Integrand size = 26, antiderivative size = 252 \[ \int \frac {1}{\left (-3 a-b x^2\right ) \sqrt [3]{-a+b x^2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} \sqrt [3]{-a} \sqrt {a} \sqrt {b}}-\frac {\arctan \left (\frac {\sqrt {3} \sqrt {a} \left (\sqrt [3]{-a}-\sqrt [3]{2} \sqrt [3]{-a+b x^2}\right )}{\sqrt [3]{-a} \sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} \sqrt [3]{-a} \sqrt {a} \sqrt {b}}+\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{6\ 2^{2/3} \sqrt [3]{-a} \sqrt {a} \sqrt {b}}-\frac {\text {arctanh}\left (\frac {\sqrt [3]{-a} \sqrt {b} x}{\sqrt {a} \left (\sqrt [3]{-a}+\sqrt [3]{2} \sqrt [3]{-a+b x^2}\right )}\right )}{2\ 2^{2/3} \sqrt [3]{-a} \sqrt {a} \sqrt {b}} \]
[Out]
Time = 0.07 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {402} \[ \int \frac {1}{\left (-3 a-b x^2\right ) \sqrt [3]{-a+b x^2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {3} \sqrt {a} \left (\sqrt [3]{-a}-\sqrt [3]{2} \sqrt [3]{b x^2-a}\right )}{\sqrt [3]{-a} \sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} \sqrt [3]{-a} \sqrt {a} \sqrt {b}}-\frac {\arctan \left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} \sqrt [3]{-a} \sqrt {a} \sqrt {b}}-\frac {\text {arctanh}\left (\frac {\sqrt [3]{-a} \sqrt {b} x}{\sqrt {a} \left (\sqrt [3]{2} \sqrt [3]{b x^2-a}+\sqrt [3]{-a}\right )}\right )}{2\ 2^{2/3} \sqrt [3]{-a} \sqrt {a} \sqrt {b}}+\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{6\ 2^{2/3} \sqrt [3]{-a} \sqrt {a} \sqrt {b}} \]
[In]
[Out]
Rule 402
Rubi steps \begin{align*} \text {integral}& = -\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} \sqrt [3]{-a} \sqrt {a} \sqrt {b}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt {a} \left (\sqrt [3]{-a}-\sqrt [3]{2} \sqrt [3]{-a+b x^2}\right )}{\sqrt [3]{-a} \sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} \sqrt [3]{-a} \sqrt {a} \sqrt {b}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{6\ 2^{2/3} \sqrt [3]{-a} \sqrt {a} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{-a} \sqrt {b} x}{\sqrt {a} \left (\sqrt [3]{-a}+\sqrt [3]{2} \sqrt [3]{-a+b x^2}\right )}\right )}{2\ 2^{2/3} \sqrt [3]{-a} \sqrt {a} \sqrt {b}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 5.41 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.65 \[ \int \frac {1}{\left (-3 a-b x^2\right ) \sqrt [3]{-a+b x^2}} \, dx=-\frac {9 a x \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )}{\sqrt [3]{-a+b x^2} \left (3 a+b x^2\right ) \left (9 a \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )+2 b x^2 \left (-\operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},2,\frac {5}{2},\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )+\operatorname {AppellF1}\left (\frac {3}{2},\frac {4}{3},1,\frac {5}{2},\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )\right )\right )} \]
[In]
[Out]
\[\int \frac {1}{\left (-b \,x^{2}-3 a \right ) \left (b \,x^{2}-a \right )^{\frac {1}{3}}}d x\]
[In]
[Out]
Timed out. \[ \int \frac {1}{\left (-3 a-b x^2\right ) \sqrt [3]{-a+b x^2}} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {1}{\left (-3 a-b x^2\right ) \sqrt [3]{-a+b x^2}} \, dx=- \int \frac {1}{3 a \sqrt [3]{- a + b x^{2}} + b x^{2} \sqrt [3]{- a + b x^{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{\left (-3 a-b x^2\right ) \sqrt [3]{-a+b x^2}} \, dx=\int { -\frac {1}{{\left (b x^{2} + 3 \, a\right )} {\left (b x^{2} - a\right )}^{\frac {1}{3}}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{\left (-3 a-b x^2\right ) \sqrt [3]{-a+b x^2}} \, dx=\int { -\frac {1}{{\left (b x^{2} + 3 \, a\right )} {\left (b x^{2} - a\right )}^{\frac {1}{3}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{\left (-3 a-b x^2\right ) \sqrt [3]{-a+b x^2}} \, dx=-\int \frac {1}{{\left (b\,x^2-a\right )}^{1/3}\,\left (b\,x^2+3\,a\right )} \,d x \]
[In]
[Out]