3.272 \(\int \frac{1}{\left (b x+c x^2\right )^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0643623, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{2 c \log (x)}{b^3}+\frac{2 c \log (b+c x)}{b^3}-\frac{c}{b^2 (b+c x)}-\frac{1}{b^2 x} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^(-2),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.69667, size = 41, normalized size = 0.98 \[ - \frac{b + 2 c x}{b^{2} \left (b x + c x^{2}\right )} - \frac{2 c \log{\left (x \right )}}{b^{3}} + \frac{2 c \log{\left (b + c x \right )}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0674355, size = 35, normalized size = 0.83 \[ -\frac{b \left (\frac{c}{b+c x}+\frac{1}{x}\right )-2 c \log (b+c x)+2 c \log (x)}{b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x + c*x^2)^(-2),x]

[Out]

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

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Maple [A]  time = 0.017, size = 43, normalized size = 1. \[ -{\frac{1}{{b}^{2}x}}-{\frac{c}{{b}^{2} \left ( cx+b \right ) }}-2\,{\frac{c\ln \left ( x \right ) }{{b}^{3}}}+2\,{\frac{c\ln \left ( cx+b \right ) }{{b}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x^2+b*x)^2,x)

[Out]

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

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Maxima [A]  time = 0.684433, size = 61, normalized size = 1.45 \[ -\frac{2 \, c x + b}{b^{2} c x^{2} + b^{3} x} + \frac{2 \, c \log \left (c x + b\right )}{b^{3}} - \frac{2 \, c \log \left (x\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.227126, size = 85, normalized size = 2.02 \[ -\frac{2 \, b c x + b^{2} - 2 \,{\left (c^{2} x^{2} + b c x\right )} \log \left (c x + b\right ) + 2 \,{\left (c^{2} x^{2} + b c x\right )} \log \left (x\right )}{b^{3} c x^{2} + b^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.65318, size = 36, normalized size = 0.86 \[ - \frac{b + 2 c x}{b^{3} x + b^{2} c x^{2}} + \frac{2 c \left (- \log{\left (x \right )} + \log{\left (\frac{b}{c} + x \right )}\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c*x**2+b*x)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.208935, size = 61, normalized size = 1.45 \[ \frac{2 \, c{\rm ln}\left ({\left | c x + b \right |}\right )}{b^{3}} - \frac{2 \, c{\rm ln}\left ({\left | x \right |}\right )}{b^{3}} - \frac{2 \, c x + b}{{\left (c x^{2} + b x\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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