Integrand size = 23, antiderivative size = 365 \[ \int \frac {\left (a-b x^4\right )^{5/2}}{\left (c-d x^4\right )^2} \, dx=\frac {b (7 b c-3 a d) x \sqrt {a-b x^4}}{12 c d^2}-\frac {(b c-a d) x \left (a-b x^4\right )^{3/2}}{4 c d \left (c-d x^4\right )}-\frac {\sqrt [4]{a} b^{3/4} \left (21 b^2 c^2-26 a b c d-3 a^2 d^2\right ) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{12 c d^3 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} (b c-a d)^2 (7 b c+3 a d) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{8 \sqrt [4]{b} c^2 d^3 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} (b c-a d)^2 (7 b c+3 a d) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{8 \sqrt [4]{b} c^2 d^3 \sqrt {a-b x^4}} \]
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Time = 0.27 (sec) , antiderivative size = 365, 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 = {424, 542, 537, 230, 227, 418, 1233, 1232} \[ \int \frac {\left (a-b x^4\right )^{5/2}}{\left (c-d x^4\right )^2} \, dx=-\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} \left (-3 a^2 d^2-26 a b c d+21 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{12 c d^3 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} (3 a d+7 b c) (b c-a d)^2 \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{8 \sqrt [4]{b} c^2 d^3 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} (3 a d+7 b c) (b c-a d)^2 \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{8 \sqrt [4]{b} c^2 d^3 \sqrt {a-b x^4}}+\frac {b x \sqrt {a-b x^4} (7 b c-3 a d)}{12 c d^2}-\frac {x \left (a-b x^4\right )^{3/2} (b c-a d)}{4 c d \left (c-d x^4\right )} \]
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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 )^{3/2}}{4 c d \left (c-d x^4\right )}-\frac {\int \frac {\sqrt {a-b x^4} \left (-a (b c+3 a d)+b (7 b c-3 a d) x^4\right )}{c-d x^4} \, dx}{4 c d} \\ & = \frac {b (7 b c-3 a d) x \sqrt {a-b x^4}}{12 c d^2}-\frac {(b c-a d) x \left (a-b x^4\right )^{3/2}}{4 c d \left (c-d x^4\right )}+\frac {\int \frac {-a \left (7 b^2 c^2-6 a b c d-9 a^2 d^2\right )+b \left (21 b^2 c^2-26 a b c d-3 a^2 d^2\right ) x^4}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx}{12 c d^2} \\ & = \frac {b (7 b c-3 a d) x \sqrt {a-b x^4}}{12 c d^2}-\frac {(b c-a d) x \left (a-b x^4\right )^{3/2}}{4 c d \left (c-d x^4\right )}+\frac {\left ((b c-a d)^2 (7 b c+3 a d)\right ) \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx}{4 c d^3}-\frac {\left (b \left (21 b^2 c^2-26 a b c d-3 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a-b x^4}} \, dx}{12 c d^3} \\ & = \frac {b (7 b c-3 a d) x \sqrt {a-b x^4}}{12 c d^2}-\frac {(b c-a d) x \left (a-b x^4\right )^{3/2}}{4 c d \left (c-d x^4\right )}+\frac {\left ((b c-a d)^2 (7 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^3}+\frac {\left ((b c-a d)^2 (7 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^3}-\frac {\left (b \left (21 b^2 c^2-26 a b c d-3 a^2 d^2\right ) \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}} \, dx}{12 c d^3 \sqrt {a-b x^4}} \\ & = \frac {b (7 b c-3 a d) x \sqrt {a-b x^4}}{12 c d^2}-\frac {(b c-a d) x \left (a-b x^4\right )^{3/2}}{4 c d \left (c-d x^4\right )}-\frac {\sqrt [4]{a} b^{3/4} \left (21 b^2 c^2-26 a b c d-3 a^2 d^2\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 )}{12 c d^3 \sqrt {a-b x^4}}+\frac {\left ((b c-a d)^2 (7 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^3 \sqrt {a-b x^4}}+\frac {\left ((b c-a d)^2 (7 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^3 \sqrt {a-b x^4}} \\ & = \frac {b (7 b c-3 a d) x \sqrt {a-b x^4}}{12 c d^2}-\frac {(b c-a d) x \left (a-b x^4\right )^{3/2}}{4 c d \left (c-d x^4\right )}-\frac {\sqrt [4]{a} b^{3/4} \left (21 b^2 c^2-26 a b c d-3 a^2 d^2\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 )}{12 c d^3 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} (b