Integrand size = 43, antiderivative size = 50 \[ \int \frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\left (2 \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a+b x^3}} \, dx=\frac {2 \text {arctanh}\left (\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt {a+b x^3}}\right )}{3 \sqrt [6]{a} \sqrt [3]{b}} \]
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Time = 0.09 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {2163, 212} \[ \int \frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\left (2 \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a+b x^3}} \, dx=\frac {2 \text {arctanh}\left (\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt {a+b x^3}}\right )}{3 \sqrt [6]{a} \sqrt [3]{b}} \]
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Rule 212
Rule 2163
Rubi steps \begin{align*} \text {integral}& = \frac {\left (2 \sqrt [3]{a}\right ) \text {Subst}\left (\int \frac {1}{9-a x^2} \, dx,x,\frac {\left (1+\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2}{\sqrt {a+b x^3}}\right )}{\sqrt [3]{b}} \\ & = \frac {2 \tanh ^{-1}\left (\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt {a+b x^3}}\right )}{3 \sqrt [6]{a} \sqrt [3]{b}} \\ \end{align*}
Time = 2.34 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\left (2 \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a+b x^3}} \, dx=\frac {2 \text {arctanh}\left (\frac {3 \sqrt [6]{a} \sqrt {a+b x^3}}{\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}\right )}{3 \sqrt [6]{a} \sqrt [3]{b}} \]
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\[\int \frac {a^{\frac {1}{3}}+b^{\frac {1}{3}} x}{\left (2 a^{\frac {1}{3}}-b^{\frac {1}{3}} x \right ) \sqrt {b \,x^{3}+a}}d x\]
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Timed out. \[ \int \frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\left (2 \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a+b x^3}} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\left (2 \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a+b x^3}} \, dx=- \int \frac {\sqrt [3]{a}}{- 2 \sqrt [3]{a} \sqrt {a + b x^{3}} + \sqrt [3]{b} x \sqrt {a + b x^{3}}}\, dx - \int \frac {\sqrt [3]{b} x}{- 2 \sqrt [3]{a} \sqrt {a + b x^{3}} + \sqrt [3]{b} x \sqrt {a + b x^{3}}}\, dx \]
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\[ \int \frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\left (2 \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a+b x^3}} \, dx=\int { -\frac {b^{\frac {1}{3}} x + a^{\frac {1}{3}}}{\sqrt {b x^{3} + a} {\left (b^{\frac {1}{3}} x - 2 \, a^{\frac {1}{3}}\right )}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\left (2 \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a+b x^3}} \, dx=\text {Timed out} \]
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Time = 9.65 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.30 \[ \int \frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\left (2 \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a+b x^3}} \, dx=\frac {\ln \left (\frac {\left (\sqrt {b\,x^3+a}+\sqrt {a}\right )\,{\left (\sqrt {b\,x^3+a}-\sqrt {a}+2\,a^{1/6}\,b^{1/3}\,x\right )}^3}{x^3\,{\left (b^{1/3}\,x-2\,a^{1/3}\right )}^3}\right )}{3\,a^{1/6}\,b^{1/3}} \]
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