Integrand size = 13, antiderivative size = 98 \[ \int \frac {1}{x^2 (a+b x)^{2/3}} \, dx=-\frac {\sqrt [3]{a+b x}}{a x}+\frac {2 b \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{5/3}}+\frac {b \log (x)}{3 a^{5/3}}-\frac {b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{a^{5/3}} \]
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Time = 0.02 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {44, 59, 631, 210, 31} \[ \int \frac {1}{x^2 (a+b x)^{2/3}} \, dx=\frac {2 b \arctan \left (\frac {2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{5/3}}+\frac {b \log (x)}{3 a^{5/3}}-\frac {b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{a^{5/3}}-\frac {\sqrt [3]{a+b x}}{a x} \]
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Rule 31
Rule 44
Rule 59
Rule 210
Rule 631
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt [3]{a+b x}}{a x}-\frac {(2 b) \int \frac {1}{x (a+b x)^{2/3}} \, dx}{3 a} \\ & = -\frac {\sqrt [3]{a+b x}}{a x}+\frac {b \log (x)}{3 a^{5/3}}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x}\right )}{a^{5/3}}+\frac {b \text {Subst}\left (\int \frac {1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x}\right )}{a^{4/3}} \\ & = -\frac {\sqrt [3]{a+b x}}{a x}+\frac {b \log (x)}{3 a^{5/3}}-\frac {b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{a^{5/3}}-\frac {(2 b) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}\right )}{a^{5/3}} \\ & = -\frac {\sqrt [3]{a+b x}}{a x}+\frac {2 b \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} a^{5/3}}+\frac {b \log (x)}{3 a^{5/3}}-\frac {b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{a^{5/3}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.21 \[ \int \frac {1}{x^2 (a+b x)^{2/3}} \, dx=\frac {-3 a^{2/3} \sqrt [3]{a+b x}+2 \sqrt {3} b x \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 b x \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )+b x \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )}{3 a^{5/3} x} \]
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Time = 0.12 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.97
method | result | size |
pseudoelliptic | \(\frac {2 \arctan \left (\frac {\left (a^{\frac {1}{3}}+2 \left (b x +a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}\, b x -2 \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right ) b x +\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right ) b x -3 \left (b x +a \right )^{\frac {1}{3}} a^{\frac {2}{3}}}{3 a^{\frac {5}{3}} x}\) | \(95\) |
derivativedivides | \(3 b \left (-\frac {\left (b x +a \right )^{\frac {1}{3}}}{3 a b x}+\frac {-\frac {2 \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{9 a^{\frac {2}{3}}}+\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{9 a^{\frac {2}{3}}}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{9 a^{\frac {2}{3}}}}{a}\right )\) | \(104\) |
default | \(3 b \left (-\frac {\left (b x +a \right )^{\frac {1}{3}}}{3 a b x}+\frac {-\frac {2 \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{9 a^{\frac {2}{3}}}+\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{9 a^{\frac {2}{3}}}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{9 a^{\frac {2}{3}}}}{a}\right )\) | \(104\) |
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Leaf count of result is larger than twice the leaf count of optimal. 166 vs. \(2 (75) = 150\).
Time = 0.22 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.69 \[ \int \frac {1}{x^2 (a+b x)^{2/3}} \, dx=\frac {2 \, \sqrt {3} a b x \sqrt {-\left (-a^{2}\right )^{\frac {1}{3}}} \arctan \left (-\frac {{\left (\sqrt {3} \left (-a^{2}\right )^{\frac {1}{3}} a - 2 \, \sqrt {3} \left (-a^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}}\right )} \sqrt {-\left (-a^{2}\right )^{\frac {1}{3}}}}{3 \, a^{2}}\right ) + \left (-a^{2}\right )^{\frac {2}{3}} b x \log \left ({\left (b x + a\right )}^{\frac {2}{3}} a - \left (-a^{2}\right )^{\frac {1}{3}} a + \left (-a^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}}\right ) - 2 \, \left (-a^{2}\right )^{\frac {2}{3}} b x \log \left ({\left (b x + a\right )}^{\frac {1}{3}} a - \left (-a^{2}\right )^{\frac {2}{3}}\right ) - 3 \, {\left (b x + a\right )}^{\frac {1}{3}} a^{2}}{3 \, a^{3} x} \]
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Result contains complex when optimal does not.
Time = 2.64 (sec) , antiderivative size = 830, normalized size of antiderivative = 8.47 \[ \int \frac {1}{x^2 (a+b x)^{2/3}} \, dx=\text {Too large to display} \]
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Time = 0.29 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^2 (a+b x)^{2/3}} \, dx=\frac {2 \, \sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{3 \, a^{\frac {5}{3}}} - \frac {{\left (b x + a\right )}^{\frac {1}{3}} b}{{\left (b x + a\right )} a - a^{2}} + \frac {b \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{3 \, a^{\frac {5}{3}}} - \frac {2 \, b \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{3 \, a^{\frac {5}{3}}} \]
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Time = 0.53 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.10 \[ \int \frac {1}{x^2 (a+b x)^{2/3}} \, dx=\frac {\frac {2 \, \sqrt {3} b^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {5}{3}}} + \frac {b^{2} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{a^{\frac {5}{3}}} - \frac {2 \, b^{2} \log \left ({\left | {\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{a^{\frac {5}{3}}} - \frac {3 \, {\left (b x + a\right )}^{\frac {1}{3}} b}{a x}}{3 \, b} \]
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Time = 0.15 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.24 \[ \int \frac {1}{x^2 (a+b x)^{2/3}} \, dx=-\frac {{\left (a+b\,x\right )}^{1/3}}{a\,x}+\frac {\ln \left (\frac {3\,\left (b-\sqrt {3}\,b\,1{}\mathrm {i}\right )}{a^{2/3}}+\frac {6\,b\,{\left (a+b\,x\right )}^{1/3}}{a}\right )\,\left (b-\sqrt {3}\,b\,1{}\mathrm {i}\right )}{3\,a^{5/3}}+\frac {\ln \left (\frac {3\,\left (b+\sqrt {3}\,b\,1{}\mathrm {i}\right )}{a^{2/3}}+\frac {6\,b\,{\left (a+b\,x\right )}^{1/3}}{a}\right )\,\left (b+\sqrt {3}\,b\,1{}\mathrm {i}\right )}{3\,a^{5/3}}-\frac {2\,b\,\ln \left ({\left (a+b\,x\right )}^{1/3}-a^{1/3}\right )}{3\,a^{5/3}} \]
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