\(\int x^{-1+4 n} (a+b x^n)^3 \, dx\) [2539]

Optimal result

Integrand size = 17, antiderivative size = 63 \[ \int x^{-1+4 n} \left (a+b x^n\right )^3 \, dx=\frac {a^3 x^{4 n}}{4 n}+\frac {3 a^2 b x^{5 n}}{5 n}+\frac {a b^2 x^{6 n}}{2 n}+\frac {b^3 x^{7 n}}{7 n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 63, 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} \[ \int x^{-1+4 n} \left (a+b x^n\right )^3 \, dx=\frac {a^3 x^{4 n}}{4 n}+\frac {3 a^2 b x^{5 n}}{5 n}+\frac {a b^2 x^{6 n}}{2 n}+\frac {b^3 x^{7 n}}{7 n} \]

[In]

Int[x^(-1 + 4*n)*(a + b*x^n)^3,x]

[Out]

(a^3*x^(4*n))/(4*n) + (3*a^2*b*x^(5*n))/(5*n) + (a*b^2*x^(6*n))/(2*n) + (b^3*x^(7*n))/(7*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)^3 \, dx,x,x^n\right )}{n} \\ & = \frac {\text {Subst}\left (\int \left (a^3 x^3+3 a^2 b x^4+3 a b^2 x^5+b^3 x^6\right ) \, dx,x,x^n\right )}{n} \\ & = \frac {a^3 x^{4 n}}{4 n}+\frac {3 a^2 b x^{5 n}}{5 n}+\frac {a b^2 x^{6 n}}{2 n}+\frac {b^3 x^{7 n}}{7 n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.76 \[ \int x^{-1+4 n} \left (a+b x^n\right )^3 \, dx=\frac {x^{4 n} \left (35 a^3+84 a^2 b x^n+70 a b^2 x^{2 n}+20 b^3 x^{3 n}\right )}{140 n} \]

[In]

Integrate[x^(-1 + 4*n)*(a + b*x^n)^3,x]

[Out]

(x^(4*n)*(35*a^3 + 84*a^2*b*x^n + 70*a*b^2*x^(2*n) + 20*b^3*x^(3*n)))/(140*n)

Maple [A] (verified)

Time = 3.88 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.89

method	result	size
risch	\(\frac {a^{3} x^{4 n}}{4 n}+\frac {3 a^{2} b \,x^{5 n}}{5 n}+\frac {a \,b^{2} x^{6 n}}{2 n}+\frac {b^{3} x^{7 n}}{7 n}\)	\(56\)
parallelrisch	\(\frac {20 x \,x^{3 n} x^{-1+4 n} b^{3}+70 x \,x^{2 n} x^{-1+4 n} a \,b^{2}+84 x \,x^{n} x^{-1+4 n} a^{2} b +35 x \,x^{-1+4 n} a^{3}}{140 n}\)	\(74\)

[In]

int(x^(-1+4*n)*(a+b*x^n)^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.76 \[ \int x^{-1+4 n} \left (a+b x^n\right )^3 \, dx=\frac {20 \, b^{3} x^{7 \, n} + 70 \, a b^{2} x^{6 \, n} + 84 \, a^{2} b x^{5 \, n} + 35 \, a^{3} x^{4 \, n}}{140 \, n} \]

[In]

integrate(x^(-1+4*n)*(a+b*x^n)^3,x, algorithm="fricas")

[Out]

1/140*(20*b^3*x^(7*n) + 70*a*b^2*x^(6*n) + 84*a^2*b*x^(5*n) + 35*a^3*x^(4*n))/n

Sympy [A] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.38 \[ \int x^{-1+4 n} \left (a+b x^n\right )^3 \, dx=\begin {cases} \frac {a^{3} x x^{4 n - 1}}{4 n} + \frac {3 a^{2} b x x^{n} x^{4 n - 1}}{5 n} + \frac {a b^{2} x x^{2 n} x^{4 n - 1}}{2 n} + \frac {b^{3} x x^{3 n} x^{4 n - 1}}{7 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{3} \log {\left (x \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(x**(-1+4*n)*(a+b*x**n)**3,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.87 \[ \int x^{-1+4 n} \left (a+b x^n\right )^3 \, dx=\frac {b^{3} x^{7 \, n}}{7 \, n} + \frac {a b^{2} x^{6 \, n}}{2 \, n} + \frac {3 \, a^{2} b x^{5 \, n}}{5 \, n} + \frac {a^{3} x^{4 \, n}}{4 \, n} \]

[In]

integrate(x^(-1+4*n)*(a+b*x^n)^3,x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int x^{-1+4 n} \left (a+b x^n\right )^3 \, dx=\int { {\left (b x^{n} + a\right )}^{3} x^{4 \, n - 1} \,d x } \]

[In]

integrate(x^(-1+4*n)*(a+b*x^n)^3,x, algorithm="giac")

[Out]

integrate((b*x^n + a)^3*x^(4*n - 1), x)

Mupad [B] (verification not implemented)

Time = 5.82 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.87 \[ \int x^{-1+4 n} \left (a+b x^n\right )^3 \, dx=\frac {a^3\,x^{4\,n}}{4\,n}+\frac {b^3\,x^{7\,n}}{7\,n}+\frac {3\,a^2\,b\,x^{5\,n}}{5\,n}+\frac {a\,b^2\,x^{6\,n}}{2\,n} \]

[In]

int(x^(4*n - 1)*(a + b*x^n)^3,x)

[Out]

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