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]

(a + b*x^n)^17/(17*b*n)

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Rubi [A]  time = 0.0044505, antiderivative size = 19, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0066572, size = 19, normalized size = 1. \[ \frac{\left (a+b x^n\right )^{17}}{17 b n} \]

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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Maple [B]  time = 0.031, size = 260, normalized size = 13.7 \begin{align*}{\frac{{b}^{16} \left ({x}^{n} \right ) ^{17}}{17\,n}}+{\frac{{b}^{15}a \left ({x}^{n} \right ) ^{16}}{n}}+8\,{\frac{{a}^{2}{b}^{14} \left ({x}^{n} \right ) ^{15}}{n}}+40\,{\frac{{a}^{3}{b}^{13} \left ({x}^{n} \right ) ^{14}}{n}}+140\,{\frac{{a}^{4}{b}^{12} \left ({x}^{n} \right ) ^{13}}{n}}+364\,{\frac{{a}^{5}{b}^{11} \left ({x}^{n} \right ) ^{12}}{n}}+728\,{\frac{{a}^{6}{b}^{10} \left ({x}^{n} \right ) ^{11}}{n}}+1144\,{\frac{{b}^{9}{a}^{7} \left ({x}^{n} \right ) ^{10}}{n}}+1430\,{\frac{{a}^{8}{b}^{8} \left ({x}^{n} \right ) ^{9}}{n}}+1430\,{\frac{{a}^{9}{b}^{7} \left ({x}^{n} \right ) ^{8}}{n}}+1144\,{\frac{{a}^{10}{b}^{6} \left ({x}^{n} \right ) ^{7}}{n}}+728\,{\frac{{a}^{11}{b}^{5} \left ({x}^{n} \right ) ^{6}}{n}}+364\,{\frac{{a}^{12}{b}^{4} \left ({x}^{n} \right ) ^{5}}{n}}+140\,{\frac{{a}^{13}{b}^{3} \left ({x}^{n} \right ) ^{4}}{n}}+40\,{\frac{{a}^{14}{b}^{2} \left ({x}^{n} \right ) ^{3}}{n}}+8\,{\frac{{a}^{15}b \left ({x}^{n} \right ) ^{2}}{n}}+{\frac{{a}^{16}{x}^{n}}{n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/17*b^16/n*(x^n)^17+b^15*a/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*b^9*a^7/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 = 0.971877, size = 23, normalized size = 1.21 \begin{align*} \frac{{\left (b x^{n} + a\right )}^{17}}{17 \, b n} \end{align*}

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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Fricas [B]  time = 1.09755, size = 537, normalized size = 28.26 \begin{align*} \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} \end{align*}

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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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)*(a+b*x**n)**16,x)

[Out]

Timed out

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Giac [B]  time = 1.21411, size = 289, normalized size = 15.21 \begin{align*} \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} \end{align*}

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