Optimal. Leaf size=43 \[ -\frac{b+2 c x}{b^2 \left (b x+c x^2\right )}-\frac{2 c \log (x)}{b^3}+\frac{2 c \log (b+c x)}{b^3} \]
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Rubi [A] time = 0.009857, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {614, 615} \[ -\frac{b+2 c x}{b^2 \left (b x+c x^2\right )}-\frac{2 c \log (x)}{b^3}+\frac{2 c \log (b+c x)}{b^3} \]
Antiderivative was successfully verified.
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Rule 614
Rule 615
Rubi steps
\begin{align*} \int \frac{1}{\left (b x+c x^2\right )^2} \, dx &=-\frac{b+2 c x}{b^2 \left (b x+c x^2\right )}-\frac{(2 c) \int \frac{1}{b x+c x^2} \, dx}{b^2}\\ &=-\frac{b+2 c x}{b^2 \left (b x+c x^2\right )}-\frac{2 c \log (x)}{b^3}+\frac{2 c \log (b+c x)}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0381159, size = 35, normalized size = 0.81 \[ -\frac{b \left (\frac{c}{b+c x}+\frac{1}{x}\right )-2 c \log (b+c x)+2 c \log (x)}{b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 43, normalized size = 1. \begin{align*} -{\frac{1}{{b}^{2}x}}-2\,{\frac{c\ln \left ( x \right ) }{{b}^{3}}}-{\frac{c}{{b}^{2} \left ( cx+b \right ) }}+2\,{\frac{c\ln \left ( cx+b \right ) }{{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13675, size = 61, normalized size = 1.42 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71884, size = 138, normalized size = 3.21 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.22442, size = 36, normalized size = 0.84 \begin{align*} - \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.3025, size = 61, normalized size = 1.42 \begin{align*} \frac{2 \, c \log \left ({\left | c x + b \right |}\right )}{b^{3}} - \frac{2 \, c \log \left ({\left | x \right |}\right )}{b^{3}} - \frac{2 \, c x + b}{{\left (c x^{2} + b x\right )} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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