Integrand size = 23, antiderivative size = 334 \[ \int \frac {1}{\left (a-b x^4\right )^{5/2} \left (c-d x^4\right )} \, dx=\frac {b x}{6 a (b c-a d) \left (a-b x^4\right )^{3/2}}+\frac {b (5 b c-11 a d) x}{12 a^2 (b c-a d)^2 \sqrt {a-b x^4}}+\frac {b^{3/4} (5 b c-11 a d) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{12 a^{7/4} (b c-a d)^2 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} d^2 \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{b} c (b c-a d)^2 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} d^2 \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{b} c (b c-a d)^2 \sqrt {a-b x^4}} \]
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Time = 0.28 (sec) , antiderivative size = 334, 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} \[ \int \frac {1}{\left (a-b x^4\right )^{5/2} \left (c-d x^4\right )} \, dx=\frac {b^{3/4} \sqrt {1-\frac {b x^4}{a}} (5 b c-11 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{12 a^{7/4} \sqrt {a-b x^4} (b c-a d)^2}+\frac {b x (5 b c-11 a d)}{12 a^2 \sqrt {a-b x^4} (b c-a d)^2}+\frac {\sqrt [4]{a} d^2 \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{b} c \sqrt {a-b x^4} (b c-a d)^2}+\frac {\sqrt [4]{a} d^2 \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{b} c \sqrt {a-b x^4} (b c-a d)^2}+\frac {b x}{6 a \left (a-b x^4\right )^{3/2} (b c-a d)} \]
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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 {b x}{6 a (b c-a d) \left (a-b x^4\right )^{3/2}}+\frac {\int \frac {5 b c-6 a d-5 b d x^4}{\left (a-b x^4\right )^{3/2} \left (c-d x^4\right )} \, dx}{6 a (b c-a d)} \\ & = \frac {b x}{6 a (b c-a d) \left (a-b x^4\right )^{3/2}}+\frac {b (5 b c-11 a d) x}{12 a^2 (b c-a d)^2 \sqrt {a-b x^4}}+\frac {\int \frac {5 b^2 c^2-11 a b c d+12 a^2 d^2-b d (5 b c-11 a d) x^4}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx}{12 a^2 (b c-a d)^2} \\ & = \frac {b x}{6 a (b c-a d) \left (a-b x^4\right )^{3/2}}+\frac {b (5 b c-11 a d) x}{12 a^2 (b c-a d)^2 \sqrt {a-b x^4}}+\frac {d^2 \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx}{(b c-a d)^2}+\frac {(b (5 b c-11 a d)) \int \frac {1}{\sqrt {a-b x^4}} \, dx}{12 a^2 (b c-a d)^2} \\ & = \frac {b x}{6 a (b c-a d) \left (a-b x^4\right )^{3/2}}+\frac {b (5 b c-11 a d) x}{12 a^2 (b c-a d)^2 \sqrt {a-b x^4}}+\frac {d^2 \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)^2}+\frac {d^2 \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)^2}+\frac {\left (b (5 b c-11 a d) \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}} \, dx}{12 a^2 (b c-a d)^2 \sqrt {a-b x^4}} \\ & = \frac {b x}{6 a (b c-a d) \left (a-b x^4\right )^{3/2}}+\frac {b (5 b c-11 a d) x}{12 a^2 (b c-a d)^2 \sqrt {a-b x^4}}+\frac {b^{3/4} (5 b c-11 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 )}{12 a^{7/4} (b c-a d)^2 \sqrt {a-b x^4}}+\frac {\left (d^2 \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)^2 \sqrt {a-b x^4}}+\frac {\left (d^2 \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)^2 \sqrt {a-b x^4}} \\ & = \frac {b x}{6 a (b c-a d) \left (a-b x^4\right )^{3/2}}+\frac {b (5 b c-11 a d) x}{12 a^2 (b c-a d)^2 \sqrt {a-b x^4}}+\frac {b^{3/4} (5 b c-11 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 )}{12 a^{7/4} (b c-a d)^2 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} d^2 \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)^2 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} d^2 \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)^2 \sqrt {a-b x^4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.72 (sec) , antiderivative size = 422, normalized