### 3.760 $$\int \frac{(a+b x)^4}{(a^2-b^2 x^2)^2} \, dx$$

Optimal. Leaf size=31 $\frac{4 a^2}{b (a-b x)}+\frac{4 a \log (a-b x)}{b}+x$

[Out]

x + (4*a^2)/(b*(a - b*x)) + (4*a*Log[a - b*x])/b

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Rubi [A]  time = 0.0214676, antiderivative size = 31, 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} $\frac{4 a^2}{b (a-b x)}+\frac{4 a \log (a-b x)}{b}+x$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x)^4/(a^2 - b^2*x^2)^2,x]

[Out]

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

Mathematica [A]  time = 0.0195169, size = 32, normalized size = 1.03 $-\frac{4 a^2}{b (b x-a)}+\frac{4 a \log (a-b x)}{b}+x$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + b*x)^4/(a^2 - b^2*x^2)^2,x]

[Out]

x - (4*a^2)/(b*(-a + b*x)) + (4*a*Log[a - b*x])/b

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Maple [A]  time = 0.043, size = 34, normalized size = 1.1 \begin{align*} x-4\,{\frac{{a}^{2}}{b \left ( bx-a \right ) }}+4\,{\frac{a\ln \left ( bx-a \right ) }{b}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^4/(-b^2*x^2+a^2)^2,x)

[Out]

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

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Maxima [A]  time = 1.02119, size = 45, normalized size = 1.45 \begin{align*} -\frac{4 \, a^{2}}{b^{2} x - a b} + x + \frac{4 \, a \log \left (b x - a\right )}{b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4/(-b^2*x^2+a^2)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.75862, size = 97, normalized size = 3.13 \begin{align*} \frac{b^{2} x^{2} - a b x - 4 \, a^{2} + 4 \,{\left (a b x - a^{2}\right )} \log \left (b x - a\right )}{b^{2} x - a b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4/(-b^2*x^2+a^2)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.341498, size = 26, normalized size = 0.84 \begin{align*} - \frac{4 a^{2}}{- a b + b^{2} x} + \frac{4 a \log{\left (- a + b x \right )}}{b} + x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**4/(-b**2*x**2+a**2)**2,x)

[Out]

-4*a**2/(-a*b + b**2*x) + 4*a*log(-a + b*x)/b + x

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Giac [A]  time = 1.24342, size = 46, normalized size = 1.48 \begin{align*} x + \frac{4 \, a \log \left ({\left | b x - a \right |}\right )}{b} - \frac{4 \, a^{2}}{{\left (b x - a\right )} b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4/(-b^2*x^2+a^2)^2,x, algorithm="giac")

[Out]

x + 4*a*log(abs(b*x - a))/b - 4*a^2/((b*x - a)*b)