Integrand size = 22, antiderivative size = 425 \[ \int \frac {1}{\left (d+e x^2\right )^3 \sqrt {a-c x^4}} \, dx=-\frac {e^2 x \sqrt {a-c x^4}}{4 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )^2}-\frac {3 e^2 \left (3 c d^2-a e^2\right ) x \sqrt {a-c x^4}}{8 d^2 \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )}-\frac {3 a^{3/4} \sqrt [4]{c} e \left (3 c d^2-a e^2\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 d^2 \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \sqrt [4]{c} \left (7 c d^2-2 \sqrt {a} \sqrt {c} d e-3 a e^2\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{8 d^2 \left (\sqrt {c} d+\sqrt {a} e\right ) \left (c d^2-a e^2\right ) \sqrt {a-c x^4}}+\frac {3 \sqrt [4]{a} \left (5 c^2 d^4-2 a c d^2 e^2+a^2 e^4\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{8 \sqrt [4]{c} d^3 \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}} \]
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Time = 0.45 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1238, 1711, 1731, 1215, 230, 227, 1214, 1213, 435, 1233, 1232} \[ \int \frac {1}{\left (d+e x^2\right )^3 \sqrt {a-c x^4}} \, dx=-\frac {3 a^{3/4} \sqrt [4]{c} e \sqrt {1-\frac {c x^4}{a}} \left (3 c d^2-a e^2\right ) E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 d^2 \sqrt {a-c x^4} \left (c d^2-a e^2\right )^2}+\frac {3 \sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (a^2 e^4-2 a c d^2 e^2+5 c^2 d^4\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{8 \sqrt [4]{c} d^3 \sqrt {a-c x^4} \left (c d^2-a e^2\right )^2}-\frac {\sqrt [4]{a} \sqrt [4]{c} \sqrt {1-\frac {c x^4}{a}} \left (-2 \sqrt {a} \sqrt {c} d e-3 a e^2+7 c d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{8 d^2 \sqrt {a-c x^4} \left (\sqrt {a} e+\sqrt {c} d\right ) \left (c d^2-a e^2\right )}-\frac {3 e^2 x \sqrt {a-c x^4} \left (3 c d^2-a e^2\right )}{8 d^2 \left (d+e x^2\right ) \left (c d^2-a e^2\right )^2}-\frac {e^2 x \sqrt {a-c x^4}}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )} \]
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Rule 227
Rule 230
Rule 435
Rule 1213
Rule 1214
Rule 1215
Rule 1232
Rule 1233
Rule 1238
Rule 1711
Rule 1731
Rubi steps \begin{align*} \text {integral}& = -\frac {e^2 x \sqrt {a-c x^4}}{4 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )^2}+\frac {\int \frac {4 c d^2-3 a e^2-4 c d e x^2+c e^2 x^4}{\left (d+e x^2\right )^2 \sqrt {a-c x^4}} \, dx}{4 d \left (c d^2-a e^2\right )} \\ & = -\frac {e^2 x \sqrt {a-c x^4}}{4 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )^2}-\frac {3 e^2 \left (3 c d^2-a e^2\right ) x \sqrt {a-c x^4}}{8 d^2 \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )}+\frac {\int \frac {8 c^2 d^4-5 a c d^2 e^2+3 a^2 e^4-4 c d e \left (4 c d^2-a e^2\right ) x^2-3 c e^2 \left (3 c d^2-a e^2\right ) x^4}{\left (d+e x^2\right ) \sqrt {a-c x^4}} \, dx}{8 d^2 \left (c d^2-a e^2\right )^2} \\ & = -\frac {e^2 x \sqrt {a-c x^4}}{4 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )^2}-\frac {3 e^2 \left (3 c d^2-a e^2\right ) x \sqrt {a-c x^4}}{8 d^2 \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )}-\frac {\int \frac {-3 c d e^2 \left (3 c d^2-a e^2\right )+4 c d e^2 \left (4 c d^2-a e^2\right )+3 c e^3 \left (3 c d^2-a e^2\right ) x^2}{\sqrt {a-c x^4}} \, dx}{8 d^2 e^2 \left (c d^2-a e^2\right )^2}+\frac {\left (3 \left (5 c^2 d^4-2 a c d^2 e^2+a^2 e^4\right )\right ) \int \frac {1}{\left (d+e x^2\right ) \sqrt {a-c x^4}} \, dx}{8 d^2 \left (c d^2-a e^2\right )^2} \\ & = -\frac {e^2 x \sqrt {a-c x^4}}{4 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )^2}-\frac {3 e^2 \left (3 c d^2-a e^2\right ) x \sqrt {a-c x^4}}{8 d^2 \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )}-\frac {\left (\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (7 c d^2-2 \sqrt {a} \sqrt {c} d e-3 a e^2\right )\right ) \int \frac {1}{\sqrt {a-c x^4}} \, dx}{8 d^2 \left (c d^2-a e^2\right )^2}-\frac {\left (3 \sqrt {a} \sqrt {c} e \left (3 c d^2-a e^2\right )\right ) \int \frac {1+\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a-c x^4}} \, dx}{8 d^2 \left (c d^2-a e^2\right )^2}+\frac {\left (3 \left (5 c^2 d^4-2 a c d^2 e^2+a^2 e^4\right ) \sqrt {1-\frac {c x^4}{a}}\right ) \int \frac {1}{\left (d+e x^2\right ) \sqrt {1-\frac {c x^4}{a}}} \, dx}{8 d^2 \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}} \\ & = -\frac {e^2 x \sqrt {a-c x^4}}{4 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )^2}-\frac {3 e^2 \left (3 c d^2-a e^2\right ) x \sqrt {a-c x^4}}{8 d^2 \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )}+\frac {3 \sqrt [4]{a} \left (5 c^2 d^4-2 a c d^2 e^2+a^2 e^4\right ) \sqrt {1-\frac {c x^4}{a}} \Pi \left (-\frac {\sqrt {a} e}{\sqrt {c} d};\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{c} d^3 \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}}-\frac {\left (\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (7 c d^2-2 \sqrt {a} \sqrt {c} d e-3 a e^2\right ) \sqrt {1-\frac {c x^4}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {c x^4}{a}}} \, dx}{8 d^2 \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}}-\frac {\left (3 \sqrt {a} \sqrt {c} e \left (3 c d^2-a e^2\right ) \sqrt {1-\frac {c x^4}{a}}\right ) \int \frac {1+\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {1-\frac {c x^4}{a}}} \, dx}{8 d^2 \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}} \\ & = -\frac {e^2 x \sqrt {a-c x^4}}{4 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )^2}-\frac {3 e^2 \left (3 c d^2-a e^2\right ) x \sqrt {a-c x^4}}{8 d^2 \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )}-\frac {\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (7 c d^2-2 \sqrt {a} \sqrt {c} d e-3 a e^2\right ) \sqrt {1-\frac {c x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 d^2 \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}}+\frac {3 \sqrt [4]{a} \left (5 c^2 d^4-2 a c d^2 e^2+a^2 e^4\right ) \sqrt {1-\frac {c x^4}{a}} \Pi \left (-\frac {\sqrt {a} e}{\sqrt {c} d};\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{c} d^3 \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}}-\frac {\left (3 \sqrt {a} \sqrt {c} e \left (3 c d^2-a e^2\right ) \sqrt {1-\frac {c x^4}{a}}\right ) \int \frac {\sqrt {1+\frac {\sqrt {c} x^2}{\sqrt {a}}}}{\sqrt {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}} \, dx}{8 d^2 \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}} \\ & = -\frac {e^2 x \sqrt {a-c x^4}}{4 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )^2}-\frac {3 e^2 \left (3 c d^2-a e^2\right ) x \sqrt {a-c x^4}}{8 d^2 \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )}-\frac {3 a^{3/4} \sqrt [4]{c} e \left (3 c d^2-a e^2\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 d^2 \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (7 c d^2-2 \sqrt {a} \sqrt {c} d e-3 a e^2\right ) \sqrt {1-\frac {c x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 d^2 \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}}+\frac {3 \sqrt [4]{a} \left (5 c^2 d^4-2 a c d^2 e^2+a^2 e^4\right ) \sqrt {1-\frac {c x^4}{a}} \Pi \left (-\frac {\sqrt {a} e}{\sqrt {c} d};\left .\sin ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{c} d^3 \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.98 (sec) , antiderivative size = 321, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\left (d+e x^2\right )^3 \sqrt {a-c x^4}} \, dx=\frac {\frac {d e^2 x \left (a-c x^4\right ) \left (a e^2 \left (5 d+3 e x^2\right )-c d^2 \left (11 d+9 e x^2\right )\right )}{\left (d+e x^2\right )^2}-\frac {i \sqrt {1-\frac {c x^4}{a}} \left (3 \sqrt {a} \sqrt {c} d e \left (-3 c d^2+a e^2\right ) E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+\left (-7 c^2 d^4+9 \sqrt {a} c^{3/2} d^3 e+a c d^2 e^2-3 a^{3/2} \sqrt {c} d e^3\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+3 \left (5 c^2 d^4-2 a c d^2 e^2+a^2 e^4\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}}{8 d^3 \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 960 vs. \(2 (363 ) = 726\).
