Integrand size = 21, antiderivative size = 264 \[ \int \frac {\left (d+e x^2\right )^2}{\sqrt {a+c x^4}} \, dx=\frac {e^2 x \sqrt {a+c x^4}}{3 c}+\frac {2 d e x \sqrt {a+c x^4}}{\sqrt {c} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {2 \sqrt [4]{a} d e \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{c^{3/4} \sqrt {a+c x^4}}+\frac {\left (3 c d^2+6 \sqrt {a} \sqrt {c} d e-a e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{6 \sqrt [4]{a} c^{5/4} \sqrt {a+c x^4}} \]
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Time = 0.08 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {1221, 1212, 226, 1210} \[ \int \frac {\left (d+e x^2\right )^2}{\sqrt {a+c x^4}} \, dx=\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (6 \sqrt {a} \sqrt {c} d e-a e^2+3 c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{6 \sqrt [4]{a} c^{5/4} \sqrt {a+c x^4}}-\frac {2 \sqrt [4]{a} d e \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{c^{3/4} \sqrt {a+c x^4}}+\frac {2 d e x \sqrt {a+c x^4}}{\sqrt {c} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {e^2 x \sqrt {a+c x^4}}{3 c} \]
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Rule 226
Rule 1210
Rule 1212
Rule 1221
Rubi steps \begin{align*} \text {integral}& = \frac {e^2 x \sqrt {a+c x^4}}{3 c}+\frac {\int \frac {3 c d^2-a e^2+6 c d e x^2}{\sqrt {a+c x^4}} \, dx}{3 c} \\ & = \frac {e^2 x \sqrt {a+c x^4}}{3 c}-\frac {\left (2 \sqrt {a} d e\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+c x^4}} \, dx}{\sqrt {c}}+\frac {\left (3 c d^2+6 \sqrt {a} \sqrt {c} d e-a e^2\right ) \int \frac {1}{\sqrt {a+c x^4}} \, dx}{3 c} \\ & = \frac {e^2 x \sqrt {a+c x^4}}{3 c}+\frac {2 d e x \sqrt {a+c x^4}}{\sqrt {c} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {2 \sqrt [4]{a} d 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 )}{c^{3/4} \sqrt {a+c x^4}}+\frac {\left (3 c d^2+6 \sqrt {a} \sqrt {c} d e-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}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{6 \sqrt [4]{a} c^{5/4} \sqrt {a+c x^4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.07 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.45 \[ \int \frac {\left (d+e x^2\right )^2}{\sqrt {a+c x^4}} \, dx=\frac {\left (3 c d^2-a e^2\right ) x \sqrt {1+\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {c x^4}{a}\right )+e x \left (e \left (a+c x^4\right )+2 c d x^2 \sqrt {1+\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {c x^4}{a}\right )\right )}{3 c \sqrt {a+c x^4}} \]
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Result contains complex when optimal does not.
Time = 1.33 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.76
method | result | size |
elliptic | \(\frac {e^{2} x \sqrt {c \,x^{4}+a}}{3 c}+\frac {\left (d^{2}-\frac {a \,e^{2}}{3 c}\right ) \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {2 i e d \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\) | \(200\) |
default | \(\frac {d^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+e^{2} \left (\frac {x \sqrt {c \,x^{4}+a}}{3 c}-\frac {a \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{3 c \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )+\frac {2 i e d \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\) | \(266\) |
risch | \(\frac {e^{2} x \sqrt {c \,x^{4}+a}}{3 c}-\frac {\frac {a \,e^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {3 c \,d^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {6 i d \sqrt {c}\, e \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}}{3 c}\) | \(271\) |
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Time = 0.09 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.43 \[ \int \frac {\left (d+e x^2\right )^2}{\sqrt {a+c x^4}} \, dx=\frac {6 \, a \sqrt {c} d e x \left (-\frac {a}{c}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (-\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (3 \, c d^{2} - 6 \, a d e - a e^{2}\right )} \sqrt {c} x \left (-\frac {a}{c}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (a e^{2} x^{2} + 6 \, a d e\right )} \sqrt {c x^{4} + a}}{3 \, a c x} \]
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Result contains complex when optimal does not.
Time = 1.30 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.47 \[ \int \frac {\left (d+e x^2\right )^2}{\sqrt {a+c x^4}} \, dx=\frac {d^{2} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {5}{4}\right )} + \frac {d e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} \Gamma \left (\frac {7}{4}\right )} + \frac {e^{2} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {9}{4}\right )} \]
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\[ \int \frac {\left (d+e x^2\right )^2}{\sqrt {a+c x^4}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{2}}{\sqrt {c x^{4} + a}} \,d x } \]
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\[ \int \frac {\left (d+e x^2\right )^2}{\sqrt {a+c x^4}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{2}}{\sqrt {c x^{4} + a}} \,d x } \]
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Timed out. \[ \int \frac {\left (d+e x^2\right )^2}{\sqrt {a+c x^4}} \, dx=\int \frac {{\left (e\,x^2+d\right )}^2}{\sqrt {c\,x^4+a}} \,d x \]
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