Integrand size = 13, antiderivative size = 80 \[ \int \frac {1}{x (a+b x)^{2/3}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{2/3}}-\frac {\log (x)}{2 a^{2/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{2/3}} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {59, 631, 210, 31} \[ \int \frac {1}{x (a+b x)^{2/3}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{2/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{2/3}}-\frac {\log (x)}{2 a^{2/3}} \]
[In]
[Out]
Rule 31
Rule 59
Rule 210
Rule 631
Rubi steps \begin{align*} \text {integral}& = -\frac {\log (x)}{2 a^{2/3}}-\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x}\right )}{2 a^{2/3}}-\frac {3 \text {Subst}\left (\int \frac {1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x}\right )}{2 \sqrt [3]{a}} \\ & = -\frac {\log (x)}{2 a^{2/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{2/3}}+\frac {3 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}\right )}{a^{2/3}} \\ & = -\frac {\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{2/3}}-\frac {\log (x)}{2 a^{2/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{2/3}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.16 \[ \int \frac {1}{x (a+b x)^{2/3}} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )+\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )}{2 a^{2/3}} \]
[In]
[Out]
Time = 0.08 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.94
method | result | size |
pseudoelliptic | \(\frac {-2 \arctan \left (\frac {\left (a^{\frac {1}{3}}+2 \left (b x +a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}+2 \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )-\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{2 a^{\frac {2}{3}}}\) | \(75\) |
derivativedivides | \(\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{a^{\frac {2}{3}}}-\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{2 a^{\frac {2}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{a^{\frac {2}{3}}}\) | \(76\) |
default | \(\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{a^{\frac {2}{3}}}-\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{2 a^{\frac {2}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{a^{\frac {2}{3}}}\) | \(76\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (57) = 114\).
Time = 0.22 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.44 \[ \int \frac {1}{x (a+b x)^{2/3}} \, dx=-\frac {2 \, \sqrt {3} {\left (a^{2}\right )}^{\frac {1}{6}} a \arctan \left (\frac {\sqrt {3} {\left (a^{2}\right )}^{\frac {1}{6}} {\left ({\left (a^{2}\right )}^{\frac {1}{3}} a + 2 \, {\left (a^{2}\right )}^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}}\right )}}{3 \, a^{2}}\right ) + {\left (a^{2}\right )}^{\frac {2}{3}} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} a + {\left (a^{2}\right )}^{\frac {1}{3}} a + {\left (a^{2}\right )}^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}}\right ) - 2 \, {\left (a^{2}\right )}^{\frac {2}{3}} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} a - {\left (a^{2}\right )}^{\frac {2}{3}}\right )}{2 \, a^{2}} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 1.69 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.88 \[ \int \frac {1}{x (a+b x)^{2/3}} \, dx=\frac {\log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {1}{3}\right )}{3 a^{\frac {2}{3}} \Gamma \left (\frac {4}{3}\right )} + \frac {e^{- \frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x} e^{\frac {2 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {1}{3}\right )}{3 a^{\frac {2}{3}} \Gamma \left (\frac {4}{3}\right )} + \frac {e^{\frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b} + x} e^{\frac {4 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {1}{3}\right )}{3 a^{\frac {2}{3}} \Gamma \left (\frac {4}{3}\right )} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x (a+b x)^{2/3}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {2}{3}}} - \frac {\log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{2 \, a^{\frac {2}{3}}} + \frac {\log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{a^{\frac {2}{3}}} \]
[In]
[Out]
none
Time = 0.54 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.98 \[ \int \frac {1}{x (a+b x)^{2/3}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {2}{3}}} - \frac {\log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{2 \, a^{\frac {2}{3}}} + \frac {\log \left ({\left | {\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{a^{\frac {2}{3}}} \]
[In]
[Out]
Time = 0.20 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.19 \[ \int \frac {1}{x (a+b x)^{2/3}} \, dx=\frac {\ln \left (9\,{\left (a+b\,x\right )}^{1/3}-9\,a^{1/3}\right )}{a^{2/3}}+\frac {\ln \left (\frac {9\,a^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}-9\,{\left (a+b\,x\right )}^{1/3}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,a^{2/3}}-\frac {\ln \left (\frac {9\,a^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}+9\,{\left (a+b\,x\right )}^{1/3}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,a^{2/3}} \]
[In]
[Out]