Integrand size = 29, antiderivative size = 339 \[ \int \frac {1}{x^2 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\frac {e (c d-b e) x}{a d \left (c d^2+e (-b d+a e)\right ) \sqrt {d+e x^2}}+\frac {-d-2 e x^2}{a d^2 x \sqrt {d+e x^2}}-\frac {2 c^2 \left (1+\frac {b}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a \sqrt {b-\sqrt {b^2-4 a c}} \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right )^{3/2}}-\frac {2 c^2 \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a \sqrt {b+\sqrt {b^2-4 a c}} \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right )^{3/2}} \]
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Time = 2.00 (sec) , antiderivative size = 462, normalized size of antiderivative = 1.36, number of steps used = 12, 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} \[ \int \frac {1}{x^2 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=-\frac {c \left (\frac {2 a c e+b^2 (-e)+b c d}{\sqrt {b^2-4 a c}}-b e+c d\right ) \arctan \left (\frac {x \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )}-\frac {c \left (-\frac {2 a c e+b^2 (-e)+b c d}{\sqrt {b^2-4 a c}}-b e+c d\right ) \arctan \left (\frac {x \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {d+e x^2}}\right )}{a \sqrt {\sqrt {b^2-4 a c}+b} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )}-\frac {e^2}{d x \sqrt {d+e x^2} \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {d+e x^2} (c d-b e)}{a d x \left (a e^2-b d e+c d^2\right )}-\frac {2 e^3 x}{d^2 \sqrt {d+e x^2} \left (a e^2-b d e+c d^2\right )} \]
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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^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}{x^2 \left (d+e x^2\right )^{3/2}} \, dx}{c d^2-b d e+a e^2} \\ & = -\frac {e^2}{d \left (c d^2-b d e+a e^2\right ) x \sqrt {d+e x^2}}+\frac {\int \left (\frac {c d-b e}{a x^2 \sqrt {d+e x^2}}+\frac {-b c d+b^2 e-a c e-c (c d-b e) x^2}{a \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 (2 e^3\right ) \int \frac {1}{\left (d+e x^2\right )^{3/2}} \, dx}{d \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {e^2}{d \left (c d^2-b d e+a e^2\right ) x \sqrt {d+e x^2}}-\frac {2 e^3 x}{d^2 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x^2}}+\frac {\int \frac {-b c d+b^2 e-a c e-c (c d-b e) x^2}{\sqrt {d+e x^2} \left (a+b x^2+c x^4\right )} \, dx}{a \left (c d^2-b d e+a e^2\right )}+\frac {(c d-b e) \int \frac {1}{x^2 \sqrt {d+e x^2}} \, dx}{a \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {e^2}{d \left (c d^2-b d e+a e^2\right ) x \sqrt {d+e x^2}}-\frac {2 e^3 x}{d^2 \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}}{a d \left (c d^2-b d e+a e^2\right ) x}+\frac {\int \left (\frac {-c (c d-b e)+\frac {c \left (-b c d+b^2 e-2 a 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 (c d-b e)-\frac {c \left (-b c d+b^2 e-2 a 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 \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {e^2}{d \left (c d^2-b d e+a e^2\right ) x \sqrt {d+e x^2}}-\frac {2 e^3 x}{d^2 \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}}{a d \left (c d^2-b d e+a e^2\right ) x}-\frac {\left (c \left (c d-b e-\frac {b c d-b^2 e+2 a 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 \left (c d^2-b d e+a e^2\right )}-\frac {\left (c \left (c d-b e+\frac {b c d-b^2 e+2 a 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 \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {e^2}{d \left (c d^2-b d e+a e^2\right ) x \sqrt {d+e x^2}}-\frac {2 e^3 x}{d^2 \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}}{a d \left (c d^2-b d e+a e^2\right ) x}-\frac {\left (c \left (c d-b e-\frac {b c d-b^2 e+2 a 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 \left (c d^2-b d e+a e^2\right )}-\frac {\left (c \left (c d-b e+\frac {b c d-b^2 e+2 a 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 \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {e^2}{d \left (c d^2-b d e+a e^2\right ) x \sqrt {d+e x^2}}-\frac {2 e^3 x}{d^2 \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}}{a d \left (c d^2-b d e+a e^2\right ) x}-\frac {c \left (c d-b e+\frac {b c d-b^2 e+2 a 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 \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 (c d-b e-\frac {b c d-b^2 e+2 a 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 \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 = 17.54 (sec) , antiderivative size = 2158, normalized size of antiderivative = 6.37 \[ \int \frac {1}{x^2 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\text {Result too large to show} \]
