\(\int x^{-1+4 n} (a+b x^n)^5 \, dx\) [2551]

Optimal result

Integrand size = 17, antiderivative size = 84 \[ \int x^{-1+4 n} \left (a+b x^n\right )^5 \, dx=-\frac {a^3 \left (a+b x^n\right )^6}{6 b^4 n}+\frac {3 a^2 \left (a+b x^n\right )^7}{7 b^4 n}-\frac {3 a \left (a+b x^n\right )^8}{8 b^4 n}+\frac {\left (a+b x^n\right )^9}{9 b^4 n} \]

[Out]

-1/6*a^3*(a+b*x^n)^6/b^4/n+3/7*a^2*(a+b*x^n)^7/b^4/n-3/8*a*(a+b*x^n)^8/b^4/n+1/9*(a+b*x^n)^9/b^4/n

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {272, 45} \[ \int x^{-1+4 n} \left (a+b x^n\right )^5 \, dx=-\frac {a^3 \left (a+b x^n\right )^6}{6 b^4 n}+\frac {3 a^2 \left (a+b x^n\right )^7}{7 b^4 n}+\frac {\left (a+b x^n\right )^9}{9 b^4 n}-\frac {3 a \left (a+b x^n\right )^8}{8 b^4 n} \]

[In]

Int[x^(-1 + 4*n)*(a + b*x^n)^5,x]

[Out]

-1/6*(a^3*(a + b*x^n)^6)/(b^4*n) + (3*a^2*(a + b*x^n)^7)/(7*b^4*n) - (3*a*(a + b*x^n)^8)/(8*b^4*n) + (a + b*x^
n)^9/(9*b^4*n)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int x^3 (a+b x)^5 \, dx,x,x^n\right )}{n} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {a^3 (a+b x)^5}{b^3}+\frac {3 a^2 (a+b x)^6}{b^3}-\frac {3 a (a+b x)^7}{b^3}+\frac {(a+b x)^8}{b^3}\right ) \, dx,x,x^n\right )}{n} \\ & = -\frac {a^3 \left (a+b x^n\right )^6}{6 b^4 n}+\frac {3 a^2 \left (a+b x^n\right )^7}{7 b^4 n}-\frac {3 a \left (a+b x^n\right )^8}{8 b^4 n}+\frac {\left (a+b x^n\right )^9}{9 b^4 n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.88 \[ \int x^{-1+4 n} \left (a+b x^n\right )^5 \, dx=\frac {x^{4 n} \left (126 a^5+504 a^4 b x^n+840 a^3 b^2 x^{2 n}+720 a^2 b^3 x^{3 n}+315 a b^4 x^{4 n}+56 b^5 x^{5 n}\right )}{504 n} \]

[In]

Integrate[x^(-1 + 4*n)*(a + b*x^n)^5,x]

[Out]

(x^(4*n)*(126*a^5 + 504*a^4*b*x^n + 840*a^3*b^2*x^(2*n) + 720*a^2*b^3*x^(3*n) + 315*a*b^4*x^(4*n) + 56*b^5*x^(
5*n)))/(504*n)

Maple [A] (verified)

Time = 4.34 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.04

method	result	size
risch	\(\frac {b^{5} x^{9 n}}{9 n}+\frac {5 a \,b^{4} x^{8 n}}{8 n}+\frac {10 a^{2} b^{3} x^{7 n}}{7 n}+\frac {5 a^{3} b^{2} x^{6 n}}{3 n}+\frac {a^{4} b \,x^{5 n}}{n}+\frac {a^{5} x^{4 n}}{4 n}\)	\(87\)
parallelrisch	\(\frac {56 x \,x^{5 n} x^{-1+4 n} b^{5}+315 x \,x^{4 n} x^{-1+4 n} a \,b^{4}+720 x \,x^{3 n} x^{-1+4 n} a^{2} b^{3}+840 x \,x^{2 n} x^{-1+4 n} a^{3} b^{2}+504 x \,x^{n} x^{-1+4 n} a^{4} b +126 x \,x^{-1+4 n} a^{5}}{504 n}\)	\(116\)

[In]

int(x^(-1+4*n)*(a+b*x^n)^5,x,method=_RETURNVERBOSE)

[Out]

1/9*b^5/n*(x^n)^9+5/8*a*b^4/n*(x^n)^8+10/7*a^2*b^3/n*(x^n)^7+5/3*a^3*b^2/n*(x^n)^6+a^4*b/n*(x^n)^5+1/4*a^5/n*(
x^n)^4

