3.13.88 \(\int \frac {1}{x (2 b+b x^5)} \, dx\) [1288]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, 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 = {272, 36, 29, 31} \begin {gather*} \frac {\log (x)}{2 b}-\frac {\log \left (x^5+2\right )}{10 b} \end {gather*}

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

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Mathematica [A]
time = 0.00, size = 23, normalized size = 1.00 \begin {gather*} \frac {\log (x)}{2 b}-\frac {\log \left (2+x^5\right )}{10 b} \end {gather*}

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 = 0.17, size = 18, normalized size = 0.78

method	result	size
default	\(\frac {\frac {\ln \left (x \right )}{2}-\frac {\ln \left (x^{5}+2\right )}{10}}{b}\)	\(18\)
norman	\(\frac {\ln \left (x \right )}{2 b}-\frac {\ln \left (x^{5}+2\right )}{10 b}\)	\(20\)
risch	\(\frac {\ln \left (x \right )}{2 b}-\frac {\ln \left (x^{5}+2\right )}{10 b}\)	\(20\)
meijerg	\(\frac {-\ln \left (1+\frac {x^{5}}{2}\right )+5 \ln \left (x \right )-\ln \left (2\right )}{10 b}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 21, normalized size = 0.91 \begin {gather*} -\frac {\log \left (x^{5} + 2\right )}{10 \, b} + \frac {\log \left (x^{5}\right )}{10 \, b} \end {gather*}

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 = 0.37, size = 16, normalized size = 0.70 \begin {gather*} -\frac {\log \left (x^{5} + 2\right ) - 5 \, \log \left (x\right )}{10 \, b} \end {gather*}

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 = 0.14, size = 15, normalized size = 0.65 \begin {gather*} \frac {\log {\left (x \right )}}{2 b} - \frac {\log {\left (x^{5} + 2 \right )}}{10 b} \end {gather*}

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 = 2.01, size = 21, normalized size = 0.91 \begin {gather*} -\frac {\log \left ({\left | x^{5} + 2 \right |}\right )}{10 \, b} + \frac {\log \left ({\left | x \right |}\right )}{2 \, b} \end {gather*}

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

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Mupad [B]
time = 1.09, size = 19, normalized size = 0.83 \begin {gather*} \frac {\ln \left (x\right )}{2\,b}-\frac {\ln \left (x^5+2\right )}{10\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(2*b + b*x^5)),x)

[Out]

log(x)/(2*b) - log(x^5 + 2)/(10*b)

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