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

Optimal. Leaf size=87 \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{4 a^5 b}+\frac{1}{16 a^4 b (a-b x)}-\frac{3}{16 a^4 b (a+b x)}-\frac{1}{8 a^3 b (a+b x)^2}-\frac{1}{12 a^2 b (a+b x)^3} \]

[Out]

1/(16*a^4*b*(a - b*x)) - 1/(12*a^2*b*(a + b*x)^3) - 1/(8*a^3*b*(a + b*x)^2) - 3/
(16*a^4*b*(a + b*x)) + ArcTanh[(b*x)/a]/(4*a^5*b)

_______________________________________________________________________________________

Rubi [A]  time = 0.133175, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{4 a^5 b}+\frac{1}{16 a^4 b (a-b x)}-\frac{3}{16 a^4 b (a+b x)}-\frac{1}{8 a^3 b (a+b x)^2}-\frac{1}{12 a^2 b (a+b x)^3} \]

Antiderivative was successfully verified.

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

[Out]

1/(16*a^4*b*(a - b*x)) - 1/(12*a^2*b*(a + b*x)^3) - 1/(8*a^3*b*(a + b*x)^2) - 3/
(16*a^4*b*(a + b*x)) + ArcTanh[(b*x)/a]/(4*a^5*b)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 27.3508, size = 70, normalized size = 0.8 \[ - \frac{1}{12 a^{2} b \left (a + b x\right )^{3}} - \frac{1}{8 a^{3} b \left (a + b x\right )^{2}} - \frac{3}{16 a^{4} b \left (a + b x\right )} + \frac{1}{16 a^{4} b \left (a - b x\right )} + \frac{\operatorname{atanh}{\left (\frac{b x}{a} \right )}}{4 a^{5} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**2/(-b**2*x**2+a**2)**2,x)

[Out]

-1/(12*a**2*b*(a + b*x)**3) - 1/(8*a**3*b*(a + b*x)**2) - 3/(16*a**4*b*(a + b*x)
) + 1/(16*a**4*b*(a - b*x)) + atanh(b*x/a)/(4*a**5*b)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0794748, size = 75, normalized size = 0.86 \[ \frac{\frac{2 a \left (-4 a^3+a^2 b x+6 a b^2 x^2+3 b^3 x^3\right )}{(a-b x) (a+b x)^3}-3 \log (a-b x)+3 \log (a+b x)}{24 a^5 b} \]

Antiderivative was successfully verified.

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

[Out]

((2*a*(-4*a^3 + a^2*b*x + 6*a*b^2*x^2 + 3*b^3*x^3))/((a - b*x)*(a + b*x)^3) - 3*
Log[a - b*x] + 3*Log[a + b*x])/(24*a^5*b)

_______________________________________________________________________________________

Maple [A]  time = 0.016, size = 94, normalized size = 1.1 \[ -{\frac{\ln \left ( bx-a \right ) }{8\,{a}^{5}b}}-{\frac{1}{16\,{a}^{4}b \left ( bx-a \right ) }}+{\frac{\ln \left ( bx+a \right ) }{8\,{a}^{5}b}}-{\frac{3}{16\,{a}^{4}b \left ( bx+a \right ) }}-{\frac{1}{8\,{a}^{3}b \left ( bx+a \right ) ^{2}}}-{\frac{1}{12\,{a}^{2}b \left ( bx+a \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8/a^5/b*ln(b*x-a)-1/16/a^4/b/(b*x-a)+1/8/a^5/b*ln(b*x+a)-3/16/a^4/b/(b*x+a)-1
/8/a^3/b/(b*x+a)^2-1/12/a^2/b/(b*x+a)^3

_______________________________________________________________________________________

Maxima [A]  time = 0.703372, size = 136, normalized size = 1.56 \[ -\frac{3 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + a^{2} b x - 4 \, a^{3}}{12 \,{\left (a^{4} b^{5} x^{4} + 2 \, a^{5} b^{4} x^{3} - 2 \, a^{7} b^{2} x - a^{8} b\right )}} + \frac{\log \left (b x + a\right )}{8 \, a^{5} b} - \frac{\log \left (b x - a\right )}{8 \, a^{5} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*(3*b^3*x^3 + 6*a*b^2*x^2 + a^2*b*x - 4*a^3)/(a^4*b^5*x^4 + 2*a^5*b^4*x^3 -
2*a^7*b^2*x - a^8*b) + 1/8*log(b*x + a)/(a^5*b) - 1/8*log(b*x - a)/(a^5*b)

_______________________________________________________________________________________

Fricas [A]  time = 0.212706, size = 203, normalized size = 2.33 \[ -\frac{6 \, a b^{3} x^{3} + 12 \, a^{2} b^{2} x^{2} + 2 \, a^{3} b x - 8 \, a^{4} - 3 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{3} - 2 \, a^{3} b x - a^{4}\right )} \log \left (b x + a\right ) + 3 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{3} - 2 \, a^{3} b x - a^{4}\right )} \log \left (b x - a\right )}{24 \,{\left (a^{5} b^{5} x^{4} + 2 \, a^{6} b^{4} x^{3} - 2 \, a^{8} b^{2} x - a^{9} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24*(6*a*b^3*x^3 + 12*a^2*b^2*x^2 + 2*a^3*b*x - 8*a^4 - 3*(b^4*x^4 + 2*a*b^3*x
^3 - 2*a^3*b*x - a^4)*log(b*x + a) + 3*(b^4*x^4 + 2*a*b^3*x^3 - 2*a^3*b*x - a^4)
*log(b*x - a))/(a^5*b^5*x^4 + 2*a^6*b^4*x^3 - 2*a^8*b^2*x - a^9*b)

_______________________________________________________________________________________

Sympy [A]  time = 2.53714, size = 92, normalized size = 1.06 \[ - \frac{- 4 a^{3} + a^{2} b x + 6 a b^{2} x^{2} + 3 b^{3} x^{3}}{- 12 a^{8} b - 24 a^{7} b^{2} x + 24 a^{5} b^{4} x^{3} + 12 a^{4} b^{5} x^{4}} + \frac{- \frac{\log{\left (- \frac{a}{b} + x \right )}}{8} + \frac{\log{\left (\frac{a}{b} + x \right )}}{8}}{a^{5} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-4*a**3 + a**2*b*x + 6*a*b**2*x**2 + 3*b**3*x**3)/(-12*a**8*b - 24*a**7*b**2*x
+ 24*a**5*b**4*x**3 + 12*a**4*b**5*x**4) + (-log(-a/b + x)/8 + log(a/b + x)/8)/
(a**5*b)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.218057, size = 134, normalized size = 1.54 \[ -\frac{{\rm ln}\left ({\left | -\frac{2 \, a}{b x + a} + 1 \right |}\right )}{8 \, a^{5} b} + \frac{1}{32 \, a^{5} b{\left (\frac{2 \, a}{b x + a} - 1\right )}} - \frac{\frac{9 \, a^{2} b^{5}}{b x + a} + \frac{6 \, a^{3} b^{5}}{{\left (b x + a\right )}^{2}} + \frac{4 \, a^{4} b^{5}}{{\left (b x + a\right )}^{3}}}{48 \, a^{6} b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*ln(abs(-2*a/(b*x + a) + 1))/(a^5*b) + 1/32/(a^5*b*(2*a/(b*x + a) - 1)) - 1/
48*(9*a^2*b^5/(b*x + a) + 6*a^3*b^5/(b*x + a)^2 + 4*a^4*b^5/(b*x + a)^3)/(a^6*b^
6)