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.88 \[ \int x^{-1+4 n} \left (a+b x^n\right )^5 \, dx=\frac {56 \, b^{5} x^{9 \, n} + 315 \, a b^{4} x^{8 \, n} + 720 \, a^{2} b^{3} x^{7 \, n} + 840 \, a^{3} b^{2} x^{6 \, n} + 504 \, a^{4} b x^{5 \, n} + 126 \, a^{5} x^{4 \, n}}{504 \, n} \]

[In]

integrate(x^(-1+4*n)*(a+b*x^n)^5,x, algorithm="fricas")

[Out]

1/504*(56*b^5*x^(9*n) + 315*a*b^4*x^(8*n) + 720*a^2*b^3*x^(7*n) + 840*a^3*b^2*x^(6*n) + 504*a^4*b*x^(5*n) + 12
6*a^5*x^(4*n))/n

Sympy [A] (verification not implemented)

Time = 0.75 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.62 \[ \int x^{-1+4 n} \left (a+b x^n\right )^5 \, dx=\begin {cases} \frac {a^{5} x x^{4 n - 1}}{4 n} + \frac {a^{4} b x x^{n} x^{4 n - 1}}{n} + \frac {5 a^{3} b^{2} x x^{2 n} x^{4 n - 1}}{3 n} + \frac {10 a^{2} b^{3} x x^{3 n} x^{4 n - 1}}{7 n} + \frac {5 a b^{4} x x^{4 n} x^{4 n - 1}}{8 n} + \frac {b^{5} x x^{5 n} x^{4 n - 1}}{9 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{5} \log {\left (x \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(x**(-1+4*n)*(a+b*x**n)**5,x)

[Out]

Piecewise((a**5*x*x**(4*n - 1)/(4*n) + a**4*b*x*x**n*x**(4*n - 1)/n + 5*a**3*b**2*x*x**(2*n)*x**(4*n - 1)/(3*n
) + 10*a**2*b**3*x*x**(3*n)*x**(4*n - 1)/(7*n) + 5*a*b**4*x*x**(4*n)*x**(4*n - 1)/(8*n) + b**5*x*x**(5*n)*x**(
4*n - 1)/(9*n), Ne(n, 0)), ((a + b)**5*log(x), True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.02 \[ \int x^{-1+4 n} \left (a+b x^n\right )^5 \, dx=\frac {b^{5} x^{9 \, n}}{9 \, n} + \frac {5 \, a b^{4} x^{8 \, n}}{8 \, n} + \frac {10 \, a^{2} b^{3} x^{7 \, n}}{7 \, n} + \frac {5 \, a^{3} b^{2} x^{6 \, n}}{3 \, n} + \frac {a^{4} b x^{5 \, n}}{n} + \frac {a^{5} x^{4 \, n}}{4 \, n} \]

[In]

integrate(x^(-1+4*n)*(a+b*x^n)^5,x, algorithm="maxima")

[Out]

1/9*b^5*x^(9*n)/n + 5/8*a*b^4*x^(8*n)/n + 10/7*a^2*b^3*x^(7*n)/n + 5/3*a^3*b^2*x^(6*n)/n + a^4*b*x^(5*n)/n + 1
/4*a^5*x^(4*n)/n

Giac [F]

\[ \int x^{-1+4 n} \left (a+b x^n\right )^5 \, dx=\int { {\left (b x^{n} + a\right )}^{5} x^{4 \, n - 1} \,d x } \]

[In]

integrate(x^(-1+4*n)*(a+b*x^n)^5,x, algorithm="giac")

[Out]

integrate((b*x^n + a)^5*x^(4*n - 1), x)

Mupad [B] (verification not implemented)

Time = 5.83 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.02 \[ \int x^{-1+4 n} \left (a+b x^n\right )^5 \, dx=\frac {a^5\,x^{4\,n}}{4\,n}+\frac {b^5\,x^{9\,n}}{9\,n}+\frac {5\,a^3\,b^2\,x^{6\,n}}{3\,n}+\frac {10\,a^2\,b^3\,x^{7\,n}}{7\,n}+\frac {a^4\,b\,x^{5\,n}}{n}+\frac {5\,a\,b^4\,x^{8\,n}}{8\,n} \]

[In]

int(x^(4*n - 1)*(a + b*x^n)^5,x)

[Out]

(a^5*x^(4*n))/(4*n) + (b^5*x^(9*n))/(9*n) + (5*a^3*b^2*x^(6*n))/(3*n) + (10*a^2*b^3*x^(7*n))/(7*n) + (a^4*b*x^
(5*n))/n + (5*a*b^4*x^(8*n))/(8*n)