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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]