Integrand size = 22, antiderivative size = 288 \[ \int \frac {x^2}{\left (b+a x^2\right ) \sqrt [3]{x+x^3}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+x^3}}\right )}{2 a}+\frac {\sqrt {3} \sqrt [3]{b} \arctan \left (\frac {\sqrt {3} \sqrt [3]{a-b} x}{\sqrt [3]{a-b} x-2 \sqrt [3]{b} \sqrt [3]{x+x^3}}\right )}{2 a \sqrt [3]{a-b}}-\frac {\log \left (-x+\sqrt [3]{x+x^3}\right )}{2 a}-\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a-b} x+\sqrt [3]{b} \sqrt [3]{x+x^3}\right )}{2 a \sqrt [3]{a-b}}+\frac {\log \left (x^2+x \sqrt [3]{x+x^3}+\left (x+x^3\right )^{2/3}\right )}{4 a}+\frac {\sqrt [3]{b} \log \left ((a-b)^{2/3} x^2-\sqrt [3]{a-b} \sqrt [3]{b} x \sqrt [3]{x+x^3}+b^{2/3} \left (x+x^3\right )^{2/3}\right )}{4 a \sqrt [3]{a-b}} \]
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Time = 0.37 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.13, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2067, 477, 476, 494, 245, 384} \[ \int \frac {x^2}{\left (b+a x^2\right ) \sqrt [3]{x+x^3}} \, dx=\frac {\sqrt {3} \sqrt [3]{b} \sqrt [3]{x} \sqrt [3]{x^2+1} \arctan \left (\frac {1-\frac {2 x^{2/3} \sqrt [3]{a-b}}{\sqrt [3]{b} \sqrt [3]{x^2+1}}}{\sqrt {3}}\right )}{2 a \sqrt [3]{x^3+x} \sqrt [3]{a-b}}+\frac {\sqrt {3} \sqrt [3]{x} \sqrt [3]{x^2+1} \arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{2 a \sqrt [3]{x^3+x}}+\frac {\sqrt [3]{b} \sqrt [3]{x} \sqrt [3]{x^2+1} \log \left (a x^2+b\right )}{4 a \sqrt [3]{x^3+x} \sqrt [3]{a-b}}-\frac {3 \sqrt [3]{b} \sqrt [3]{x} \sqrt [3]{x^2+1} \log \left (x^{2/3} \sqrt [3]{a-b}+\sqrt [3]{b} \sqrt [3]{x^2+1}\right )}{4 a \sqrt [3]{x^3+x} \sqrt [3]{a-b}}-\frac {3 \sqrt [3]{x} \sqrt [3]{x^2+1} \log \left (x^{2/3}-\sqrt [3]{x^2+1}\right )}{4 a \sqrt [3]{x^3+x}} \]
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Rule 245
Rule 384
Rule 476
Rule 477
Rule 494
Rule 2067
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \int \frac {x^{5/3}}{\sqrt [3]{1+x^2} \left (b+a x^2\right )} \, dx}{\sqrt [3]{x+x^3}} \\ & = \frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {x^7}{\sqrt [3]{1+x^6} \left (b+a x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x+x^3}} \\ & = \frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt [3]{1+x^3} \left (b+a x^3\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x+x^3}} \\ & = \frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{2 a \sqrt [3]{x+x^3}}-\frac {\left (3 b \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^3} \left (b+a x^3\right )} \, dx,x,x^{2/3}\right )}{2 a \sqrt [3]{x+x^3}} \\ & = \frac {\sqrt {3} \sqrt [3]{x} \sqrt [3]{1+x^2} \arctan \left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{2 a \sqrt [3]{x+x^3}}+\frac {\sqrt {3} \sqrt [3]{b} \sqrt [3]{x} \sqrt [3]{1+x^2} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a-b} x^{2/3}}{\sqrt [3]{b} \sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{2 a \sqrt [3]{a-b} \sqrt [3]{x+x^3}}+\frac {\sqrt [3]{b} \sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (b+a x^2\right )}{4 a \sqrt [3]{a-b} \sqrt [3]{x+x^3}}-\frac {3 \sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (x^{2/3}-\sqrt [3]{1+x^2}\right )}{4 a \sqrt [3]{x+x^3}}-\frac {3 \sqrt [3]{b} \sqrt [3]{x} \sqrt [3]{1+x^2} \log \left (\sqrt [3]{a-b} x^{2/3}+\sqrt [3]{b} \sqrt [3]{1+x^2}\right )}{4 a \sqrt [3]{a-b} \sqrt [3]{x+x^3}} \\ \end{align*}
Time = 8.32 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.16 \[ \int \frac {x^2}{\left (b+a x^2\right ) \sqrt [3]{x+x^3}} \, dx=\frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \left (2 \sqrt {3} \sqrt [3]{a-b} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{1+x^2}}\right )+2 \sqrt {3} \sqrt [3]{b} \arctan \left (\frac {\sqrt {3} \sqrt [3]{a-b} x^{2/3}}{\sqrt [3]{a-b} x^{2/3}-2 \sqrt [3]{b} \sqrt [3]{1+x^2}}\right )-2 \sqrt [3]{a-b} \log \left (a \left (-x^{2/3}+\sqrt [3]{1+x^2}\right )\right )-2 \sqrt [3]{b} \log \left (\sqrt [3]{a-b} x^{2/3}+\sqrt [3]{b} \sqrt [3]{1+x^2}\right )+\sqrt [3]{a-b} \log \left (x^{4/3}+x^{2/3} \sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right )+\sqrt [3]{b} \log \left ((a-b)^{2/3} x^{4/3}-\sqrt [3]{a-b} \sqrt [3]{b} x^{2/3} \sqrt [3]{1+x^2}+b^{2/3} \left (1+x^2\right )^{2/3}\right )\right )}{4 a \sqrt [3]{a-b} \sqrt [3]{x+x^3}} \]
