Integrand size = 11, antiderivative size = 42 \[ \int \frac {1}{x^3 (a+b x)} \, dx=-\frac {1}{2 a x^2}+\frac {b}{a^2 x}+\frac {b^2 \log (x)}{a^3}-\frac {b^2 \log (a+b x)}{a^3} \]
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Time = 0.01 (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} \[ \int \frac {1}{x^3 (a+b x)} \, dx=\frac {b^2 \log (x)}{a^3}-\frac {b^2 \log (a+b x)}{a^3}+\frac {b}{a^2 x}-\frac {1}{2 a x^2} \]
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Rule 46
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{a x^3}-\frac {b}{a^2 x^2}+\frac {b^2}{a^3 x}-\frac {b^3}{a^3 (a+b x)}\right ) \, dx \\ & = -\frac {1}{2 a x^2}+\frac {b}{a^2 x}+\frac {b^2 \log (x)}{a^3}-\frac {b^2 \log (a+b x)}{a^3} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^3 (a+b x)} \, dx=-\frac {1}{2 a x^2}+\frac {b}{a^2 x}+\frac {b^2 \log (x)}{a^3}-\frac {b^2 \log (a+b x)}{a^3} \]
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Time = 0.18 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.98
method | result | size |
default | \(-\frac {1}{2 a \,x^{2}}+\frac {b}{a^{2} x}+\frac {b^{2} \ln \left (x \right )}{a^{3}}-\frac {b^{2} \ln \left (b x +a \right )}{a^{3}}\) | \(41\) |
norman | \(\frac {\frac {b x}{a^{2}}-\frac {1}{2 a}}{x^{2}}+\frac {b^{2} \ln \left (x \right )}{a^{3}}-\frac {b^{2} \ln \left (b x +a \right )}{a^{3}}\) | \(41\) |
risch | \(\frac {\frac {b x}{a^{2}}-\frac {1}{2 a}}{x^{2}}+\frac {b^{2} \ln \left (-x \right )}{a^{3}}-\frac {b^{2} \ln \left (b x +a \right )}{a^{3}}\) | \(43\) |
parallelrisch | \(\frac {2 b^{2} \ln \left (x \right ) x^{2}-2 b^{2} \ln \left (b x +a \right ) x^{2}+2 a b x -a^{2}}{2 a^{3} x^{2}}\) | \(44\) |
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Time = 0.23 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.98 \[ \int \frac {1}{x^3 (a+b x)} \, dx=-\frac {2 \, b^{2} x^{2} \log \left (b x + a\right ) - 2 \, b^{2} x^{2} \log \left (x\right ) - 2 \, a b x + a^{2}}{2 \, a^{3} x^{2}} \]
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Time = 0.11 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.74 \[ \int \frac {1}{x^3 (a+b x)} \, dx=\frac {- a + 2 b x}{2 a^{2} x^{2}} + \frac {b^{2} \left (\log {\left (x \right )} - \log {\left (\frac {a}{b} + x \right )}\right )}{a^{3}} \]
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Time = 0.21 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x^3 (a+b x)} \, dx=-\frac {b^{2} \log \left (b x + a\right )}{a^{3}} + \frac {b^{2} \log \left (x\right )}{a^{3}} + \frac {2 \, b x - a}{2 \, a^{2} x^{2}} \]
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Time = 0.31 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x^3 (a+b x)} \, dx=-\frac {b^{2} \log \left ({\left | b x + a \right |}\right )}{a^{3}} + \frac {b^{2} \log \left ({\left | x \right |}\right )}{a^{3}} + \frac {2 \, a b x - a^{2}}{2 \, a^{3} x^{2}} \]
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Time = 0.07 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x^3 (a+b x)} \, dx=-\frac {\frac {a^2}{2}-a\,b\,x}{a^3\,x^2}-\frac {2\,b^2\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^3} \]
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