\(\int x^{-1-3 n} (a+b x^n) \, dx\) [2525]

Optimal result

Integrand size = 15, antiderivative size = 27 \[ \int x^{-1-3 n} \left (a+b x^n\right ) \, dx=-\frac {a x^{-3 n}}{3 n}-\frac {b x^{-2 n}}{2 n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {14} \[ \int x^{-1-3 n} \left (a+b x^n\right ) \, dx=-\frac {a x^{-3 n}}{3 n}-\frac {b x^{-2 n}}{2 n} \]

[In]

Int[x^(-1 - 3*n)*(a + b*x^n),x]

[Out]

-1/3*a/(n*x^(3*n)) - b/(2*n*x^(2*n))

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps \begin{align*} \text {integral}& = \int \left (a x^{-1-3 n}+b x^{-1-2 n}\right ) \, dx \\ & = -\frac {a x^{-3 n}}{3 n}-\frac {b x^{-2 n}}{2 n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int x^{-1-3 n} \left (a+b x^n\right ) \, dx=\frac {x^{-3 n} \left (-2 a-3 b x^n\right )}{6 n} \]

[In]

Integrate[x^(-1 - 3*n)*(a + b*x^n),x]

[Out]

(-2*a - 3*b*x^n)/(6*n*x^(3*n))

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89

method	result	size
risch	\(-\frac {b \,x^{-2 n}}{2 n}-\frac {a \,x^{-3 n}}{3 n}\)	\(24\)
norman	\(\left (-\frac {a}{3 n}-\frac {b \,{\mathrm e}^{n \ln \left (x \right )}}{2 n}\right ) {\mathrm e}^{-3 n \ln \left (x \right )}\)	\(27\)
parallelrisch	\(-\frac {3 x \,x^{n} x^{-1-3 n} b +2 x \,x^{-1-3 n} a}{6 n}\)	\(32\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int x^{-1-3 n} \left (a+b x^n\right ) \, dx=-\frac {3 \, b x^{n} + 2 \, a}{6 \, n x^{3 \, n}} \]

[In]

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

[Out]

-1/6*(3*b*x^n + 2*a)/(n*x^(3*n))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (20) = 40\).

Time = 0.17 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.52 \[ \int x^{-1-3 n} \left (a+b x^n\right ) \, dx=\begin {cases} - \frac {a x x^{- 3 n - 1}}{3 n} - \frac {b x x^{n} x^{- 3 n - 1}}{2 n} & \text {for}\: n \neq 0 \\\left (a + b\right ) \log {\left (x \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int x^{-1-3 n} \left (a+b x^n\right ) \, dx=-\frac {a}{3 \, n x^{3 \, n}} - \frac {b}{2 \, n x^{2 \, n}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int x^{-1-3 n} \left (a+b x^n\right ) \, dx=-\frac {3 \, b x^{n} + 2 \, a}{6 \, n x^{3 \, n}} \]

[In]

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

[Out]

-1/6*(3*b*x^n + 2*a)/(n*x^(3*n))

Mupad [B] (verification not implemented)

Time = 5.81 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int x^{-1-3 n} \left (a+b x^n\right ) \, dx=-\frac {2\,a+3\,b\,x^n}{6\,n\,x^{3\,n}} \]

[In]

int((a + b*x^n)/x^(3*n + 1),x)

[Out]

-(2*a + 3*b*x^n)/(6*n*x^(3*n))