Integrand size = 55, antiderivative size = 65 \[ \int \frac {2^{2/3} \sqrt [3]{a}+2 \sqrt [3]{b} x}{\left (2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx=-\frac {2\ 2^{2/3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{b} x\right )}{\sqrt {a-b x^3}}\right )}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{b}} \]
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Time = 0.13 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {2162, 209} \[ \int \frac {2^{2/3} \sqrt [3]{a}+2 \sqrt [3]{b} x}{\left (2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx=-\frac {2\ 2^{2/3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{b} x\right )}{\sqrt {a-b x^3}}\right )}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{b}} \]
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Rule 209
Rule 2162
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (2\ 2^{2/3} \sqrt [3]{a}\right ) \text {Subst}\left (\int \frac {1}{1+3 a x^2} \, dx,x,\frac {1-\frac {\sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {a-b x^3}}\right )}{\sqrt [3]{b}} \\ & = -\frac {2\ 2^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{b} x\right )}{\sqrt {a-b x^3}}\right )}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{b}} \\ \end{align*}
Time = 5.39 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.03 \[ \int \frac {2^{2/3} \sqrt [3]{a}+2 \sqrt [3]{b} x}{\left (2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx=\frac {2\ 2^{2/3} \arctan \left (\frac {\sqrt {a-b x^3}}{\sqrt {3} \left (\sqrt {a}-\sqrt [3]{2} \sqrt [6]{a} \sqrt [3]{b} x\right )}\right )}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{b}} \]
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\[\int \frac {2^{\frac {2}{3}} a^{\frac {1}{3}}+2 b^{\frac {1}{3}} x}{\left (2^{\frac {2}{3}} a^{\frac {1}{3}}-b^{\frac {1}{3}} x \right ) \sqrt {-b \,x^{3}+a}}d x\]
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Timed out. \[ \int \frac {2^{2/3} \sqrt [3]{a}+2 \sqrt [3]{b} x}{\left (2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx=\text {Timed out} \]
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\[ \int \frac {2^{2/3} \sqrt [3]{a}+2 \sqrt [3]{b} x}{\left (2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx=- \int \frac {2^{\frac {2}{3}} \sqrt [3]{a}}{- 2^{\frac {2}{3}} \sqrt [3]{a} \sqrt {a - b x^{3}} + \sqrt [3]{b} x \sqrt {a - b x^{3}}}\, dx - \int \frac {2 \sqrt [3]{b} x}{- 2^{\frac {2}{3}} \sqrt [3]{a} \sqrt {a - b x^{3}} + \sqrt [3]{b} x \sqrt {a - b x^{3}}}\, dx \]
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\[ \int \frac {2^{2/3} \sqrt [3]{a}+2 \sqrt [3]{b} x}{\left (2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx=\int { -\frac {2 \, b^{\frac {1}{3}} x + 2^{\frac {2}{3}} a^{\frac {1}{3}}}{\sqrt {-b x^{3} + a} {\left (b^{\frac {1}{3}} x - 2^{\frac {2}{3}} a^{\frac {1}{3}}\right )}} \,d x } \]
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Timed out. \[ \int \frac {2^{2/3} \sqrt [3]{a}+2 \sqrt [3]{b} x}{\left (2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx=\text {Timed out} \]
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Time = 10.97 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.65 \[ \int \frac {2^{2/3} \sqrt [3]{a}+2 \sqrt [3]{b} x}{\left (2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx=\frac {2^{2/3}\,\sqrt {3}\,\ln \left (\frac {\left (\sqrt {a-b\,x^3}-\sqrt {3}\,\sqrt {a}\,1{}\mathrm {i}+2^{1/3}\,\sqrt {3}\,a^{1/6}\,b^{1/3}\,x\,1{}\mathrm {i}\right )\,{\left (\sqrt {3}\,\sqrt {a}\,1{}\mathrm {i}+\sqrt {a-b\,x^3}-2^{1/3}\,\sqrt {3}\,a^{1/6}\,b^{1/3}\,x\,1{}\mathrm {i}\right )}^3}{{\left (2^{2/3}\,a^{1/3}-b^{1/3}\,x\right )}^6}\right )\,1{}\mathrm {i}}{3\,a^{1/6}\,b^{1/3}} \]
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