c-a d)^2 (7 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^3 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} (b c-a d)^2 (7 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^3 \sqrt {a-b x^4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.58 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a-b x^4\right )^{5/2}}{\left (c-d x^4\right )^2} \, dx=-\frac {b \left (-21 b^2 c^2+26 a b c d+3 a^2 d^2\right ) x^5 \sqrt {1-\frac {b x^4}{a}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+\frac {5 c \left (5 a c x \left (12 a^3 d^2+2 a b^2 c d x^4-3 a^2 b d^2 x^4+b^3 c x^4 \left (-7 c+4 d x^4\right )\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+2 x^5 \left (a-b x^4\right ) \left (-6 a b c d+3 a^2 d^2+b^2 c \left (7 c-4 d x^4\right )\right ) \left (2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )\right )}{\left (-c+d x^4\right ) \left (5 a c \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+2 x^4 \left (2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )\right )}}{60 c^2 d^2 \sqrt {a-b x^4}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 7.16 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.13
method | result | size |
default | \(\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x \sqrt {-b \,x^{4}+a}}{4 d^{2} c \left (-d \,x^{4}+c \right )}+\frac {b^{2} x \sqrt {-b \,x^{4}+a}}{3 d^{2}}+\frac {\left (\frac {b^{2} \left (3 a d -2 b c \right )}{d^{3}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b}{4 d^{3} c}-\frac {b^{2} a}{3 d^{2}}\right ) \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}-c \right )}{\sum }\frac {\left (3 a^{3} d^{3}+a^{2} b c \,d^{2}-11 a \,b^{2} c^{2} d +7 b^{3} c^{3}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{32 c \,d^{4}}\) | \(411\) |
elliptic | \(\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x \sqrt {-b \,x^{4}+a}}{4 d^{2} c \left (-d \,x^{4}+c \right )}+\frac {b^{2} x \sqrt {-b \,x^{4}+a}}{3 d^{2}}+\frac {\left (\frac {b^{2} \left (3 a d -2 b c \right )}{d^{3}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b}{4 d^{3} c}-\frac {b^{2} a}{3 d^{2}}\right ) \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}-c \right )}{\sum }\frac {\left (3 a^{3} d^{3}+a^{2} b c \,d^{2}-11 a \,b^{2} c^{2} d +7 b^{3} c^{3}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{32 c \,d^{4}}\) | \(411\) |
risch | \(\frac {b^{2} x \sqrt {-b \,x^{4}+a}}{3 d^{2}}+\frac {\frac {2 b^{2} \left (4 a d -3 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 )}{d \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {9 b \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\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 )}{8 d^{2}}+\frac {\left (3 a^{3} d^{3}-9 a^{2} b c \,d^{2}+9 a \,b^{2} c^{2} d -3 b^{3} c^{3}\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}}{3 d^{2}}\) | \(672\) |
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Timed out. \[ \int \frac {\left (a-b x^4\right )^{5/2}}{\left (c-d x^4\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (a-b x^4\right )^{5/2}}{\left (c-d x^4\right )^2} \, dx=\int \frac {\left (a - b x^{4}\right )^{\frac {5}{2}}}{\left (- c + d x^{4}\right )^{2}}\, dx \]
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\[ \int \frac {\left (a-b x^4\right )^{5/2}}{\left (c-d x^4\right )^2} \, dx=\int { \frac {{\left (-b x^{4} + a\right )}^{\frac {5}{2}}}{{\left (d x^{4} - c\right )}^{2}} \,d x } \]
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\[ \int \frac {\left (a-b x^4\right )^{5/2}}{\left (c-d x^4\right )^2} \, dx=\int { \frac {{\left (-b x^{4} + a\right )}^{\frac {5}{2}}}{{\left (d x^{4} - c\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (a-b x^4\right )^{5/2}}{\left (c-d x^4\right )^2} \, dx=\int \frac {{\left (a-b\,x^4\right )}^{5/2}}{{\left (c-d\,x^4\right )}^2} \,d x \]
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