3.1971 \(\int \frac{1}{(a+\frac{b}{x^3}) x} \, dx\)

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

[Out]

Log[b + a*x^3]/(3*a)

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Rubi [A]  time = 0.0058785, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {263, 260} \[ \frac{\log \left (a x^3+b\right )}{3 a} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^3)*x),x]

[Out]

Log[b + a*x^3]/(3*a)

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin{align*} \int \frac{1}{\left (a+\frac{b}{x^3}\right ) x} \, dx &=\int \frac{x^2}{b+a x^3} \, dx\\ &=\frac{\log \left (b+a x^3\right )}{3 a}\\ \end{align*}

Mathematica [A]  time = 0.0027306, size = 15, normalized size = 1. \[ \frac{\log \left (a x^3+b\right )}{3 a} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^3)*x),x]

[Out]

Log[b + a*x^3]/(3*a)

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Maple [A]  time = 0.001, size = 14, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( a{x}^{3}+b \right ) }{3\,a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x^3)/x,x)

[Out]

1/3*ln(a*x^3+b)/a

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Maxima [A]  time = 0.97607, size = 18, normalized size = 1.2 \begin{align*} \frac{\log \left (a x^{3} + b\right )}{3 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*log(a*x^3 + b)/a

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Fricas [A]  time = 1.36813, size = 30, normalized size = 2. \begin{align*} \frac{\log \left (a x^{3} + b\right )}{3 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*log(a*x^3 + b)/a

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Sympy [A]  time = 0.160637, size = 10, normalized size = 0.67 \begin{align*} \frac{\log{\left (a x^{3} + b \right )}}{3 a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(a*x**3 + b)/(3*a)

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Giac [A]  time = 1.22726, size = 19, normalized size = 1.27 \begin{align*} \frac{\log \left ({\left | a x^{3} + b \right |}\right )}{3 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*log(abs(a*x^3 + b))/a