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Time = 0.75 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.90
method | result | size |
pseudoelliptic | \(-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right ) \left (\frac {a -b}{b}\right )^{\frac {1}{3}}+\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {a -b}{b}\right )^{\frac {1}{3}} x -2 {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}\right )}{3 \left (\frac {a -b}{b}\right )^{\frac {1}{3}} x}\right )+\ln \left (\frac {{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}-x}{x}\right ) \left (\frac {a -b}{b}\right )^{\frac {1}{3}}-\frac {\ln \left (\frac {{\left (x \left (x^{2}+1\right )\right )}^{\frac {2}{3}}+{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}} x +x^{2}}{x^{2}}\right ) \left (\frac {a -b}{b}\right )^{\frac {1}{3}}}{2}+\ln \left (\frac {\left (\frac {a -b}{b}\right )^{\frac {1}{3}} x +{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}}{x}\right )-\frac {\ln \left (\frac {\left (\frac {a -b}{b}\right )^{\frac {2}{3}} x^{2}-\left (\frac {a -b}{b}\right )^{\frac {1}{3}} {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}} x +{\left (x \left (x^{2}+1\right )\right )}^{\frac {2}{3}}}{x^{2}}\right )}{2}}{2 \left (\frac {a -b}{b}\right )^{\frac {1}{3}} a}\) | \(259\) |
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Timed out. \[ \int \frac {x^2}{\left (b+a x^2\right ) \sqrt [3]{x+x^3}} \, dx=\text {Timed out} \]
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\[ \int \frac {x^2}{\left (b+a x^2\right ) \sqrt [3]{x+x^3}} \, dx=\int \frac {x^{2}}{\sqrt [3]{x \left (x^{2} + 1\right )} \left (a x^{2} + b\right )}\, dx \]
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\[ \int \frac {x^2}{\left (b+a x^2\right ) \sqrt [3]{x+x^3}} \, dx=\int { \frac {x^{2}}{{\left (a x^{2} + b\right )} {\left (x^{3} + x\right )}^{\frac {1}{3}}} \,d x } \]
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Time = 0.33 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.89 \[ \int \frac {x^2}{\left (b+a x^2\right ) \sqrt [3]{x+x^3}} \, dx=-\frac {b \left (-\frac {a - b}{b}\right )^{\frac {2}{3}} \log \left ({\left | -\left (-\frac {a - b}{b}\right )^{\frac {1}{3}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} \right |}\right )}{2 \, {\left (a^{2} - a b\right )}} - \frac {3 \, {\left (-a b^{2} + b^{3}\right )}^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (\left (-\frac {a - b}{b}\right )^{\frac {1}{3}} + 2 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a - b}{b}\right )^{\frac {1}{3}}}\right )}{2 \, {\left (\sqrt {3} a^{2} b - \sqrt {3} a b^{2}\right )}} + \frac {{\left (-a b^{2} + b^{3}\right )}^{\frac {2}{3}} \log \left (\left (-\frac {a - b}{b}\right )^{\frac {2}{3}} + \left (-\frac {a - b}{b}\right )^{\frac {1}{3}} {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}}\right )}{4 \, {\left (a^{2} b - a b^{2}\right )}} - \frac {\sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right )}\right )}{2 \, a} + \frac {\log \left ({\left (\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right )}{4 \, a} - \frac {\log \left ({\left | {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right )}{2 \, a} \]
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Timed out. \[ \int \frac {x^2}{\left (b+a x^2\right ) \sqrt [3]{x+x^3}} \, dx=\int \frac {x^2}{\left (a\,x^2+b\right )\,{\left (x^3+x\right )}^{1/3}} \,d x \]
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