Integrand size = 11, antiderivative size = 42 \[ \int \frac {1}{x^2 (a+b x)^2} \, dx=-\frac {1}{a^2 x}-\frac {b}{a^2 (a+b x)}-\frac {2 b \log (x)}{a^3}+\frac {2 b \log (a+b x)}{a^3} \]
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Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {46} \[ \int \frac {1}{x^2 (a+b x)^2} \, dx=-\frac {2 b \log (x)}{a^3}+\frac {2 b \log (a+b x)}{a^3}-\frac {b}{a^2 (a+b x)}-\frac {1}{a^2 x} \]
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Rule 46
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{a^2 x^2}-\frac {2 b}{a^3 x}+\frac {b^2}{a^2 (a+b x)^2}+\frac {2 b^2}{a^3 (a+b x)}\right ) \, dx \\ & = -\frac {1}{a^2 x}-\frac {b}{a^2 (a+b x)}-\frac {2 b \log (x)}{a^3}+\frac {2 b \log (a+b x)}{a^3} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x^2 (a+b x)^2} \, dx=-\frac {a \left (\frac {1}{x}+\frac {b}{a+b x}\right )+2 b \log (x)-2 b \log (a+b x)}{a^3} \]
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Time = 0.08 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.02
method | result | size |
default | \(-\frac {1}{a^{2} x}-\frac {b}{a^{2} \left (b x +a \right )}-\frac {2 b \ln \left (x \right )}{a^{3}}+\frac {2 b \ln \left (b x +a \right )}{a^{3}}\) | \(43\) |
risch | \(\frac {-\frac {2 b x}{a^{2}}-\frac {1}{a}}{x \left (b x +a \right )}+\frac {2 b \ln \left (-b x -a \right )}{a^{3}}-\frac {2 b \ln \left (x \right )}{a^{3}}\) | \(49\) |
norman | \(\frac {\frac {2 b^{2} x^{2}}{a^{3}}-\frac {1}{a}}{x \left (b x +a \right )}-\frac {2 b \ln \left (x \right )}{a^{3}}+\frac {2 b \ln \left (b x +a \right )}{a^{3}}\) | \(50\) |
parallelrisch | \(-\frac {2 \ln \left (x \right ) x^{2} b^{2}-2 \ln \left (b x +a \right ) x^{2} b^{2}+2 \ln \left (x \right ) x a b -2 \ln \left (b x +a \right ) x a b -2 x^{2} b^{2}+a^{2}}{a^{3} x \left (b x +a \right )}\) | \(70\) |
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Time = 0.25 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.50 \[ \int \frac {1}{x^2 (a+b x)^2} \, dx=-\frac {2 \, a b x + a^{2} - 2 \, {\left (b^{2} x^{2} + a b x\right )} \log \left (b x + a\right ) + 2 \, {\left (b^{2} x^{2} + a b x\right )} \log \left (x\right )}{a^{3} b x^{2} + a^{4} x} \]
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Time = 0.12 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^2 (a+b x)^2} \, dx=\frac {- a - 2 b x}{a^{3} x + a^{2} b x^{2}} + \frac {2 b \left (- \log {\left (x \right )} + \log {\left (\frac {a}{b} + x \right )}\right )}{a^{3}} \]
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Time = 0.18 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x^2 (a+b x)^2} \, dx=-\frac {2 \, b x + a}{a^{2} b x^{2} + a^{3} x} + \frac {2 \, b \log \left (b x + a\right )}{a^{3}} - \frac {2 \, b \log \left (x\right )}{a^{3}} \]
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Time = 0.28 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.24 \[ \int \frac {1}{x^2 (a+b x)^2} \, dx=-\frac {2 \, b \log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{3}} - \frac {b}{{\left (b x + a\right )} a^{2}} + \frac {b}{a^{3} {\left (\frac {a}{b x + a} - 1\right )}} \]
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Time = 0.19 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x^2 (a+b x)^2} \, dx=\frac {2\,b\,\ln \left (\frac {a+b\,x}{x}\right )}{a^3}-\frac {1}{a\,x\,\left (a+b\,x\right )}-\frac {2\,b}{a^2\,\left (a+b\,x\right )} \]
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