Integrand size = 18, antiderivative size = 16 \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3}}{x^6} \, dx=\frac {3 \left (1+x^4\right )^{5/3}}{5 x^5} \]
[Out]
Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {460} \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3}}{x^6} \, dx=\frac {3 \left (x^4+1\right )^{5/3}}{5 x^5} \]
[In]
[Out]
Rule 460
Rubi steps \begin{align*} \text {integral}& = \frac {3 \left (1+x^4\right )^{5/3}}{5 x^5} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3}}{x^6} \, dx=\frac {3 \left (1+x^4\right )^{5/3}}{5 x^5} \]
[In]
[Out]
Time = 0.88 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81
method | result | size |
gosper | \(\frac {3 \left (x^{4}+1\right )^{\frac {5}{3}}}{5 x^{5}}\) | \(13\) |
trager | \(\frac {3 \left (x^{4}+1\right )^{\frac {5}{3}}}{5 x^{5}}\) | \(13\) |
pseudoelliptic | \(\frac {3 \left (x^{4}+1\right )^{\frac {5}{3}}}{5 x^{5}}\) | \(13\) |
risch | \(\frac {\frac {3}{5} x^{8}+\frac {6}{5} x^{4}+\frac {3}{5}}{x^{5} \left (x^{4}+1\right )^{\frac {1}{3}}}\) | \(23\) |
meijerg | \(\frac {3 \operatorname {hypergeom}\left (\left [-\frac {5}{4}, -\frac {2}{3}\right ], \left [-\frac {1}{4}\right ], -x^{4}\right )}{5 x^{5}}-\frac {\operatorname {hypergeom}\left (\left [-\frac {2}{3}, -\frac {1}{4}\right ], \left [\frac {3}{4}\right ], -x^{4}\right )}{x}\) | \(34\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3}}{x^6} \, dx=\frac {3 \, {\left (x^{4} + 1\right )}^{\frac {5}{3}}}{5 \, x^{5}} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 1.08 (sec) , antiderivative size = 73, normalized size of antiderivative = 4.56 \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3}}{x^6} \, dx=\frac {\Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 x \Gamma \left (\frac {3}{4}\right )} - \frac {3 \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {2}{3} \\ - \frac {1}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 x^{5} \Gamma \left (- \frac {1}{4}\right )} \]
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3}}{x^6} \, dx=\frac {3 \, {\left (x^{4} + 1\right )}^{\frac {5}{3}}}{5 \, x^{5}} \]
[In]
[Out]
\[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3}}{x^6} \, dx=\int { \frac {{\left (x^{4} + 1\right )}^{\frac {2}{3}} {\left (x^{4} - 3\right )}}{x^{6}} \,d x } \]
[In]
[Out]
Time = 5.13 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.69 \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3}}{x^6} \, dx=\frac {3\,{\left (x^4+1\right )}^{2/3}+3\,x^4\,{\left (x^4+1\right )}^{2/3}}{5\,x^5} \]
[In]
[Out]