Integrand size = 21, antiderivative size = 298 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{\left (c+d x^4\right )^2} \, dx=-\frac {(b c-a d) x \sqrt [4]{a+b x^4}}{4 c d \left (c+d x^4\right )}+\frac {\sqrt {a} b^{3/2} \left (1+\frac {a}{b x^4}\right )^{3/4} x^3 \operatorname {EllipticF}\left (\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{4 c d \left (a+b x^4\right )^{3/4}}+\frac {(2 b c+3 a d) \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} \operatorname {EllipticPi}\left (-\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right ),-1\right )}{8 \sqrt [4]{b} c^2 d}+\frac {(2 b c+3 a d) \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} \operatorname {EllipticPi}\left (\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right ),-1\right )}{8 \sqrt [4]{b} c^2 d} \]
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Time = 0.18 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {424, 543, 243, 342, 281, 237, 416, 418, 1232} \[ \int \frac {\left (a+b x^4\right )^{5/4}}{\left (c+d x^4\right )^2} \, dx=\frac {\sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} (3 a d+2 b c) \operatorname {EllipticPi}\left (-\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right ),-1\right )}{8 \sqrt [4]{b} c^2 d}+\frac {\sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} (3 a d+2 b c) \operatorname {EllipticPi}\left (\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right ),-1\right )}{8 \sqrt [4]{b} c^2 d}+\frac {\sqrt {a} b^{3/2} x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{4 c d \left (a+b x^4\right )^{3/4}}-\frac {x \sqrt [4]{a+b x^4} (b c-a d)}{4 c d \left (c+d x^4\right )} \]
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Rule 237
Rule 243
Rule 281
Rule 342
Rule 416
Rule 418
Rule 424
Rule 543
Rule 1232
Rubi steps \begin{align*} \text {integral}& = -\frac {(b c-a d) x \sqrt [4]{a+b x^4}}{4 c d \left (c+d x^4\right )}+\frac {\int \frac {a (b c+3 a d)+2 b (b c+a d) x^4}{\left (a+b x^4\right )^{3/4} \left (c+d x^4\right )} \, dx}{4 c d} \\ & = -\frac {(b c-a d) x \sqrt [4]{a+b x^4}}{4 c d \left (c+d x^4\right )}-\frac {(a b) \int \frac {1}{\left (a+b x^4\right )^{3/4}} \, dx}{4 c d}-\frac {(-2 b c-3 a d) \int \frac {\sqrt [4]{a+b x^4}}{c+d x^4} \, dx}{4 c d} \\ & = -\frac {(b c-a d) x \sqrt [4]{a+b x^4}}{4 c d \left (c+d x^4\right )}-\frac {\left (a b \left (1+\frac {a}{b x^4}\right )^{3/4} x^3\right ) \int \frac {1}{\left (1+\frac {a}{b x^4}\right )^{3/4} x^3} \, dx}{4 c d \left (a+b x^4\right )^{3/4}}-\frac {\left ((-2 b c-3 a d) \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-b x^4} \left (c-(b c-a d) x^4\right )} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{4 c d} \\ & = -\frac {(b c-a d) x \sqrt [4]{a+b x^4}}{4 c d \left (c+d x^4\right )}+\frac {\left (a b \left (1+\frac {a}{b x^4}\right )^{3/4} x^3\right ) \text {Subst}\left (\int \frac {x}{\left (1+\frac {a x^4}{b}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )}{4 c d \left (a+b x^4\right )^{3/4}}-\frac {\left ((-2 b c-3 a d) \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt {b c-a d} x^2}{\sqrt {c}}\right ) \sqrt {1-b x^4}} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{8 c^2 d}-\frac {\left ((-2 b c-3 a d) \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt {b c-a d} x^2}{\sqrt {c}}\right ) \sqrt {1-b x^4}} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )}{8 c^2 d} \\ & = -\frac {(b c-a d) x \sqrt [4]{a+b x^4}}{4 c d \left (c+d x^4\right )}+\frac {(2 b c+3 a d) \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} \Pi \left (-\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d}+\frac {(2 b c+3 a d) \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} \Pi \left (\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d}+\frac {\left (a b \left (1+\frac {a}{b x^4}\right )^{3/4} x^3\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {a x^2}{b}\right )^{3/4}} \, dx,x,\frac {1}{x^2}\right )}{8 c d \left (a+b x^4\right )^{3/4}} \\ & = -\frac {(b c-a d) x \sqrt [4]{a+b x^4}}{4 c d \left (c+d x^4\right )}+\frac {\sqrt {a} b^{3/2} \left (1+\frac {a}{b x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{4 c d \left (a+b x^4\right )^{3/4}}+\frac {(2 b c+3 a d) \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} \Pi \left (-\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d}+\frac {(2 b c+3 a d) \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} \Pi \left (\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right |-1\right )}{8 \sqrt [4]{b} c^2 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.35 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{\left (c+d x^4\right )^2} \, dx=\frac {x \left (2 b (b c+a d) x^4 \left (1+\frac {b x^4}{a}\right )^{3/4} \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{4},1,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+\frac {5 c \left (-5 a c \left (4 a^2 d-b^2 c x^4+a b d x^4\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {3}{4},1,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+(-b c+a d) x^4 \left (a+b x^4\right ) \left (4 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{4},2,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+3 b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {7}{4},1,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\right )\right )}{\left (c+d x^4\right ) \left (-5 a c \operatorname {AppellF1}\left (\frac {1}{4},\frac {3}{4},1,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+x^4 \left (4 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{4},2,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+3 b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {7}{4},1,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\right )\right )}\right )}{20 c^2 d \left (a+b x^4\right )^{3/4}} \]
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\[\int \frac {\left (b \,x^{4}+a \right )^{\frac {5}{4}}}{\left (d \,x^{4}+c \right )^{2}}d x\]
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Timed out. \[ \int \frac {\left (a+b x^4\right )^{5/4}}{\left (c+d x^4\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (a+b x^4\right )^{5/4}}{\left (c+d x^4\right )^2} \, dx=\int \frac {\left (a + b x^{4}\right )^{\frac {5}{4}}}{\left (c + d x^{4}\right )^{2}}\, dx \]
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\[ \int \frac {\left (a+b x^4\right )^{5/4}}{\left (c+d x^4\right )^2} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {5}{4}}}{{\left (d x^{4} + c\right )}^{2}} \,d x } \]
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\[ \int \frac {\left (a+b x^4\right )^{5/4}}{\left (c+d x^4\right )^2} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {5}{4}}}{{\left (d x^{4} + c\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b x^4\right )^{5/4}}{\left (c+d x^4\right )^2} \, dx=\int \frac {{\left (b\,x^4+a\right )}^{5/4}}{{\left (d\,x^4+c\right )}^2} \,d x \]
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