\(\int \frac {1}{(a+\frac {b}{x^3}) x} \, dx\) [1971]

Optimal result

Integrand size = 13, antiderivative size = 15 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x} \, dx=\frac {\log \left (b+a x^3\right )}{3 a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {269, 266} \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x} \, dx=\frac {\log \left (a x^3+b\right )}{3 a} \]

[In]

Int[1/((a + b/x^3)*x),x]

[Out]

Log[b + a*x^3]/(3*a)

Rule 266

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]

Rule 269

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]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x} \, dx=\frac {\log \left (b+a x^3\right )}{3 a} \]

[In]

Integrate[1/((a + b/x^3)*x),x]

[Out]

Log[b + a*x^3]/(3*a)

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93

method	result	size
default	\(\frac {\ln \left (a \,x^{3}+b \right )}{3 a}\)	\(14\)
norman	\(\frac {\ln \left (a \,x^{3}+b \right )}{3 a}\)	\(14\)
risch	\(\frac {\ln \left (a \,x^{3}+b \right )}{3 a}\)	\(14\)
parallelrisch	\(\frac {\ln \left (a \,x^{3}+b \right )}{3 a}\)	\(14\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x} \, dx=\frac {\log \left (a x^{3} + b\right )}{3 \, a} \]

[In]

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

[Out]

1/3*log(a*x^3 + b)/a

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x} \, dx=\frac {\log {\left (a x^{3} + b \right )}}{3 a} \]

[In]

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

[Out]

log(a*x**3 + b)/(3*a)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x} \, dx=\frac {\log \left (a x^{3} + b\right )}{3 \, a} \]

[In]

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

[Out]

1/3*log(a*x^3 + b)/a

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x} \, dx=\frac {\log \left ({\left | a x^{3} + b \right |}\right )}{3 \, a} \]

[In]

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

[Out]

1/3*log(abs(a*x^3 + b))/a

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x} \, dx=\frac {\ln \left (a\,x^3+b\right )}{3\,a} \]

[In]

int(1/(x*(a + b/x^3)),x)

[Out]

log(b + a*x^3)/(3*a)