Integrand size = 23, antiderivative size = 426 \[ \int \frac {\left (a-b x^4\right )^{7/2}}{\left (c-d x^4\right )^2} \, dx=-\frac {b \left (77 b^2 c^2-122 a b c d+21 a^2 d^2\right ) x \sqrt {a-b x^4}}{84 c d^3}+\frac {b (11 b c-7 a d) x \left (a-b x^4\right )^{3/2}}{28 c d^2}-\frac {(b c-a d) x \left (a-b x^4\right )^{5/2}}{4 c d \left (c-d x^4\right )}+\frac {\sqrt [4]{a} b^{3/4} \left (231 b^3 c^3-553 a b^2 c^2 d+349 a^2 b c d^2+21 a^3 d^3\right ) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{84 c d^4 \sqrt {a-b x^4}}-\frac {\sqrt [4]{a} (b c-a d)^3 (11 b c+3 a d) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{8 \sqrt [4]{b} c^2 d^4 \sqrt {a-b x^4}}-\frac {\sqrt [4]{a} (b c-a d)^3 (11 b c+3 a d) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{8 \sqrt [4]{b} c^2 d^4 \sqrt {a-b x^4}} \]
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Time = 0.35 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {424, 542, 537, 230, 227, 418, 1233, 1232} \[ \int \frac {\left (a-b x^4\right )^{7/2}}{\left (c-d x^4\right )^2} \, dx=-\frac {b x \sqrt {a-b x^4} \left (21 a^2 d^2-122 a b c d+77 b^2 c^2\right )}{84 c d^3}+\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (21 a^3 d^3+349 a^2 b c d^2-553 a b^2 c^2 d+231 b^3 c^3\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{84 c d^4 \sqrt {a-b x^4}}-\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} (3 a d+11 b c) (b c-a d)^3 \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{8 \sqrt [4]{b} c^2 d^4 \sqrt {a-b x^4}}-\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} (3 a d+11 b c) (b c-a d)^3 \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{8 \sqrt [4]{b} c^2 d^4 \sqrt {a-b x^4}}+\frac {b x \left (a-b x^4\right )^{3/2} (11 b c-7 a d)}{28 c d^2}-\frac {x \left (a-b x^4\right )^{5/2} (b c-a d)}{4 c d \left (c-d x^4\right )} \]
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Rule 227
Rule 230
Rule 418
Rule 424
Rule 537
Rule 542
Rule 1232
Rule 1233
Rubi steps \begin{align*} \text {integral}& = -\frac {(b c-a d) x \left (a-b x^4\right )^{5/2}}{4 c d \left (c-d x^4\right )}-\frac {\int \frac {\left (a-b x^4\right )^{3/2} \left (-a (b c+3 a d)+b (11 b c-7 a d) x^4\right )}{c-d x^4} \, dx}{4 c d} \\ & = \frac {b (11 b c-7 a d) x \left (a-b x^4\right )^{3/2}}{28 c d^2}-\frac {(b c-a d) x \left (a-b x^4\right )^{5/2}}{4 c d \left (c-d x^4\right )}+\frac {\int \frac {\sqrt {a-b x^4} \left (-a \left (11 b^2 c^2-14 a b c d-21 a^2 d^2\right )+b \left (77 b^2 c^2-122 a b c d+21 a^2 d^2\right ) x^4\right )}{c-d x^4} \, dx}{28 c d^2} \\ & = -\frac {b \left (77 b^2 c^2-122 a b c d+21 a^2 d^2\right ) x \sqrt {a-b x^4}}{84 c d^3}+\frac {b (11 b c-7 a d) x \left (a-b x^4\right )^{3/2}}{28 c d^2}-\frac {(b c-a d) x \left (a-b x^4\right )^{5/2}}{4 c d \left (c-d x^4\right )}-\frac {\int \frac {-a \left (77 b^3 c^3-155 a b^2 c^2 d+63 a^2 b c d^2+63 a^3 d^3\right )+b \left (231 b^3 c^3-553 a b^2 c^2 d+349 a^2 b c d^2+21 a^3 d^3\right ) x^4}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx}{84 c d^3} \\ & = -\frac {b \left (77 b^2 c^2-122 a b c d+21 a^2 d^2\right ) x \sqrt {a-b x^4}}{84 c d^3}+\frac {b (11 b c-7 a d) x \left (a-b x^4\right )^{3/2}}{28 c d^2}-\frac {(b c-a d) x \left (a-b x^4\right )^{5/2}}{4 c d \left (c-d x^4\right )}-\frac {\left ((b c-a d)^3 (11 b c+3 a d)\right ) \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx}{4 c d^4}+\frac {\left (b \left (231 b^3 c^3-553 a b^2 c^2 d+349 a^2 b c d^2+21 a^3 