\(\int \frac {1}{\frac {b}{x^3}+a x} \, dx\) [340]

Optimal result

Integrand size = 11, antiderivative size = 15 \[ \int \frac {1}{\frac {b}{x^3}+a x} \, dx=\frac {\log \left (b+a x^4\right )}{4 a} \]

[Out]

1/4*ln(a*x^4+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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1607, 266} \[ \int \frac {1}{\frac {b}{x^3}+a x} \, dx=\frac {\log \left (a x^4+b\right )}{4 a} \]

[In]

Int[(b/x^3 + a*x)^(-1),x]

[Out]

Log[b + a*x^4]/(4*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 1607

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^3}{b+a x^4} \, dx \\ & = \frac {\log \left (b+a x^4\right )}{4 a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\frac {b}{x^3}+a x} \, dx=\frac {\log \left (b+a x^4\right )}{4 a} \]

[In]

Integrate[(b/x^3 + a*x)^(-1),x]

[Out]

Log[b + a*x^4]/(4*a)

Maple [A] (verified)

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

method	result	size
default	\(\frac {\ln \left (a \,x^{4}+b \right )}{4 a}\)	\(14\)
norman	\(\frac {\ln \left (a \,x^{4}+b \right )}{4 a}\)	\(14\)
risch	\(\frac {\ln \left (a \,x^{4}+b \right )}{4 a}\)	\(14\)
parallelrisch	\(\frac {\ln \left (a \,x^{4}+b \right )}{4 a}\)	\(14\)

[In]

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

[Out]

1/4*ln(a*x^4+b)/a

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\frac {b}{x^3}+a x} \, dx=\frac {\log \left (a x^{4} + b\right )}{4 \, a} \]

[In]

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

[Out]

1/4*log(a*x^4 + b)/a

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\frac {b}{x^3}+a x} \, dx=\frac {\log {\left (a x^{4} + b \right )}}{4 a} \]

[In]

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

[Out]

log(a*x**4 + b)/(4*a)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\frac {b}{x^3}+a x} \, dx=\frac {\log \left (a x^{4} + b\right )}{4 \, a} \]

[In]

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

[Out]

1/4*log(a*x^4 + b)/a

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\frac {b}{x^3}+a x} \, dx=\frac {\log \left ({\left | a x^{4} + b \right |}\right )}{4 \, a} \]

[In]

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

[Out]

1/4*log(abs(a*x^4 + b))/a

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\frac {b}{x^3}+a x} \, dx=\frac {\ln \left (a\,x^4+b\right )}{4\,a} \]

[In]

int(1/(a*x + b/x^3),x)

[Out]

log(b + a*x^4)/(4*a)