Integrand size = 23, antiderivative size = 362 \[ \int \frac {1}{\left (a-b x^4\right )^{3/2} \left (c-d x^4\right )^2} \, dx=\frac {b (2 b c+a d) x}{4 a c (b c-a d)^2 \sqrt {a-b x^4}}-\frac {d x}{4 c (b c-a d) \sqrt {a-b x^4} \left (c-d x^4\right )}+\frac {b^{3/4} (2 b c+a d) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{4 a^{3/4} c (b c-a d)^2 \sqrt {a-b x^4}}-\frac {3 \sqrt [4]{a} d (3 b c-a d) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{8 \sqrt [4]{b} c^2 (b c-a d)^2 \sqrt {a-b x^4}}-\frac {3 \sqrt [4]{a} d (3 b c-a d) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{8 \sqrt [4]{b} c^2 (b c-a d)^2 \sqrt {a-b x^4}} \]
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Time = 0.28 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {425, 541, 537, 230, 227, 418, 1233, 1232} \[ \int \frac {1}{\left (a-b x^4\right )^{3/2} \left (c-d x^4\right )^2} \, dx=\frac {b^{3/4} \sqrt {1-\frac {b x^4}{a}} (a d+2 b c) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{4 a^{3/4} c \sqrt {a-b x^4} (b c-a d)^2}-\frac {3 \sqrt [4]{a} d \sqrt {1-\frac {b x^4}{a}} (3 b c-a d) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{8 \sqrt [4]{b} c^2 \sqrt {a-b x^4} (b c-a d)^2}-\frac {3 \sqrt [4]{a} d \sqrt {1-\frac {b x^4}{a}} (3 b c-a d) \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{8 \sqrt [4]{b} c^2 \sqrt {a-b x^4} (b c-a d)^2}+\frac {b x (a d+2 b c)}{4 a c \sqrt {a-b x^4} (b c-a d)^2}-\frac {d x}{4 c \sqrt {a-b x^4} \left (c-d x^4\right ) (b c-a d)} \]
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Rule 227
Rule 230
Rule 418
Rule 425
Rule 537
Rule 541
Rule 1232
Rule 1233
Rubi steps \begin{align*} \text {integral}& = -\frac {d x}{4 c (b c-a d) \sqrt {a-b x^4} \left (c-d x^4\right )}-\frac {\int \frac {-4 b c+3 a d-5 b d x^4}{\left (a-b x^4\right )^{3/2} \left (c-d x^4\right )} \, dx}{4 c (b c-a d)} \\ & = \frac {b (2 b c+a d) x}{4 a c (b c-a d)^2 \sqrt {a-b x^4}}-\frac {d x}{4 c (b c-a d) \sqrt {a-b x^4} \left (c-d x^4\right )}-\frac {\int \frac {-2 \left (2 b^2 c^2-8 a b c d+3 a^2 d^2\right )+2 b d (2 b c+a d) x^4}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx}{8 a c (b c-a d)^2} \\ & = \frac {b (2 b c+a d) x}{4 a c (b c-a d)^2 \sqrt {a-b x^4}}-\frac {d x}{4 c (b c-a d) \sqrt {a-b x^4} \left (c-d x^4\right )}-\frac {(3 d (3 b c-a d)) \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx}{4 c (b c-a d)^2}+\frac {(b (2 b c+a d)) \int \frac {1}{\sqrt {a-b x^4}} \, dx}{4 a c (b c-a d)^2} \\ & = \frac {b (2 b c+a d) x}{4 a c (b c-a d)^2 \sqrt {a-b x^4}}-\frac {d x}{4 c (b c-a d) \sqrt {a-b x^4} \left (c-d x^4\right )}-\frac {(3 d (3 b c-a d)) \int \frac {1}{\left (1-\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {a-b x^4}} \, dx}{8 c^2 (b c-a d)^2}-\frac {(3 d (3 b c-a d)) \int \frac {1}{\left (1+\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {a-b x^4}} \, dx}{8 c^2 (b c-a d)^2}+\frac {\left (b (2 b c+a d) \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}} \, dx}{4 a c (b c-a d)^2 \sqrt {a-b x^4}} \\ & = \frac {b (2 b c+a d) x}{4 a c (b c-a d)^2 \sqrt {a-b x^4}}-\frac {d x}{4 c (b c-a d) \sqrt {a-b x^4} \left (c-d x^4\right )}+\frac {b^{3/4} (2 b c+a d) \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{4 a^{3/4} c (b c-a d)^2 \sqrt {a-b x^4}}-\frac {\left (3 d (3 b c-a d) \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1}{\left (1-\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {1-\frac {b x^4}{a}}} \, dx}{8 c^2 (b c-a d)^2 \sqrt {a-b x^4}}-\frac {\left (3 d (3 b c-a d) \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1}{\left (1+\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {1-\frac {b x^4}{a}}} \, dx}{8 c^2 (b c-a d)^2 \sqrt {a-b x^4}} \\ & = \frac {b (2 b c+a d) x}{4 a c (b c-a d)^2 \sqrt {a-b x^4}}-\frac {d x}{4 c (b c-a