3.764 \(\int \frac {1}{(a+b x) (a^2-b^2 x^2)^2} \, dx\)

Optimal. Leaf size=70 \[ \frac {3 \tanh ^{-1}\left (\frac {b x}{a}\right )}{8 a^4 b}+\frac {1}{8 a^3 b (a-b x)}-\frac {1}{4 a^3 b (a+b x)}-\frac {1}{8 a^2 b (a+b x)^2} \]

[Out]

1/8/a^3/b/(-b*x+a)-1/8/a^2/b/(b*x+a)^2-1/4/a^3/b/(b*x+a)+3/8*arctanh(b*x/a)/a^4/b

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Rubi [A]  time = 0.05, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {627, 44, 208} \[ \frac {1}{8 a^3 b (a-b x)}-\frac {1}{4 a^3 b (a+b x)}-\frac {1}{8 a^2 b (a+b x)^2}+\frac {3 \tanh ^{-1}\left (\frac {b x}{a}\right )}{8 a^4 b} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)*(a^2 - b^2*x^2)^2),x]

[Out]

1/(8*a^3*b*(a - b*x)) - 1/(8*a^2*b*(a + b*x)^2) - 1/(4*a^3*b*(a + b*x)) + (3*ArcTanh[(b*x)/a])/(8*a^4*b)

Rule 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/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]))

Rubi steps

\begin {align*} \int \frac {1}{(a+b x) \left (a^2-b^2 x^2\right )^2} \, dx &=\int \frac {1}{(a-b x)^2 (a+b x)^3} \, dx\\ &=\int \left (\frac {1}{8 a^3 (a-b x)^2}+\frac {1}{4 a^2 (a+b x)^3}+\frac {1}{4 a^3 (a+b x)^2}+\frac {3}{8 a^3 \left (a^2-b^2 x^2\right )}\right ) \, dx\\ &=\frac {1}{8 a^3 b (a-b x)}-\frac {1}{8 a^2 b (a+b x)^2}-\frac {1}{4 a^3 b (a+b x)}+\frac {3 \int \frac {1}{a^2-b^2 x^2} \, dx}{8 a^3}\\ &=\frac {1}{8 a^3 b (a-b x)}-\frac {1}{8 a^2 b (a+b x)^2}-\frac {1}{4 a^3 b (a+b x)}+\frac {3 \tanh ^{-1}\left (\frac {b x}{a}\right )}{8 a^4 b}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 87, normalized size = 1.24 \[ -\frac {3 \log (a-b x)}{16 a^4 b}+\frac {3 \log (a+b x)}{16 a^4 b}-\frac {1}{8 a^3 b (b x-a)}-\frac {1}{4 a^3 b (a+b x)}-\frac {1}{8 a^2 b (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)*(a^2 - b^2*x^2)^2),x]

[Out]

-1/8*1/(a^3*b*(-a + b*x)) - 1/(8*a^2*b*(a + b*x)^2) - 1/(4*a^3*b*(a + b*x)) - (3*Log[a - b*x])/(16*a^4*b) + (3
*Log[a + b*x])/(16*a^4*b)

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fricas [B]  time = 1.17, size = 136, normalized size = 1.94 \[ -\frac {6 \, a b^{2} x^{2} + 6 \, a^{2} b x - 4 \, a^{3} - 3 \, {\left (b^{3} x^{3} + a b^{2} x^{2} - a^{2} b x - a^{3}\right )} \log \left (b x + a\right ) + 3 \, {\left (b^{3} x^{3} + a b^{2} x^{2} - a^{2} b x - a^{3}\right )} \log \left (b x - a\right )}{16 \, {\left (a^{4} b^{4} x^{3} + a^{5} b^{3} x^{2} - a^{6} b^{2} x - a^{7} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)/(-b^2*x^2+a^2)^2,x, algorithm="fricas")

[Out]

-1/16*(6*a*b^2*x^2 + 6*a^2*b*x - 4*a^3 - 3*(b^3*x^3 + a*b^2*x^2 - a^2*b*x - a^3)*log(b*x + a) + 3*(b^3*x^3 + a
*b^2*x^2 - a^2*b*x - a^3)*log(b*x - a))/(a^4*b^4*x^3 + a^5*b^3*x^2 - a^6*b^2*x - a^7*b)

