Integrand size = 22, antiderivative size = 213 \[ \int \frac {\left (d+e x^2\right )^3}{\sqrt {a-c x^4}} \, dx=-\frac {d e^2 x \sqrt {a-c x^4}}{c}-\frac {e^3 x^3 \sqrt {a-c x^4}}{5 c}+\frac {3 a^{3/4} e \left (5 c d^2+a e^2\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 c^{7/4} \sqrt {a-c x^4}}+\frac {a^{3/4} \left (\frac {5 \sqrt {c} d \left (c d^2+a e^2\right )}{\sqrt {a}}-3 e \left (5 c d^2+a e^2\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{5 c^{7/4} \sqrt {a-c x^4}} \]
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Time = 0.17 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {1221, 1902, 1215, 230, 227, 1214, 1213, 435} \[ \int \frac {\left (d+e x^2\right )^3}{\sqrt {a-c x^4}} \, dx=\frac {a^{3/4} \sqrt {1-\frac {c x^4}{a}} \left (\frac {5 \sqrt {c} d \left (a e^2+c d^2\right )}{\sqrt {a}}-3 e \left (a e^2+5 c d^2\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{5 c^{7/4} \sqrt {a-c x^4}}+\frac {3 a^{3/4} e \sqrt {1-\frac {c x^4}{a}} \left (a e^2+5 c d^2\right ) E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 c^{7/4} \sqrt {a-c x^4}}-\frac {d e^2 x \sqrt {a-c x^4}}{c}-\frac {e^3 x^3 \sqrt {a-c x^4}}{5 c} \]
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Rule 227
Rule 230
Rule 435
Rule 1213
Rule 1214
Rule 1215
Rule 1221
Rule 1902
Rubi steps \begin{align*} \text {integral}& = -\frac {e^3 x^3 \sqrt {a-c x^4}}{5 c}-\frac {\int \frac {-5 c d^3-3 e \left (5 c d^2+a e^2\right ) x^2-15 c d e^2 x^4}{\sqrt {a-c x^4}} \, dx}{5 c} \\ & = -\frac {d e^2 x \sqrt {a-c x^4}}{c}-\frac {e^3 x^3 \sqrt {a-c x^4}}{5 c}+\frac {\int \frac {15 c d \left (c d^2+a e^2\right )+9 c e \left (5 c d^2+a e^2\right ) x^2}{\sqrt {a-c x^4}} \, dx}{15 c^2} \\ & = -\frac {d e^2 x \sqrt {a-c x^4}}{c}-\frac {e^3 x^3 \sqrt {a-c x^4}}{5 c}+\frac {\left (3 \sqrt {a} e \left (5 c d^2+a e^2\right )\right ) \int \frac {1+\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a-c x^4}} \, dx}{5 c^{3/2}}+\frac {\left (5 \sqrt {c} d \left (c d^2+a e^2\right )-3 \sqrt {a} e \left (5 c d^2+a e^2\right )\right ) \int \frac {1}{\sqrt {a-c x^4}} \, dx}{5 c^{3/2}} \\ & = -\frac {d e^2 x \sqrt {a-c x^4}}{c}-\frac {e^3 x^3 \sqrt {a-c x^4}}{5 c}+\frac {\left (3 \sqrt {a} e \left (5 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}{5 c^{3/2} \sqrt {a-c x^4}}+\frac {\left (\left (5 \sqrt {c} d \left (c d^2+a e^2\right )-3 \sqrt {a} e \left (5 c d^2+a e^2\right )\right ) \sqrt {1-\frac {c x^4}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {c x^4}{a}}} \, dx}{5 c^{3/2} \sqrt {a-c x^4}} \\ & = -\frac {d e^2 x \sqrt {a-c x^4}}{c}-\frac {e^3 x^3 \sqrt {a-c x^4}}{5 c}+\frac {\sqrt [4]{a} \left (5 \sqrt {c} d \left (c d^2+a e^2\right )-3 \sqrt {a} e \left (5 c d^2+a e^2\right )\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 )}{5 c^{7/4} \sqrt {a-c x^4}}+\frac {\left (3 \sqrt {a} e \left (5 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}{5 c^{3/2} \sqrt {a-c x^4}} \\ & = -\frac {d e^2 x \sqrt {a-c x^4}}{c}-\frac {e^3 x^3 \sqrt {a-c x^4}}{5 c}+\frac {3 a^{3/4} e \left (5 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 )}{5 c^{7/4} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \left (5 \sqrt {c} d \left (c d^2+a e^2\right )-3 \sqrt {a} e \left (5 c d^2+a e^2\right )\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 )}{5 c^{7/4} \sqrt {a-c x^4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.12 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.66 \[ \int \frac {\left (d+e x^2\right )^3}{\sqrt {a-c x^4}} \, dx=\frac {5 d \left (c d^2+a e^2\right ) x \sqrt {1-\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {c x^4}{a}\right )+e x \left (e \left (5 d+e x^2\right ) \left (-a+c x^4\right )+\left (5 c d^2+a e^2\right ) x^2 \sqrt {1-\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {c x^4}{a}\right )\right )}{5 c \sqrt {a-c x^4}} \]
