\(\int \frac {1}{x (3+b x^5)} \, dx\) [1293]

Optimal result

Integrand size = 13, antiderivative size = 19 \[ \int \frac {1}{x \left (3+b x^5\right )} \, dx=\frac {\log (x)}{3}-\frac {1}{15} \log \left (3+b x^5\right ) \]

[Out]

1/3*ln(x)-1/15*ln(b*x^5+3)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 19, 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} \[ \int \frac {1}{x \left (3+b x^5\right )} \, dx=\frac {\log (x)}{3}-\frac {1}{15} \log \left (b x^5+3\right ) \]

[In]

Int[1/(x*(3 + b*x^5)),x]

[Out]

Log[x]/3 - Log[3 + b*x^5]/15

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 {1}{5} \text {Subst}\left (\int \frac {1}{x (3+b x)} \, dx,x,x^5\right ) \\ & = \frac {1}{15} \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^5\right )-\frac {1}{15} b \text {Subst}\left (\int \frac {1}{3+b x} \, dx,x,x^5\right ) \\ & = \frac {\log (x)}{3}-\frac {1}{15} \log \left (3+b x^5\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (3+b x^5\right )} \, dx=\frac {\log (x)}{3}-\frac {1}{15} \log \left (3+b x^5\right ) \]

[In]

Integrate[1/(x*(3 + b*x^5)),x]

[Out]

Log[x]/3 - Log[3 + b*x^5]/15

Maple [A] (verified)

Time = 4.41 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84

method	result	size
default	\(\frac {\ln \left (x \right )}{3}-\frac {\ln \left (b \,x^{5}+3\right )}{15}\)	\(16\)
norman	\(\frac {\ln \left (x \right )}{3}-\frac {\ln \left (b \,x^{5}+3\right )}{15}\)	\(16\)
risch	\(\frac {\ln \left (x \right )}{3}-\frac {\ln \left (b \,x^{5}+3\right )}{15}\)	\(16\)
parallelrisch	\(\frac {\ln \left (x \right )}{3}-\frac {\ln \left (b \,x^{5}+3\right )}{15}\)	\(16\)
meijerg	\(\frac {\ln \left (x \right )}{3}-\frac {\ln \left (3\right )}{15}+\frac {\ln \left (b \right )}{15}-\frac {\ln \left (1+\frac {b \,x^{5}}{3}\right )}{15}\)	\(25\)

[In]

int(1/x/(b*x^5+3),x,method=_RETURNVERBOSE)

[Out]

1/3*ln(x)-1/15*ln(b*x^5+3)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {1}{x \left (3+b x^5\right )} \, dx=-\frac {1}{15} \, \log \left (b x^{5} + 3\right ) + \frac {1}{3} \, \log \left (x\right ) \]

[In]

integrate(1/x/(b*x^5+3),x, algorithm="fricas")

[Out]

-1/15*log(b*x^5 + 3) + 1/3*log(x)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74 \[ \int \frac {1}{x \left (3+b x^5\right )} \, dx=\frac {\log {\left (x \right )}}{3} - \frac {\log {\left (x^{5} + \frac {3}{b} \right )}}{15} \]

[In]

integrate(1/x/(b*x**5+3),x)

[Out]

log(x)/3 - log(x**5 + 3/b)/15

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x \left (3+b x^5\right )} \, dx=-\frac {1}{15} \, \log \left (b x^{5} + 3\right ) + \frac {1}{15} \, \log \left (x^{5}\right ) \]

[In]

integrate(1/x/(b*x^5+3),x, algorithm="maxima")

[Out]

-1/15*log(b*x^5 + 3) + 1/15*log(x^5)

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x \left (3+b x^5\right )} \, dx=-\frac {1}{15} \, \log \left ({\left | b x^{5} + 3 \right |}\right ) + \frac {1}{3} \, \log \left ({\left | x \right |}\right ) \]

[In]

integrate(1/x/(b*x^5+3),x, algorithm="giac")

[Out]

-1/15*log(abs(b*x^5 + 3)) + 1/3*log(abs(x))

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x \left (3+b x^5\right )} \, dx=\frac {\ln \left (x\right )}{3}-\frac {\ln \left (\frac {2\,b\,x^5}{5}+\frac {6}{5}\right )}{15} \]

[In]

int(1/(x*(b*x^5 + 3)),x)

[Out]

log(x)/3 - log((2*b*x^5)/5 + 6/5)/15