Integrand size = 21, antiderivative size = 274 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{c+d x^4} \, dx=\frac {b x \sqrt [4]{a+b x^4}}{2 d}-\frac {\sqrt {a} b^{3/2} \left (1+\frac {a}{b x^4}\right )^{3/4} x^3 \operatorname {EllipticF}\left (\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{2 d \left (a+b x^4\right )^{3/4}}-\frac {(b c-a d) \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} \operatorname {EllipticPi}\left (-\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right ),-1\right )}{2 \sqrt [4]{b} c d}-\frac {(b c-a d) \sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} \operatorname {EllipticPi}\left (\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right ),-1\right )}{2 \sqrt [4]{b} c d} \]
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Time = 0.11 (sec) , antiderivative size = 274, 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 = {417, 201, 243, 342, 281, 237, 416, 418, 1232} \[ \int \frac {\left (a+b x^4\right )^{5/4}}{c+d x^4} \, dx=-\frac {\sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} (b c-a d) \operatorname {EllipticPi}\left (-\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right ),-1\right )}{2 \sqrt [4]{b} c d}-\frac {\sqrt {\frac {a}{a+b x^4}} \sqrt {a+b x^4} (b c-a d) \operatorname {EllipticPi}\left (\frac {\sqrt {b c-a d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{b x^4+a}}\right ),-1\right )}{2 \sqrt [4]{b} c d}-\frac {\sqrt {a} b^{3/2} x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{2 d \left (a+b x^4\right )^{3/4}}+\frac {b x \sqrt [4]{a+b x^4}}{2 d} \]
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Rule 201
Rule 237
Rule 243
Rule 281
Rule 342
Rule 416
Rule 417
Rule 418
Rule 1232
Rubi steps \begin{align*} \text {integral}& = \frac {b \int \sqrt [4]{a+b x^4} \, dx}{d}-\frac {(b c-a d) \int \frac {\sqrt [4]{a+b x^4}}{c+d x^4} \, dx}{d} \\ & = \frac {b x \sqrt [4]{a+b x^4}}{2 d}+\frac {(a b) \int \frac {1}{\left (a+b x^4\right )^{3/4}} \, dx}{2 d}-\frac {\left ((b c-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 )}{d} \\ & = \frac {b x \sqrt [4]{a+b x^4}}{2 d}+\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}{2 d \left (a+b x^4\right )^{3/4}}-\frac {\left ((b c-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 )}{2 c d}-\frac {\left ((b c-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 )}{2 c d} \\ & = \frac {b x \sqrt [4]{a+b x^4}}{2 d}-\frac {(b c-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 )}{2 \sqrt [4]{b} c d}-\frac {(b c-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 )}{2 \sqrt [4]{b} c d}-\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 )}{2 d \left (a+b x^4\right )^{3/4}} \\ & = \frac {b x \sqrt [4]{a+b x^4}}{2 d}-\frac {(b c-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 )}{2 \sqrt [4]{b} c d}-\frac {(b c-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 )}{2 \sqrt [4]{b} c 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 )}{4 d \left (a+b x^4\right )^{3/4}} \\ & = \frac {b x \sqrt [4]{a+b x^4}}{2 d}-\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 )}{2 d \left (a+b x^4\right )^{3/4}}-\frac {(b c-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 )}{2 \sqrt [4]{b} c d}-\frac {(b c-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 )}{2 \sqrt [4]{b} c d} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.34 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.26 \[ \int \frac {\left (a+b x^4\right )^{5/4}}{c+d x^4} \, dx=\frac {x \left (\frac {b (-2 b c+3 a d) x^4 \left (1+\frac {b x^4}{a}\right )^{3/4} \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{4},1,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{c}+\frac {5 \left (-5 a c \left (2 a^2 d+a b d x^4+b^2 x^4 \left (c+d x^4\right )\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {3}{4},1,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+b x^4 \left (a+b x^4\right ) \left (c+d x^4\right ) \left (4 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{4},2,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+3 b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {7}{4},1,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\right )\right )}{\left (c+d x^4\right ) \left (-5 a c \operatorname {AppellF1}\left (\frac {1}{4},\frac {3}{4},1,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+x^4 \left (4 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{4},2,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+3 b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {7}{4},1,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\right )\right )}\right )}{10 d \left (a+b x^4\right )^{3/4}} \]
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\[\int \frac {\left (b \,x^{4}+a \right )^{\frac {5}{4}}}{d \,x^{4}+c}d x\]
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Timed out. \[ \int \frac {\left (a+b x^4\right )^{5/4}}{c+d x^4} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (a+b x^4\right )^{5/4}}{c+d x^4} \, dx=\int \frac {\left (a + b x^{4}\right )^{\frac {5}{4}}}{c + d x^{4}}\, dx \]
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\[ \int \frac {\left (a+b x^4\right )^{5/4}}{c+d x^4} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {5}{4}}}{d x^{4} + c} \,d x } \]
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\[ \int \frac {\left (a+b x^4\right )^{5/4}}{c+d x^4} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {5}{4}}}{d x^{4} + c} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b x^4\right )^{5/4}}{c+d x^4} \, dx=\int \frac {{\left (b\,x^4+a\right )}^{5/4}}{d\,x^4+c} \,d x \]
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