Integrand size = 23, antiderivative size = 31 \[ \int \frac {2+x}{\sqrt {13+16 x^2+8 x^3+x^4}} \, dx=\frac {1}{2} \log \left (4 x+x^2+\sqrt {13+16 x^2+8 x^3+x^4}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.65, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {1694, 1121, 633, 221} \[ \int \frac {2+x}{\sqrt {13+16 x^2+8 x^3+x^4}} \, dx=-\frac {1}{2} \text {arcsinh}\left (\frac {4-(x+2)^2}{\sqrt {13}}\right ) \]
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Rule 221
Rule 633
Rule 1121
Rule 1694
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {x}{\sqrt {29-8 x^2+x^4}} \, dx,x,2+x\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {29-8 x+x^2}} \, dx,x,(2+x)^2\right ) \\ & = \frac {\text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{52}}} \, dx,x,2 x (4+x)\right )}{4 \sqrt {13}} \\ & = \frac {1}{2} \text {arcsinh}\left (\frac {x (4+x)}{\sqrt {13}}\right ) \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {2+x}{\sqrt {13+16 x^2+8 x^3+x^4}} \, dx=\frac {1}{2} \log \left (4 x+x^2+\sqrt {13+16 x^2+8 x^3+x^4}\right ) \]
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Time = 1.72 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.42
method | result | size |
default | \(\frac {\operatorname {arcsinh}\left (\frac {\sqrt {13}\, x \left (x +4\right )}{13}\right )}{2}\) | \(13\) |
pseudoelliptic | \(\frac {\operatorname {arcsinh}\left (\frac {\sqrt {13}\, x \left (x +4\right )}{13}\right )}{2}\) | \(13\) |
trager | \(-\frac {\ln \left (-x^{2}+\sqrt {x^{4}+8 x^{3}+16 x^{2}+13}-4 x \right )}{2}\) | \(30\) |
elliptic | \(\text {Expression too large to display}\) | \(1258\) |
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Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {2+x}{\sqrt {13+16 x^2+8 x^3+x^4}} \, dx=\frac {1}{2} \, \log \left (x^{2} + 4 \, x + \sqrt {x^{4} + 8 \, x^{3} + 16 \, x^{2} + 13}\right ) \]
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\[ \int \frac {2+x}{\sqrt {13+16 x^2+8 x^3+x^4}} \, dx=\int \frac {x + 2}{\sqrt {x^{4} + 8 x^{3} + 16 x^{2} + 13}}\, dx \]
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\[ \int \frac {2+x}{\sqrt {13+16 x^2+8 x^3+x^4}} \, dx=\int { \frac {x + 2}{\sqrt {x^{4} + 8 \, x^{3} + 16 \, x^{2} + 13}} \,d x } \]
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Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.55 \[ \int \frac {2+x}{\sqrt {13+16 x^2+8 x^3+x^4}} \, dx=\frac {1}{4} \, \sqrt {{\left (x^{2} + 4 \, x\right )}^{2} + 13} {\left (x^{2} + 4 \, x\right )} - \frac {13}{4} \, \log \left (-x^{2} - 4 \, x + \sqrt {{\left (x^{2} + 4 \, x\right )}^{2} + 13}\right ) \]
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Timed out. \[ \int \frac {2+x}{\sqrt {13+16 x^2+8 x^3+x^4}} \, dx=\int \frac {x+2}{\sqrt {x^4+8\,x^3+16\,x^2+13}} \,d x \]
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