### 3.762 $$\int \frac{(a+b x)^2}{(a^2-b^2 x^2)^2} \, dx$$

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

[Out]

1/(b*(a - b*x))

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Rubi [A]  time = 0.0061974, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.091, Rules used = {627, 32} $\frac{1}{b (a-b x)}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x)^2/(a^2 - b^2*x^2)^2,x]

[Out]

1/(b*(a - b*x))

Rule 627

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c*x)/e)^
p, x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && I
ntegerQ[m + 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{(a+b x)^2}{\left (a^2-b^2 x^2\right )^2} \, dx &=\int \frac{1}{(a-b x)^2} \, dx\\ &=\frac{1}{b (a-b x)}\\ \end{align*}

Mathematica [A]  time = 0.0020324, size = 12, normalized size = 1. $\frac{1}{b (a-b x)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

1/(b*(a - b*x))

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Maple [A]  time = 0.037, size = 15, normalized size = 1.3 \begin{align*} -{\frac{1}{b \left ( bx-a \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.32449, size = 19, normalized size = 1.58 \begin{align*} -\frac{1}{b^{2} x - a b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(b^2*x - a*b)

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Fricas [A]  time = 1.70612, size = 24, normalized size = 2. \begin{align*} -\frac{1}{b^{2} x - a b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(b^2*x - a*b)

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Sympy [A]  time = 0.296404, size = 10, normalized size = 0.83 \begin{align*} - \frac{1}{- a b + b^{2} x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(-a*b + b**2*x)

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Giac [A]  time = 1.19993, size = 19, normalized size = 1.58 \begin{align*} -\frac{1}{{\left (b x - a\right )} b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/((b*x - a)*b)