d) \sqrt {a-b x^4} \left (c-d x^4\right )}+\frac {b^{3/4} (2 b c+a d) \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{4 a^{3/4} c (b c-a d)^2 \sqrt {a-b x^4}}-\frac {3 \sqrt [4]{a} d (3 b c-a d) \sqrt {1-\frac {b x^4}{a}} \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 (b c-a d)^2 \sqrt {a-b x^4}}-\frac {3 \sqrt [4]{a} d (3 b c-a d) \sqrt {1-\frac {b x^4}{a}} \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 (b c-a d)^2 \sqrt {a-b x^4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.52 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.03 \[ \int \frac {1}{\left (a-b x^4\right )^{3/2} \left (c-d x^4\right )^2} \, dx=\frac {x \left (-b d (2 b c+a d) x^4 \sqrt {1-\frac {b x^4}{a}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+\frac {c \left (25 a c \left (4 a^2 d^2+2 b^2 c \left (2 c-d x^4\right )-a b d \left (8 c+d x^4\right )\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )-10 x^4 \left (-a^2 d^2+a b d^2 x^4-2 b^2 c \left (c-d x^4\right )\right ) \left (2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )\right )}{\left (c-d x^4\right ) \left (5 a c \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+2 x^4 \left (2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )\right )}\right )}{20 a c^2 (b c-a d)^2 \sqrt {a-b x^4}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 4.75 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.03
method | result | size |
default | \(\frac {d^{2} x \sqrt {-b \,x^{4}+a}}{4 c \left (a d -b c \right )^{2} \left (-d \,x^{4}+c \right )}+\frac {b^{2} x}{2 a \left (a d -b c \right )^{2} \sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}+\frac {\left (\frac {b d}{4 \left (a d -b c \right )^{2} c}+\frac {b^{2}}{2 a \left (a d -b c \right )^{2}}\right ) \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {3 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}-c \right )}{\sum }\frac {\left (a d -3 b c \right ) \left (-\frac {\operatorname {arctanh}\left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}\right )}{\left (a d -b c \right )^{2} \underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{32 c}\) | \(374\) |
elliptic | \(\frac {d^{2} x \sqrt {-b \,x^{4}+a}}{4 c \left (a d -b c \right )^{2} \left (-d \,x^{4}+c \right )}+\frac {b^{2} x}{2 a \left (a d -b c \right )^{2} \sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}+\frac {\left (\frac {b d}{4 \left (a d -b c \right )^{2} c}+\frac {b^{2}}{2 a \left (a d -b c \right )^{2}}\right ) \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {3 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}-c \right )}{\sum }\frac {\left (a d -3 b c \right ) \left (-\frac {\operatorname {arctanh}\left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}\right )}{\left (a d -b c \right )^{2} \underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{32 c}\) | \(374\) |
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Timed out. \[ \int \frac {1}{\left (a-b x^4\right )^{3/2} \left (c-d x^4\right )^2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1}{\left (a-b x^4\right )^{3/2} \left (c-d x^4\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\left (a-b x^4\right )^{3/2} \left (c-d x^4\right )^2} \, dx=\int { \frac {1}{{\left (-b x^{4} + a\right )}^{\frac {3}{2}} {\left (d x^{4} - c\right )}^{2}} \,d x } \]
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\[ \int \frac {1}{\left (a-b x^4\right )^{3/2} \left (c-d x^4\right )^2} \, dx=\int { \frac {1}{{\left (-b x^{4} + a\right )}^{\frac {3}{2}} {\left (d x^{4} - c\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (a-b x^4\right )^{3/2} \left (c-d x^4\right )^2} \, dx=\int \frac {1}{{\left (a-b\,x^4\right )}^{3/2}\,{\left (c-d\,x^4\right )}^2} \,d x \]
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