\(\int \frac {1}{a x+b x^3} \, dx\) [16]

Optimal result

Integrand size = 11, antiderivative size = 22 \[ \int \frac {1}{a x+b x^3} \, dx=\frac {\log (x)}{a}-\frac {\log \left (a+b x^2\right )}{2 a} \]

[Out]

ln(x)/a-1/2*ln(b*x^2+a)/a

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {1607, 272, 36, 29, 31} \[ \int \frac {1}{a x+b x^3} \, dx=\frac {\log (x)}{a}-\frac {\log \left (a+b x^2\right )}{2 a} \]

[In]

Int[(a*x + b*x^3)^(-1),x]

[Out]

Log[x]/a - Log[a + b*x^2]/(2*a)

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]]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a x+b x^3} \, dx=\frac {\log (x)}{a}-\frac {\log \left (a+b x^2\right )}{2 a} \]

[In]

Integrate[(a*x + b*x^3)^(-1),x]

[Out]

Log[x]/a - Log[a + b*x^2]/(2*a)

Maple [A] (verified)

Time = 2.09 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95

method	result	size
default	\(\frac {\ln \left (x \right )}{a}-\frac {\ln \left (b \,x^{2}+a \right )}{2 a}\)	\(21\)
norman	\(\frac {\ln \left (x \right )}{a}-\frac {\ln \left (b \,x^{2}+a \right )}{2 a}\)	\(21\)
risch	\(\frac {\ln \left (x \right )}{a}-\frac {\ln \left (b \,x^{2}+a \right )}{2 a}\)	\(21\)
parallelrisch	\(\frac {2 \ln \left (x \right )-\ln \left (b \,x^{2}+a \right )}{2 a}\)	\(21\)

[In]

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

[Out]

ln(x)/a-1/2*ln(b*x^2+a)/a

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {1}{a x+b x^3} \, dx=-\frac {\log \left (b x^{2} + a\right ) - 2 \, \log \left (x\right )}{2 \, a} \]

[In]

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

[Out]

-1/2*(log(b*x^2 + a) - 2*log(x))/a

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int \frac {1}{a x+b x^3} \, dx=\frac {\log {\left (x \right )}}{a} - \frac {\log {\left (\frac {a}{b} + x^{2} \right )}}{2 a} \]

[In]

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

[Out]

log(x)/a - log(a/b + x**2)/(2*a)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {1}{a x+b x^3} \, dx=-\frac {\log \left (b x^{2} + a\right )}{2 \, a} + \frac {\log \left (x\right )}{a} \]

[In]

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

[Out]

-1/2*log(b*x^2 + a)/a + log(x)/a

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{a x+b x^3} \, dx=\frac {\log \left (x^{2}\right )}{2 \, a} - \frac {\log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a} \]

[In]

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

[Out]

1/2*log(x^2)/a - 1/2*log(abs(b*x^2 + a))/a

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {1}{a x+b x^3} \, dx=-\frac {\ln \left (b\,x^2+a\right )-2\,\ln \left (x\right )}{2\,a} \]

[In]

int(1/(a*x + b*x^3),x)

[Out]

-(log(a + b*x^2) - 2*log(x))/(2*a)