3.134 \(\int \frac{x}{a+b x^2} \, dx\)

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

[Out]

Log[a + b*x^2]/(2*b)

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Rubi [A]  time = 0.0030103, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {260} \[ \frac{\log \left (a+b x^2\right )}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*x^2),x]

[Out]

Log[a + b*x^2]/(2*b)

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

Mathematica [A]  time = 0.0021747, size = 15, normalized size = 1. \[ \frac{\log \left (a+b x^2\right )}{2 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*x^2),x]

[Out]

Log[a + b*x^2]/(2*b)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(b*x^2+a),x)

[Out]

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

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Maxima [A]  time = 1.96971, size = 18, normalized size = 1.2 \begin{align*} \frac{\log \left (b x^{2} + a\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x^2+a),x, algorithm="maxima")

[Out]

1/2*log(b*x^2 + a)/b

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Fricas [A]  time = 1.25512, size = 30, normalized size = 2. \begin{align*} \frac{\log \left (b x^{2} + a\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x^2+a),x, algorithm="fricas")

[Out]

1/2*log(b*x^2 + a)/b

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Sympy [A]  time = 0.101219, size = 10, normalized size = 0.67 \begin{align*} \frac{\log{\left (a + b x^{2} \right )}}{2 b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x**2+a),x)

[Out]

log(a + b*x**2)/(2*b)

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Giac [A]  time = 1.73615, size = 19, normalized size = 1.27 \begin{align*} \frac{\log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x^2+a),x, algorithm="giac")

[Out]

1/2*log(abs(b*x^2 + a))/b