Integrand size = 21, antiderivative size = 388 \[ \int \frac {\left (d+e x^2\right )^4}{\sqrt {a+c x^4}} \, dx=\frac {e^2 \left (42 c d^2-5 a e^2\right ) x \sqrt {a+c x^4}}{21 c^2}+\frac {4 d e^3 x^3 \sqrt {a+c x^4}}{5 c}+\frac {e^4 x^5 \sqrt {a+c x^4}}{7 c}+\frac {4 d e \left (5 c d^2-3 a e^2\right ) x \sqrt {a+c x^4}}{5 c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {4 \sqrt [4]{a} d e \left (5 c d^2-3 a e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 c^{7/4} \sqrt {a+c x^4}}+\frac {\left (105 c^2 d^4+420 \sqrt {a} c^{3/2} d^3 e-210 a c d^2 e^2-252 a^{3/2} \sqrt {c} d e^3+25 a^2 e^4\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{210 \sqrt [4]{a} c^{9/4} \sqrt {a+c x^4}} \]
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Time = 0.26 (sec) , antiderivative size = 386, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {1221, 1902, 1212, 226, 1210} \[ \int \frac {\left (d+e x^2\right )^4}{\sqrt {a+c x^4}} \, dx=\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (5 \left (5 a^2 e^4-42 a c d^2 e^2+21 c^2 d^4\right )+84 \sqrt {a} \sqrt {c} d e \left (5 c d^2-3 a e^2\right )\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{210 \sqrt [4]{a} c^{9/4} \sqrt {a+c x^4}}-\frac {4 \sqrt [4]{a} d e \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (5 c d^2-3 a e^2\right ) E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 c^{7/4} \sqrt {a+c x^4}}+\frac {4 d e x \sqrt {a+c x^4} \left (5 c d^2-3 a e^2\right )}{5 c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {e^2 x \sqrt {a+c x^4} \left (42 c d^2-5 a e^2\right )}{21 c^2}+\frac {4 d e^3 x^3 \sqrt {a+c x^4}}{5 c}+\frac {e^4 x^5 \sqrt {a+c x^4}}{7 c} \]
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Rule 226
Rule 1210
Rule 1212
Rule 1221
Rule 1902
Rubi steps \begin{align*} \text {integral}& = \frac {e^4 x^5 \sqrt {a+c x^4}}{7 c}+\frac {\int \frac {7 c d^4+28 c d^3 e x^2+e^2 \left (42 c d^2-5 a e^2\right ) x^4+28 c d e^3 x^6}{\sqrt {a+c x^4}} \, dx}{7 c} \\ & = \frac {4 d e^3 x^3 \sqrt {a+c x^4}}{5 c}+\frac {e^4 x^5 \sqrt {a+c x^4}}{7 c}+\frac {\int \frac {35 c^2 d^4+28 c d e \left (5 c d^2-3 a e^2\right ) x^2+5 c e^2 \left (42 c d^2-5 a e^2\right ) x^4}{\sqrt {a+c x^4}} \, dx}{35 c^2} \\ & = \frac {e^2 \left (42 c d^2-5 a e^2\right ) x \sqrt {a+c x^4}}{21 c^2}+\frac {4 d e^3 x^3 \sqrt {a+c x^4}}{5 c}+\frac {e^4 x^5 \sqrt {a+c x^4}}{7 c}+\frac {\int \frac {5 c \left (21 c^2 d^4-42 a c d^2 e^2+5 a^2 e^4\right )+84 c^2 d e \left (5 c d^2-3 a e^2\right ) x^2}{\sqrt {a+c x^4}} \, dx}{105 c^3} \\ & = \frac {e^2 \left (42 c d^2-5 a e^2\right ) x \sqrt {a+c x^4}}{21 c^2}+\frac {4 d e^3 x^3 \sqrt {a+c x^4}}{5 c}+\frac {e^4 x^5 \sqrt {a+c x^4}}{7 c}-\frac {\left (4 \sqrt {a} d e \left (5 c d^2-3 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 (105 c^2 d^4+420 \sqrt {a} c^{3/2} d^3 e-210 a c d^2 e^2-252 a^{3/2} \sqrt {c} d e^3+25 a^2 e^4\right ) \int \frac {1}{\sqrt {a+c x^4}} \, dx}{105 c^2} \\ & = \frac {e^2 \left (42 c d^2-5 a e^2\right ) x \sqrt {a+c x^4}}{21 c^2}+\frac {4 d e^3 x^3 \sqrt {a+c x^4}}{5 c}+\frac {e^4 x^5 \sqrt {a+c x^4}}{7 c}+\frac {4 d e \left (5 c d^2-3 a e^2\right ) x \sqrt {a+c x^4}}{5 c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {4 \sqrt [4]{a} d e \left (5 c d^2-3 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 (105 c^2 d^4+420 \sqrt {a} c^{3/2} d^3 e-210 a c d^2 e^2-252 a^{3/2} \sqrt {c} d e^3+25 a^2 e^4\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 )}{210 \sqrt [4]{a} c^{9/4} \sqrt {a+c x^4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.18 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.46 \[ \int \frac {\left (d+e x^2\right )^4}{\sqrt {a+c x^4}} \, dx=\frac {5 \left (21 c^2 d^4-42 a c d^2 e^2+5 a^2 e^4\right ) x \sqrt {1+\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {c x^4}{a}\right )+e x \left (-e \left (a+c x^4\right ) \left (25 a e^2-3 c \left (70 d^2+28 d e x^2+5 e^2 x^4\right )\right )+28 c d \left (5 c d^2-3 a e^2\right ) x^2 \sqrt {1+\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {c x^4}{a}\right )\right )}{105 c^2 \sqrt {a+c x^4}} \]
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Result contains complex when optimal does not.
