Integrand size = 13, antiderivative size = 105 \[ \int \frac {1}{x^2 \sqrt [3]{a+b x}} \, dx=-\frac {\sqrt [3]{(a+b x)^2}}{a x}-\frac {b \left (\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{a+b x}}{2 \sqrt [3]{a}+\sqrt [3]{a+b x}}\right )+\frac {3}{2} \log \left (\frac {-\sqrt [3]{a}+\sqrt [3]{a+b x}}{\sqrt [3]{x}}\right )\right )}{3 a \sqrt [3]{a^2}} \]
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Time = 0.04 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {44, 57, 631, 210, 31} \[ \int \frac {1}{x^2 \sqrt [3]{a+b x}} \, dx=-\frac {b \arctan \left (\frac {2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3}}+\frac {b \log (x)}{6 a^{4/3}}-\frac {b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{4/3}}-\frac {(a+b x)^{2/3}}{a x} \]
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Rule 31
Rule 44
Rule 57
Rule 210
Rule 631
Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b x)^{2/3}}{a x}-\frac {b \int \frac {1}{x \sqrt [3]{a+b x}} \, dx}{3 a} \\ & = -\frac {(a+b x)^{2/3}}{a x}+\frac {b \log (x)}{6 a^{4/3}}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x}\right )}{2 a^{4/3}}-\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 )}{2 a} \\ & = -\frac {(a+b x)^{2/3}}{a x}+\frac {b \log (x)}{6 a^{4/3}}-\frac {b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{4/3}}+\frac {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^{4/3}} \\ & = -\frac {(a+b x)^{2/3}}{a x}-\frac {b \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} a^{4/3}}+\frac {b \log (x)}{6 a^{4/3}}-\frac {b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 a^{4/3}} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x^2 \sqrt [3]{a+b x}} \, dx=-\frac {6 \sqrt [3]{a} (a+b x)^{2/3}+2 \sqrt {3} b x \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 b x \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )-b x \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )}{6 a^{4/3} x} \]
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Time = 0.12 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.90
method | result | size |
risch | \(-\frac {\left (b x +a \right )^{\frac {2}{3}}}{a x}-\frac {b \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{3 a^{\frac {4}{3}}}+\frac {b \ln \left (\left (b x +a \right )^{\frac {2}{3}}+\left (b x +a \right )^{\frac {1}{3}} a^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {4}{3}}}-\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{3 a^{\frac {4}{3}}}\) | \(95\) |
pseudoelliptic | \(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (2 \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {1}{3}}\right )}{3 a^{\frac {1}{3}}}\right ) b x -6 a^{\frac {1}{3}} \left (b x +a \right )^{\frac {2}{3}}-2 \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right ) b x +\ln \left (\left (b x +a \right )^{\frac {2}{3}}+\left (b x +a \right )^{\frac {1}{3}} a^{\frac {1}{3}}+a^{\frac {2}{3}}\right ) b x}{6 a^{\frac {4}{3}} x}\) | \(95\) |
derivativedivides | \(3 b \left (-\frac {\left (b x +a \right )^{\frac {2}{3}}}{3 a b x}+\frac {-\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{3 a^{\frac {1}{3}}}+\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+\left (b x +a \right )^{\frac {1}{3}} a^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {1}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{3 a^{\frac {1}{3}}}}{3 a}\right )\) | \(104\) |
default | \(3 b \left (-\frac {\left (b x +a \right )^{\frac {2}{3}}}{3 a b x}+\frac {-\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{3 a^{\frac {1}{3}}}+\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+\left (b x +a \right )^{\frac {1}{3}} a^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {1}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{3 a^{\frac {1}{3}}}}{3 a}\right )\) | \(104\) |
