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

Optimal. Leaf size=12 \[ \frac{x}{a-b x^2} \]

[Out]

x/(a - b*x^2)

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Rubi [A]  time = 0.0043831, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {383} \[ \frac{x}{a-b x^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)/(-a + b*x^2)^2,x]

[Out]

x/(a - b*x^2)

Rule 383

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*x*(a + b*x^n)^(p + 1))/a, x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[a*d - b*c*(n*(p + 1) + 1), 0]

Rubi steps

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

Mathematica [A]  time = 0.007032, size = 14, normalized size = 1.17 \[ -\frac{x}{b x^2-a} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)/(-a + b*x^2)^2,x]

[Out]

-(x/(-a + b*x^2))

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Maple [A]  time = 0.006, size = 15, normalized size = 1.3 \begin{align*} -{\frac{x}{b{x}^{2}-a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.966137, size = 19, normalized size = 1.58 \begin{align*} -\frac{x}{b x^{2} - a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/(b*x^2 - a)

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Fricas [A]  time = 1.30814, size = 22, normalized size = 1.83 \begin{align*} -\frac{x}{b x^{2} - a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/(b*x^2 - a)

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Sympy [A]  time = 0.317147, size = 8, normalized size = 0.67 \begin{align*} - \frac{x}{- a + b x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/(-a + b*x**2)

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Giac [A]  time = 1.17046, size = 19, normalized size = 1.58 \begin{align*} -\frac{x}{b x^{2} - a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/(b*x^2 - a)