Integrand size = 13, antiderivative size = 38 \[ \int \frac {1}{x \left (a+b x^3\right )^2} \, dx=\frac {1}{3 a \left (a+b x^3\right )}+\frac {\log \left (\frac {x^3}{a+b x^3}\right )}{3 a^2} \]
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Time = 0.03 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 46} \[ \int \frac {1}{x \left (a+b x^3\right )^2} \, dx=-\frac {\log \left (a+b x^3\right )}{3 a^2}+\frac {\log (x)}{a^2}+\frac {1}{3 a \left (a+b x^3\right )} \]
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Rule 46
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {1}{x (a+b x)^2} \, dx,x,x^3\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \left (\frac {1}{a^2 x}-\frac {b}{a (a+b x)^2}-\frac {b}{a^2 (a+b x)}\right ) \, dx,x,x^3\right ) \\ & = \frac {1}{3 a \left (a+b x^3\right )}+\frac {\log (x)}{a^2}-\frac {\log \left (a+b x^3\right )}{3 a^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x \left (a+b x^3\right )^2} \, dx=\frac {\frac {a}{a+b x^3}+3 \log (x)-\log \left (a+b x^3\right )}{3 a^2} \]
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Time = 0.30 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.92
method | result | size |
risch | \(\frac {1}{3 a \left (b \,x^{3}+a \right )}+\frac {\ln \left (x \right )}{a^{2}}-\frac {\ln \left (b \,x^{3}+a \right )}{3 a^{2}}\) | \(35\) |
norman | \(-\frac {b \,x^{3}}{3 a^{2} \left (b \,x^{3}+a \right )}+\frac {\ln \left (x \right )}{a^{2}}-\frac {\ln \left (b \,x^{3}+a \right )}{3 a^{2}}\) | \(39\) |
default | \(-\frac {b \left (-\frac {a}{b \left (b \,x^{3}+a \right )}+\frac {\ln \left (b \,x^{3}+a \right )}{b}\right )}{3 a^{2}}+\frac {\ln \left (x \right )}{a^{2}}\) | \(42\) |
parallelrisch | \(\frac {3 \ln \left (x \right ) x^{3} b -\ln \left (b \,x^{3}+a \right ) x^{3} b -b \,x^{3}+3 \ln \left (x \right ) a -\ln \left (b \,x^{3}+a \right ) a}{3 a^{2} \left (b \,x^{3}+a \right )}\) | \(60\) |
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Time = 0.24 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.24 \[ \int \frac {1}{x \left (a+b x^3\right )^2} \, dx=-\frac {{\left (b x^{3} + a\right )} \log \left (b x^{3} + a\right ) - 3 \, {\left (b x^{3} + a\right )} \log \left (x\right ) - a}{3 \, {\left (a^{2} b x^{3} + a^{3}\right )}} \]
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Time = 0.16 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x \left (a+b x^3\right )^2} \, dx=\frac {1}{3 a^{2} + 3 a b x^{3}} + \frac {\log {\left (x \right )}}{a^{2}} - \frac {\log {\left (\frac {a}{b} + x^{3} \right )}}{3 a^{2}} \]
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Time = 0.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.97 \[ \int \frac {1}{x \left (a+b x^3\right )^2} \, dx=\frac {1}{3 \, {\left (a b x^{3} + a^{2}\right )}} - \frac {\log \left (b x^{3} + a\right )}{3 \, a^{2}} + \frac {\log \left (x^{3}\right )}{3 \, a^{2}} \]
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Time = 0.26 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.18 \[ \int \frac {1}{x \left (a+b x^3\right )^2} \, dx=-\frac {\log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{2}} + \frac {\log \left ({\left | x \right |}\right )}{a^{2}} + \frac {b x^{3} + 2 \, a}{3 \, {\left (b x^{3} + a\right )} a^{2}} \]
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Time = 14.73 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x \left (a+b x^3\right )^2} \, dx=\frac {\ln \left (x\right )}{a^2}+\frac {1}{3\,a\,\left (b\,x^3+a\right )}-\frac {\ln \left (b\,x^3+a\right )}{3\,a^2} \]
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