### 3.772 $$\int \frac{(a+b x)^3}{(a^2-b^2 x^2)^3} \, dx$$

Optimal. Leaf size=15 $\frac{1}{2 b (a-b x)^2}$

[Out]

1/(2*b*(a - b*x)^2)

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Rubi [A]  time = 0.0066662, antiderivative size = 15, 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} $\frac{1}{2 b (a-b x)^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x)^3/(a^2 - b^2*x^2)^3,x]

[Out]

1/(2*b*(a - b*x)^2)

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

Mathematica [A]  time = 0.001539, size = 15, normalized size = 1. $\frac{1}{2 b (a-b x)^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

1/(2*b*(a - b*x)^2)

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Maple [A]  time = 0.039, size = 15, normalized size = 1. \begin{align*}{\frac{1}{2\,b \left ( bx-a \right ) ^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3/(-b^2*x^2+a^2)^3,x)

[Out]

1/2/b/(b*x-a)^2

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Maxima [A]  time = 1.02489, size = 32, normalized size = 2.13 \begin{align*} \frac{1}{2 \,{\left (b^{3} x^{2} - 2 \, a b^{2} x + a^{2} b\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/(-b^2*x^2+a^2)^3,x, algorithm="maxima")

[Out]

1/2/(b^3*x^2 - 2*a*b^2*x + a^2*b)

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Fricas [A]  time = 1.6542, size = 47, normalized size = 3.13 \begin{align*} \frac{1}{2 \,{\left (b^{3} x^{2} - 2 \, a b^{2} x + a^{2} b\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/(-b^2*x^2+a^2)^3,x, algorithm="fricas")

[Out]

1/2/(b^3*x^2 - 2*a*b^2*x + a^2*b)

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Sympy [B]  time = 0.347631, size = 24, normalized size = 1.6 \begin{align*} \frac{1}{2 a^{2} b - 4 a b^{2} x + 2 b^{3} x^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3/(-b**2*x**2+a**2)**3,x)

[Out]

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

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Giac [A]  time = 1.19786, size = 19, normalized size = 1.27 \begin{align*} \frac{1}{2 \,{\left (b x - a\right )}^{2} b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3/(-b^2*x^2+a^2)^3,x, algorithm="giac")

[Out]

1/2/((b*x - a)^2*b)