3.175 \(\int \frac {x}{(a+b x^2)^3} \, dx\)

Optimal. Leaf size=16 \[ -\frac {1}{4 b \left (a+b x^2\right )^2} \]

[Out]

-1/4/b/(b*x^2+a)^2

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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 = {261} \[ -\frac {1}{4 b \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*x^2)^3,x]

[Out]

-1/(4*b*(a + b*x^2)^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 )^3} \, dx &=-\frac {1}{4 b \left (a+b x^2\right )^2}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 1.00 \[ -\frac {1}{4 b \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*x^2)^3,x]

[Out]

-1/4*1/(b*(a + b*x^2)^2)

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fricas [A]  time = 0.90, size = 26, normalized size = 1.62 \[ -\frac {1}{4 \, {\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x^2+a)^3,x, algorithm="fricas")

[Out]

-1/4/(b^3*x^4 + 2*a*b^2*x^2 + a^2*b)

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giac [A]  time = 0.63, size = 14, normalized size = 0.88 \[ -\frac {1}{4 \, {\left (b x^{2} + a\right )}^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x^2+a)^3,x, algorithm="giac")

[Out]

-1/4/((b*x^2 + a)^2*b)

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maple [A]  time = 0.00, size = 15, normalized size = 0.94 \[ -\frac {1}{4 \left (b \,x^{2}+a \right )^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(b*x^2+a)^3,x)

[Out]

-1/4/b/(b*x^2+a)^2

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maxima [A]  time = 1.33, size = 14, normalized size = 0.88 \[ -\frac {1}{4 \, {\left (b x^{2} + a\right )}^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x^2+a)^3,x, algorithm="maxima")

[Out]

-1/4/((b*x^2 + a)^2*b)

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mupad [B]  time = 4.62, size = 28, normalized size = 1.75 \[ -\frac {1}{4\,a^2\,b+8\,a\,b^2\,x^2+4\,b^3\,x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b*x^2)^3,x)

[Out]

-1/(4*a^2*b + 4*b^3*x^4 + 8*a*b^2*x^2)

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sympy [A]  time = 0.24, size = 27, normalized size = 1.69 \[ - \frac {1}{4 a^{2} b + 8 a b^{2} x^{2} + 4 b^{3} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x**2+a)**3,x)

[Out]

-1/(4*a**2*b + 8*a*b**2*x**2 + 4*b**3*x**4)

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