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giac [A]  time = 0.16, size = 79, normalized size = 1.13 \[ \frac {3 \, \log \left ({\left | b x + a \right |}\right )}{16 \, a^{4} b} - \frac {3 \, \log \left ({\left | b x - a \right |}\right )}{16 \, a^{4} b} - \frac {3 \, a b^{2} x^{2} + 3 \, a^{2} b x - 2 \, a^{3}}{8 \, {\left (b x + a\right )}^{2} {\left (b x - a\right )} a^{4} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)/(-b^2*x^2+a^2)^2,x, algorithm="giac")

[Out]

3/16*log(abs(b*x + a))/(a^4*b) - 3/16*log(abs(b*x - a))/(a^4*b) - 1/8*(3*a*b^2*x^2 + 3*a^2*b*x - 2*a^3)/((b*x
+ a)^2*(b*x - a)*a^4*b)

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maple [A]  time = 0.05, size = 79, normalized size = 1.13 \[ -\frac {1}{8 \left (b x +a \right )^{2} a^{2} b}-\frac {1}{8 \left (b x -a \right ) a^{3} b}-\frac {1}{4 \left (b x +a \right ) a^{3} b}-\frac {3 \ln \left (b x -a \right )}{16 a^{4} b}+\frac {3 \ln \left (b x +a \right )}{16 a^{4} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)/(-b^2*x^2+a^2)^2,x)

[Out]

-3/16/a^4/b*ln(b*x-a)-1/8/a^3/b/(b*x-a)+3/16/a^4/b*ln(b*x+a)-1/4/(b*x+a)/a^3/b-1/8/(b*x+a)^2/a^2/b

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maxima [A]  time = 1.45, size = 90, normalized size = 1.29 \[ -\frac {3 \, b^{2} x^{2} + 3 \, a b x - 2 \, a^{2}}{8 \, {\left (a^{3} b^{4} x^{3} + a^{4} b^{3} x^{2} - a^{5} b^{2} x - a^{6} b\right )}} + \frac {3 \, \log \left (b x + a\right )}{16 \, a^{4} b} - \frac {3 \, \log \left (b x - a\right )}{16 \, a^{4} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)/(-b^2*x^2+a^2)^2,x, algorithm="maxima")

[Out]

-1/8*(3*b^2*x^2 + 3*a*b*x - 2*a^2)/(a^3*b^4*x^3 + a^4*b^3*x^2 - a^5*b^2*x - a^6*b) + 3/16*log(b*x + a)/(a^4*b)
- 3/16*log(b*x - a)/(a^4*b)

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mupad [B]  time = 0.08, size = 70, normalized size = 1.00 \[ \frac {\frac {3\,x}{8\,a^2}-\frac {1}{4\,a\,b}+\frac {3\,b\,x^2}{8\,a^3}}{a^3+a^2\,b\,x-a\,b^2\,x^2-b^3\,x^3}+\frac {3\,\mathrm {atanh}\left (\frac {b\,x}{a}\right )}{8\,a^4\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a^2 - b^2*x^2)^2*(a + b*x)),x)

[Out]

((3*x)/(8*a^2) - 1/(4*a*b) + (3*b*x^2)/(8*a^3))/(a^3 - b^3*x^3 - a*b^2*x^2 + a^2*b*x) + (3*atanh((b*x)/a))/(8*
a^4*b)

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sympy [A]  time = 0.47, size = 85, normalized size = 1.21 \[ \frac {2 a^{2} - 3 a b x - 3 b^{2} x^{2}}{- 8 a^{6} b - 8 a^{5} b^{2} x + 8 a^{4} b^{3} x^{2} + 8 a^{3} b^{4} x^{3}} + \frac {- \frac {3 \log {\left (- \frac {a}{b} + x \right )}}{16} + \frac {3 \log {\left (\frac {a}{b} + x \right )}}{16}}{a^{4} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)/(-b**2*x**2+a**2)**2,x)

[Out]

(2*a**2 - 3*a*b*x - 3*b**2*x**2)/(-8*a**6*b - 8*a**5*b**2*x + 8*a**4*b**3*x**2 + 8*a**3*b**4*x**3) + (-3*log(-
a/b + x)/16 + 3*log(a/b + x)/16)/(a**4*b)

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