Integrand size = 11, antiderivative size = 57 \[ \int \frac {1}{x^3 (a+b x)^2} \, dx=\frac {\frac {3 b^2}{a^3}-\frac {1}{2 a x^2}+\frac {3 b}{2 a^2 x}}{a+b x}-\frac {3 b^2 \log \left (\frac {a+b x}{x}\right )}{a^4} \]
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Time = 0.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.02, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {46} \[ \int \frac {1}{x^3 (a+b x)^2} \, dx=\frac {3 b^2 \log (x)}{a^4}-\frac {3 b^2 \log (a+b x)}{a^4}+\frac {b^2}{a^3 (a+b x)}+\frac {2 b}{a^3 x}-\frac {1}{2 a^2 x^2} \]
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Rule 46
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{a^2 x^3}-\frac {2 b}{a^3 x^2}+\frac {3 b^2}{a^4 x}-\frac {b^3}{a^3 (a+b x)^2}-\frac {3 b^3}{a^4 (a+b x)}\right ) \, dx \\ & = -\frac {1}{2 a^2 x^2}+\frac {2 b}{a^3 x}+\frac {b^2}{a^3 (a+b x)}+\frac {3 b^2 \log (x)}{a^4}-\frac {3 b^2 \log (a+b x)}{a^4} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x^3 (a+b x)^2} \, dx=\frac {a \left (-\frac {a}{x^2}+\frac {4 b}{x}+\frac {2 b^2}{a+b x}\right )+6 b^2 \log (x)-6 b^2 \log (a+b x)}{2 a^4} \]
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Time = 0.20 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00
method | result | size |
default | \(-\frac {3 b^{2} \ln \left (b x +a \right )}{a^{4}}+\frac {b^{2}}{a^{3} \left (b x +a \right )}-\frac {1}{2 a^{2} x^{2}}+\frac {2 b}{a^{3} x}+\frac {3 b^{2} \ln \left (x \right )}{a^{4}}\) | \(57\) |
norman | \(\frac {-\frac {3 b^{3} x^{3}}{a^{4}}-\frac {1}{2 a}+\frac {3 b x}{2 a^{2}}}{x^{2} \left (b x +a \right )}+\frac {3 b^{2} \ln \left (x \right )}{a^{4}}-\frac {3 b^{2} \ln \left (b x +a \right )}{a^{4}}\) | \(61\) |
risch | \(\frac {\frac {3 b^{2} x^{2}}{a^{3}}+\frac {3 b x}{2 a^{2}}-\frac {1}{2 a}}{x^{2} \left (b x +a \right )}+\frac {3 b^{2} \ln \left (-x \right )}{a^{4}}-\frac {3 b^{2} \ln \left (b x +a \right )}{a^{4}}\) | \(63\) |
parallelrisch | \(\frac {6 \ln \left (x \right ) x^{3} b^{3}-6 \ln \left (b x +a \right ) x^{3} b^{3}+6 \ln \left (x \right ) x^{2} a \,b^{2}-6 \ln \left (b x +a \right ) x^{2} a \,b^{2}-6 b^{3} x^{3}+3 a^{2} b x -a^{3}}{2 a^{4} x^{2} \left (b x +a \right )}\) | \(87\) |
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none
Time = 0.24 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.51 \[ \int \frac {1}{x^3 (a+b x)^2} \, dx=\frac {6 \, a b^{2} x^{2} + 3 \, a^{2} b x - a^{3} - 6 \, {\left (b^{3} x^{3} + a b^{2} x^{2}\right )} \log \left (b x + a\right ) + 6 \, {\left (b^{3} x^{3} + a b^{2} x^{2}\right )} \log \left (x\right )}{2 \, {\left (a^{4} b x^{3} + a^{5} x^{2}\right )}} \]
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Time = 0.13 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x^3 (a+b x)^2} \, dx=\frac {- a^{2} + 3 a b x + 6 b^{2} x^{2}}{2 a^{4} x^{2} + 2 a^{3} b x^{3}} + \frac {3 b^{2} \left (\log {\left (x \right )} - \log {\left (\frac {a}{b} + x \right )}\right )}{a^{4}} \]
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none
Time = 0.19 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.12 \[ \int \frac {1}{x^3 (a+b x)^2} \, dx=\frac {6 \, b^{2} x^{2} + 3 \, a b x - a^{2}}{2 \, {\left (a^{3} b x^{3} + a^{4} x^{2}\right )}} - \frac {3 \, b^{2} \log \left (b x + a\right )}{a^{4}} + \frac {3 \, b^{2} \log \left (x\right )}{a^{4}} \]
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Time = 0.26 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.30 \[ \int \frac {1}{x^3 (a+b x)^2} \, dx=\frac {3 \, b^{2} \log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{4}} + \frac {b^{2}}{{\left (b x + a\right )} a^{3}} - \frac {\frac {6 \, a b^{2}}{b x + a} - 5 \, b^{2}}{2 \, a^{4} {\left (\frac {a}{b x + a} - 1\right )}^{2}} \]
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Time = 0.08 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^3 (a+b x)^2} \, dx=\frac {\frac {3\,b^2\,x^2}{a^3}-\frac {1}{2\,a}+\frac {3\,b\,x}{2\,a^2}}{b\,x^3+a\,x^2}-\frac {6\,b^2\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^4} \]
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