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

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

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, 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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.45, size = 14, normalized size = 1.17 \[ -\frac {x}{b x^{2} - a} \]

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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giac [A]  time = 0.35, size = 14, normalized size = 1.17 \[ -\frac {x}{b x^{2} - a} \]

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)

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maple [A]  time = 0.01, size = 15, normalized size = 1.25 \[ -\frac {x}{b \,x^{2}-a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.05, size = 14, normalized size = 1.17 \[ -\frac {x}{b x^{2} - a} \]

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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mupad [B]  time = 0.00, size = 12, normalized size = 1.00 \[ \frac {x}{a-b\,x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)/(a - b*x^2)^2,x)

[Out]

x/(a - b*x^2)

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sympy [A]  time = 0.18, size = 8, normalized size = 0.67 \[ - \frac {x}{- a + b x^{2}} \]

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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