3.2.57 \(\int \frac {x}{(a+b x^2)^2} \, dx\) [157]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 16, 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 = {267} \begin {gather*} -\frac {1}{2 b \left (a+b x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(a + b*x^2)^2,x]

[Out]

-1/2*1/(b*(a + b*x^2))

Rule 267

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*}

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Mathematica [A]
time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} -\frac {1}{2 b \left (a+b x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*x^2)^2,x]

[Out]

-1/2*1/(b*(a + b*x^2))

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Maple [A]
time = 0.02, size = 15, normalized size = 0.94

method	result	size
gosper	\(-\frac {1}{2 b \left (b \,x^{2}+a \right )}\)	\(15\)
derivativedivides	\(-\frac {1}{2 b \left (b \,x^{2}+a \right )}\)	\(15\)
default	\(-\frac {1}{2 b \left (b \,x^{2}+a \right )}\)	\(15\)
norman	\(-\frac {1}{2 b \left (b \,x^{2}+a \right )}\)	\(15\)
risch	\(-\frac {1}{2 b \left (b \,x^{2}+a \right )}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 14, normalized size = 0.88 \begin {gather*} -\frac {1}{2 \, {\left (b x^{2} + a\right )} b} \end {gather*}

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 = 1.79, size = 15, normalized size = 0.94 \begin {gather*} -\frac {1}{2 \, {\left (b^{2} x^{2} + a b\right )}} \end {gather*}

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 = 0.05, size = 15, normalized size = 0.94 \begin {gather*} - \frac {1}{2 a b + 2 b^{2} x^{2}} \end {gather*}

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 = 0.72, size = 14, normalized size = 0.88 \begin {gather*} -\frac {1}{2 \, {\left (b x^{2} + a\right )} b} \end {gather*}

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)

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Mupad [B]
time = 0.03, size = 14, normalized size = 0.88 \begin {gather*} -\frac {1}{2\,b\,\left (b\,x^2+a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b*x^2)^2,x)

[Out]

-1/(2*b*(a + b*x^2))

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