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

Optimal. Leaf size=42 \[ -\frac{2 b \log (x)}{a^3}+\frac{2 b \log (a+b x)}{a^3}-\frac{b}{a^2 (a+b x)}-\frac{1}{a^2 x} \]

[Out]

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

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Rubi [A]  time = 0.0489097, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{2 b \log (x)}{a^3}+\frac{2 b \log (a+b x)}{a^3}-\frac{b}{a^2 (a+b x)}-\frac{1}{a^2 x} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a + b*x)^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 8.45152, size = 39, normalized size = 0.93 \[ - \frac{b}{a^{2} \left (a + b x\right )} - \frac{1}{a^{2} x} - \frac{2 b \log{\left (x \right )}}{a^{3}} + \frac{2 b \log{\left (a + b x \right )}}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0814997, size = 35, normalized size = 0.83 \[ -\frac{a \left (\frac{b}{a+b x}+\frac{1}{x}\right )-2 b \log (a+b x)+2 b \log (x)}{a^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.014, size = 43, normalized size = 1. \[ -{\frac{1}{{a}^{2}x}}-{\frac{b}{{a}^{2} \left ( bx+a \right ) }}-2\,{\frac{b\ln \left ( x \right ) }{{a}^{3}}}+2\,{\frac{b\ln \left ( bx+a \right ) }{{a}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(b*x+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.36734, size = 61, normalized size = 1.45 \[ -\frac{2 \, b x + a}{a^{2} b x^{2} + a^{3} x} + \frac{2 \, b \log \left (b x + a\right )}{a^{3}} - \frac{2 \, b \log \left (x\right )}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.213742, size = 85, normalized size = 2.02 \[ -\frac{2 \, a b x + a^{2} - 2 \,{\left (b^{2} x^{2} + a b x\right )} \log \left (b x + a\right ) + 2 \,{\left (b^{2} x^{2} + a b x\right )} \log \left (x\right )}{a^{3} b x^{2} + a^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.57837, size = 36, normalized size = 0.86 \[ - \frac{a + 2 b x}{a^{3} x + a^{2} b x^{2}} + \frac{2 b \left (- \log{\left (x \right )} + \log{\left (\frac{a}{b} + x \right )}\right )}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(b*x+a)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.212444, size = 70, normalized size = 1.67 \[ -\frac{2 \, b{\rm ln}\left ({\left | -\frac{a}{b x + a} + 1 \right |}\right )}{a^{3}} - \frac{b}{{\left (b x + a\right )} a^{2}} + \frac{b}{a^{3}{\left (\frac{a}{b x + a} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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