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

Optimal. Leaf size=12 \[ -\frac{1}{b (a+b x)} \]

[Out]

-(1/(b*(a + b*x)))

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Rubi [A]  time = 0.00547363, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{1}{b (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(-2),x]

[Out]

-(1/(b*(a + b*x)))

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Rubi in Sympy [A]  time = 0.638036, size = 8, normalized size = 0.67 \[ - \frac{1}{b \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**2,x)

[Out]

-1/(b*(a + b*x))

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Mathematica [A]  time = 0.00382892, size = 12, normalized size = 1. \[ -\frac{1}{b (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(-2),x]

[Out]

-(1/(b*(a + b*x)))

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Maple [A]  time = 0., size = 13, normalized size = 1.1 \[ -{\frac{1}{b \left ( bx+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)^2,x)

[Out]

-1/b/(b*x+a)

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Maxima [A]  time = 1.39585, size = 16, normalized size = 1.33 \[ -\frac{1}{{\left (b x + a\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/((b*x + a)*b)

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Fricas [A]  time = 0.213944, size = 18, normalized size = 1.5 \[ -\frac{1}{b^{2} x + a b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(b^2*x + a*b)

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Sympy [A]  time = 0.51837, size = 10, normalized size = 0.83 \[ - \frac{1}{a b + b^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+a)**2,x)

[Out]

-1/(a*b + b**2*x)

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GIAC/XCAS [A]  time = 0.202497, size = 16, normalized size = 1.33 \[ -\frac{1}{{\left (b x + a\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/((b*x + a)*b)