### 3.1521 $$\int \frac{1}{(a^2+2 a b x+b^2 x^2)^2} \, dx$$

Optimal. Leaf size=14 $-\frac{1}{3 b (a+b x)^3}$

[Out]

-1/(3*b*(a + b*x)^3)

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Rubi [A]  time = 0.0023577, antiderivative size = 14, normalized size of antiderivative = 1., 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} $-\frac{1}{3 b (a+b x)^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^(-2),x]

[Out]

-1/(3*b*(a + b*x)^3)

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}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{1}{(a+b x)^4} \, dx\\ &=-\frac{1}{3 b (a+b x)^3}\\ \end{align*}

Mathematica [A]  time = 0.0027517, size = 14, normalized size = 1. $-\frac{1}{3 b (a+b x)^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-1/(3*b*(a + b*x)^3)

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Maple [A]  time = 0.043, size = 13, normalized size = 0.9 \begin{align*} -{\frac{1}{3\,b \left ( bx+a \right ) ^{3}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/b/(b*x+a)^3

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Maxima [B]  time = 1.38165, size = 47, normalized size = 3.36 \begin{align*} -\frac{1}{3 \,{\left (b^{4} x^{3} + 3 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} x + a^{3} b\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/(b^4*x^3 + 3*a*b^3*x^2 + 3*a^2*b^2*x + a^3*b)

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Fricas [B]  time = 1.70062, size = 70, normalized size = 5. \begin{align*} -\frac{1}{3 \,{\left (b^{4} x^{3} + 3 \, a b^{3} x^{2} + 3 \, a^{2} b^{2} x + a^{3} b\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/(b^4*x^3 + 3*a*b^3*x^2 + 3*a^2*b^2*x + a^3*b)

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Sympy [B]  time = 0.395112, size = 37, normalized size = 2.64 \begin{align*} - \frac{1}{3 a^{3} b + 9 a^{2} b^{2} x + 9 a b^{3} x^{2} + 3 b^{4} x^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(3*a**3*b + 9*a**2*b**2*x + 9*a*b**3*x**2 + 3*b**4*x**3)

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Giac [A]  time = 1.20085, size = 16, normalized size = 1.14 \begin{align*} -\frac{1}{3 \,{\left (b x + a\right )}^{3} b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/((b*x + a)^3*b)