Integrand size = 23, antiderivative size = 281 \[ \int \frac {1}{\left (a-b x^4\right )^{3/2} \left (c-d x^4\right )} \, dx=\frac {b x}{2 a (b c-a d) \sqrt {a-b x^4}}+\frac {b^{3/4} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 a^{3/4} (b c-a d) \sqrt {a-b x^4}}-\frac {\sqrt [4]{a} d \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{b} c (b c-a d) \sqrt {a-b x^4}}-\frac {\sqrt [4]{a} d \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{b} c (b c-a d) \sqrt {a-b x^4}} \]
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Time = 0.16 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {425, 537, 230, 227, 418, 1233, 1232} \[ \int \frac {1}{\left (a-b x^4\right )^{3/2} \left (c-d x^4\right )} \, dx=\frac {b^{3/4} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 a^{3/4} \sqrt {a-b x^4} (b c-a d)}-\frac {\sqrt [4]{a} d \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{b} c \sqrt {a-b x^4} (b c-a d)}-\frac {\sqrt [4]{a} d \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{b} c \sqrt {a-b x^4} (b c-a d)}+\frac {b x}{2 a \sqrt {a-b x^4} (b c-a d)} \]
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Rule 227
Rule 230
Rule 418
Rule 425
Rule 537
Rule 1232
Rule 1233
Rubi steps \begin{align*} \text {integral}& = \frac {b x}{2 a (b c-a d) \sqrt {a-b x^4}}+\frac {\int \frac {b c-2 a d-b d x^4}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx}{2 a (b c-a d)} \\ & = \frac {b x}{2 a (b c-a d) \sqrt {a-b x^4}}+\frac {b \int \frac {1}{\sqrt {a-b x^4}} \, dx}{2 a (b c-a d)}-\frac {d \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx}{b c-a d} \\ & = \frac {b x}{2 a (b c-a d) \sqrt {a-b x^4}}-\frac {d \int \frac {1}{\left (1-\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {a-b x^4}} \, dx}{2 c (b c-a d)}-\frac {d \int \frac {1}{\left (1+\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {a-b x^4}} \, dx}{2 c (b c-a d)}+\frac {\left (b \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}} \, dx}{2 a (b c-a d) \sqrt {a-b x^4}} \\ & = \frac {b x}{2 a (b c-a d) \sqrt {a-b x^4}}+\frac {b^{3/4} \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 )}{2 a^{3/4} (b c-a d) \sqrt {a-b x^4}}-\frac {\left (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}{2 c (b c-a d) \sqrt {a-b x^4}}-\frac {\left (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}{2 c (b c-a d) \sqrt {a-b x^4}} \\ & = \frac {b x}{2 a (b c-a d) \sqrt {a-b x^4}}+\frac {b^{3/4} \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 )}{2 a^{3/4} (b c-a d) \sqrt {a-b x^4}}-\frac {\sqrt [4]{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 )}{2 \sqrt [4]{b} c (b c-a d) \sqrt {a-b x^4}}-\frac {\sqrt [4]{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 )}{2 \sqrt [4]{b} c (b c-a d) \sqrt {a-b x^4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.26 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.36 \[ \int \frac {1}{\left (a-b x^4\right )^{3/2} \left (c-d x^4\right )} \, dx=\frac {5 a c x \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right ) \left (-5 c \left (-2 b c+2 a d+b d x^4\right )+b d x^4 \sqrt {1-\frac {b x^4}{a}} \left (-c+d x^4\right ) \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )+2 b x^5 \left (c-d x^4\right ) \left (5 c-d x^4 \sqrt {1-\frac {b x^4}{a}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )\right ) \left (2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )}{10 a c (-b c+a d) \sqrt {a-b x^4} \left (-c+d x^4\right ) \left (5 a c \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+2 x^4 \left (2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )\right )} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 4.15 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.07
method | result | size |
default | \(-\frac {b x}{2 a \left (a d -b c \right ) \sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}-\frac {b \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{2 \left (a d -b c \right ) a \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}-c \right )}{\sum }\frac {-\frac {\operatorname {arctanh}\left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}}{\left (a d -b c \right ) \underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{8}\) | \(301\) |
elliptic | \(-\frac {b x}{2 a \left (a d -b c \right ) \sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}-\frac {b \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{2 \left (a d -b c \right ) a \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}-c \right )}{\sum }\frac {-\frac {\operatorname {arctanh}\left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}}{\left (a d -b c \right ) \underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{8}\) | \(301\) |
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Timed out. \[ \int \frac {1}{\left (a-b x^4\right )^{3/2} \left (c-d x^4\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\left (a-b x^4\right )^{3/2} \left (c-d x^4\right )} \, dx=- \int \frac {1}{- a c \sqrt {a - b x^{4}} + a d x^{4} \sqrt {a - b x^{4}} + b c x^{4} \sqrt {a - b x^{4}} - b d x^{8} \sqrt {a - b x^{4}}}\, dx \]
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\[ \int \frac {1}{\left (a-b x^4\right )^{3/2} \left (c-d x^4\right )} \, dx=\int { -\frac {1}{{\left (-b x^{4} + a\right )}^{\frac {3}{2}} {\left (d x^{4} - c\right )}} \,d x } \]
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\[ \int \frac {1}{\left (a-b x^4\right )^{3/2} \left (c-d x^4\right )} \, dx=\int { -\frac {1}{{\left (-b x^{4} + a\right )}^{\frac {3}{2}} {\left (d x^{4} - c\right )}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (a-b x^4\right )^{3/2} \left (c-d x^4\right )} \, dx=\int \frac {1}{{\left (a-b\,x^4\right )}^{3/2}\,\left (c-d\,x^4\right )} \,d x \]
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