Integrand size = 23, antiderivative size = 276 \[ \int \frac {\sqrt {a-b x^4}}{\left (c-d x^4\right )^2} \, dx=\frac {x \sqrt {a-b x^4}}{4 c \left (c-d x^4\right )}+\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{4 c d \sqrt {a-b x^4}}-\frac {\sqrt [4]{a} (b c-3 a d) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{8 \sqrt [4]{b} c^2 d \sqrt {a-b x^4}}-\frac {\sqrt [4]{a} (b c-3 a d) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{8 \sqrt [4]{b} c^2 d \sqrt {a-b x^4}} \]
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Time = 0.15 (sec) , antiderivative size = 276, 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 = {423, 537, 230, 227, 418, 1233, 1232} \[ \int \frac {\sqrt {a-b x^4}}{\left (c-d x^4\right )^2} \, dx=\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{4 c d \sqrt {a-b x^4}}-\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} (b c-3 a d) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{8 \sqrt [4]{b} c^2 d \sqrt {a-b x^4}}-\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} (b c-3 a d) \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{8 \sqrt [4]{b} c^2 d \sqrt {a-b x^4}}+\frac {x \sqrt {a-b x^4}}{4 c \left (c-d x^4\right )} \]
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Rule 227
Rule 230
Rule 418
Rule 423
Rule 537
Rule 1232
Rule 1233
Rubi steps \begin{align*} \text {integral}& = \frac {x \sqrt {a-b x^4}}{4 c \left (c-d x^4\right )}-\frac {\int \frac {-3 a+b x^4}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx}{4 c} \\ & = \frac {x \sqrt {a-b x^4}}{4 c \left (c-d x^4\right )}+\frac {b \int \frac {1}{\sqrt {a-b x^4}} \, dx}{4 c d}+\frac {(-b c+3 a d) \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx}{4 c d} \\ & = \frac {x \sqrt {a-b x^4}}{4 c \left (c-d x^4\right )}-\frac {(b c-3 a d) \int \frac {1}{\left (1-\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {a-b x^4}} \, dx}{8 c^2 d}-\frac {(b c-3 a d) \int \frac {1}{\left (1+\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {a-b x^4}} \, dx}{8 c^2 d}+\frac {\left (b \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}} \, dx}{4 c d \sqrt {a-b x^4}} \\ & = \frac {x \sqrt {a-b x^4}}{4 c \left (c-d x^4\right )}+\frac {\sqrt [4]{a} 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 )}{4 c d \sqrt {a-b x^4}}-\frac {\left ((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 \sqrt {a-b x^4}}-\frac {\left ((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 \sqrt {a-b x^4}} \\ & = \frac {x \sqrt {a-b x^4}}{4 c \left (c-d x^4\right )}+\frac {\sqrt [4]{a} 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 )}{4 c d \sqrt {a-b x^4}}-\frac {\sqrt [4]{a} (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 \sqrt {a-b x^4}}-\frac {\sqrt [4]{a} (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 \sqrt {a-b x^4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.17 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {a-b x^4}}{\left (c-d x^4\right )^2} \, dx=\frac {x \left (-\frac {5 \left (a-b x^4\right )}{c}+\frac {b x^4 \sqrt {1-\frac {b x^4}{a}} \left (c-d x^4\right ) \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )}{c^2}-\frac {75 a^2 \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )}{5 a c \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+2 x^4 \left (2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )}\right )}{20 \sqrt {a-b x^4} \left (-c+d x^4\right )} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 4.35 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.06
method | result | size |
default | \(\frac {x \sqrt {-b \,x^{4}+a}}{4 c \left (-d \,x^{4}+c \right )}+\frac {b \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{4 c d \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}-c \right )}{\sum }\frac {\left (3 a d -b c \right ) \left (-\frac {\operatorname {arctanh}\left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{32 c \,d^{2}}\) | \(293\) |
elliptic | \(\frac {x \sqrt {-b \,x^{4}+a}}{4 c \left (-d \,x^{4}+c \right )}+\frac {b \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{4 c d \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}-c \right )}{\sum }\frac {\left (3 a d -b c \right ) \left (-\frac {\operatorname {arctanh}\left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{32 c \,d^{2}}\) | \(293\) |
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Timed out. \[ \int \frac {\sqrt {a-b x^4}}{\left (c-d x^4\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {a-b x^4}}{\left (c-d x^4\right )^2} \, dx=\int \frac {\sqrt {a - b x^{4}}}{\left (- c + d x^{4}\right )^{2}}\, dx \]
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\[ \int \frac {\sqrt {a-b x^4}}{\left (c-d x^4\right )^2} \, dx=\int { \frac {\sqrt {-b x^{4} + a}}{{\left (d x^{4} - c\right )}^{2}} \,d x } \]
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\[ \int \frac {\sqrt {a-b x^4}}{\left (c-d x^4\right )^2} \, dx=\int { \frac {\sqrt {-b x^{4} + a}}{{\left (d x^{4} - c\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {a-b x^4}}{\left (c-d x^4\right )^2} \, dx=\int \frac {\sqrt {a-b\,x^4}}{{\left (c-d\,x^4\right )}^2} \,d x \]
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