Time = 4.80 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.74
method | result | size |
elliptic | \(\frac {e^{4} x^{5} \sqrt {c \,x^{4}+a}}{7 c}+\frac {4 d \,e^{3} x^{3} \sqrt {c \,x^{4}+a}}{5 c}+\frac {\left (6 e^{2} d^{2}-\frac {5 e^{4} a}{7 c}\right ) x \sqrt {c \,x^{4}+a}}{3 c}+\frac {\left (d^{4}-\frac {\left (6 e^{2} d^{2}-\frac {5 e^{4} a}{7 c}\right ) a}{3 c}\right ) \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {i \left (4 d^{3} e -\frac {12 d \,e^{3} a}{5 c}\right ) \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\) | \(286\) |
risch | \(-\frac {e^{2} x \left (-15 e^{2} x^{4} c -84 d e \,x^{2} c +25 a \,e^{2}-210 c \,d^{2}\right ) \sqrt {c \,x^{4}+a}}{105 c^{2}}+\frac {\frac {25 a^{2} e^{4} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {105 c^{2} d^{4} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {210 a c \,d^{2} e^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {i \left (252 a c d \,e^{3}-420 c^{2} d^{3} e \right ) \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}}{105 c^{2}}\) | \(400\) |
default | \(\frac {d^{4} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+e^{4} \left (\frac {x^{5} \sqrt {c \,x^{4}+a}}{7 c}-\frac {5 a x \sqrt {c \,x^{4}+a}}{21 c^{2}}+\frac {5 a^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{21 c^{2} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )+4 d \,e^{3} \left (\frac {x^{3} \sqrt {c \,x^{4}+a}}{5 c}-\frac {3 i a^{\frac {3}{2}} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{5 c^{\frac {3}{2}} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )+6 e^{2} d^{2} \left (\frac {x \sqrt {c \,x^{4}+a}}{3 c}-\frac {a \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{3 c \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )+\frac {4 i d^{3} e \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\) | \(506\) |
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none
Time = 0.09 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.53 \[ \int \frac {\left (d+e x^2\right )^4}{\sqrt {a+c x^4}} \, dx=\frac {84 \, {\left (5 \, a c d^{3} e - 3 \, a^{2} d e^{3}\right )} \sqrt {c} x \left (-\frac {a}{c}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (-\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (105 \, c^{2} d^{4} - 420 \, a c d^{3} e - 210 \, a c d^{2} e^{2} + 252 \, a^{2} d e^{3} + 25 \, a^{2} e^{4}\right )} \sqrt {c} x \left (-\frac {a}{c}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (15 \, a c e^{4} x^{6} + 84 \, a c d e^{3} x^{4} + 420 \, a c d^{3} e - 252 \, a^{2} d e^{3} + 5 \, {\left (42 \, a c d^{2} e^{2} - 5 \, a^{2} e^{4}\right )} x^{2}\right )} \sqrt {c x^{4} + a}}{105 \, a c^{2} x} \]
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Result contains complex when optimal does not.
Time = 2.11 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.55 \[ \int \frac {\left (d+e x^2\right )^4}{\sqrt {a+c x^4}} \, dx=\frac {d^{4} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {5}{4}\right )} + \frac {d^{3} e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{\sqrt {a} \Gamma \left (\frac {7}{4}\right )} + \frac {3 d^{2} e^{2} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} \Gamma \left (\frac {9}{4}\right )} + \frac {d e^{3} x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{\sqrt {a} \Gamma \left (\frac {11}{4}\right )} + \frac {e^{4} x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {13}{4}\right )} \]
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\[ \int \frac {\left (d+e x^2\right )^4}{\sqrt {a+c x^4}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{4}}{\sqrt {c x^{4} + a}} \,d x } \]
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\[ \int \frac {\left (d+e x^2\right )^4}{\sqrt {a+c x^4}} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{4}}{\sqrt {c x^{4} + a}} \,d x } \]
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Timed out. \[ \int \frac {\left (d+e x^2\right )^4}{\sqrt {a+c x^4}} \, dx=\int \frac {{\left (e\,x^2+d\right )}^4}{\sqrt {c\,x^4+a}} \,d x \]
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