3.227 \(\int \frac{x}{a-b x^2} \, dx\)

Optimal. Leaf size=16 \[ -\frac{\log \left (a-b x^2\right )}{2 b} \]

[Out]

-Log[a - b*x^2]/(2*b)

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Rubi [A]  time = 0.0028838, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, 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 = 0.0022229, size = 16, 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., size = 16, normalized size = 1. \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/b*ln(b*x^2-a)

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Maxima [A]  time = 1.55631, size = 20, normalized size = 1.25 \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 = 1.2447, size = 31, normalized size = 1.94 \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 = 0.108907, size = 12, normalized size = 0.75 \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 = 1.694, size = 22, normalized size = 1.38 \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