\(\int \frac {1}{a^2+2 a b x+b^2 x^2} \, dx\) [1510]

Optimal result

Integrand size = 18, antiderivative size = 12 \[ \int \frac {1}{a^2+2 a b x+b^2 x^2} \, dx=-\frac {1}{b (a+b x)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {27, 32} \[ \int \frac {1}{a^2+2 a b x+b^2 x^2} \, dx=-\frac {1}{b (a+b x)} \]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a^2+2 a b x+b^2 x^2} \, dx=-\frac {1}{b (a+b x)} \]

[In]

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

[Out]

-(1/(b*(a + b*x)))

Maple [A] (verified)

Time = 2.18 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.08

method	result	size
gosper	\(-\frac {1}{b \left (b x +a \right )}\)	\(13\)
default	\(-\frac {1}{b \left (b x +a \right )}\)	\(13\)
norman	\(\frac {x}{a \left (b x +a \right )}\)	\(13\)
risch	\(-\frac {1}{b \left (b x +a \right )}\)	\(13\)
parallelrisch	\(-\frac {1}{b \left (b x +a \right )}\)	\(13\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.08 \[ \int \frac {1}{a^2+2 a b x+b^2 x^2} \, dx=-\frac {1}{b^{2} x + a b} \]

[In]

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

[Out]

-1/(b^2*x + a*b)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {1}{a^2+2 a b x+b^2 x^2} \, dx=- \frac {1}{a b + b^{2} x} \]

[In]

integrate(1/(b**2*x**2+2*a*b*x+a**2),x)

[Out]

-1/(a*b + b**2*x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.08 \[ \int \frac {1}{a^2+2 a b x+b^2 x^2} \, dx=-\frac {1}{b^{2} x + a b} \]

[In]

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

[Out]

-1/(b^2*x + a*b)

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a^2+2 a b x+b^2 x^2} \, dx=-\frac {1}{{\left (b x + a\right )} b} \]

[In]

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

[Out]

-1/((b*x + a)*b)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a^2+2 a b x+b^2 x^2} \, dx=-\frac {1}{b\,\left (a+b\,x\right )} \]

[In]

int(1/(a^2 + b^2*x^2 + 2*a*b*x),x)

[Out]

-1/(b*(a + b*x))