Integrand size = 11, antiderivative size = 76 \[ \int \frac {1}{x^3 (a+b x)^3} \, dx=-\frac {1}{2 a^3 x^2}+\frac {3 b}{a^4 x}+\frac {b^2}{2 a^3 (a+b x)^2}+\frac {3 b^2}{a^4 (a+b x)}+\frac {6 b^2 \log (x)}{a^5}-\frac {6 b^2 \log (a+b x)}{a^5} \]
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Time = 0.03 (sec) , antiderivative size = 76, 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} \[ \int \frac {1}{x^3 (a+b x)^3} \, dx=\frac {6 b^2 \log (x)}{a^5}-\frac {6 b^2 \log (a+b x)}{a^5}+\frac {3 b^2}{a^4 (a+b x)}+\frac {3 b}{a^4 x}+\frac {b^2}{2 a^3 (a+b x)^2}-\frac {1}{2 a^3 x^2} \]
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Rule 46
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{a^3 x^3}-\frac {3 b}{a^4 x^2}+\frac {6 b^2}{a^5 x}-\frac {b^3}{a^3 (a+b x)^3}-\frac {3 b^3}{a^4 (a+b x)^2}-\frac {6 b^3}{a^5 (a+b x)}\right ) \, dx \\ & = -\frac {1}{2 a^3 x^2}+\frac {3 b}{a^4 x}+\frac {b^2}{2 a^3 (a+b x)^2}+\frac {3 b^2}{a^4 (a+b x)}+\frac {6 b^2 \log (x)}{a^5}-\frac {6 b^2 \log (a+b x)}{a^5} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^3 (a+b x)^3} \, dx=\frac {\frac {a \left (-a^3+4 a^2 b x+18 a b^2 x^2+12 b^3 x^3\right )}{x^2 (a+b x)^2}+12 b^2 \log (x)-12 b^2 \log (a+b x)}{2 a^5} \]
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Time = 0.18 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.95
method | result | size |
norman | \(\frac {-\frac {9 b^{4} x^{4}}{a^{5}}-\frac {1}{2 a}+\frac {2 b x}{a^{2}}-\frac {12 b^{3} x^{3}}{a^{4}}}{x^{2} \left (b x +a \right )^{2}}+\frac {6 b^{2} \ln \left (x \right )}{a^{5}}-\frac {6 b^{2} \ln \left (b x +a \right )}{a^{5}}\) | \(72\) |
default | \(-\frac {1}{2 a^{3} x^{2}}+\frac {3 b}{a^{4} x}+\frac {b^{2}}{2 a^{3} \left (b x +a \right )^{2}}+\frac {3 b^{2}}{a^{4} \left (b x +a \right )}+\frac {6 b^{2} \ln \left (x \right )}{a^{5}}-\frac {6 b^{2} \ln \left (b x +a \right )}{a^{5}}\) | \(73\) |
risch | \(\frac {\frac {6 b^{3} x^{3}}{a^{4}}+\frac {9 b^{2} x^{2}}{a^{3}}+\frac {2 b x}{a^{2}}-\frac {1}{2 a}}{x^{2} \left (b x +a \right )^{2}}-\frac {6 b^{2} \ln \left (b x +a \right )}{a^{5}}+\frac {6 b^{2} \ln \left (-x \right )}{a^{5}}\) | \(74\) |
parallelrisch | \(\frac {12 \ln \left (x \right ) x^{4} b^{6}-12 \ln \left (b x +a \right ) x^{4} b^{6}+24 \ln \left (x \right ) x^{3} a \,b^{5}-24 \ln \left (b x +a \right ) x^{3} a \,b^{5}+12 \ln \left (x \right ) x^{2} a^{2} b^{4}-12 \ln \left (b x +a \right ) x^{2} a^{2} b^{4}+12 x^{3} a \,b^{5}+18 x^{2} a^{2} b^{4}+4 x \,a^{3} b^{3}-a^{4} b^{2}}{2 a^{5} b^{2} x^{2} \left (b x +a \right )^{2}}\) | \(137\) |
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none
Time = 0.22 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.71 \[ \int \frac {1}{x^3 (a+b x)^3} \, dx=\frac {12 \, a b^{3} x^{3} + 18 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x - a^{4} - 12 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{3} + a^{2} b^{2} x^{2}\right )} \log \left (b x + a\right ) + 12 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{3} + a^{2} b^{2} x^{2}\right )} \log \left (x\right )}{2 \, {\left (a^{5} b^{2} x^{4} + 2 \, a^{6} b x^{3} + a^{7} x^{2}\right )}} \]
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Time = 0.21 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.03 \[ \int \frac {1}{x^3 (a+b x)^3} \, dx=\frac {- a^{3} + 4 a^{2} b x + 18 a b^{2} x^{2} + 12 b^{3} x^{3}}{2 a^{6} x^{2} + 4 a^{5} b x^{3} + 2 a^{4} b^{2} x^{4}} + \frac {6 b^{2} \left (\log {\left (x \right )} - \log {\left (\frac {a}{b} + x \right )}\right )}{a^{5}} \]
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Time = 0.21 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.13 \[ \int \frac {1}{x^3 (a+b x)^3} \, dx=\frac {12 \, b^{3} x^{3} + 18 \, a b^{2} x^{2} + 4 \, a^{2} b x - a^{3}}{2 \, {\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}} - \frac {6 \, b^{2} \log \left (b x + a\right )}{a^{5}} + \frac {6 \, b^{2} \log \left (x\right )}{a^{5}} \]
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Time = 0.31 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x^3 (a+b x)^3} \, dx=-\frac {6 \, b^{2} \log \left ({\left | b x + a \right |}\right )}{a^{5}} + \frac {6 \, b^{2} \log \left ({\left | x \right |}\right )}{a^{5}} + \frac {12 \, b^{3} x^{3} + 18 \, a b^{2} x^{2} + 4 \, a^{2} b x - a^{3}}{2 \, {\left (b x^{2} + a x\right )}^{2} a^{4}} \]
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Time = 0.30 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x^3 (a+b x)^3} \, dx=\frac {\frac {9\,b^2\,x^2}{a^3}-\frac {1}{2\,a}+\frac {6\,b^3\,x^3}{a^4}+\frac {2\,b\,x}{a^2}}{a^2\,x^2+2\,a\,b\,x^3+b^2\,x^4}-\frac {12\,b^2\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^5} \]
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