\(\int x^{-1+n} (a+b x^n)^5 \, dx\) [2554]

Optimal result

Integrand size = 15, antiderivative size = 19 \[ \int x^{-1+n} \left (a+b x^n\right )^5 \, dx=\frac {\left (a+b x^n\right )^6}{6 b n} \]

[Out]

1/6*(a+b*x^n)^6/b/n

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {267} \[ \int x^{-1+n} \left (a+b x^n\right )^5 \, dx=\frac {\left (a+b x^n\right )^6}{6 b n} \]

[In]

Int[x^(-1 + n)*(a + b*x^n)^5,x]

[Out]

(a + b*x^n)^6/(6*b*n)

Rule 267

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*} \text {integral}& = \frac {\left (a+b x^n\right )^6}{6 b n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int x^{-1+n} \left (a+b x^n\right )^5 \, dx=\frac {\left (a+b x^n\right )^6}{6 b n} \]

[In]

Integrate[x^(-1 + n)*(a + b*x^n)^5,x]

[Out]

(a + b*x^n)^6/(6*b*n)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(83\) vs. \(2(17)=34\).

Time = 4.41 (sec) , antiderivative size = 84, normalized size of antiderivative = 4.42

method	result	size
risch	\(\frac {b^{5} x^{6 n}}{6 n}+\frac {a \,b^{4} x^{5 n}}{n}+\frac {5 a^{2} b^{3} x^{4 n}}{2 n}+\frac {10 a^{3} b^{2} x^{3 n}}{3 n}+\frac {5 a^{4} b \,x^{2 n}}{2 n}+\frac {a^{5} x^{n}}{n}\)	\(84\)
parallelrisch	\(\frac {x \,x^{5 n} x^{-1+n} b^{5}+6 x \,x^{4 n} x^{-1+n} a \,b^{4}+15 x \,x^{3 n} x^{-1+n} a^{2} b^{3}+20 x \,x^{2 n} x^{-1+n} a^{3} b^{2}+15 x \,x^{n} x^{-1+n} a^{4} b +6 x \,x^{-1+n} a^{5}}{6 n}\)	\(103\)

[In]

int(x^(-1+n)*(a+b*x^n)^5,x,method=_RETURNVERBOSE)

[Out]

1/6*b^5/n*(x^n)^6+a*b^4/n*(x^n)^5+5/2*a^2*b^3/n*(x^n)^4+10/3*a^3*b^2/n*(x^n)^3+5/2*a^4*b/n*(x^n)^2+a^5/n*x^n

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (17) = 34\).

Time = 0.27 (sec) , antiderivative size = 71, normalized size of antiderivative = 3.74 \[ \int x^{-1+n} \left (a+b x^n\right )^5 \, dx=\frac {b^{5} x^{6 \, n} + 6 \, a b^{4} x^{5 \, n} + 15 \, a^{2} b^{3} x^{4 \, n} + 20 \, a^{3} b^{2} x^{3 \, n} + 15 \, a^{4} b x^{2 \, n} + 6 \, a^{5} x^{n}}{6 \, n} \]

[In]

integrate(x^(-1+n)*(a+b*x^n)^5,x, algorithm="fricas")

[Out]

1/6*(b^5*x^(6*n) + 6*a*b^4*x^(5*n) + 15*a^2*b^3*x^(4*n) + 20*a^3*b^2*x^(3*n) + 15*a^4*b*x^(2*n) + 6*a^5*x^n)/n

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (12) = 24\).

Time = 0.75 (sec) , antiderivative size = 124, normalized size of antiderivative = 6.53 \[ \int x^{-1+n} \left (a+b x^n\right )^5 \, dx=\begin {cases} \frac {a^{5} x x^{n - 1}}{n} + \frac {5 a^{4} b x x^{n} x^{n - 1}}{2 n} + \frac {10 a^{3} b^{2} x x^{2 n} x^{n - 1}}{3 n} + \frac {5 a^{2} b^{3} x x^{3 n} x^{n - 1}}{2 n} + \frac {a b^{4} x x^{4 n} x^{n - 1}}{n} + \frac {b^{5} x x^{5 n} x^{n - 1}}{6 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{5} \log {\left (x \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(x**(-1+n)*(a+b*x**n)**5,x)

[Out]

Piecewise((a**5*x*x**(n - 1)/n + 5*a**4*b*x*x**n*x**(n - 1)/(2*n) + 10*a**3*b**2*x*x**(2*n)*x**(n - 1)/(3*n) +
5*a**2*b**3*x*x**(3*n)*x**(n - 1)/(2*n) + a*b**4*x*x**(4*n)*x**(n - 1)/n + b**5*x*x**(5*n)*x**(n - 1)/(6*n),
Ne(n, 0)), ((a + b)**5*log(x), True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int x^{-1+n} \left (a+b x^n\right )^5 \, dx=\frac {{\left (b x^{n} + a\right )}^{6}}{6 \, b n} \]

[In]

integrate(x^(-1+n)*(a+b*x^n)^5,x, algorithm="maxima")

[Out]

1/6*(b*x^n + a)^6/(b*n)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (17) = 34\).

Time = 0.28 (sec) , antiderivative size = 71, normalized size of antiderivative = 3.74 \[ \int x^{-1+n} \left (a+b x^n\right )^5 \, dx=\frac {b^{5} x^{6 \, n} + 6 \, a b^{4} x^{5 \, n} + 15 \, a^{2} b^{3} x^{4 \, n} + 20 \, a^{3} b^{2} x^{3 \, n} + 15 \, a^{4} b x^{2 \, n} + 6 \, a^{5} x^{n}}{6 \, n} \]

[In]

integrate(x^(-1+n)*(a+b*x^n)^5,x, algorithm="giac")

[Out]

1/6*(b^5*x^(6*n) + 6*a*b^4*x^(5*n) + 15*a^2*b^3*x^(4*n) + 20*a^3*b^2*x^(3*n) + 15*a^4*b*x^(2*n) + 6*a^5*x^n)/n

Mupad [B] (verification not implemented)

Time = 5.76 (sec) , antiderivative size = 83, normalized size of antiderivative = 4.37 \[ \int x^{-1+n} \left (a+b x^n\right )^5 \, dx=\frac {a^5\,x^n}{n}+\frac {b^5\,x^{6\,n}}{6\,n}+\frac {10\,a^3\,b^2\,x^{3\,n}}{3\,n}+\frac {5\,a^2\,b^3\,x^{4\,n}}{2\,n}+\frac {5\,a^4\,b\,x^{2\,n}}{2\,n}+\frac {a\,b^4\,x^{5\,n}}{n} \]

[In]

int(x^(n - 1)*(a + b*x^n)^5,x)

[Out]

(a^5*x^n)/n + (b^5*x^(6*n))/(6*n) + (10*a^3*b^2*x^(3*n))/(3*n) + (5*a^2*b^3*x^(4*n))/(2*n) + (5*a^4*b*x^(2*n))
/(2*n) + (a*b^4*x^(5*n))/n