d^3\right )\right ) \int \frac {1}{\sqrt {a-b x^4}} \, dx}{84 c d^4} \\ & = -\frac {b \left (77 b^2 c^2-122 a b c d+21 a^2 d^2\right ) x \sqrt {a-b x^4}}{84 c d^3}+\frac {b (11 b c-7 a d) x \left (a-b x^4\right )^{3/2}}{28 c d^2}-\frac {(b c-a d) x \left (a-b x^4\right )^{5/2}}{4 c d \left (c-d x^4\right )}-\frac {\left ((b c-a d)^3 (11 b c+3 a d)\right ) \int \frac {1}{\left (1-\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {a-b x^4}} \, dx}{8 c^2 d^4}-\frac {\left ((b c-a d)^3 (11 b c+3 a d)\right ) \int \frac {1}{\left (1+\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {a-b x^4}} \, dx}{8 c^2 d^4}+\frac {\left (b \left (231 b^3 c^3-553 a b^2 c^2 d+349 a^2 b c d^2+21 a^3 d^3\right ) \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}} \, dx}{84 c d^4 \sqrt {a-b x^4}} \\ & = -\frac {b \left (77 b^2 c^2-122 a b c d+21 a^2 d^2\right ) x \sqrt {a-b x^4}}{84 c d^3}+\frac {b (11 b c-7 a d) x \left (a-b x^4\right )^{3/2}}{28 c d^2}-\frac {(b c-a d) x \left (a-b x^4\right )^{5/2}}{4 c d \left (c-d x^4\right )}+\frac {\sqrt [4]{a} b^{3/4} \left (231 b^3 c^3-553 a b^2 c^2 d+349 a^2 b c d^2+21 a^3 d^3\right ) \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 )}{84 c d^4 \sqrt {a-b x^4}}-\frac {\left ((b c-a d)^3 (11 b c+3 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 d^4 \sqrt {a-b x^4}}-\frac {\left ((b c-a d)^3 (11 b c+3 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 d^4 \sqrt {a-b x^4}} \\ & = -\frac {b \left (77 b^2 c^2-122 a b c d+21 a^2 d^2\right ) x \sqrt {a-b x^4}}{84 c d^3}+\frac {b (11 b c-7 a d) x \left (a-b x^4\right )^{3/2}}{28 c d^2}-\frac {(b c-a d) x \left (a-b x^4\right )^{5/2}}{4 c d \left (c-d x^4\right )}+\frac {\sqrt [4]{a} b^{3/4} \left (231 b^3 c^3-553 a b^2 c^2 d+349 a^2 b c d^2+21 a^3 d^3\right ) \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 )}{84 c d^4 \sqrt {a-b x^4}}-\frac {\sqrt [4]{a} (b c-a d)^3 (11 b c+3 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 d^4 \sqrt {a-b x^4}}-\frac {\sqrt [4]{a} (b c-a d)^3 (11 b c+3 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 d^4 \sqrt {a-b x^4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.82 (sec) , antiderivative size = 477, normalized size of antiderivative = 1.12 \[ \int \frac {\left (a-b x^4\right )^{7/2}}{\left (c-d x^4\right )^2} \, dx=-\frac {b \left (231 b^3 c^3-553 a b^2 c^2 d+349 a^2 b c d^2+21 a^3 d^3\right ) x^5 \sqrt {1-\frac {b x^4}{a}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+\frac {5 c \left (5 a c x \left (-84 a^4 d^3+29 a^2 b^2 c d^2 x^4+21 a^3 b d^3 x^4+a b^3 c d x^4 \left (111 c-104 d x^4\right )+b^4 c x^4 \left (-77 c^2+44 c d x^4+12 d^2 x^8\right )\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+2 x^5 \left (-a+b x^4\right ) \left (-63 a^2 b c d^2+21 a^3 d^3+a b^2 c d \left (155 c-92 d x^4\right )+b^3 c \left (-77 c^2+44 c d x^4+12 d^2 x^8\right )\right ) \left (2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )\right )}{\left (c-d x^4\right ) \left (5 a c \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+2 x^4 \left (2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )\right )}}{420 c^2 d^3 \sqrt {a-b x^4}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 7.94 (sec) , antiderivative size = 539, normalized size of antiderivative = 1.27
method | result | size |
