\(\int \frac {1}{x (a+b x^n)} \, dx\) [2728]

Optimal result

Integrand size = 13, antiderivative size = 23 \[ \int \frac {1}{x \left (a+b x^n\right )} \, dx=\frac {\log (x)}{a}-\frac {\log \left (a+b x^n\right )}{a n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {272, 36, 29, 31} \[ \int \frac {1}{x \left (a+b x^n\right )} \, dx=\frac {\log (x)}{a}-\frac {\log \left (a+b x^n\right )}{a n} \]

[In]

Int[1/(x*(a + b*x^n)),x]

[Out]

Log[x]/a - Log[a + b*x^n]/(a*n)

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

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

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 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 {1}{x (a+b x)} \, dx,x,x^n\right )}{n} \\ & = \frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,x^n\right )}{a n}-\frac {b \text {Subst}\left (\int \frac {1}{a+b x} \, dx,x,x^n\right )}{a n} \\ & = \frac {\log (x)}{a}-\frac {\log \left (a+b x^n\right )}{a n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x \left (a+b x^n\right )} \, dx=\frac {\log \left (x^n\right )-\log \left (a n \left (a+b x^n\right )\right )}{a n} \]

[In]

Integrate[1/(x*(a + b*x^n)),x]

[Out]

(Log[x^n] - Log[a*n*(a + b*x^n)])/(a*n)

Maple [A] (verified)

Time = 3.66 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00

method	result	size
parallelrisch	\(\frac {n \ln \left (x \right )-\ln \left (a +b \,x^{n}\right )}{a n}\)	\(23\)
norman	\(\frac {\ln \left (x \right )}{a}-\frac {\ln \left (a +b \,{\mathrm e}^{n \ln \left (x \right )}\right )}{a n}\)	\(26\)
risch	\(\frac {\ln \left (x \right )}{a}-\frac {\ln \left (x^{n}+\frac {a}{b}\right )}{a n}\)	\(26\)
derivativedivides	\(\frac {\frac {\ln \left (x^{n}\right )}{a}-\frac {\ln \left (a +b \,x^{n}\right )}{a}}{n}\)	\(27\)
default	\(\frac {\frac {\ln \left (x^{n}\right )}{a}-\frac {\ln \left (a +b \,x^{n}\right )}{a}}{n}\)	\(27\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x \left (a+b x^n\right )} \, dx=\frac {n \log \left (x\right ) - \log \left (b x^{n} + a\right )}{a n} \]

[In]

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

[Out]

(n*log(x) - log(b*x^n + a))/(a*n)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (15) = 30\).

Time = 0.31 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.78 \[ \int \frac {1}{x \left (a+b x^n\right )} \, dx=\begin {cases} \tilde {\infty } \log {\left (x \right )} & \text {for}\: a = 0 \wedge b = 0 \wedge n = 0 \\- \frac {x^{- n}}{b n} & \text {for}\: a = 0 \\\frac {\log {\left (x \right )}}{a} & \text {for}\: b = 0 \\\frac {\log {\left (x \right )}}{a + b} & \text {for}\: n = 0 \\\frac {\log {\left (x \right )}}{a} - \frac {\log {\left (\frac {a}{b} + x^{n} \right )}}{a n} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22 \[ \int \frac {1}{x \left (a+b x^n\right )} \, dx=-\frac {\log \left (b x^{n} + a\right )}{a n} + \frac {\log \left (x^{n}\right )}{a n} \]

[In]

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

[Out]

-log(b*x^n + a)/(a*n) + log(x^n)/(a*n)

Giac [F]

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

[In]

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

[Out]

integrate(1/((b*x^n + a)*x), x)

Mupad [B] (verification not implemented)

Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x \left (a+b x^n\right )} \, dx=-\frac {\ln \left (a+b\,x^n\right )-n\,\ln \left (x\right )}{a\,n} \]

[In]

int(1/(x*(a + b*x^n)),x)

[Out]

-(log(a + b*x^n) - n*log(x))/(a*n)