size of antiderivative = 1.26 \[ \int \frac {1}{\left (a-b x^4\right )^{5/2} \left (c-d x^4\right )} \, dx=\frac {x \left (\frac {b d (-5 b c+11 a d) x^4 \sqrt {1-\frac {b x^4}{a}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )}{c}-\frac {5 \left (5 a c \left (12 a^3 d^2+a^2 b d \left (-24 c+d x^4\right )+5 b^3 c x^4 \left (-2 c+d x^4\right )+a b^2 \left (12 c^2+15 c d x^4-11 d^2 x^8\right )\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+2 b x^4 \left (-c+d x^4\right ) \left (13 a^2 d+5 b^2 c x^4-a b \left (7 c+11 d x^4\right )\right ) \left (2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )\right )}{\left (a-b x^4\right ) \left (-c+d x^4\right ) \left (5 a c \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+2 x^4 \left (2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )\right )}\right )}{60 a^2 (b c-a d)^2 \sqrt {a-b x^4}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 4.18 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.08
method | result | size |
default | \(-\frac {x \sqrt {-b \,x^{4}+a}}{6 a b \left (a d -b c \right ) \left (x^{4}-\frac {a}{b}\right )^{2}}-\frac {b x \left (11 a d -5 b c \right )}{12 a^{2} \left (a d -b c \right )^{2} \sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}-\frac {b \left (11 a d -5 b c \right ) \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{12 a^{2} \left (a d -b c \right )^{2} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {d \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}-c \right )}{\sum }\frac {-\frac {\operatorname {arctanh}\left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}}{\left (a d -b c \right )^{2} \underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{8}\) | \(361\) |
elliptic | \(-\frac {x \sqrt {-b \,x^{4}+a}}{6 a b \left (a d -b c \right ) \left (x^{4}-\frac {a}{b}\right )^{2}}-\frac {b x \left (11 a d -5 b c \right )}{12 a^{2} \left (a d -b c \right )^{2} \sqrt {-\left (x^{4}-\frac {a}{b}\right ) b}}-\frac {b \left (11 a d -5 b c \right ) \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{12 a^{2} \left (a d -b c \right )^{2} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {d \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}-c \right )}{\sum }\frac {-\frac {\operatorname {arctanh}\left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}}{\left (a d -b c \right )^{2} \underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{8}\) | \(361\) |
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Timed out. \[ \int \frac {1}{\left (a-b x^4\right )^{5/2} \left (c-d x^4\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\left (a-b x^4\right )^{5/2} \left (c-d x^4\right )} \, dx=- \int \frac {1}{- a^{2} c \sqrt {a - b x^{4}} + a^{2} d x^{4} \sqrt {a - b x^{4}} + 2 a b c x^{4} \sqrt {a - b x^{4}} - 2 a b d x^{8} \sqrt {a - b x^{4}} - b^{2} c x^{8} \sqrt {a - b x^{4}} + b^{2} d x^{12} \sqrt {a - b x^{4}}}\, dx \]
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\[ \int \frac {1}{\left (a-b x^4\right )^{5/2} \left (c-d x^4\right )} \, dx=\int { -\frac {1}{{\left (-b x^{4} + a\right )}^{\frac {5}{2}} {\left (d x^{4} - c\right )}} \,d x } \]
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\[ \int \frac {1}{\left (a-b x^4\right )^{5/2} \left (c-d x^4\right )} \, dx=\int { -\frac {1}{{\left (-b x^{4} + a\right )}^{\frac {5}{2}} {\left (d x^{4} - c\right )}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (a-b x^4\right )^{5/2} \left (c-d x^4\right )} \, dx=\int \frac {1}{{\left (a-b\,x^4\right )}^{5/2}\,\left (c-d\,x^4\right )} \,d x \]
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