3.1288 \(\int \frac{1}{x (2 b+b x^5)} \, dx\)

Optimal. Leaf size=23 \[ \frac{\log (x)}{2 b}-\frac{\log \left (x^5+2\right )}{10 b} \]

[Out]

Log[x]/(2*b) - Log[2 + x^5]/(10*b)

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Rubi [A]  time = 0.0087873, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 36, 29, 31} \[ \frac{\log (x)}{2 b}-\frac{\log \left (x^5+2\right )}{10 b} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(2*b + b*x^5)),x]

[Out]

Log[x]/(2*b) - Log[2 + x^5]/(10*b)

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 (2 b+b x^5\right )} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{1}{x (2 b+b x)} \, dx,x,x^5\right )\\ &=-\left (\frac{1}{10} \operatorname{Subst}\left (\int \frac{1}{2 b+b x} \, dx,x,x^5\right )\right )+\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^5\right )}{10 b}\\ &=\frac{\log (x)}{2 b}-\frac{\log \left (2+x^5\right )}{10 b}\\ \end{align*}

Mathematica [A]  time = 0.005415, size = 23, normalized size = 1. \[ \frac{\log (x)}{2 b}-\frac{\log \left (x^5+2\right )}{10 b} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(2*b + b*x^5)),x]

[Out]

Log[x]/(2*b) - Log[2 + x^5]/(10*b)

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Maple [A]  time = 0.004, size = 20, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( x \right ) }{2\,b}}-{\frac{\ln \left ({x}^{5}+2 \right ) }{10\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(b*x^5+2*b),x)

[Out]

1/2*ln(x)/b-1/10*ln(x^5+2)/b

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Maxima [A]  time = 0.994236, size = 28, normalized size = 1.22 \begin{align*} -\frac{\log \left (x^{5} + 2\right )}{10 \, b} + \frac{\log \left (x^{5}\right )}{10 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x^5+2*b),x, algorithm="maxima")

[Out]

-1/10*log(x^5 + 2)/b + 1/10*log(x^5)/b

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Fricas [A]  time = 1.70775, size = 47, normalized size = 2.04 \begin{align*} -\frac{\log \left (x^{5} + 2\right ) - 5 \, \log \left (x\right )}{10 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x^5+2*b),x, algorithm="fricas")

[Out]

-1/10*(log(x^5 + 2) - 5*log(x))/b

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Sympy [A]  time = 0.345973, size = 15, normalized size = 0.65 \begin{align*} \frac{\log{\left (x \right )}}{2 b} - \frac{\log{\left (x^{5} + 2 \right )}}{10 b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x**5+2*b),x)

[Out]

log(x)/(2*b) - log(x**5 + 2)/(10*b)

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Giac [A]  time = 1.19682, size = 28, normalized size = 1.22 \begin{align*} -\frac{\log \left ({\left | x^{5} + 2 \right |}\right )}{10 \, b} + \frac{\log \left ({\left | x \right |}\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x^5+2*b),x, algorithm="giac")

[Out]

-1/10*log(abs(x^5 + 2))/b + 1/2*log(abs(x))/b