Integrand size = 22, antiderivative size = 149 \[ \int \left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4} \, dx=\frac {5 x \left (2+x^2\right )}{\sqrt {2+3 x^2+x^4}}+\frac {1}{3} x \left (10+3 x^2\right ) \sqrt {2+3 x^2+x^4}-\frac {5 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{\sqrt {2+3 x^2+x^4}}+\frac {11 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{3 \sqrt {2+3 x^2+x^4}} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1190, 1203, 1113, 1149} \[ \int \left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4} \, dx=\frac {11 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{3 \sqrt {x^4+3 x^2+2}}-\frac {5 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{\sqrt {x^4+3 x^2+2}}+\frac {5 x \left (x^2+2\right )}{\sqrt {x^4+3 x^2+2}}+\frac {1}{3} x \left (3 x^2+10\right ) \sqrt {x^4+3 x^2+2} \]
[In]
[Out]
Rule 1113
Rule 1149
Rule 1190
Rule 1203
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x \left (10+3 x^2\right ) \sqrt {2+3 x^2+x^4}+\frac {1}{15} \int \frac {110+75 x^2}{\sqrt {2+3 x^2+x^4}} \, dx \\ & = \frac {1}{3} x \left (10+3 x^2\right ) \sqrt {2+3 x^2+x^4}+5 \int \frac {x^2}{\sqrt {2+3 x^2+x^4}} \, dx+\frac {22}{3} \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx \\ & = \frac {5 x \left (2+x^2\right )}{\sqrt {2+3 x^2+x^4}}+\frac {1}{3} x \left (10+3 x^2\right ) \sqrt {2+3 x^2+x^4}-\frac {5 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {2+3 x^2+x^4}}+\frac {11 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{3 \sqrt {2+3 x^2+x^4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 4.60 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.73 \[ \int \left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4} \, dx=\frac {20 x+36 x^3+19 x^5+3 x^7-15 i \sqrt {1+x^2} \sqrt {2+x^2} E\left (\left .i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )-7 i \sqrt {1+x^2} \sqrt {2+x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right ),2\right )}{3 \sqrt {2+3 x^2+x^4}} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 1.18 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.86
method | result | size |
risch | \(\frac {x \left (3 x^{2}+10\right ) \sqrt {x^{4}+3 x^{2}+2}}{3}-\frac {11 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{3 \sqrt {x^{4}+3 x^{2}+2}}+\frac {5 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{2 \sqrt {x^{4}+3 x^{2}+2}}\) | \(128\) |
default | \(\frac {10 x \sqrt {x^{4}+3 x^{2}+2}}{3}-\frac {11 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{3 \sqrt {x^{4}+3 x^{2}+2}}+\frac {5 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{2 \sqrt {x^{4}+3 x^{2}+2}}+x^{3} \sqrt {x^{4}+3 x^{2}+2}\) | \(137\) |
elliptic | \(\frac {10 x \sqrt {x^{4}+3 x^{2}+2}}{3}-\frac {11 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{3 \sqrt {x^{4}+3 x^{2}+2}}+\frac {5 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{2 \sqrt {x^{4}+3 x^{2}+2}}+x^{3} \sqrt {x^{4}+3 x^{2}+2}\) | \(137\) |
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.36 \[ \int \left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4} \, dx=\frac {-15 i \, x E(\arcsin \left (\frac {i}{x}\right )\,|\,2) + 37 i \, x F(\arcsin \left (\frac {i}{x}\right )\,|\,2) + {\left (3 \, x^{4} + 10 \, x^{2} + 15\right )} \sqrt {x^{4} + 3 \, x^{2} + 2}}{3 \, x} \]
[In]
[Out]
\[ \int \left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4} \, dx=\int \sqrt {\left (x^{2} + 1\right ) \left (x^{2} + 2\right )} \left (5 x^{2} + 7\right )\, dx \]
[In]
[Out]
\[ \int \left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4} \, dx=\int { \sqrt {x^{4} + 3 \, x^{2} + 2} {\left (5 \, x^{2} + 7\right )} \,d x } \]
[In]
[Out]
\[ \int \left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4} \, dx=\int { \sqrt {x^{4} + 3 \, x^{2} + 2} {\left (5 \, x^{2} + 7\right )} \,d x } \]
[In]
[Out]
Timed out. \[ \int \left (7+5 x^2\right ) \sqrt {2+3 x^2+x^4} \, dx=\int \left (5\,x^2+7\right )\,\sqrt {x^4+3\,x^2+2} \,d x \]
[In]
[Out]