3.338 \(\int \frac{1}{\frac{b}{x}+a x} \, dx\)

Optimal. Leaf size=15 \[ \frac{\log \left (a x^2+b\right )}{2 a} \]

[Out]

Log[b + a*x^2]/(2*a)

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Rubi [A]  time = 0.004795, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {1593, 260} \[ \frac{\log \left (a x^2+b\right )}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[(b/x + a*x)^(-1),x]

[Out]

Log[b + a*x^2]/(2*a)

Rule 1593

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]

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

Mathematica [A]  time = 0.0026931, size = 15, normalized size = 1. \[ \frac{\log \left (a x^2+b\right )}{2 a} \]

Antiderivative was successfully verified.

[In]

Integrate[(b/x + a*x)^(-1),x]

[Out]

Log[b + a*x^2]/(2*a)

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Maple [A]  time = 0.002, size = 14, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( a{x}^{2}+b \right ) }{2\,a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b/x+a*x),x)

[Out]

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

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Maxima [A]  time = 1.26634, size = 18, normalized size = 1.2 \begin{align*} \frac{\log \left (a x^{2} + b\right )}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(a*x^2 + b)/a

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Fricas [A]  time = 0.651248, size = 30, normalized size = 2. \begin{align*} \frac{\log \left (a x^{2} + b\right )}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(a*x^2 + b)/a

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Sympy [A]  time = 0.1141, size = 10, normalized size = 0.67 \begin{align*} \frac{\log{\left (a x^{2} + b \right )}}{2 a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(a*x**2 + b)/(2*a)

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Giac [A]  time = 1.12146, size = 19, normalized size = 1.27 \begin{align*} \frac{\log \left ({\left | a x^{2} + b \right |}\right )}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(abs(a*x^2 + b))/a