17*a^16*x^n)/n

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giac [B]  time = 0.18, size = 214, normalized size = 11.26 \[ \frac {b^{16} x^{17 \, n} + 17 \, a b^{15} x^{16 \, n} + 136 \, a^{2} b^{14} x^{15 \, n} + 680 \, a^{3} b^{13} x^{14 \, n} + 2380 \, a^{4} b^{12} x^{13 \, n} + 6188 \, a^{5} b^{11} x^{12 \, n} + 12376 \, a^{6} b^{10} x^{11 \, n} + 19448 \, a^{7} b^{9} x^{10 \, n} + 24310 \, a^{8} b^{8} x^{9 \, n} + 24310 \, a^{9} b^{7} x^{8 \, n} + 19448 \, a^{10} b^{6} x^{7 \, n} + 12376 \, a^{11} b^{5} x^{6 \, n} + 6188 \, a^{12} b^{4} x^{5 \, n} + 2380 \, a^{13} b^{3} x^{4 \, n} + 680 \, a^{14} b^{2} x^{3 \, n} + 136 \, a^{15} b x^{2 \, n} + 17 \, a^{16} x^{n}}{17 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)*(a+b*x^n)^16,x, algorithm="giac")

[Out]

1/17*(b^16*x^(17*n) + 17*a*b^15*x^(16*n) + 136*a^2*b^14*x^(15*n) + 680*a^3*b^13*x^(14*n) + 2380*a^4*b^12*x^(13
*n) + 6188*a^5*b^11*x^(12*n) + 12376*a^6*b^10*x^(11*n) + 19448*a^7*b^9*x^(10*n) + 24310*a^8*b^8*x^(9*n) + 2431
0*a^9*b^7*x^(8*n) + 19448*a^10*b^6*x^(7*n) + 12376*a^11*b^5*x^(6*n) + 6188*a^12*b^4*x^(5*n) + 2380*a^13*b^3*x^
(4*n) + 680*a^14*b^2*x^(3*n) + 136*a^15*b*x^(2*n) + 17*a^16*x^n)/n

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maple [B]  time = 0.04, size = 260, normalized size = 13.68 \[ \frac {a^{16} x^{n}}{n}+\frac {8 a^{15} b \,x^{2 n}}{n}+\frac {40 a^{14} b^{2} x^{3 n}}{n}+\frac {140 a^{13} b^{3} x^{4 n}}{n}+\frac {364 a^{12} b^{4} x^{5 n}}{n}+\frac {728 a^{11} b^{5} x^{6 n}}{n}+\frac {1144 a^{10} b^{6} x^{7 n}}{n}+\frac {1430 a^{9} b^{7} x^{8 n}}{n}+\frac {1430 a^{8} b^{8} x^{9 n}}{n}+\frac {1144 a^{7} b^{9} x^{10 n}}{n}+\frac {728 a^{6} b^{10} x^{11 n}}{n}+\frac {364 a^{5} b^{11} x^{12 n}}{n}+\frac {140 a^{4} b^{12} x^{13 n}}{n}+\frac {40 a^{3} b^{13} x^{14 n}}{n}+\frac {8 a^{2} b^{14} x^{15 n}}{n}+\frac {a \,b^{15} x^{16 n}}{n}+\frac {b^{16} x^{17 n}}{17 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(n-1)*(b*x^n+a)^16,x)

[Out]

1/17*b^16/n*(x^n)^17+a*b^15/n*(x^n)^16+8*a^2*b^14/n*(x^n)^15+40*a^3*b^13/n*(x^n)^14+140*a^4*b^12/n*(x^n)^13+36
4*a^5*b^11/n*(x^n)^12+728*a^6*b^10/n*(x^n)^11+1144*a^7*b^9/n*(x^n)^10+1430*a^8*b^8/n*(x^n)^9+1430*a^9*b^7/n*(x
^n)^8+1144*a^10*b^6/n*(x^n)^7+728*a^11*b^5/n*(x^n)^6+364*a^12*b^4/n*(x^n)^5+140*a^13*b^3/n*(x^n)^4+40*a^14*b^2
/n*(x^n)^3+8*a^15*b/n*(x^n)^2+a^16/n*x^n

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maxima [A]  time = 0.55, size = 17, normalized size = 0.89 \[ \frac {{\left (b x^{n} + a\right )}^{17}}{17 \, b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)*(a+b*x^n)^16,x, algorithm="maxima")

[Out]

1/17*(b*x^n + a)^17/(b*n)

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mupad [B]  time = 1.87, size = 259, normalized size = 13.63 \[ \frac {a^{16}\,x^n}{n}+\frac {b^{16}\,x^{17\,n}}{17\,n}+\frac {40\,a^{14}\,b^2\,x^{3\,n}}{n}+\frac {140\,a^{13}\,b^3\,x^{4\,n}}{n}+\frac {364\,a^{12}\,b^4\,x^{5\,n}}{n}+\frac {728\,a^{11}\,b^5\,x^{6\,n}}{n}+\frac {1144\,a^{10}\,b^6\,x^{7\,n}}{n}+\frac {1430\,a^9\,b^7\,x^{8\,n}}{n}+\frac {1430\,a^8\,b^8\,x^{9\,n}}{n}+\frac {1144\,a^7\,b^9\,x^{10\,n}}{n}+\frac {728\,a^6\,b^{10}\,x^{11\,n}}{n}+\frac {364\,a^5\,b^{11}\,x^{12\,n}}{n}+\frac {140\,a^4\,b^{12}\,x^{13\,n}}{n}+\frac {40\,a^3\,b^{13}\,x^{14\,n}}{n}+\frac {8\,a^2\,b^{14}\,x^{15\,n}}{n}+\frac {8\,a^{15}\,b\,x^{2\,n}}{n}+\frac {a\,b^{15}\,x^{16\,n}}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(n - 1)*(a + b*x^n)^16,x)

[Out]

(a^16*x^n)/n + (b^16*x^(17*n))/(17*n) + (40*a^14*b^2*x^(3*n))/n + (140*a^13*b^3*x^(4*n))/n + (364*a^12*b^4*x^(
5*n))/n + (728*a^11*b^5*x^(6*n))/n + (1144*a^10*b^6*x^(7*n))/n + (1430*a^9*b^7*x^(8*n))/n + (1430*a^8*b^8*x^(9
*n))/n + (1144*a^7*b^9*x^(10*n))/n + (728*a^6*b^10*x^(11*n))/n + (364*a^5*b^11*x^(12*n))/n + (140*a^4*b^12*x^(
13*n))/n + (40*a^3*b^13*x^(14*n))/n + (8*a^2*b^14*x^(15*n))/n + (8*a^15*b*x^(2*n))/n + (a*b^15*x^(16*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**(-1+n)*(a+b*x**n)**16,x)

[Out]

Timed out

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