Integrand size = 29, antiderivative size = 419 \[ \int \frac {1}{x^4 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=-\frac {1}{3 a d x^3 \sqrt {d+e x^2}}+\frac {3 b d+4 a e}{3 a^2 d^2 x \sqrt {d+e x^2}}+\frac {2 e (3 b d+4 a e) x}{3 a^2 d^3 \sqrt {d+e x^2}}-\frac {e \left (b c d-b^2 e+a c e\right ) x}{a^2 d \left (c d^2+e (-b d+a e)\right ) \sqrt {d+e x^2}}+\frac {2 c^2 \left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a^2 \sqrt {b-\sqrt {b^2-4 a c}} \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right )^{3/2}}+\frac {2 c^2 \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a^2 \sqrt {b+\sqrt {b^2-4 a c}} \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right )^{3/2}} \]
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Time = 3.37 (sec) , antiderivative size = 647, normalized size of antiderivative = 1.54, number of steps used = 15, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {1315, 277, 197, 6860, 270, 1706, 385, 211} \[ \int \frac {1}{x^4 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\frac {c \left (\frac {3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt {b^2-4 a c}}+a c e+b^2 (-e)+b c d\right ) \arctan \left (\frac {x \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a^2 \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )}+\frac {c \left (-\frac {3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d}{\sqrt {b^2-4 a c}}+a c e+b^2 (-e)+b c d\right ) \arctan \left (\frac {x \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {d+e x^2}}\right )}{a^2 \sqrt {\sqrt {b^2-4 a c}+b} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {d+e x^2} \left (a c e+b^2 (-e)+b c d\right )}{a^2 d x \left (a e^2-b d e+c d^2\right )}+\frac {2 e \sqrt {d+e x^2} (c d-b e)}{3 a d^2 x \left (a e^2-b d e+c d^2\right )}-\frac {e^2}{3 d x^3 \sqrt {d+e x^2} \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {d+e x^2} (c d-b e)}{3 a d x^3 \left (a e^2-b d e+c d^2\right )}+\frac {4 e^3}{3 d^2 x \sqrt {d+e x^2} \left (a e^2-b d e+c d^2\right )}+\frac {8 e^4 x}{3 d^3 \sqrt {d+e x^2} \left (a e^2-b d e+c d^2\right )} \]
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Rule 197
Rule 211
Rule 270
Rule 277
Rule 385
Rule 1315
Rule 1706
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {c d-b e-c e x^2}{x^4 \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}{x^4 \left (d+e x^2\right )^{3/2}} \, dx}{c d^2-b d e+a e^2} \\ & = -\frac {e^2}{3 d \left (c d^2-b d e+a e^2\right ) x^3 \sqrt {d+e x^2}}+\frac {\int \left (\frac {c d-b e}{a x^4 \sqrt {d+e x^2}}+\frac {-b c d+b^2 e-a c e}{a^2 x^2 \sqrt {d+e x^2}}+\frac {b^2 c d-a c^2 d-b^3 e+2 a b c e+c \left (b c d-b^2 e+a c e\right ) x^2}{a^2 \sqrt {d+e x^2} \left (a+b x^2+c x^4\right )}\right ) \, dx}{c d^2-b d e+a e^2}-\frac {\left (4 e^3\right ) \int \frac {1}{x^2 \left (d+e x^2\right )^{3/2}} \, dx}{3 d \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {e^2}{3 d \left (c d^2-b d e+a e^2\right ) x^3 \sqrt {d+e x^2}}+\frac {4 e^3}{3 d^2 \left (c d^2-b d e+a e^2\right ) x \sqrt {d+e x^2}}+\frac {\int \frac {b^2 c d-a c^2 d-b^3 e+2 a b c e+c \left (b c d-b^2 e+a c e\right ) x^2}{\sqrt {d+e x^2} \left (a+b x^2+c x^4\right )} \, dx}{a^2 \left (c d^2-b d e+a e^2\right )}+\frac {\left (8 e^4\right ) \int \frac {1}{\left (d+e x^2\right )^{3/2}} \, dx}{3 d^2 \left (c d^2-b d e+a e^2\right )}+\frac {(c d-b e) \int \frac {1}{x^4 \sqrt {d+e x^2}} \, dx}{a \left (c d^2-b d e+a e^2\right )}-\frac {\left (b c d-b^2 e+a c e\right ) \int \frac {1}{x^2 \sqrt {d+e x^2}} \, dx}{a^2 \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {e^2}{3 d \left (c d^2-b d e+a e^2\right ) x^3 \sqrt {d+e x^2}}+\frac {4 e^3}{3 d^2 \left (c d^2-b d e+a e^2\right ) x \sqrt {d+e x^2}}+\frac {8 e^4 x}{3 d^3 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x^2}}-\frac {(c d-b e) \sqrt {d+e x^2}}{3 a d \left (c d^2-b d e+a e^2\right ) x^3}+\frac {\left (b c d-b^2 e+a c e\right ) \sqrt {d+e x^2}}{a^2 d \left (c d^2-b d e+a e^2\right ) x}+\frac {\int \left (\frac {c \left (b c d-b^2 e+a c e\right )-\frac {c \left (-b^2 c d+2 a c^2 d+b^3 e-3 a b c e\right )}{\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 \left (b c d-b^2 e+a c e\right )+\frac {c \left (-b^2 c d+2 a c^2 