\(\int x^{-1-4 n} (a+b x^n)^2 \, dx\) [2536]

Optimal result

Integrand size = 17, antiderivative size = 45 \[ \int x^{-1-4 n} \left (a+b x^n\right )^2 \, dx=-\frac {a^2 x^{-4 n}}{4 n}-\frac {2 a b x^{-3 n}}{3 n}-\frac {b^2 x^{-2 n}}{2 n} \]

[Out]

-1/4*a^2/n/(x^(4*n))-2/3*a*b/n/(x^(3*n))-1/2*b^2/n/(x^(2*n))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 45, 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} \[ \int x^{-1-4 n} \left (a+b x^n\right )^2 \, dx=-\frac {a^2 x^{-4 n}}{4 n}-\frac {2 a b x^{-3 n}}{3 n}-\frac {b^2 x^{-2 n}}{2 n} \]

[In]

Int[x^(-1 - 4*n)*(a + b*x^n)^2,x]

[Out]

-1/4*a^2/(n*x^(4*n)) - (2*a*b)/(3*n*x^(3*n)) - b^2/(2*n*x^(2*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 \frac {(a+b x)^2}{x^5} \, dx,x,x^n\right )}{n} \\ & = \frac {\text {Subst}\left (\int \left (\frac {a^2}{x^5}+\frac {2 a b}{x^4}+\frac {b^2}{x^3}\right ) \, dx,x,x^n\right )}{n} \\ & = -\frac {a^2 x^{-4 n}}{4 n}-\frac {2 a b x^{-3 n}}{3 n}-\frac {b^2 x^{-2 n}}{2 n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.78 \[ \int x^{-1-4 n} \left (a+b x^n\right )^2 \, dx=\frac {x^{-4 n} \left (-3 a^2-8 a b x^n-6 b^2 x^{2 n}\right )}{12 n} \]

[In]

Integrate[x^(-1 - 4*n)*(a + b*x^n)^2,x]

[Out]

(-3*a^2 - 8*a*b*x^n - 6*b^2*x^(2*n))/(12*n*x^(4*n))

Maple [A] (verified)

Time = 3.59 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.89

method	result	size
risch	\(-\frac {b^{2} x^{-2 n}}{2 n}-\frac {2 a b \,x^{-3 n}}{3 n}-\frac {a^{2} x^{-4 n}}{4 n}\)	\(40\)
norman	\(\left (-\frac {a^{2}}{4 n}-\frac {b^{2} {\mathrm e}^{2 n \ln \left (x \right )}}{2 n}-\frac {2 a b \,{\mathrm e}^{n \ln \left (x \right )}}{3 n}\right ) {\mathrm e}^{-4 n \ln \left (x \right )}\)	\(45\)
parallelrisch	\(\frac {-6 x \,x^{2 n} x^{-1-4 n} b^{2}-8 x \,x^{n} x^{-1-4 n} a b -3 x \,x^{-1-4 n} a^{2}}{12 n}\)	\(53\)

[In]

int(x^(-1-4*n)*(a+b*x^n)^2,x,method=_RETURNVERBOSE)

[Out]

-1/2*b^2/n/(x^n)^2-2/3*a*b/n/(x^n)^3-1/4*a^2/n/(x^n)^4

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.78 \[ \int x^{-1-4 n} \left (a+b x^n\right )^2 \, dx=-\frac {6 \, b^{2} x^{2 \, n} + 8 \, a b x^{n} + 3 \, a^{2}}{12 \, n x^{4 \, n}} \]

[In]

integrate(x^(-1-4*n)*(a+b*x^n)^2,x, algorithm="fricas")

[Out]

-1/12*(6*b^2*x^(2*n) + 8*a*b*x^n + 3*a^2)/(n*x^(4*n))

Sympy [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.56 \[ \int x^{-1-4 n} \left (a+b x^n\right )^2 \, dx=\begin {cases} - \frac {a^{2} x x^{- 4 n - 1}}{4 n} - \frac {2 a b x x^{n} x^{- 4 n - 1}}{3 n} - \frac {b^{2} x x^{2 n} x^{- 4 n - 1}}{2 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{2} \log {\left (x \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(x**(-1-4*n)*(a+b*x**n)**2,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int x^{-1-4 n} \left (a+b x^n\right )^2 \, dx=-\frac {a^{2}}{4 \, n x^{4 \, n}} - \frac {2 \, a b}{3 \, n x^{3 \, n}} - \frac {b^{2}}{2 \, n x^{2 \, n}} \]

[In]

integrate(x^(-1-4*n)*(a+b*x^n)^2,x, algorithm="maxima")

[Out]

-1/4*a^2/(n*x^(4*n)) - 2/3*a*b/(n*x^(3*n)) - 1/2*b^2/(n*x^(2*n))

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.78 \[ \int x^{-1-4 n} \left (a+b x^n\right )^2 \, dx=-\frac {6 \, b^{2} x^{2 \, n} + 8 \, a b x^{n} + 3 \, a^{2}}{12 \, n x^{4 \, n}} \]

[In]

integrate(x^(-1-4*n)*(a+b*x^n)^2,x, algorithm="giac")

[Out]

-1/12*(6*b^2*x^(2*n) + 8*a*b*x^n + 3*a^2)/(n*x^(4*n))

Mupad [B] (verification not implemented)

Time = 5.90 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.78 \[ \int x^{-1-4 n} \left (a+b x^n\right )^2 \, dx=-\frac {3\,a^2+6\,b^2\,x^{2\,n}+8\,a\,b\,x^n}{12\,n\,x^{4\,n}} \]

[In]

int((a + b*x^n)^2/x^(4*n + 1),x)

[Out]

-(3*a^2 + 6*b^2*x^(2*n) + 8*a*b*x^n)/(12*n*x^(4*n))