\(\int \frac {x^{-1+n}}{a+b x^n} \, dx\) [2610]

Optimal result

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

[Out]

ln(a+b*x^n)/b/n

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, 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 = {266} \[ \int \frac {x^{-1+n}}{a+b x^n} \, dx=\frac {\log \left (a+b x^n\right )}{b n} \]

[In]

Int[x^(-1 + n)/(a + b*x^n),x]

[Out]

Log[a + b*x^n]/(b*n)

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {x^{-1+n}}{a+b x^n} \, dx=\frac {\log \left (a+b x^n\right )}{b n} \]

[In]

Integrate[x^(-1 + n)/(a + b*x^n),x]

[Out]

Log[a + b*x^n]/(b*n)

Maple [A] (verified)

Time = 3.90 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.20

method	result	size
norman	\(\frac {\ln \left (a +b \,{\mathrm e}^{n \ln \left (x \right )}\right )}{b n}\)	\(18\)
risch	\(\frac {\ln \left (x^{n}+\frac {a}{b}\right )}{b n}\)	\(18\)

[In]

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

[Out]

1/b/n*ln(a+b*exp(n*ln(x)))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {x^{-1+n}}{a+b x^n} \, dx=\frac {\log \left (b x^{n} + a\right )}{b n} \]

[In]

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

[Out]

log(b*x^n + a)/(b*n)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (10) = 20\).

Time = 0.72 (sec) , antiderivative size = 31, normalized size of antiderivative = 2.07 \[ \int \frac {x^{-1+n}}{a+b x^n} \, dx=\begin {cases} \frac {\log {\left (x \right )}}{a} & \text {for}\: b = 0 \wedge n = 0 \\\frac {x x^{n - 1}}{a n} & \text {for}\: b = 0 \\\frac {\log {\left (x \right )}}{a + b} & \text {for}\: n = 0 \\\frac {\log {\left (\frac {a}{b} + x^{n} \right )}}{b n} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((log(x)/a, Eq(b, 0) & Eq(n, 0)), (x*x**(n - 1)/(a*n), Eq(b, 0)), (log(x)/(a + b), Eq(n, 0)), (log(a/
b + x**n)/(b*n), True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {x^{-1+n}}{a+b x^n} \, dx=\frac {\log \left (b x^{n} + a\right )}{b n} \]

[In]

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

[Out]

log(b*x^n + a)/(b*n)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07 \[ \int \frac {x^{-1+n}}{a+b x^n} \, dx=\frac {\log \left ({\left | b x^{n} + a \right |}\right )}{b n} \]

[In]

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

[Out]

log(abs(b*x^n + a))/(b*n)

Mupad [B] (verification not implemented)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {x^{-1+n}}{a+b x^n} \, dx=\frac {\ln \left (a+b\,x^n\right )}{b\,n} \]

[In]

int(x^(n - 1)/(a + b*x^n),x)

[Out]

log(a + b*x^n)/(b*n)