d+b^3 e-3 a b c e\right )}{\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}{a^2 \left (c d^2-b d e+a e^2\right )}-\frac {(2 e (c d-b e)) \int \frac {1}{x^2 \sqrt {d+e x^2}} \, dx}{3 a d \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {e^2}{3 d \left (c d^2-b d e+a e^2\right ) x^3 \sqrt {d+e x^2}}+\frac {4 e^3}{3 d^2 \left (c d^2-b d e+a e^2\right ) x \sqrt {d+e x^2}}+\frac {8 e^4 x}{3 d^3 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x^2}}-\frac {(c d-b e) \sqrt {d+e x^2}}{3 a d \left (c d^2-b d e+a e^2\right ) x^3}+\frac {2 e (c d-b e) \sqrt {d+e x^2}}{3 a d^2 \left (c d^2-b d e+a e^2\right ) x}+\frac {\left (b c d-b^2 e+a c e\right ) \sqrt {d+e x^2}}{a^2 d \left (c d^2-b d e+a e^2\right ) x}+\frac {\left (c \left (b c d-b^2 e+a c e-\frac {b^2 c d-2 a c^2 d-b^3 e+3 a b c 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}{a^2 \left (c d^2-b d e+a e^2\right )}+\frac {\left (c \left (b c d-b^2 e+a c e+\frac {b^2 c d-2 a c^2 d-b^3 e+3 a b c 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}{a^2 \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {e^2}{3 d \left (c d^2-b d e+a e^2\right ) x^3 \sqrt {d+e x^2}}+\frac {4 e^3}{3 d^2 \left (c d^2-b d e+a e^2\right ) x \sqrt {d+e x^2}}+\frac {8 e^4 x}{3 d^3 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x^2}}-\frac {(c d-b e) \sqrt {d+e x^2}}{3 a d \left (c d^2-b d e+a e^2\right ) x^3}+\frac {2 e (c d-b e) \sqrt {d+e x^2}}{3 a d^2 \left (c d^2-b d e+a e^2\right ) x}+\frac {\left (b c d-b^2 e+a c e\right ) \sqrt {d+e x^2}}{a^2 d \left (c d^2-b d e+a e^2\right ) x}+\frac {\left (c \left (b c d-b^2 e+a c e-\frac {b^2 c d-2 a c^2 d-b^3 e+3 a b c 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 )}{a^2 \left (c d^2-b d e+a e^2\right )}+\frac {\left (c \left (b c d-b^2 e+a c e+\frac {b^2 c d-2 a c^2 d-b^3 e+3 a b c 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 )}{a^2 \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {e^2}{3 d \left (c d^2-b d e+a e^2\right ) x^3 \sqrt {d+e x^2}}+\frac {4 e^3}{3 d^2 \left (c d^2-b d e+a e^2\right ) x \sqrt {d+e x^2}}+\frac {8 e^4 x}{3 d^3 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x^2}}-\frac {(c d-b e) \sqrt {d+e x^2}}{3 a d \left (c d^2-b d e+a e^2\right ) x^3}+\frac {2 e (c d-b e) \sqrt {d+e x^2}}{3 a d^2 \left (c d^2-b d e+a e^2\right ) x}+\frac {\left (b c d-b^2 e+a c e\right ) \sqrt {d+e x^2}}{a^2 d \left (c d^2-b d e+a e^2\right ) x}+\frac {c \left (b c d-b^2 e+a c e+\frac {b^2 c d-2 a c^2 d-b^3 e+3 a b c 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 )}{a^2 \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 (b c d-b^2 e+a c e-\frac {b^2 c d-2 a c^2 d-b^3 e+3 a b c 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 )}{a^2 \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 = 17.82 (sec) , antiderivative size = 2218, normalized size of antiderivative = 5.29 \[ \int \frac {1}{x^4 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\text {Result too large to show} \]
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Time = 1.32 (sec) , antiderivative size = 610, normalized size of antiderivative = 1.46
method | result | size |
pseudoelliptic | \(-\frac {-3 \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {2}\, d^{3} \left (\left (c \left (b e -\frac {c d}{2}\right ) a -\frac {b^{2} \left (b e -c d \right )}{2}\right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}+d \left (a^{2} c^{2} e +\left (-2 b^{2} c e +\frac {3}{2} b \,c^{2} d \right ) a +\frac {b^{3} \left (b e -c d \right )}{2}\right )\right ) x^{3} \sqrt {e \,x^{2}+d}\, \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )+\sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \left (-3 \sqrt {2}\, d^{3} \left (\left (\left (-e b c +\frac {1}{2} c^{2} d \right ) a +\frac {b^{2} \left (b e -c d \right )}{2}\right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}+d \left (a^{2} c^{2} e +\left (-2 b^{2} c e +\frac {3}{2} b \,c^{2} d \right ) a +\frac {b^{3} \left (b e -c d \right )}{2}\right )\right ) x^{3} \sqrt {e \,x^{2}+d}\, \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )+\sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \left (e^{2} \left (-8 e^{2} x^{4}-4 e d \,x^{2}+d^{2}\right ) a^{2}-\left (-c \,d^{2}+e \left (5 c \,x^{2}+b \right ) d -2 b \,e^{2} x^{2}\right ) \left (e \,x^{2}+d \right ) d a +3 b \,d^{2} x^{2} \left (e \,x^{2}+d \right ) \left (b e -c d \right )\right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right )}{3 \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {e \,x^{2}+d}\, \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, a^{2} x^{3} \left (a \,e^{2}-b d e +c \,d^{2}\right ) d^{3}}\) | \(610\) |
