3.1293 \(\int \frac{1}{x (3+b x^5)} \, dx\)

Optimal. Leaf size=19 \[ \frac{\log (x)}{3}-\frac{1}{15} \log \left (b x^5+3\right ) \]

[Out]

Log[x]/3 - Log[3 + b*x^5]/15

________________________________________________________________________________________

Rubi [A]  time = 0.0094368, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {266, 36, 29, 31} \[ \frac{\log (x)}{3}-\frac{1}{15} \log \left (b x^5+3\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(3 + b*x^5)),x]

[Out]

Log[x]/3 - Log[3 + b*x^5]/15

Rule 266

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]]

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 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]

Rubi steps

\begin{align*} \int \frac{1}{x \left (3+b x^5\right )} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{1}{x (3+b x)} \, dx,x,x^5\right )\\ &=\frac{1}{15} \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^5\right )-\frac{1}{15} b \operatorname{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]  time = 0.0036338, size = 19, normalized size = 1. \[ \frac{\log (x)}{3}-\frac{1}{15} \log \left (b x^5+3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(3 + b*x^5)),x]

[Out]

Log[x]/3 - Log[3 + b*x^5]/15

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 16, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( x \right ) }{3}}-{\frac{\ln \left ( b{x}^{5}+3 \right ) }{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.970603, size = 23, normalized size = 1.21 \begin{align*} -\frac{1}{15} \, \log \left (b x^{5} + 3\right ) + \frac{1}{15} \, \log \left (x^{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*log(b*x^5 + 3) + 1/15*log(x^5)

________________________________________________________________________________________

Fricas [A]  time = 1.68837, size = 47, normalized size = 2.47 \begin{align*} -\frac{1}{15} \, \log \left (b x^{5} + 3\right ) + \frac{1}{3} \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]  time = 0.183386, size = 14, normalized size = 0.74 \begin{align*} \frac{\log{\left (x \right )}}{3} - \frac{\log{\left (x^{5} + \frac{3}{b} \right )}}{15} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x)/3 - log(x**5 + 3/b)/15

________________________________________________________________________________________

Giac [A]  time = 1.44475, size = 23, normalized size = 1.21 \begin{align*} -\frac{1}{15} \, \log \left ({\left | b x^{5} + 3 \right |}\right ) + \frac{1}{3} \, \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))