Integrand size = 22, antiderivative size = 162 \[ \int \frac {\left (d+e x^2\right )^2}{\sqrt {a-c x^4}} \, dx=-\frac {e^2 x \sqrt {a-c x^4}}{3 c}+\frac {2 a^{3/4} d e \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \left (3 c d^2-6 \sqrt {a} \sqrt {c} d e+a e^2\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{3 c^{5/4} \sqrt {a-c x^4}} \]
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Time = 0.09 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {1221, 1215, 230, 227, 1214, 1213, 435} \[ \int \frac {\left (d+e x^2\right )^2}{\sqrt {a-c x^4}} \, dx=\frac {2 a^{3/4} d e \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (-6 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{3 c^{5/4} \sqrt {a-c x^4}}-\frac {e^2 x \sqrt {a-c x^4}}{3 c} \]
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Rule 227
Rule 230
Rule 435
Rule 1213
Rule 1214
Rule 1215
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 {\left (2 \sqrt {a} d e \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}{\sqrt {c} \sqrt {a-c x^4}}-\frac {\left (\left (-3 c d^2+6 \sqrt {a} \sqrt {c} d e-a e^2\right ) \sqrt {1-\frac {c x^4}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {c x^4}{a}}} \, dx}{3 c \sqrt {a-c x^4}} \\ & = -\frac {e^2 x \sqrt {a-c x^4}}{3 c}+\frac {\sqrt [4]{a} \left (3 c d^2-6 \sqrt {a} \sqrt {c} d e+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 )}{3 c^{5/4} \sqrt {a-c x^4}}+\frac {\left (2 \sqrt {a} d e \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}{\sqrt {c} \sqrt {a-c x^4}} \\ & = -\frac {e^2 x \sqrt {a-c x^4}}{3 c}+\frac {2 a^{3/4} d e \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 )}{c^{3/4} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \left (3 c d^2-6 \sqrt {a} \sqrt {c} d e+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 )}{3 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 = 121, normalized size of antiderivative = 0.75 \[ \int \frac {\left (d+e x^2\right )^2}{\sqrt {a-c x^4}} \, dx=\frac {\left (3 c d^2+a e^2\right ) x \sqrt {1-\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {c x^4}{a}\right )+e x \left (-a e+c e x^4+2 c d x^2 \sqrt {1-\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {c x^4}{a}\right )\right )}{3 c \sqrt {a-c x^4}} \]
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Time = 1.02 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.15
method | result | size |
elliptic | \(-\frac {e^{2} x \sqrt {-c \,x^{4}+a}}{3 c}+\frac {\left (d^{2}+\frac {a \,e^{2}}{3 c}\right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {2 e d \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\) | \(186\) |
default | \(\frac {d^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+e^{2} \left (-\frac {x \sqrt {-c \,x^{4}+a}}{3 c}+\frac {a \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{3 c \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )-\frac {2 e d \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\) | \(246\) |
risch | \(-\frac {e^{2} x \sqrt {-c \,x^{4}+a}}{3 c}+\frac {\frac {a \,e^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {3 c \,d^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {6 d \sqrt {c}\, e \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}}{3 c}\) | \(251\) |
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Time = 0.10 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.71 \[ \int \frac {\left (d+e x^2\right )^2}{\sqrt {a-c x^4}} \, dx=-\frac {6 \, a \sqrt {-c} d e x \left (\frac {a}{c}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left (3 \, c d^{2} + 6 \, a d e + a e^{2}\right )} \sqrt {-c} x \left (\frac {a}{c}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (a e^{2} x^{2} + 6 \, a d e\right )} \sqrt {-c x^{4} + a}}{3 \, a c x} \]
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Time = 1.39 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.80 \[ \int \frac {\left (d+e x^2\right )^2}{\sqrt {a-c x^4}} \, dx=\frac {d^{2} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {5}{4}\right )} + \frac {d e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{2 \sqrt {a} \Gamma \left (\frac {7}{4}\right )} + \frac {e^{2} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {9}{4}\right )} \]
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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. \[ \int \frac {\left (d+e x^2\right )^2}{\sqrt {a-c x^4}} \, dx=\int \frac {{\left (e\,x^2+d\right )}^2}{\sqrt {a-c\,x^4}} \,d x \]
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