3.324 \(\int \frac{1}{a x^2+b x^2} \, dx\)

Optimal. Leaf size=10 \[ -\frac{1}{x (a+b)} \]

[Out]

-(1/((a + b)*x))

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Rubi [A]  time = 0.00727225, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{1}{x (a+b)} \]

Antiderivative was successfully verified.

[In]  Int[(a*x^2 + b*x^2)^(-1),x]

[Out]

-(1/((a + b)*x))

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Rubi in Sympy [A]  time = 2.29567, size = 7, normalized size = 0.7 \[ - \frac{1}{x \left (a + b\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a*x**2+b*x**2),x)

[Out]

-1/(x*(a + b))

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Mathematica [A]  time = 0.00131545, size = 10, normalized size = 1. \[ -\frac{1}{x (a+b)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*x^2 + b*x^2)^(-1),x]

[Out]

-(1/((a + b)*x))

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Maple [A]  time = 0.001, size = 11, normalized size = 1.1 \[ -{\frac{1}{ \left ( a+b \right ) x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a*x^2+b*x^2),x)

[Out]

-1/(a+b)/x

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Maxima [A]  time = 1.39618, size = 14, normalized size = 1.4 \[ -\frac{1}{{\left (a + b\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a*x^2 + b*x^2),x, algorithm="maxima")

[Out]

-1/((a + b)*x)

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Fricas [A]  time = 0.200741, size = 14, normalized size = 1.4 \[ -\frac{1}{{\left (a + b\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a*x^2 + b*x^2),x, algorithm="fricas")

[Out]

-1/((a + b)*x)

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Sympy [A]  time = 0.095939, size = 7, normalized size = 0.7 \[ - \frac{1}{x \left (a + b\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a*x**2+b*x**2),x)

[Out]

-1/(x*(a + b))

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GIAC/XCAS [A]  time = 0.217472, size = 14, normalized size = 1.4 \[ -\frac{1}{{\left (a + b\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a*x^2 + b*x^2),x, algorithm="giac")

[Out]

-1/((a + b)*x)