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]

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

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Rubi [A]  time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 1.00 \[ \frac {\log \left (a+b x^2\right )}{2 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*x^2),x]

[Out]

Log[a + b*x^2]/(2*b)

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fricas [A]  time = 0.68, size = 13, normalized size = 0.87 \[ \frac {\log \left (b x^{2} + a\right )}{2 \, b} \]

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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giac [A]  time = 1.07, size = 14, normalized size = 0.93 \[ \frac {\log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b} \]

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

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maple [A]  time = 0.00, size = 14, normalized size = 0.93 \[ \frac {\ln \left (b \,x^{2}+a \right )}{2 b} \]

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 = 1.34, size = 13, normalized size = 0.87 \[ \frac {\log \left (b x^{2} + a\right )}{2 \, b} \]

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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mupad [B]  time = 4.63, size = 13, normalized size = 0.87 \[ \frac {\ln \left (b\,x^2+a\right )}{2\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b*x^2),x)

[Out]

log(a + b*x^2)/(2*b)

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sympy [A]  time = 0.11, size = 10, normalized size = 0.67 \[ \frac {\log {\left (a + b x^{2} \right )}}{2 b} \]

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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