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Time = 0.25 (sec) , antiderivative size = 306, normalized size of antiderivative = 2.91 \[ \int \frac {1}{x^2 \sqrt [3]{a+b x}} \, dx=\left [\frac {3 \, \sqrt {\frac {1}{3}} a b x \sqrt {\frac {\left (-a\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b x - 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (b x + a\right )}^{\frac {2}{3}} \left (-a\right )^{\frac {2}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} a + \left (-a\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (-a\right )^{\frac {1}{3}}}{a}} - 3 \, {\left (b x + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {2}{3}} + 3 \, a}{x}\right ) + \left (-a\right )^{\frac {2}{3}} b x \log \left ({\left (b x + a\right )}^{\frac {2}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + \left (-a\right )^{\frac {2}{3}}\right ) - 2 \, \left (-a\right )^{\frac {2}{3}} b x \log \left ({\left (b x + a\right )}^{\frac {1}{3}} + \left (-a\right )^{\frac {1}{3}}\right ) - 6 \, {\left (b x + a\right )}^{\frac {2}{3}} a}{6 \, a^{2} x}, -\frac {6 \, \sqrt {\frac {1}{3}} a b x \sqrt {-\frac {\left (-a\right )^{\frac {1}{3}}}{a}} \arctan \left (\sqrt {\frac {1}{3}} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} - \left (-a\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a\right )^{\frac {1}{3}}}{a}}\right ) - \left (-a\right )^{\frac {2}{3}} b x \log \left ({\left (b x + a\right )}^{\frac {2}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + \left (-a\right )^{\frac {2}{3}}\right ) + 2 \, \left (-a\right )^{\frac {2}{3}} b x \log \left ({\left (b x + a\right )}^{\frac {1}{3}} + \left (-a\right )^{\frac {1}{3}}\right ) + 6 \, {\left (b x + a\right )}^{\frac {2}{3}} a}{6 \, a^{2} x}\right ] \]
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Result contains complex when optimal does not.
Time = 1.46 (sec) , antiderivative size = 831, normalized size of antiderivative = 7.91 \[ \int \frac {1}{x^2 \sqrt [3]{a+b x}} \, dx=\text {Too large to display} \]
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Time = 0.30 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.01 \[ \int \frac {1}{x^2 \sqrt [3]{a+b x}} \, dx=-\frac {\sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{3 \, a^{\frac {4}{3}}} - \frac {{\left (b x + a\right )}^{\frac {2}{3}} b}{{\left (b x + a\right )} a - a^{2}} + \frac {b \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{6 \, a^{\frac {4}{3}}} - \frac {b \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{3 \, a^{\frac {4}{3}}} \]
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Time = 0.48 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x^2 \sqrt [3]{a+b x}} \, dx=-\frac {\frac {2 \, \sqrt {3} b^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {4}{3}}} - \frac {b^{2} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{a^{\frac {4}{3}}} + \frac {2 \, b^{2} \log \left ({\left | {\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{a^{\frac {4}{3}}} + \frac {6 \, {\left (b x + a\right )}^{\frac {2}{3}} b}{a x}}{6 \, b} \]
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Time = 15.94 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.24 \[ \int \frac {1}{x^2 \sqrt [3]{a+b x}} \, dx=-\frac {{\left (a+b\,x\right )}^{2/3}}{a\,x}+\frac {\ln \left (\frac {{\left (b-\sqrt {3}\,b\,1{}\mathrm {i}\right )}^2}{4\,a^{5/3}}-\frac {b^2\,{\left (a+b\,x\right )}^{1/3}}{a^2}\right )\,\left (b-\sqrt {3}\,b\,1{}\mathrm {i}\right )}{6\,a^{4/3}}+\frac {\ln \left (\frac {{\left (b+\sqrt {3}\,b\,1{}\mathrm {i}\right )}^2}{4\,a^{5/3}}-\frac {b^2\,{\left (a+b\,x\right )}^{1/3}}{a^2}\right )\,\left (b+\sqrt {3}\,b\,1{}\mathrm {i}\right )}{6\,a^{4/3}}-\frac {b\,\ln \left ({\left (a+b\,x\right )}^{1/3}-a^{1/3}\right )}{3\,a^{4/3}} \]
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