Integrand size = 11, antiderivative size = 43 \[ \int \frac {1}{x (a+b x)^3} \, dx=\frac {1}{2 a (a+b x)^2}+\frac {1}{a^2 (a+b x)}+\frac {\log (x)}{a^3}-\frac {\log (a+b x)}{a^3} \]
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Time = 0.02 (sec) , antiderivative size = 43, 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} \[ \int \frac {1}{x (a+b x)^3} \, dx=-\frac {\log (a+b x)}{a^3}+\frac {\log (x)}{a^3}+\frac {1}{a^2 (a+b x)}+\frac {1}{2 a (a+b x)^2} \]
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Rule 46
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{a^3 x}-\frac {b}{a (a+b x)^3}-\frac {b}{a^2 (a+b x)^2}-\frac {b}{a^3 (a+b x)}\right ) \, dx \\ & = \frac {1}{2 a (a+b x)^2}+\frac {1}{a^2 (a+b x)}+\frac {\log (x)}{a^3}-\frac {\log (a+b x)}{a^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x (a+b x)^3} \, dx=\frac {\frac {a (3 a+2 b x)}{(a+b x)^2}+2 \log (x)-2 \log (a+b x)}{2 a^3} \]
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Time = 0.18 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.95
method | result | size |
risch | \(\frac {\frac {b x}{a^{2}}+\frac {3}{2 a}}{\left (b x +a \right )^{2}}-\frac {\ln \left (b x +a \right )}{a^{3}}+\frac {\ln \left (-x \right )}{a^{3}}\) | \(41\) |
default | \(\frac {1}{2 a \left (b x +a \right )^{2}}+\frac {1}{a^{2} \left (b x +a \right )}+\frac {\ln \left (x \right )}{a^{3}}-\frac {\ln \left (b x +a \right )}{a^{3}}\) | \(42\) |
norman | \(\frac {-\frac {2 b x}{a^{2}}-\frac {3 b^{2} x^{2}}{2 a^{3}}}{\left (b x +a \right )^{2}}+\frac {\ln \left (x \right )}{a^{3}}-\frac {\ln \left (b x +a \right )}{a^{3}}\) | \(46\) |
parallelrisch | \(\frac {2 b^{2} \ln \left (x \right ) x^{2}-2 b^{2} \ln \left (b x +a \right ) x^{2}+4 a b \ln \left (x \right ) x -4 \ln \left (b x +a \right ) x a b -3 b^{2} x^{2}+2 a^{2} \ln \left (x \right )-2 a^{2} \ln \left (b x +a \right )-4 a b x}{2 a^{3} \left (b x +a \right )^{2}}\) | \(87\) |
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Time = 0.22 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.86 \[ \int \frac {1}{x (a+b x)^3} \, dx=\frac {2 \, a b x + 3 \, a^{2} - 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \log \left (b x + a\right ) + 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \log \left (x\right )}{2 \, {\left (a^{3} b^{2} x^{2} + 2 \, a^{4} b x + a^{5}\right )}} \]
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Time = 0.15 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x (a+b x)^3} \, dx=\frac {3 a + 2 b x}{2 a^{4} + 4 a^{3} b x + 2 a^{2} b^{2} x^{2}} + \frac {\log {\left (x \right )} - \log {\left (\frac {a}{b} + x \right )}}{a^{3}} \]
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Time = 0.22 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.19 \[ \int \frac {1}{x (a+b x)^3} \, dx=\frac {2 \, b x + 3 \, a}{2 \, {\left (a^{2} b^{2} x^{2} + 2 \, a^{3} b x + a^{4}\right )}} - \frac {\log \left (b x + a\right )}{a^{3}} + \frac {\log \left (x\right )}{a^{3}} \]
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Time = 0.30 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x (a+b x)^3} \, dx=-\frac {\log \left ({\left | b x + a \right |}\right )}{a^{3}} + \frac {\log \left ({\left | x \right |}\right )}{a^{3}} + \frac {2 \, a b x + 3 \, a^{2}}{2 \, {\left (b x + a\right )}^{2} a^{3}} \]
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Time = 0.12 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x (a+b x)^3} \, dx=\frac {\frac {1}{a^2+b\,x\,a}-\frac {\ln \left (\frac {a+b\,x}{x}\right )}{a^2}}{a}+\frac {1}{2\,a\,{\left (a+b\,x\right )}^2} \]
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