3.1510 \(\int \frac {1}{a^2+2 a b x+b^2 x^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {27, 32} \[ -\frac {1}{b (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^(-1),x]

[Out]

-(1/(b*(a + b*x)))

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin {align*} \int \frac {1}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac {1}{(a+b x)^2} \, dx\\ &=-\frac {1}{b (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 12, normalized size = 1.00 \[ -\frac {1}{b (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(1/(b*(a + b*x)))

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fricas [A]  time = 0.95, size = 13, normalized size = 1.08 \[ -\frac {1}{b^{2} x + a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(b^2*x + a*b)

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giac [A]  time = 0.15, size = 12, normalized size = 1.00 \[ -\frac {1}{{\left (b x + a\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/((b*x + a)*b)

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maple [A]  time = 0.05, size = 13, normalized size = 1.08 \[ -\frac {1}{\left (b x +a \right ) b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b^2*x^2+2*a*b*x+a^2),x)

[Out]

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

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maxima [A]  time = 1.33, size = 13, normalized size = 1.08 \[ -\frac {1}{b^{2} x + a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(b^2*x + a*b)

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mupad [B]  time = 0.03, size = 12, normalized size = 1.00 \[ -\frac {1}{b\,\left (a+b\,x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a^2 + b^2*x^2 + 2*a*b*x),x)

[Out]

-1/(b*(a + b*x))

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sympy [A]  time = 0.13, 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**2*x**2+2*a*b*x+a**2),x)

[Out]

-1/(a*b + b**2*x)

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