### 3.761 $$\int \frac{(a+b x)^3}{(a^2-b^2 x^2)^2} \, dx$$

Optimal. Leaf size=26 $\frac{2 a}{b (a-b x)}+\frac{\log (a-b x)}{b}$

[Out]

(2*a)/(b*(a - b*x)) + Log[a - b*x]/b

________________________________________________________________________________________

Rubi [A]  time = 0.017393, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.091, Rules used = {627, 43} $\frac{2 a}{b (a-b x)}+\frac{\log (a-b x)}{b}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x)^3/(a^2 - b^2*x^2)^2,x]

[Out]

(2*a)/(b*(a - b*x)) + Log[a - b*x]/b

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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0077067, size = 23, normalized size = 0.88 $\frac{\frac{2 a}{a-b x}+\log (a-b x)}{b}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((2*a)/(a - b*x) + Log[a - b*x])/b

________________________________________________________________________________________

Maple [A]  time = 0.043, size = 29, normalized size = 1.1 \begin{align*} -2\,{\frac{a}{b \left ( bx-a \right ) }}+{\frac{\ln \left ( bx-a \right ) }{b}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/b*a/(b*x-a)+1/b*ln(b*x-a)

________________________________________________________________________________________

Maxima [A]  time = 1.07968, size = 38, normalized size = 1.46 \begin{align*} -\frac{2 \, a}{b^{2} x - a b} + \frac{\log \left (b x - a\right )}{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)^2,x, algorithm="maxima")

[Out]

-2*a/(b^2*x - a*b) + log(b*x - a)/b

________________________________________________________________________________________

Fricas [A]  time = 1.74987, size = 62, normalized size = 2.38 \begin{align*} \frac{{\left (b x - a\right )} \log \left (b x - a\right ) - 2 \, a}{b^{2} x - a 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)^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.320725, size = 19, normalized size = 0.73 \begin{align*} - \frac{2 a}{- a b + b^{2} x} + \frac{\log{\left (- a + b x \right )}}{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)**2,x)

[Out]

-2*a/(-a*b + b**2*x) + log(-a + b*x)/b

________________________________________________________________________________________

Giac [A]  time = 1.25697, size = 39, normalized size = 1.5 \begin{align*} \frac{\log \left ({\left | b x - a \right |}\right )}{b} - \frac{2 \, a}{{\left (b x - a\right )} 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)^2,x, algorithm="giac")

[Out]

log(abs(b*x - a))/b - 2*a/((b*x - a)*b)