Integrand size = 11, antiderivative size = 62 \[ \int x \sqrt {1+x^8} \, dx=\frac {1}{6} x^2 \sqrt {1+x^8}+\frac {\left (1+x^4\right ) \sqrt {\frac {1+x^8}{\left (1+x^4\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (x^2\right ),\frac {1}{2}\right )}{6 \sqrt {1+x^8}} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {281, 201, 226} \[ \int x \sqrt {1+x^8} \, dx=\frac {\left (x^4+1\right ) \sqrt {\frac {x^8+1}{\left (x^4+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (x^2\right ),\frac {1}{2}\right )}{6 \sqrt {x^8+1}}+\frac {1}{6} \sqrt {x^8+1} x^2 \]
[In]
[Out]
Rule 201
Rule 226
Rule 281
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \sqrt {1+x^4} \, dx,x,x^2\right ) \\ & = \frac {1}{6} x^2 \sqrt {1+x^8}+\frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,x^2\right ) \\ & = \frac {1}{6} x^2 \sqrt {1+x^8}+\frac {\left (1+x^4\right ) \sqrt {\frac {1+x^8}{\left (1+x^4\right )^2}} F\left (2 \tan ^{-1}\left (x^2\right )|\frac {1}{2}\right )}{6 \sqrt {1+x^8}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.35 \[ \int x \sqrt {1+x^8} \, dx=\frac {1}{2} x^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},-x^8\right ) \]
[In]
[Out]
Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 4.69 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.27
method | result | size |
meijerg | \(\frac {x^{2} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (-\frac {1}{2},\frac {1}{4};\frac {5}{4};-x^{8}\right )}{2}\) | \(17\) |
risch | \(\frac {x^{2} \sqrt {x^{8}+1}}{6}+\frac {x^{2} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-x^{8}\right )}{3}\) | \(30\) |
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.45 \[ \int x \sqrt {1+x^8} \, dx=\frac {1}{6} \, \sqrt {x^{8} + 1} x^{2} + \frac {1}{3} i \, \sqrt {i} F(\arcsin \left (\frac {\sqrt {i}}{x^{2}}\right )\,|\,-1) \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.41 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.50 \[ \int x \sqrt {1+x^8} \, dx=\frac {x^{2} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {x^{8} e^{i \pi }} \right )}}{8 \Gamma \left (\frac {5}{4}\right )} \]
[In]
[Out]
\[ \int x \sqrt {1+x^8} \, dx=\int { \sqrt {x^{8} + 1} x \,d x } \]
[In]
[Out]
\[ \int x \sqrt {1+x^8} \, dx=\int { \sqrt {x^{8} + 1} x \,d x } \]
[In]
[Out]
Timed out. \[ \int x \sqrt {1+x^8} \, dx=\int x\,\sqrt {x^8+1} \,d x \]
[In]
[Out]