Integrand size = 15, antiderivative size = 32 \[ \int \frac {1}{\sqrt {\frac {a+b x^3}{x}}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {\frac {a}{x}+b x^2}}\right )}{3 \sqrt {b}} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2004, 2033, 212} \[ \int \frac {1}{\sqrt {\frac {a+b x^3}{x}}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {\frac {a}{x}+b x^2}}\right )}{3 \sqrt {b}} \]
[In]
[Out]
Rule 212
Rule 2004
Rule 2033
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sqrt {\frac {a}{x}+b x^2}} \, dx \\ & = \frac {2}{3} \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {\frac {a}{x}+b x^2}}\right ) \\ & = \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {\frac {a}{x}+b x^2}}\right )}{3 \sqrt {b}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.00 \[ \int \frac {1}{\sqrt {\frac {a+b x^3}{x}}} \, dx=\frac {2 \sqrt {a+b x^3} \log \left (\sqrt {b} x^{3/2}+\sqrt {a+b x^3}\right )}{3 \sqrt {b} \sqrt {x} \sqrt {\frac {a+b x^3}{x}}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(55\) vs. \(2(24)=48\).
Time = 0.29 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.75
method | result | size |
default | \(\frac {2 \left (b \,x^{3}+a \right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (b \,x^{3}+a \right )}}{x^{2} \sqrt {b}}\right )}{3 \sqrt {\frac {b \,x^{3}+a}{x}}\, \sqrt {x \left (b \,x^{3}+a \right )}\, \sqrt {b}}\) | \(56\) |
[In]
[Out]
none
Time = 0.34 (sec) , antiderivative size = 102, normalized size of antiderivative = 3.19 \[ \int \frac {1}{\sqrt {\frac {a+b x^3}{x}}} \, dx=\left [\frac {\log \left (-8 \, b^{2} x^{6} - 8 \, a b x^{3} - a^{2} - 4 \, {\left (2 \, b x^{5} + a x^{2}\right )} \sqrt {b} \sqrt {\frac {b x^{3} + a}{x}}\right )}{6 \, \sqrt {b}}, -\frac {\sqrt {-b} \arctan \left (\frac {2 \, \sqrt {-b} x^{2} \sqrt {\frac {b x^{3} + a}{x}}}{2 \, b x^{3} + a}\right )}{3 \, b}\right ] \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {\frac {a+b x^3}{x}}} \, dx=\int \frac {1}{\sqrt {\frac {a + b x^{3}}{x}}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {\frac {a+b x^3}{x}}} \, dx=\int { \frac {1}{\sqrt {\frac {b x^{3} + a}{x}}} \,d x } \]
[In]
[Out]
Exception generated. \[ \int \frac {1}{\sqrt {\frac {a+b x^3}{x}}} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int \frac {1}{\sqrt {\frac {a+b x^3}{x}}} \, dx=\int \frac {1}{\sqrt {\frac {b\,x^3+a}{x}}} \,d x \]
[In]
[Out]