### 3.1533 $$\int \frac{1}{(a^2+2 a b x+b^2 x^2)^3} \, dx$$

Optimal. Leaf size=14 $-\frac{1}{5 b (a+b x)^5}$

[Out]

-1/(5*b*(a + b*x)^5)

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Rubi [A]  time = 0.0022241, 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} $-\frac{1}{5 b (a+b x)^5}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^(-3),x]

[Out]

-1/(5*b*(a + b*x)^5)

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 )^3} \, dx &=\int \frac{1}{(a+b x)^6} \, dx\\ &=-\frac{1}{5 b (a+b x)^5}\\ \end{align*}

Mathematica [A]  time = 0.0032114, size = 14, normalized size = 1. $-\frac{1}{5 b (a+b x)^5}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(-3),x]

[Out]

-1/(5*b*(a + b*x)^5)

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Maple [A]  time = 0.041, size = 13, normalized size = 0.9 \begin{align*} -{\frac{1}{5\,b \left ( bx+a \right ) ^{5}}} \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)^3,x)

[Out]

-1/5/b/(b*x+a)^5

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Maxima [B]  time = 1.11459, size = 77, normalized size = 5.5 \begin{align*} -\frac{1}{5 \,{\left (b^{6} x^{5} + 5 \, a b^{5} x^{4} + 10 \, a^{2} b^{4} x^{3} + 10 \, a^{3} b^{3} x^{2} + 5 \, a^{4} b^{2} x + a^{5} 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)^3,x, algorithm="maxima")

[Out]

-1/5/(b^6*x^5 + 5*a*b^5*x^4 + 10*a^2*b^4*x^3 + 10*a^3*b^3*x^2 + 5*a^4*b^2*x + a^5*b)

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Fricas [B]  time = 1.49589, size = 116, normalized size = 8.29 \begin{align*} -\frac{1}{5 \,{\left (b^{6} x^{5} + 5 \, a b^{5} x^{4} + 10 \, a^{2} b^{4} x^{3} + 10 \, a^{3} b^{3} x^{2} + 5 \, a^{4} b^{2} x + a^{5} 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)^3,x, algorithm="fricas")

[Out]

-1/5/(b^6*x^5 + 5*a*b^5*x^4 + 10*a^2*b^4*x^3 + 10*a^3*b^3*x^2 + 5*a^4*b^2*x + a^5*b)

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Sympy [B]  time = 0.558109, size = 61, normalized size = 4.36 \begin{align*} - \frac{1}{5 a^{5} b + 25 a^{4} b^{2} x + 50 a^{3} b^{3} x^{2} + 50 a^{2} b^{4} x^{3} + 25 a b^{5} x^{4} + 5 b^{6} x^{5}} \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)**3,x)

[Out]

-1/(5*a**5*b + 25*a**4*b**2*x + 50*a**3*b**3*x**2 + 50*a**2*b**4*x**3 + 25*a*b**5*x**4 + 5*b**6*x**5)

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Giac [A]  time = 1.12057, size = 16, normalized size = 1.14 \begin{align*} -\frac{1}{5 \,{\left (b x + a\right )}^{5} 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)^3,x, algorithm="giac")

[Out]

-1/5/((b*x + a)^5*b)