default | \(\frac {-\frac {1}{3 d \,x^{3} \sqrt {e \,x^{2}+d}}-\frac {4 e \left (-\frac {1}{d x \sqrt {e \,x^{2}+d}}-\frac {2 e x}{d^{2} \sqrt {e \,x^{2}+d}}\right )}{3 d}}{a}-\frac {b \left (-\frac {1}{d x \sqrt {e \,x^{2}+d}}-\frac {2 e x}{d^{2} \sqrt {e \,x^{2}+d}}\right )}{a^{2}}+\frac {\sqrt {2}\, d \left (\left (-\frac {d c \left (a c -b^{2}\right )}{2}+b e \left (a c -\frac {b^{2}}{2}\right )\right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}+d \left (\frac {\left (3 a b \,c^{2}-b^{3} c \right ) d}{2}+\left (-2 a \,b^{2} c +\frac {1}{2} b^{4}+a^{2} c^{2}\right ) e \right )\right ) \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}\, d \left (\left (\frac {d c \left (a c -b^{2}\right )}{2}-b e \left (a c -\frac {b^{2}}{2}\right )\right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}+d \left (\frac {\left (3 a b \,c^{2}-b^{3} c \right ) d}{2}+\left (-2 a \,b^{2} c +\frac {1}{2} b^{4}+a^{2} c^{2}\right ) e \right )\right ) \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 \left (\left (a c -b^{2}\right ) e +b c d \right ) \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 )}\, x \right ) \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}{a^{2} \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}\) | \(629\) |
risch | \(-\frac {\sqrt {e \,x^{2}+d}\, \left (-5 a e \,x^{2}-3 b d \,x^{2}+d a \right )}{3 d^{3} a^{2} x^{3}}+\frac {\frac {e^{3} a^{2} \sqrt {\left (x -\frac {\sqrt {-e d}}{e}\right )^{2} e +2 \sqrt {-e d}\, \left (x -\frac {\sqrt {-e d}}{e}\right )}}{2 d \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (x -\frac {\sqrt {-e d}}{e}\right )}+\frac {e^{3} a^{2} \sqrt {\left (x +\frac {\sqrt {-e d}}{e}\right )^{2} e -2 \sqrt {-e d}\, \left (x +\frac {\sqrt {-e d}}{e}\right )}}{2 d \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (x +\frac {\sqrt {-e d}}{e}\right )}+\frac {d^{2} \sqrt {2}\, \left (\sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \left (\left (c \left (b e -\frac {c d}{2}\right ) a -\frac {b^{2} \left (b e -c d \right )}{2}\right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}+d \left (a^{2} c^{2} e +\left (-2 b^{2} c e +\frac {3}{2} b \,c^{2} d \right ) a +\frac {b^{3} \left (b e -c d \right )}{2}\right )\right ) \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 )+\sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \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 ) \left (\left (\left (-e b c +\frac {1}{2} c^{2} d \right ) a +\frac {b^{2} \left (b e -c d \right )}{2}\right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}+d \left (a^{2} c^{2} e +\left (-2 b^{2} c e +\frac {3}{2} b \,c^{2} d \right ) a +\frac {b^{3} \left (b e -c d \right )}{2}\right )\right )\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\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 {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}}}{a^{2} d^{2}}\) | \(637\) |
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Timed out. \[ \int \frac {1}{x^4 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{x^4 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\int \frac {1}{x^{4} \left (d + e x^{2}\right )^{\frac {3}{2}} \left (a + b x^{2} + c x^{4}\right )}\, dx \]
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\[ \int \frac {1}{x^4 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\int { \frac {1}{{\left (c x^{4} + b x^{2} + a\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{4}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^4 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1}{x^4 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\int \frac {1}{x^4\,{\left (e\,x^2+d\right )}^{3/2}\,\left (c\,x^4+b\,x^2+a\right )} \,d x \]
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