3.1061 \(\int \frac{1}{(a+b x)^2 (a c-b c x)^2} \, dx\)

Optimal. Leaf size=46 \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{2 a^3 b c^2}+\frac{x}{2 a^2 c^2 \left (a^2-b^2 x^2\right )} \]

[Out]

x/(2*a^2*c^2*(a^2 - b^2*x^2)) + ArcTanh[(b*x)/a]/(2*a^3*b*c^2)

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Rubi [A]  time = 0.0448485, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{2 a^3 b c^2}+\frac{x}{2 a^2 c^2 \left (a^2-b^2 x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

x/(2*a^2*c^2*(a^2 - b^2*x^2)) + ArcTanh[(b*x)/a]/(2*a^3*b*c^2)

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Rubi in Sympy [A]  time = 14.1912, size = 36, normalized size = 0.78 \[ \frac{x}{2 a^{2} c^{2} \left (a^{2} - b^{2} x^{2}\right )} + \frac{\operatorname{atanh}{\left (\frac{b x}{a} \right )}}{2 a^{3} b c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/(2*a**2*c**2*(a**2 - b**2*x**2)) + atanh(b*x/a)/(2*a**3*b*c**2)

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Mathematica [A]  time = 0.0366393, size = 74, normalized size = 1.61 \[ \frac{\left (b^2 x^2-a^2\right ) \log (a-b x)+\left (a^2-b^2 x^2\right ) \log (a+b x)+2 a b x}{4 a^3 b c^2 (a-b x) (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.015, size = 76, normalized size = 1.7 \[{\frac{\ln \left ( bx+a \right ) }{4\,{c}^{2}{a}^{3}b}}-{\frac{1}{4\,{c}^{2}{a}^{2}b \left ( bx+a \right ) }}-{\frac{\ln \left ( bx-a \right ) }{4\,{c}^{2}{a}^{3}b}}-{\frac{1}{4\,{c}^{2}{a}^{2}b \left ( bx-a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.33726, size = 86, normalized size = 1.87 \[ -\frac{x}{2 \,{\left (a^{2} b^{2} c^{2} x^{2} - a^{4} c^{2}\right )}} + \frac{\log \left (b x + a\right )}{4 \, a^{3} b c^{2}} - \frac{\log \left (b x - a\right )}{4 \, a^{3} b c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.203154, size = 103, normalized size = 2.24 \[ -\frac{2 \, a b x -{\left (b^{2} x^{2} - a^{2}\right )} \log \left (b x + a\right ) +{\left (b^{2} x^{2} - a^{2}\right )} \log \left (b x - a\right )}{4 \,{\left (a^{3} b^{3} c^{2} x^{2} - a^{5} b c^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.58745, size = 49, normalized size = 1.07 \[ - \frac{x}{- 2 a^{4} c^{2} + 2 a^{2} b^{2} c^{2} x^{2}} + \frac{- \frac{\log{\left (- \frac{a}{b} + x \right )}}{4} + \frac{\log{\left (\frac{a}{b} + x \right )}}{4}}{a^{3} b c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.209528, size = 112, normalized size = 2.43 \[ -\frac{1}{4 \,{\left (b c x - a c\right )} a^{2} b c} + \frac{{\rm ln}\left ({\left | -\frac{2 \, a c}{b c x - a c} - 1 \right |}\right )}{4 \, a^{3} b c^{2}} + \frac{1}{8 \, a^{3} b{\left (\frac{2 \, a c}{b c x - a c} + 1\right )} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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