Integrand size = 13, antiderivative size = 132 \[ \int \frac {x^3}{a+\frac {b}{x^3}} \, dx=-\frac {b x}{a^2}+\frac {x^4}{4 a}-\frac {b^{4/3} \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} a^{7/3}}+\frac {b^{4/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 a^{7/3}}-\frac {b^{4/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 a^{7/3}} \]
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Time = 0.07 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {269, 308, 206, 31, 648, 631, 210, 642} \[ \int \frac {x^3}{a+\frac {b}{x^3}} \, dx=-\frac {b^{4/3} \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} a^{7/3}}-\frac {b^{4/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 a^{7/3}}+\frac {b^{4/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{7/3}}-\frac {b x}{a^2}+\frac {x^4}{4 a} \]
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Rule 31
Rule 206
Rule 210
Rule 269
Rule 308
Rule 631
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^6}{b+a x^3} \, dx \\ & = \int \left (-\frac {b}{a^2}+\frac {x^3}{a}+\frac {b^2}{a^2 \left (b+a x^3\right )}\right ) \, dx \\ & = -\frac {b x}{a^2}+\frac {x^4}{4 a}+\frac {b^2 \int \frac {1}{b+a x^3} \, dx}{a^2} \\ & = -\frac {b x}{a^2}+\frac {x^4}{4 a}+\frac {b^{4/3} \int \frac {1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{3 a^2}+\frac {b^{4/3} \int \frac {2 \sqrt [3]{b}-\sqrt [3]{a} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{3 a^2} \\ & = -\frac {b x}{a^2}+\frac {x^4}{4 a}+\frac {b^{4/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 a^{7/3}}-\frac {b^{4/3} \int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 a^{2/3} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{6 a^{7/3}}+\frac {b^{5/3} \int \frac {1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{2 a^2} \\ & = -\frac {b x}{a^2}+\frac {x^4}{4 a}+\frac {b^{4/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 a^{7/3}}-\frac {b^{4/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 a^{7/3}}+\frac {b^{4/3} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{a^{7/3}} \\ & = -\frac {b x}{a^2}+\frac {x^4}{4 a}-\frac {b^{4/3} \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} a^{7/3}}+\frac {b^{4/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 a^{7/3}}-\frac {b^{4/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 a^{7/3}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.91 \[ \int \frac {x^3}{a+\frac {b}{x^3}} \, dx=\frac {-12 \sqrt [3]{a} b x+3 a^{4/3} x^4-4 \sqrt {3} b^{4/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )+4 b^{4/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )-2 b^{4/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{12 a^{7/3}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.35
method | result | size |
risch | \(\frac {x^{4}}{4 a}-\frac {b x}{a^{2}}+\frac {b^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{3}+b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{3 a^{3}}\) | \(46\) |
default | \(\frac {\frac {1}{4} a \,x^{4}-b x}{a^{2}}+\frac {\left (\frac {\ln \left (x +\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {b}{a}\right )^{\frac {1}{3}} x +\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}\right ) b^{2}}{a^{2}}\) | \(114\) |
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Time = 0.32 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.82 \[ \int \frac {x^3}{a+\frac {b}{x^3}} \, dx=\frac {3 \, a x^{4} + 4 \, \sqrt {3} b \left (\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (\frac {b}{a}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) - 2 \, b \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (x^{2} - x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \left (\frac {b}{a}\right )^{\frac {2}{3}}\right ) + 4 \, b \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (x + \left (\frac {b}{a}\right )^{\frac {1}{3}}\right ) - 12 \, b x}{12 \, a^{2}} \]
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Time = 0.09 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.28 \[ \int \frac {x^3}{a+\frac {b}{x^3}} \, dx=\operatorname {RootSum} {\left (27 t^{3} a^{7} - b^{4}, \left ( t \mapsto t \log {\left (\frac {3 t a^{2}}{b} + x \right )} \right )\right )} + \frac {x^{4}}{4 a} - \frac {b x}{a^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.92 \[ \int \frac {x^3}{a+\frac {b}{x^3}} \, dx=\frac {\sqrt {3} b^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{3 \, a^{3} \left (\frac {b}{a}\right )^{\frac {2}{3}}} - \frac {b^{2} \log \left (x^{2} - x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \, a^{3} \left (\frac {b}{a}\right )^{\frac {2}{3}}} + \frac {b^{2} \log \left (x + \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 \, a^{3} \left (\frac {b}{a}\right )^{\frac {2}{3}}} + \frac {a x^{4} - 4 \, b x}{4 \, a^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.98 \[ \int \frac {x^3}{a+\frac {b}{x^3}} \, dx=-\frac {b \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {b}{a}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a^{2}} + \frac {\sqrt {3} \left (-a^{2} b\right )^{\frac {1}{3}} b \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{3 \, a^{3}} + \frac {\left (-a^{2} b\right )^{\frac {1}{3}} b \log \left (x^{2} + x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \, a^{3}} + \frac {a^{3} x^{4} - 4 \, a^{2} b x}{4 \, a^{4}} \]
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Time = 5.87 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.90 \[ \int \frac {x^3}{a+\frac {b}{x^3}} \, dx=\frac {x^4}{4\,a}+\frac {b^{4/3}\,\ln \left (3\,b^2\,x+\frac {3\,b^{7/3}}{a^{1/3}}\right )}{3\,a^{7/3}}-\frac {b\,x}{a^2}-\frac {b^{4/3}\,\ln \left (3\,b^2\,x-\frac {3\,b^{7/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{a^{1/3}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,a^{7/3}}+\frac {b^{4/3}\,\ln \left (3\,b^2\,x+\frac {9\,b^{7/3}\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{a^{1/3}}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{a^{7/3}} \]
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