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

Optimal. Leaf size=16 \[ -\frac{1}{2 b \left (a+b x^2\right )} \]

[Out]

-1/(2*b*(a + b*x^2))

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Rubi [A]  time = 0.0029703, antiderivative size = 16, 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 = {261} \[ -\frac{1}{2 b \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*x^2)^2,x]

[Out]

-1/(2*b*(a + b*x^2))

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

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

Mathematica [A]  time = 0.0018684, size = 16, normalized size = 1. \[ -\frac{1}{2 b \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*x^2)^2,x]

[Out]

-1/(2*b*(a + b*x^2))

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Maple [A]  time = 0.001, size = 15, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,b \left ( b{x}^{2}+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 2.07251, size = 19, normalized size = 1.19 \begin{align*} -\frac{1}{2 \,{\left (b x^{2} + a\right )} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/((b*x^2 + a)*b)

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Fricas [A]  time = 1.2853, size = 30, normalized size = 1.88 \begin{align*} -\frac{1}{2 \,{\left (b^{2} x^{2} + a b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(b^2*x^2 + a*b)

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Sympy [A]  time = 0.296921, size = 15, normalized size = 0.94 \begin{align*} - \frac{1}{2 a b + 2 b^{2} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(2*a*b + 2*b**2*x**2)

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Giac [A]  time = 3.56301, size = 19, normalized size = 1.19 \begin{align*} -\frac{1}{2 \,{\left (b x^{2} + a\right )} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/((b*x^2 + a)*b)