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Time = 1.14 (sec) , antiderivative size = 513, normalized size of antiderivative = 1.51
method | result | size |
default | \(\frac {-\frac {1}{d x \sqrt {e \,x^{2}+d}}-\frac {2 e x}{d^{2} \sqrt {e \,x^{2}+d}}}{a}-\frac {\sqrt {2}\, d \left (\left (\left (a c -b^{2}\right ) e +b c d \right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}+\left (2 a \,c^{2}-b^{2} c \right ) d^{2}+e \left (-3 a b c +b^{3}\right ) d \right ) \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {e \,x^{2}+d}\, \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )-\sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \left (\sqrt {2}\, d \sqrt {e \,x^{2}+d}\, \left (\left (\left (a c -b^{2}\right ) e +b c d \right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}+\left (-2 a \,c^{2}+b^{2} c \right ) d^{2}+e \left (3 a b c -b^{3}\right ) d \right ) \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )-2 \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, e x \left (b e -c d \right )\right )}{2 a \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {e \,x^{2}+d}\, \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \left (a \,e^{2}-b d e +c \,d^{2}\right ) d}\) | \(513\) |
pseudoelliptic | \(-\frac {\frac {\sqrt {2}\, d^{2} \left (\left (\left (a c -b^{2}\right ) e +b c d \right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}+\left (2 a \,c^{2}-b^{2} c \right ) d^{2}+e \left (-3 a b c +b^{3}\right ) d \right ) \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, x \sqrt {e \,x^{2}+d}\, \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )}{2}+\left (-\frac {\sqrt {2}\, d^{2} x \sqrt {e \,x^{2}+d}\, \left (\left (\left (a c -b^{2}\right ) e +b c d \right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}+\left (-2 a \,c^{2}+b^{2} c \right ) d^{2}+e \left (3 a b c -b^{3}\right ) d \right ) \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )}{2}+\left (d^{3} c -e \left (-c \,x^{2}+b \right ) d^{2}+e^{2} \left (-b \,x^{2}+a \right ) d +2 x^{2} e^{3} a \right ) \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}{\sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {e \,x^{2}+d}\, \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, a x \left (a \,e^{2}-b d e +c \,d^{2}\right ) d^{2}}\) | \(515\) |
risch | \(-\frac {\sqrt {e \,x^{2}+d}}{d^{2} a x}-\frac {\frac {e^{2} a \sqrt {\left (x +\frac {\sqrt {-e d}}{e}\right )^{2} e -2 \sqrt {-e d}\, \left (x +\frac {\sqrt {-e d}}{e}\right )}}{2 d \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (x +\frac {\sqrt {-e d}}{e}\right )}+\frac {e^{2} a \sqrt {\left (x -\frac {\sqrt {-e d}}{e}\right )^{2} e +2 \sqrt {-e d}\, \left (x -\frac {\sqrt {-e d}}{e}\right )}}{2 d \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (x -\frac {\sqrt {-e d}}{e}\right )}-\frac {d \sqrt {2}\, \left (-\left (\left (a c e -b^{2} e +b c d \right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}+\left (-3 b c d e +2 c^{2} d^{2}\right ) a +b^{2} d \left (b e -c d \right )\right ) \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )+\sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right ) \left (\left (a c e -b^{2} e +b c d \right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}+\left (3 b c d e -2 c^{2} d^{2}\right ) a -b^{3} d e +b^{2} c \,d^{2}\right )\right )}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}}}{a d}\) | \(573\) |
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Timed out. \[ \int \frac {1}{x^2 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{x^2 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\int \frac {1}{x^{2} \left (d + e x^{2}\right )^{\frac {3}{2}} \left (a + b x^{2} + c x^{4}\right )}\, dx \]
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\[ \int \frac {1}{x^2 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\int { \frac {1}{{\left (c x^{4} + b x^{2} + a\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^2 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1}{x^2 \left (d+e x^2\right )^{3/2} \left (a+b x^2+c x^4\right )} \, dx=\int \frac {1}{x^2\,{\left (e\,x^2+d\right )}^{3/2}\,\left (c\,x^4+b\,x^2+a\right )} \,d x \]
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