Integrand size = 26, antiderivative size = 341 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\frac {e^2 x}{d \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x^2}}-\frac {c \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )}-\frac {c \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )} \]
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Time = 0.60 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1186, 197, 1706, 385, 211} \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=-\frac {c \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {x \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )}-\frac {c \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \arctan \left (\frac {x \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {d+e x^2}}\right )}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )}+\frac {e^2 x}{d \sqrt {d+e x^2} \left (a e^2-b d e+c d^2\right )} \]
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Rule 197
Rule 211
Rule 385
Rule 1186
Rule 1706
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {c d-b e-c e x^2}{\sqrt {d+e x^2} \left (a+b x^2+c x^4\right )} \, dx}{c d^2-b d e+a e^2}+\frac {e^2 \int \frac {1}{\left (d+e x^2\right )^{3/2}} \, dx}{c d^2-b d e+a e^2} \\ & = \frac {e^2 x}{d \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x^2}}+\frac {\int \left (\frac {-c e-\frac {c (-2 c d+b e)}{\sqrt {b^2-4 a c}}}{\left (b-\sqrt {b^2-4 a c}+2 c x^2\right ) \sqrt {d+e x^2}}+\frac {-c e+\frac {c (-2 c d+b e)}{\sqrt {b^2-4 a c}}}{\left (b+\sqrt {b^2-4 a c}+2 c x^2\right ) \sqrt {d+e x^2}}\right ) \, dx}{c d^2-b d e+a e^2} \\ & = \frac {e^2 x}{d \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x^2}}-\frac {\left (c \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {1}{\left (b-\sqrt {b^2-4 a c}+2 c x^2\right ) \sqrt {d+e x^2}} \, dx}{c d^2-b d e+a e^2}-\frac {\left (c \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {1}{\left (b+\sqrt {b^2-4 a c}+2 c x^2\right ) \sqrt {d+e x^2}} \, dx}{c d^2-b d e+a e^2} \\ & = \frac {e^2 x}{d \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x^2}}-\frac {\left (c \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right )\right ) \text {Subst}\left (\int \frac {1}{b-\sqrt {b^2-4 a c}-\left (-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e\right ) x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{c d^2-b d e+a e^2}-\frac {\left (c \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right )\right ) \text {Subst}\left (\int \frac {1}{b+\sqrt {b^2-4 a c}-\left (-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{c d^2-b d e+a e^2} \\ & = \frac {e^2 x}{d \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x^2}}-\frac {c \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )}-\frac {c \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 18.25 (sec) , antiderivative size = 2061, normalized size of antiderivative = 6.04 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\text {Result too large to show} \]
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Time = 0.41 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.26
method | result | size |
default | \(-\frac {\left (\frac {\left (b e -c d \right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}}{2}+\left (\frac {b c d}{2}+e \left (a c -\frac {b^{2}}{2}\right )\right ) d \right ) \sqrt {2}\, d \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {e \,x^{2}+d}\, \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )+\left (\sqrt {2}\, \left (\frac {\left (-b e +c d \right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}}{2}+\left (\frac {b c d}{2}+e \left (a c -\frac {b^{2}}{2}\right )\right ) d \right ) d \sqrt {e \,x^{2}+d}\, \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )-e^{2} x \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\right ) \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}{\sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {e \,x^{2}+d}\, \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \left (a \,e^{2}-b d e +c \,d^{2}\right ) d}\) | \(429\) |
pseudoelliptic | \(-\frac {\left (\frac {\left (b e -c d \right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}}{2}+\left (\frac {b c d}{2}+e \left (a c -\frac {b^{2}}{2}\right )\right ) d \right ) \sqrt {2}\, d \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {e \,x^{2}+d}\, \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )+\left (\sqrt {2}\, \left (\frac {\left (-b e +c d \right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}}{2}+\left (\frac {b c d}{2}+e \left (a c -\frac {b^{2}}{2}\right )\right ) d \right ) d \sqrt {e \,x^{2}+d}\, \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )-e^{2} x \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\right ) \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}{\sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {e \,x^{2}+d}\, \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \left (a \,e^{2}-b d e +c \,d^{2}\right ) d}\) | \(429\) |
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Leaf count of result is larger than twice the leaf count of optimal. 17249 vs. \(2 (299) = 598\).
Time = 217.47 (sec) , antiderivative size = 17249, normalized size of antiderivative = 50.58 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\text {Too large to display} \]
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\[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\int \frac {1}{\left (d + e x^{2}\right )^{\frac {3}{2}} \left (a + b x^{2} + c x^{4}\right )}\, dx \]
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\[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\int { \frac {1}{{\left (c x^{4} + b x^{2} + a\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\int \frac {1}{{\left (e\,x^2+d\right )}^{3/2}\,\left (c\,x^4+b\,x^2+a\right )} \,d x \]
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