Integrand size = 35, antiderivative size = 288 \[ \int \frac {1}{\left (2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{b} x\right )}{\sqrt {a-b x^3}}\right )}{3 \sqrt {3} \sqrt {a} \sqrt [3]{b}}-\frac {2 \sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt [3]{a} \sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {a-b x^3}} \]
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Time = 0.24 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {2159, 224, 2162, 209} \[ \int \frac {1}{\left (2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx=-\frac {2 \sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt [3]{a} \sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {a-b x^3}}-\frac {2 \arctan \left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{b} x\right )}{\sqrt {a-b x^3}}\right )}{3 \sqrt {3} \sqrt {a} \sqrt [3]{b}} \]
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Rule 209
Rule 224
Rule 2159
Rule 2162
Rubi steps \begin{align*} \text {integral}& = \frac {\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}{3\ 2^{2/3} \sqrt [3]{a}}+\frac {\sqrt [3]{2} \int \frac {1}{\sqrt {a-b x^3}} \, dx}{3 \sqrt [3]{a}} \\ & = -\frac {2 \sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt [3]{a} \sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {a-b x^3}}-\frac {2 \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 )}{3 \sqrt [3]{b}} \\ & = -\frac {2 \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 )}{3 \sqrt {3} \sqrt {a} \sqrt [3]{b}}-\frac {2 \sqrt [3]{2} \sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt [3]{a} \sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b} x\right )^2}} \sqrt {a-b x^3}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.16 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.58 \[ \int \frac {1}{\left (2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx=\frac {2 i \sqrt {\frac {\sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {1+\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}+\frac {b^{2/3} x^2}{a^{2/3}}} \operatorname {EllipticPi}\left (\frac {i \sqrt {3}}{\sqrt [3]{-1}+2^{2/3}},\arcsin \left (\sqrt {\frac {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right ),\sqrt [3]{-1}\right )}{\left (\sqrt [3]{-1}+2^{2/3}\right ) \sqrt [3]{b} \sqrt {a-b x^3}} \]
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\[\int \frac {1}{\left (2^{\frac {2}{3}} a^{\frac {1}{3}}-b^{\frac {1}{3}} x \right ) \sqrt {-b \,x^{3}+a}}d x\]
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Timed out. \[ \int \frac {1}{\left (2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\left (2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx=- \int \frac {1}{- 2^{\frac {2}{3}} \sqrt [3]{a} \sqrt {a - b x^{3}} + \sqrt [3]{b} x \sqrt {a - b x^{3}}}\, dx \]
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\[ \int \frac {1}{\left (2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx=\int { -\frac {1}{\sqrt {-b x^{3} + a} {\left (b^{\frac {1}{3}} x - 2^{\frac {2}{3}} a^{\frac {1}{3}}\right )}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1}{\left (2^{2/3} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx=\int \frac {1}{\sqrt {a-b\,x^3}\,\left (2^{2/3}\,a^{1/3}-b^{1/3}\,x\right )} \,d x \]
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