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

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, 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 = {272, 36, 29, 31} \begin {gather*} \frac {\log (x)}{3}-\frac {1}{15} \log \left (b x^5+3\right ) \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {1}{x \left (3+b x^5\right )} \, dx &=\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*}

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Mathematica [A]
time = 0.00, size = 19, normalized size = 1.00 \begin {gather*} \frac {\log (x)}{3}-\frac {1}{15} \log \left (3+b x^5\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(3 + b*x^5)),x]

[Out]

Log[x]/3 - Log[3 + b*x^5]/15

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Maple [A]
time = 0.17, size = 16, normalized size = 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	\(\frac {\ln \left (x \right )}{3}-\frac {\ln \left (b \,x^{5}+3\right )}{15}\)	\(16\)
meijerg	\(-\frac {\ln \left (1+\frac {b \,x^{5}}{3}\right )}{15}+\frac {\ln \left (x \right )}{3}-\frac {\ln \left (3\right )}{15}+\frac {\ln \left (b \right )}{15}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 17, normalized size = 0.89 \begin {gather*} -\frac {1}{15} \, \log \left (b x^{5} + 3\right ) + \frac {1}{15} \, \log \left (x^{5}\right ) \end {gather*}

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)

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Fricas [A]
time = 0.38, size = 15, normalized size = 0.79 \begin {gather*} -\frac {1}{15} \, \log \left (b x^{5} + 3\right ) + \frac {1}{3} \, \log \left (x\right ) \end {gather*}

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)

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Sympy [A]
time = 0.07, size = 14, normalized size = 0.74 \begin {gather*} \frac {\log {\left (x \right )}}{3} - \frac {\log {\left (x^{5} + \frac {3}{b} \right )}}{15} \end {gather*}

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

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Giac [A]
time = 2.20, size = 17, normalized size = 0.89 \begin {gather*} -\frac {1}{15} \, \log \left ({\left | b x^{5} + 3 \right |}\right ) + \frac {1}{3} \, \log \left ({\left | x \right |}\right ) \end {gather*}

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))

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Mupad [B]
time = 0.06, size = 16, normalized size = 0.84 \begin {gather*} \frac {\ln \left (x\right )}{3}-\frac {\ln \left (\frac {2\,b\,x^5}{5}+\frac {6}{5}\right )}{15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(b*x^5 + 3)),x)

[Out]

log(x)/3 - log((2*b*x^5)/5 + 6/5)/15

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