Integrand size = 16, antiderivative size = 82 \[ \int \frac {\sqrt [4]{a-b x^4}}{x^3} \, dx=-\frac {\sqrt [4]{a-b x^4}}{2 x^2}-\frac {\sqrt {a} \sqrt {b} \left (1-\frac {b x^4}{a}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{2 \left (a-b x^4\right )^{3/4}} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {281, 283, 239, 238} \[ \int \frac {\sqrt [4]{a-b x^4}}{x^3} \, dx=-\frac {\sqrt {a} \sqrt {b} \left (1-\frac {b x^4}{a}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arcsin \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{2 \left (a-b x^4\right )^{3/4}}-\frac {\sqrt [4]{a-b x^4}}{2 x^2} \]
[In]
[Out]
Rule 238
Rule 239
Rule 281
Rule 283
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {\sqrt [4]{a-b x^2}}{x^2} \, dx,x,x^2\right ) \\ & = -\frac {\sqrt [4]{a-b x^4}}{2 x^2}-\frac {1}{4} b \text {Subst}\left (\int \frac {1}{\left (a-b x^2\right )^{3/4}} \, dx,x,x^2\right ) \\ & = -\frac {\sqrt [4]{a-b x^4}}{2 x^2}-\frac {\left (b \left (1-\frac {b x^4}{a}\right )^{3/4}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {b x^2}{a}\right )^{3/4}} \, dx,x,x^2\right )}{4 \left (a-b x^4\right )^{3/4}} \\ & = -\frac {\sqrt [4]{a-b x^4}}{2 x^2}-\frac {\sqrt {a} \sqrt {b} \left (1-\frac {b x^4}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{2 \left (a-b x^4\right )^{3/4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.63 \[ \int \frac {\sqrt [4]{a-b x^4}}{x^3} \, dx=-\frac {\sqrt [4]{a-b x^4} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {1}{4},\frac {1}{2},\frac {b x^4}{a}\right )}{2 x^2 \sqrt [4]{1-\frac {b x^4}{a}}} \]
[In]
[Out]
\[\int \frac {\left (-b \,x^{4}+a \right )^{\frac {1}{4}}}{x^{3}}d x\]
[In]
[Out]
\[ \int \frac {\sqrt [4]{a-b x^4}}{x^3} \, dx=\int { \frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{x^{3}} \,d x } \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.52 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.41 \[ \int \frac {\sqrt [4]{a-b x^4}}{x^3} \, dx=- \frac {\sqrt [4]{a} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4} \\ \frac {1}{2} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{2 x^{2}} \]
[In]
[Out]
\[ \int \frac {\sqrt [4]{a-b x^4}}{x^3} \, dx=\int { \frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{x^{3}} \,d x } \]
[In]
[Out]
\[ \int \frac {\sqrt [4]{a-b x^4}}{x^3} \, dx=\int { \frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{x^{3}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\sqrt [4]{a-b x^4}}{x^3} \, dx=\int \frac {{\left (a-b\,x^4\right )}^{1/4}}{x^3} \,d x \]
[In]
[Out]