default | \(\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x \sqrt {-b \,x^{4}+a}}{4 c \,d^{3} \left (-d \,x^{4}+c \right )}-\frac {b^{3} x^{5} \sqrt {-b \,x^{4}+a}}{7 d^{2}}-\frac {\left (-\frac {2 b^{3} \left (2 a d -b c \right )}{d^{3}}+\frac {5 b^{3} a}{7 d^{2}}\right ) x \sqrt {-b \,x^{4}+a}}{3 b}+\frac {\left (\frac {b^{2} \left (6 a^{2} d^{2}-8 a b c d +3 b^{2} c^{2}\right )}{d^{4}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b}{4 d^{4} c}+\frac {\left (-\frac {2 b^{3} \left (2 a d -b c \right )}{d^{3}}+\frac {5 b^{3} a}{7 d^{2}}\right ) a}{3 b}\right ) \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}-c \right )}{\sum }\frac {\left (3 a^{4} d^{4}+2 a^{3} b c \,d^{3}-24 a^{2} b^{2} c^{2} d^{2}+30 a \,b^{3} c^{3} d -11 b^{4} c^{4}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{32 d^{5} c}\) | \(539\) |
elliptic | \(\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x \sqrt {-b \,x^{4}+a}}{4 c \,d^{3} \left (-d \,x^{4}+c \right )}-\frac {b^{3} x^{5} \sqrt {-b \,x^{4}+a}}{7 d^{2}}-\frac {\left (-\frac {2 b^{3} \left (2 a d -b c \right )}{d^{3}}+\frac {5 b^{3} a}{7 d^{2}}\right ) x \sqrt {-b \,x^{4}+a}}{3 b}+\frac {\left (\frac {b^{2} \left (6 a^{2} d^{2}-8 a b c d +3 b^{2} c^{2}\right )}{d^{4}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b}{4 d^{4} c}+\frac {\left (-\frac {2 b^{3} \left (2 a d -b c \right )}{d^{3}}+\frac {5 b^{3} a}{7 d^{2}}\right ) a}{3 b}\right ) \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}-c \right )}{\sum }\frac {\left (3 a^{4} d^{4}+2 a^{3} b c \,d^{3}-24 a^{2} b^{2} c^{2} d^{2}+30 a \,b^{3} c^{3} d -11 b^{4} c^{4}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{32 d^{5} c}\) | \(539\) |
risch | \(\frac {b^{2} x \left (-3 b d \,x^{4}+23 a d -14 b c \right ) \sqrt {-b \,x^{4}+a}}{21 d^{3}}+\frac {\frac {b^{2} \left (103 a^{2} d^{2}-154 a b c d +63 b^{2} c^{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 )}{d \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {21 b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \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}}}{\underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{2 d^{2}}+\frac {\left (21 a^{4} d^{4}-84 a^{3} b c \,d^{3}+126 a^{2} b^{2} c^{2} d^{2}-84 a \,b^{3} c^{3} d +21 b^{4} c^{4}\right ) \left (-\frac {d x \sqrt {-b \,x^{4}+a}}{4 c \left (a d -b c \right ) \left (d \,x^{4}-c \right )}+\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 )}{4 c \left (a d -b c \right ) \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}-c \right )}{\sum }\frac {\left (3 a d -5 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 ) \underline {\hspace {1.25 ex}}\alpha ^{3}}}{32 c d}\right )}{d}}{21 d^{3}}\) | \(730\) |
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Timed out. \[ \int \frac {\left (a-b x^4\right )^{7/2}}{\left (c-d x^4\right )^2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\left (a-b x^4\right )^{7/2}}{\left (c-d x^4\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (a-b x^4\right )^{7/2}}{\left (c-d x^4\right )^2} \, dx=\int { \frac {{\left (-b x^{4} + a\right )}^{\frac {7}{2}}}{{\left (d x^{4} - c\right )}^{2}} \,d x } \]
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\[ \int \frac {\left (a-b x^4\right )^{7/2}}{\left (c-d x^4\right )^2} \, dx=\int { \frac {{\left (-b x^{4} + a\right )}^{\frac {7}{2}}}{{\left (d x^{4} - c\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (a-b x^4\right )^{7/2}}{\left (c-d x^4\right )^2} \, dx=\int \frac {{\left (a-b\,x^4\right )}^{7/2}}{{\left (c-d\,x^4\right )}^2} \,d x \]
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