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

Optimal. Leaf size=10 \[ -\frac{1}{x (a+b)^2} \]

[Out]

-(1/((a + b)^2*x))

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Rubi [A]  time = 0.0103761, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{1}{x (a+b)^2} \]

Antiderivative was successfully verified.

[In]  Int[(a*x + b*x)^(-2),x]

[Out]

-(1/((a + b)^2*x))

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Rubi in Sympy [A]  time = 2.26697, size = 8, normalized size = 0.8 \[ - \frac{1}{x \left (a + b\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(x*(a + b)**2)

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Mathematica [A]  time = 0.00447624, size = 10, normalized size = 1. \[ -\frac{1}{x (a+b)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*x + b*x)^(-2),x]

[Out]

-(1/((a + b)^2*x))

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Maple [A]  time = 0.002, size = 11, normalized size = 1.1 \[ -{\frac{1}{ \left ( a+b \right ) ^{2}x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a*x+b*x)^2,x)

[Out]

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

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Maxima [A]  time = 1.37352, size = 22, normalized size = 2.2 \[ -\frac{1}{{\left (a x + b x\right )}{\left (a + b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/((a*x + b*x)*(a + b))

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Fricas [A]  time = 0.205333, size = 24, normalized size = 2.4 \[ -\frac{1}{{\left (a^{2} + 2 \, a b + b^{2}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/((a^2 + 2*a*b + b^2)*x)

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Sympy [A]  time = 0.113945, size = 15, normalized size = 1.5 \[ - \frac{1}{x \left (a^{2} + 2 a b + b^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a*x+b*x)**2,x)

[Out]

-1/(x*(a**2 + 2*a*b + b**2))

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GIAC/XCAS [A]  time = 0.218333, size = 22, normalized size = 2.2 \[ -\frac{1}{{\left (a x + b x\right )}{\left (a + b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/((a*x + b*x)*(a + b))