\(\int \frac {1}{(a+\frac {b}{x^3}) x^4} \, dx\) [1974]

Optimal result

Integrand size = 13, antiderivative size = 15 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^4} \, dx=-\frac {\log \left (a+\frac {b}{x^3}\right )}{3 b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {266} \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^4} \, dx=-\frac {\log \left (a+\frac {b}{x^3}\right )}{3 b} \]

[In]

Int[1/((a + b/x^3)*x^4),x]

[Out]

-1/3*Log[a + b/x^3]/b

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]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.47 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^4} \, dx=\frac {\log (x)}{b}-\frac {\log \left (b+a x^3\right )}{3 b} \]

[In]

Integrate[1/((a + b/x^3)*x^4),x]

[Out]

Log[x]/b - Log[b + a*x^3]/(3*b)

Maple [A] (verified)

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

method	result	size
derivativedivides	\(-\frac {\ln \left (a +\frac {b}{x^{3}}\right )}{3 b}\)	\(14\)
default	\(-\frac {\ln \left (a \,x^{3}+b \right )}{3 b}+\frac {\ln \left (x \right )}{b}\)	\(21\)
norman	\(-\frac {\ln \left (a \,x^{3}+b \right )}{3 b}+\frac {\ln \left (x \right )}{b}\)	\(21\)
risch	\(-\frac {\ln \left (a \,x^{3}+b \right )}{3 b}+\frac {\ln \left (x \right )}{b}\)	\(21\)
parallelrisch	\(\frac {3 \ln \left (x \right )-\ln \left (a \,x^{3}+b \right )}{3 b}\)	\(21\)

[In]

int(1/(a+b/x^3)/x^4,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^4} \, dx=-\frac {\log \left (a x^{3} + b\right ) - 3 \, \log \left (x\right )}{3 \, b} \]

[In]

integrate(1/(a+b/x^3)/x^4,x, algorithm="fricas")

[Out]

-1/3*(log(a*x^3 + b) - 3*log(x))/b

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^4} \, dx=\frac {\log {\left (x \right )}}{b} - \frac {\log {\left (x^{3} + \frac {b}{a} \right )}}{3 b} \]

[In]

integrate(1/(a+b/x**3)/x**4,x)

[Out]

log(x)/b - log(x**3 + b/a)/(3*b)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^4} \, dx=-\frac {\log \left (a + \frac {b}{x^{3}}\right )}{3 \, b} \]

[In]

integrate(1/(a+b/x^3)/x^4,x, algorithm="maxima")

[Out]

-1/3*log(a + b/x^3)/b

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.47 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^4} \, dx=-\frac {\log \left ({\left | a x^{3} + b \right |}\right )}{3 \, b} + \frac {\log \left ({\left | x \right |}\right )}{b} \]

[In]

integrate(1/(a+b/x^3)/x^4,x, algorithm="giac")

[Out]

-1/3*log(abs(a*x^3 + b))/b + log(abs(x))/b

Mupad [B] (verification not implemented)

Time = 5.74 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right ) x^4} \, dx=-\frac {\ln \left (a\,x^3+b\right )-3\,\ln \left (x\right )}{3\,b} \]

[In]

int(1/(x^4*(a + b/x^3)),x)

[Out]

-(log(b + a*x^3) - 3*log(x))/(3*b)