Time = 1.82 (sec) , antiderivative size = 961, normalized size of antiderivative = 2.26
method | result | size |
default | \(\frac {e^{2} x \sqrt {-c \,x^{4}+a}}{4 \left (a \,e^{2}-c \,d^{2}\right ) d \left (e \,x^{2}+d \right )^{2}}+\frac {3 e^{2} \left (a \,e^{2}-3 c \,d^{2}\right ) x \sqrt {-c \,x^{4}+a}}{8 \left (a \,e^{2}-c \,d^{2}\right )^{2} d^{2} \left (e \,x^{2}+d \right )}+\frac {c \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right ) a \,e^{2}}{8 d \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {7 c^{2} d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{8 \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {3 \sqrt {c}\, e^{3} a^{\frac {3}{2}} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{8 \left (a \,e^{2}-c \,d^{2}\right )^{2} d^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {9 c^{\frac {3}{2}} e \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{8 \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {3 \sqrt {c}\, e^{3} a^{\frac {3}{2}} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, E\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{8 \left (a \,e^{2}-c \,d^{2}\right )^{2} d^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {9 c^{\frac {3}{2}} e \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, E\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{8 \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {3 e^{4} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {e \sqrt {a}}{d \sqrt {c}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right ) a^{2}}{8 \left (a \,e^{2}-c \,d^{2}\right )^{2} d^{3} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {3 e^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {e \sqrt {a}}{d \sqrt {c}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right ) a c}{4 \left (a \,e^{2}-c \,d^{2}\right )^{2} d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {15 d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {e \sqrt {a}}{d \sqrt {c}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right ) c^{2}}{8 \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\) | \(961\) |
elliptic | \(\frac {e^{2} x \sqrt {-c \,x^{4}+a}}{4 \left (a \,e^{2}-c \,d^{2}\right ) d \left (e \,x^{2}+d \right )^{2}}+\frac {3 e^{2} \left (a \,e^{2}-3 c \,d^{2}\right ) x \sqrt {-c \,x^{4}+a}}{8 \left (a \,e^{2}-c \,d^{2}\right )^{2} d^{2} \left (e \,x^{2}+d \right )}+\frac {c \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right ) a \,e^{2}}{8 d \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {7 c^{2} d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{8 \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {3 \sqrt {c}\, e^{3} a^{\frac {3}{2}} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{8 \left (a \,e^{2}-c \,d^{2}\right )^{2} d^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {9 c^{\frac {3}{2}} e \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{8 \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {3 \sqrt {c}\, e^{3} a^{\frac {3}{2}} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, E\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{8 \left (a \,e^{2}-c \,d^{2}\right )^{2} d^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {9 c^{\frac {3}{2}} e \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, E\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{8 \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {3 e^{4} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {e \sqrt {a}}{d \sqrt {c}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right ) a^{2}}{8 \left (a \,e^{2}-c \,d^{2}\right )^{2} d^{3} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {3 e^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {e \sqrt {a}}{d \sqrt {c}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right ) a c}{4 \left (a \,e^{2}-c \,d^{2}\right )^{2} d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {15 d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {e \sqrt {a}}{d \sqrt {c}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right ) c^{2}}{8 \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\) | \(961\) |
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Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^3 \sqrt {a-c x^4}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\left (d+e x^2\right )^3 \sqrt {a-c x^4}} \, dx=\int \frac {1}{\sqrt {a - c x^{4}} \left (d + e x^{2}\right )^{3}}\, dx \]
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\[ \int \frac {1}{\left (d+e x^2\right )^3 \sqrt {a-c x^4}} \, dx=\int { \frac {1}{\sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )}^{3}} \,d x } \]
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\[ \int \frac {1}{\left (d+e x^2\right )^3 \sqrt {a-c x^4}} \, dx=\int { \frac {1}{\sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^3 \sqrt {a-c x^4}} \, dx=\int \frac {1}{\sqrt {a-c\,x^4}\,{\left (e\,x^2+d\right )}^3} \,d x \]
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