Integrand size = 21, antiderivative size = 336 \[ \int \frac {1}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )^2} \, dx=\frac {b x}{2 c (b c-a d) \sqrt [4]{a+b x^2}}-\frac {d x \left (a+b x^2\right )^{3/4}}{2 c (b c-a d) \left (c+d x^2\right )}-\frac {\sqrt {a} \sqrt {b} \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 c (b c-a d) \sqrt [4]{a+b x^2}}-\frac {\sqrt [4]{a} (3 b c-2 a d) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{4 c \sqrt {d} (-b c+a d)^{3/2} x}+\frac {\sqrt [4]{a} (3 b c-2 a d) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{4 c \sqrt {d} (-b c+a d)^{3/2} x} \]
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Time = 0.19 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {425, 544, 235, 233, 202, 408, 504, 1232} \[ \int \frac {1}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )^2} \, dx=-\frac {\sqrt [4]{a} \sqrt {-\frac {b x^2}{a}} (3 b c-2 a d) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}},\arcsin \left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right ),-1\right )}{4 c \sqrt {d} x (a d-b c)^{3/2}}+\frac {\sqrt [4]{a} \sqrt {-\frac {b x^2}{a}} (3 b c-2 a d) \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}},\arcsin \left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right ),-1\right )}{4 c \sqrt {d} x (a d-b c)^{3/2}}-\frac {\sqrt {a} \sqrt {b} \sqrt [4]{\frac {b x^2}{a}+1} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 c \sqrt [4]{a+b x^2} (b c-a d)}+\frac {b x}{2 c \sqrt [4]{a+b x^2} (b c-a d)}-\frac {d x \left (a+b x^2\right )^{3/4}}{2 c \left (c+d x^2\right ) (b c-a d)} \]
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Rule 202
Rule 233
Rule 235
Rule 408
Rule 425
Rule 504
Rule 544
Rule 1232
Rubi steps \begin{align*} \text {integral}& = -\frac {d x \left (a+b x^2\right )^{3/4}}{2 c (b c-a d) \left (c+d x^2\right )}+\frac {\int \frac {2 b c-a d+\frac {1}{2} b d x^2}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )} \, dx}{2 c (b c-a d)} \\ & = -\frac {d x \left (a+b x^2\right )^{3/4}}{2 c (b c-a d) \left (c+d x^2\right )}+\frac {b \int \frac {1}{\sqrt [4]{a+b x^2}} \, dx}{4 c (b c-a d)}+\frac {(3 b c-2 a d) \int \frac {1}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )} \, dx}{4 c (b c-a d)} \\ & = -\frac {d x \left (a+b x^2\right )^{3/4}}{2 c (b c-a d) \left (c+d x^2\right )}+\frac {\left ((3 b c-2 a d) \sqrt {-\frac {b x^2}{a}}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-\frac {x^4}{a}} \left (b c-a d+d x^4\right )} \, dx,x,\sqrt [4]{a+b x^2}\right )}{2 c (b c-a d) x}+\frac {\left (b \sqrt [4]{1+\frac {b x^2}{a}}\right ) \int \frac {1}{\sqrt [4]{1+\frac {b x^2}{a}}} \, dx}{4 c (b c-a d) \sqrt [4]{a+b x^2}} \\ & = \frac {b x}{2 c (b c-a d) \sqrt [4]{a+b x^2}}-\frac {d x \left (a+b x^2\right )^{3/4}}{2 c (b c-a d) \left (c+d x^2\right )}-\frac {\left ((3 b c-2 a d) \sqrt {-\frac {b x^2}{a}}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-b c+a d}-\sqrt {d} x^2\right ) \sqrt {1-\frac {x^4}{a}}} \, dx,x,\sqrt [4]{a+b x^2}\right )}{4 c \sqrt {d} (b c-a d) x}+\frac {\left ((3 b c-2 a d) \sqrt {-\frac {b x^2}{a}}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-b c+a d}+\sqrt {d} x^2\right ) \sqrt {1-\frac {x^4}{a}}} \, dx,x,\sqrt [4]{a+b x^2}\right )}{4 c \sqrt {d} (b c-a d) x}-\frac {\left (b \sqrt [4]{1+\frac {b x^2}{a}}\right ) \int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{5/4}} \, dx}{4 c (b c-a d) \sqrt [4]{a+b x^2}} \\ & = \frac {b x}{2 c (b c-a d) \sqrt [4]{a+b x^2}}-\frac {d x \left (a+b x^2\right )^{3/4}}{2 c (b c-a d) \left (c+d x^2\right )}-\frac {\sqrt {a} \sqrt {b} \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 c (b c-a d) \sqrt [4]{a+b x^2}}-\frac {\sqrt [4]{a} (3 b c-2 a d) \sqrt {-\frac {b x^2}{a}} \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c \sqrt {d} (-b c+a d)^{3/2} x}+\frac {\sqrt [4]{a} (3 b c-2 a d) \sqrt {-\frac {b x^2}{a}} \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right )\right |-1\right )}{4 c \sqrt {d} (-b c+a d)^{3/2} x} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.18 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.17 \[ \int \frac {1}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )^2} \, dx=\frac {-6 a c x \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right ) \left (-6 c \left (-2 b c+2 a d+b d x^2\right )+b d x^2 \sqrt [4]{1+\frac {b x^2}{a}} \left (c+d x^2\right ) \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )-d x^3 \left (6 c \left (a+b x^2\right )-b x^2 \sqrt [4]{1+\frac {b x^2}{a}} \left (c+d x^2\right ) \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right ) \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )}{12 c^2 (b c-a d) \sqrt [4]{a+b x^2} \left (c+d x^2\right ) \left (-6 a c \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+x^2 \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )} \]
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\[\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} \left (d \,x^{2}+c \right )^{2}}d x\]
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Timed out. \[ \int \frac {1}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )^2} \, dx=\int \frac {1}{\sqrt [4]{a + b x^{2}} \left (c + d x^{2}\right )^{2}}\, dx \]
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\[ \int \frac {1}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )^2} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {1}{4}} {\left (d x^{2} + c\right )}^{2}} \,d x } \]
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\[ \int \frac {1}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )^2} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {1}{4}} {\left (d x^{2} + c\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )^2} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{1/4}\,{\left (d\,x^2+c\right )}^2} \,d x \]
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