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

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

[Out]

x/(a - b*x^2)

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Rubi [A]  time = 0.00915279, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{x}{a-b x^2} \]

Antiderivative was successfully verified.

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

[Out]

x/(a - b*x^2)

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Rubi in Sympy [A]  time = 6.11514, size = 7, normalized size = 0.58 \[ \frac{x}{a - b x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)/(-b*x**2+a)**2,x)

[Out]

x/(a - b*x**2)

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Mathematica [A]  time = 0.00900784, 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 \[ -{\frac{x}{b{x}^{2}-a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)/(-b*x^2+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.34544, size = 19, normalized size = 1.58 \[ -\frac{x}{b x^{2} - a} \]

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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Fricas [A]  time = 0.214954, size = 19, normalized size = 1.58 \[ -\frac{x}{b x^{2} - a} \]

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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Sympy [A]  time = 1.30033, 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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GIAC/XCAS [A]  time = 0.219432, size = 19, normalized size = 1.58 \[ -\frac{x}{b x^{2} - a} \]

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)