Integrand size = 21, antiderivative size = 326 \[ \int \frac {\left (d+e x^2\right )^3}{\sqrt {a+c x^4}} \, dx=\frac {d e^2 x \sqrt {a+c x^4}}{c}+\frac {e^3 x^3 \sqrt {a+c x^4}}{5 c}+\frac {3 e \left (5 c d^2-a e^2\right ) x \sqrt {a+c x^4}}{5 c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {3 \sqrt [4]{a} e \left (5 c d^2-a e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 c^{7/4} \sqrt {a+c x^4}}+\frac {\sqrt [4]{a} \left (15 c d^2 e-3 a e^3+\frac {5 \sqrt {c} d \left (c d^2-a e^2\right )}{\sqrt {a}}\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{10 c^{7/4} \sqrt {a+c x^4}} \]
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Time = 0.18 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {1221, 1902, 1212, 226, 1210} \[ \int \frac {\left (d+e x^2\right )^3}{\sqrt {a+c x^4}} \, dx=-\frac {3 \sqrt [4]{a} e \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (5 c d^2-a e^2\right ) E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 c^{7/4} \sqrt {a+c x^4}}+\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\frac {5 \sqrt {c} d \left (c d^2-a e^2\right )}{\sqrt {a}}-3 a e^3+15 c d^2 e\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{10 c^{7/4} \sqrt {a+c x^4}}+\frac {3 e x \sqrt {a+c x^4} \left (5 c d^2-a e^2\right )}{5 c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {d e^2 x \sqrt {a+c x^4}}{c}+\frac {e^3 x^3 \sqrt {a+c x^4}}{5 c} \]
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Rule 226
Rule 1210
Rule 1212
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 {3 e \left (5 c d^2-a e^2\right ) x \sqrt {a+c x^4}}{5 c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {3 \sqrt [4]{a} e \left (5 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}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 c^{7/4} \sqrt {a+c x^4}}+\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 ) \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 )}{10 \sqrt [4]{a} 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 = 10.10 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.43 \[ \int \frac {\left (d+e x^2\right )^3}{\sqrt {a+c x^4}} \, dx=\frac {5 d \left (c d^2-a e^2\right ) x \sqrt {1+\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {c x^4}{a}\right )+e x \left (e \left (5 d+e x^2\right ) \left (a+c x^4\right )+\left (5 c d^2-a e^2\right ) x^2 \sqrt {1+\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {c x^4}{a}\right )\right )}{5 c \sqrt {a+c x^4}} \]
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Result contains complex when optimal does not.
Time = 2.70 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.72
method | result | size |
elliptic | \(\frac {e^{3} x^{3} \sqrt {c \,x^{4}+a}}{5 c}+\frac {d \,e^{2} x \sqrt {c \,x^{4}+a}}{c}+\frac {\left (d^{3}-\frac {d \,e^{2} a}{c}\right ) \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {i \left (3 d^{2} e -\frac {3 e^{3} a}{5 c}\right ) \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\) | \(235\) |
risch | \(\frac {e^{2} x \left (e \,x^{2}+5 d \right ) \sqrt {c \,x^{4}+a}}{5 c}-\frac {-\frac {5 d^{3} c \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {5 d \,e^{2} a \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {i \left (3 a \,e^{3}-15 c \,d^{2} e \right ) \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}}{5 c}\) | \(294\) |
default | \(\frac {d^{3} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+e^{3} \left (\frac {x^{3} \sqrt {c \,x^{4}+a}}{5 c}-\frac {3 i a^{\frac {3}{2}} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{5 c^{\frac {3}{2}} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )+3 d \,e^{2} \left (\frac {x \sqrt {c \,x^{4}+a}}{3 c}-\frac {a \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{3 c \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )+\frac {3 i d^{2} e \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\) | \(388\) |
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Time = 0.09 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.51 \[ \int \frac {\left (d+e x^2\right )^3}{\sqrt {a+c x^4}} \, dx=\frac {3 \, {\left (5 \, a c d^{2} e - a^{2} e^{3}\right )} \sqrt {c} x \left (-\frac {a}{c}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (-\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (5 \, c^{2} d^{3} - 15 \, a c d^{2} e - 5 \, a c d e^{2} + 3 \, a^{2} e^{3}\right )} \sqrt {c} x \left (-\frac {a}{c}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (a c e^{3} x^{4} + 5 \, a c d e^{2} x^{2} + 15 \, a c d^{2} e - 3 \, a^{2} e^{3}\right )} \sqrt {c x^{4} + a}}{5 \, a c^{2} x} \]
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Result contains complex when optimal does not.
Time = 1.72 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.53 \[ \int \frac {\left (d+e x^2\right )^3}{\sqrt {a+c x^4}} \, dx=\frac {d^{3} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {5}{4}\right )} + \frac {3 d^{2} e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {7}{4}\right )} + \frac {3 d e^{2} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {9}{4}\right )} + \frac {e^{3} x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {11}{4}\right )} \]
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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. \[ \int \frac {\left (d+e x^2\right )^3}{\sqrt {a+c x^4}} \, dx=\int \frac {{\left (e\,x^2+d\right )}^3}{\sqrt {c\,x^4+a}} \,d x \]
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