Integrand size = 21, antiderivative size = 581 \[ \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a+c x^4}} \, dx=-\frac {\sqrt {c} e x \sqrt {a+c x^4}}{2 d \left (c d^2+a e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {e^2 x \sqrt {a+c x^4}}{2 d \left (c d^2+a e^2\right ) \left (d+e x^2\right )}+\frac {\sqrt {e} \left (3 c d^2+a e^2\right ) \arctan \left (\frac {\sqrt {c d^2+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+c x^4}}\right )}{4 d^{3/2} \left (c d^2+a e^2\right )^{3/2}}+\frac {\sqrt [4]{a} \sqrt [4]{c} e \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 d \left (c d^2+a e^2\right ) \sqrt {a+c x^4}}+\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} d \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {a+c x^4}}-\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \left (3 c d^2+a e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{c} d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right ) \sqrt {a+c x^4}} \]
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Time = 0.47 (sec) , antiderivative size = 581, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1238, 1729, 1210, 1723, 226, 1721} \[ \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a+c x^4}} \, dx=\frac {\sqrt [4]{a} \sqrt [4]{c} e \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 d \sqrt {a+c x^4} \left (a e^2+c d^2\right )}-\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\sqrt {a} e+\sqrt {c} d\right ) \left (a e^2+3 c d^2\right ) \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{c} d^2 \sqrt {a+c x^4} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (a e^2+c d^2\right )}+\frac {\sqrt {e} \left (a e^2+3 c d^2\right ) \arctan \left (\frac {x \sqrt {a e^2+c d^2}}{\sqrt {d} \sqrt {e} \sqrt {a+c x^4}}\right )}{4 d^{3/2} \left (a e^2+c d^2\right )^{3/2}}+\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} d \sqrt {a+c x^4} \left (\sqrt {c} d-\sqrt {a} e\right )}+\frac {e^2 x \sqrt {a+c x^4}}{2 d \left (d+e x^2\right ) \left (a e^2+c d^2\right )}-\frac {\sqrt {c} e x \sqrt {a+c x^4}}{2 d \left (\sqrt {a}+\sqrt {c} x^2\right ) \left (a e^2+c d^2\right )} \]
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Rule 226
Rule 1210
Rule 1238
Rule 1721
Rule 1723
Rule 1729
Rubi steps \begin{align*} \text {integral}& = \frac {e^2 x \sqrt {a+c x^4}}{2 d \left (c d^2+a e^2\right ) \left (d+e x^2\right )}-\frac {\int \frac {-2 c d^2-a e^2+2 c d e x^2+c e^2 x^4}{\left (d+e x^2\right ) \sqrt {a+c x^4}} \, dx}{2 d \left (c d^2+a e^2\right )} \\ & = \frac {e^2 x \sqrt {a+c x^4}}{2 d \left (c d^2+a e^2\right ) \left (d+e x^2\right )}-\frac {\int \frac {\sqrt {a} c^{3/2} d e^2+c e \left (-2 c d^2-a e^2\right )+\left (2 c^2 d e^2-c e^2 \left (c d-\sqrt {a} \sqrt {c} e\right )\right ) x^2}{\left (d+e x^2\right ) \sqrt {a+c x^4}} \, dx}{2 c d e \left (c d^2+a e^2\right )}+\frac {\left (\sqrt {a} \sqrt {c} e\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+c x^4}} \, dx}{2 d \left (c d^2+a e^2\right )} \\ & = -\frac {\sqrt {c} e x \sqrt {a+c x^4}}{2 d \left (c d^2+a e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {e^2 x \sqrt {a+c x^4}}{2 d \left (c d^2+a e^2\right ) \left (d+e x^2\right )}+\frac {\sqrt [4]{a} \sqrt [4]{c} e \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 d \left (c d^2+a e^2\right ) \sqrt {a+c x^4}}+\frac {\sqrt {c} \int \frac {1}{\sqrt {a+c x^4}} \, dx}{d \left (\sqrt {c} d-\sqrt {a} e\right )}-\frac {\left (\sqrt {a} e \left (3 c d^2+a e^2\right )\right ) \int \frac {1+\frac {\sqrt {c} x^2}{\sqrt {a}}}{\left (d+e x^2\right ) \sqrt {a+c x^4}} \, dx}{2 d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )} \\ & = -\frac {\sqrt {c} e x \sqrt {a+c x^4}}{2 d \left (c d^2+a e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {e^2 x \sqrt {a+c x^4}}{2 d \left (c d^2+a e^2\right ) \left (d+e x^2\right )}+\frac {\sqrt {e} \left (3 c d^2+a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c d^2+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+c x^4}}\right )}{4 d^{3/2} \left (c d^2+a e^2\right )^{3/2}}+\frac {\sqrt [4]{a} \sqrt [4]{c} e \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 d \left (c d^2+a e^2\right ) \sqrt {a+c x^4}}+\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} d \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {a+c x^4}}-\frac {\sqrt [4]{a} \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right ) \left (3 c d^2+a e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \Pi \left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e};2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{8 \sqrt [4]{c} d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right ) \sqrt {a+c x^4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.59 (sec) , antiderivative size = 522, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a+c x^4}} \, dx=\frac {a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} d e^2 x+\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} c d e^2 x^5-\sqrt {a} \sqrt {c} d e \left (d+e x^2\right ) \sqrt {1+\frac {c x^4}{a}} E\left (\left .i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+\sqrt {c} d \left (i \sqrt {c} d+\sqrt {a} e\right ) \left (d+e x^2\right ) \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-3 i c d^3 \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {i \sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-i a d e^2 \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {i \sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-3 i c d^2 e x^2 \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {i \sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-i a e^3 x^2 \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {i \sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )}{2 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} d^2 \left (c d^2+a e^2\right ) \left (d+e x^2\right ) \sqrt {a+c x^4}} \]
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Result contains complex when optimal does not.
Time = 0.75 (sec) , antiderivative size = 556, normalized size of antiderivative = 0.96
method | result | size |
default | \(\frac {e^{2} x \sqrt {c \,x^{4}+a}}{2 d \left (a \,e^{2}+c \,d^{2}\right ) \left (e \,x^{2}+d \right )}-\frac {c \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{2 \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {i \sqrt {c}\, e \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{2 d \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {i \sqrt {c}\, e \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, E\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{2 d \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {e^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, \frac {i \sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right ) a}{2 d^{2} \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {3 \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, \frac {i \sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right ) c}{2 \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\) | \(556\) |
elliptic | \(\frac {e^{2} x \sqrt {c \,x^{4}+a}}{2 d \left (a \,e^{2}+c \,d^{2}\right ) \left (e \,x^{2}+d \right )}-\frac {c \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{2 \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {i \sqrt {c}\, e \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{2 d \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {i \sqrt {c}\, e \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, E\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{2 d \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {e^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, \frac {i \sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right ) a}{2 d^{2} \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {3 \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, \frac {i \sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right ) c}{2 \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\) | \(556\) |
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Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a+c x^4}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a+c x^4}} \, dx=\int \frac {1}{\sqrt {a + c x^{4}} \left (d + e x^{2}\right )^{2}}\, dx \]
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\[ \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a+c x^4}} \, dx=\int { \frac {1}{\sqrt {c x^{4} + a} {\left (e x^{2} + d\right )}^{2}} \,d x } \]
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\[ \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a+c x^4}} \, dx=\int { \frac {1}{\sqrt {c x^{4} + a} {\left (e x^{2} + d\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a+c x^4}} \, dx=\int \frac {1}{\sqrt {c\,x^4+a}\,{\left (e\,x^2+d\right )}^2} \,d x \]
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