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Time = 4.50 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.04
method | result | size |
elliptic | \(-\frac {e^{3} x^{3} \sqrt {-c \,x^{4}+a}}{5 c}-\frac {d \,e^{2} x \sqrt {-c \,x^{4}+a}}{c}+\frac {\left (d^{3}+\frac {d \,e^{2} a}{c}\right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {\left (3 d^{2} e +\frac {3 e^{3} a}{5 c}\right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\) | \(222\) |
risch | \(-\frac {e^{2} x \left (e \,x^{2}+5 d \right ) \sqrt {-c \,x^{4}+a}}{5 c}+\frac {\frac {5 d^{3} c \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {5 d \,e^{2} a \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {\left (3 a \,e^{3}+15 c \,d^{2} e \right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}}{5 c}\) | \(274\) |
default | \(\frac {d^{3} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+e^{3} \left (-\frac {x^{3} \sqrt {-c \,x^{4}+a}}{5 c}-\frac {3 a^{\frac {3}{2}} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{5 c^{\frac {3}{2}} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )+3 d \,e^{2} \left (-\frac {x \sqrt {-c \,x^{4}+a}}{3 c}+\frac {a \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{3 c \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )-\frac {3 d^{2} e \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\) | \(360\) |
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Time = 0.10 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.78 \[ \int \frac {\left (d+e x^2\right )^3}{\sqrt {a-c x^4}} \, dx=-\frac {3 \, {\left (5 \, a c d^{2} e + a^{2} e^{3}\right )} \sqrt {-c} x \left (\frac {a}{c}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left (5 \, c^{2} d^{3} + 15 \, a c d^{2} e + 5 \, a c d e^{2} + 3 \, a^{2} e^{3}\right )} \sqrt {-c} x \left (\frac {a}{c}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (a c e^{3} x^{4} + 5 \, a c d e^{2} x^{2} + 15 \, a c d^{2} e + 3 \, a^{2} e^{3}\right )} \sqrt {-c x^{4} + a}}{5 \, a c^{2} x} \]
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Time = 1.88 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.85 \[ \int \frac {\left (d+e x^2\right )^3}{\sqrt {a-c x^4}} \, dx=\frac {d^{3} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {5}{4}\right )} + \frac {3 d^{2} e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {7}{4}\right )} + \frac {3 d e^{2} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {9}{4}\right )} + \frac {e^{3} x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {11}{4}\right )} \]
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\[ \int \frac {\left (d+e x^2\right )^3}{\sqrt {a-c x^4}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{3}}{\sqrt {-c x^{4} + a}} \,d x } \]
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\[ \int \frac {\left (d+e x^2\right )^3}{\sqrt {a-c x^4}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{3}}{\sqrt {-c x^{4} + a}} \,d x } \]
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Timed out. \[ \int \frac {\left (d+e x^2\right )^3}{\sqrt {a-c x^4}} \, dx=\int \frac {{\left (e\,x^2+d\right )}^3}{\sqrt {a-c\,x^4}} \,d x \]
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