Integrand size = 21, antiderivative size = 292 \[ \int \frac {1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^2} \, dx=-\frac {d x \sqrt [4]{a+b x^2}}{2 c (b c-a d) \left (c+d x^2\right )}-\frac {\sqrt {a} \sqrt {b} \left (1+\frac {b x^2}{a}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{2 c (b c-a d) \left (a+b x^2\right )^{3/4}}+\frac {\sqrt [4]{a} (5 b c-2 a d) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{4 c (b c-a d)^2 x}+\frac {\sqrt [4]{a} (5 b c-2 a d) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{4 c (b c-a d)^2 x} \]
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Time = 0.17 (sec) , antiderivative size = 292, 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, 239, 237, 410, 109, 418, 1232} \[ \int \frac {1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^2} \, dx=\frac {\sqrt [4]{a} \sqrt {-\frac {b x^2}{a}} (5 b c-2 a d) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}},\arcsin \left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right ),-1\right )}{4 c x (b c-a d)^2}+\frac {\sqrt [4]{a} \sqrt {-\frac {b x^2}{a}} (5 b c-2 a d) \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}},\arcsin \left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right ),-1\right )}{4 c x (b c-a d)^2}-\frac {\sqrt {a} \sqrt {b} \left (\frac {b x^2}{a}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{2 c \left (a+b x^2\right )^{3/4} (b c-a d)}-\frac {d x \sqrt [4]{a+b x^2}}{2 c \left (c+d x^2\right ) (b c-a d)} \]
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Rule 109
Rule 237
Rule 239
Rule 410
Rule 418
Rule 425
Rule 544
Rule 1232
Rubi steps \begin{align*} \text {integral}& = -\frac {d x \sqrt [4]{a+b x^2}}{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}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )} \, dx}{2 c (b c-a d)} \\ & = -\frac {d x \sqrt [4]{a+b x^2}}{2 c (b c-a d) \left (c+d x^2\right )}-\frac {b \int \frac {1}{\left (a+b x^2\right )^{3/4}} \, dx}{4 c (b c-a d)}+\frac {(5 b c-2 a d) \int \frac {1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )} \, dx}{4 c (b c-a d)} \\ & = -\frac {d x \sqrt [4]{a+b x^2}}{2 c (b c-a d) \left (c+d x^2\right )}+\frac {\left ((5 b c-2 a d) \sqrt {-\frac {b x^2}{a}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-\frac {b x}{a}} (a+b x)^{3/4} (c+d x)} \, dx,x,x^2\right )}{8 c (b c-a d) x}-\frac {\left (b \left (1+\frac {b x^2}{a}\right )^{3/4}\right ) \int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{3/4}} \, dx}{4 c (b c-a d) \left (a+b x^2\right )^{3/4}} \\ & = -\frac {d x \sqrt [4]{a+b x^2}}{2 c (b c-a d) \left (c+d x^2\right )}-\frac {\sqrt {a} \sqrt {b} \left (1+\frac {b x^2}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 c (b c-a d) \left (a+b x^2\right )^{3/4}}-\frac {\left ((5 b c-2 a d) \sqrt {-\frac {b x^2}{a}}\right ) \text {Subst}\left (\int \frac {1}{\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 {d x \sqrt [4]{a+b x^2}}{2 c (b c-a d) \left (c+d x^2\right )}-\frac {\sqrt {a} \sqrt {b} \left (1+\frac {b x^2}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 c (b c-a d) \left (a+b x^2\right )^{3/4}}+\frac {\left ((5 b c-2 a d) \sqrt {-\frac {b x^2}{a}}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt {d} x^2}{\sqrt {-b c+a d}}\right ) \sqrt {1-\frac {x^4}{a}}} \, dx,x,\sqrt [4]{a+b x^2}\right )}{4 c (b c-a d)^2 x}+\frac {\left ((5 b c-2 a d) \sqrt {-\frac {b x^2}{a}}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt {d} x^2}{\sqrt {-b c+a d}}\right ) \sqrt {1-\frac {x^4}{a}}} \, dx,x,\sqrt [4]{a+b x^2}\right )}{4 c (b c-a d)^2 x} \\ & = -\frac {d x \sqrt [4]{a+b x^2}}{2 c (b c-a d) \left (c+d x^2\right )}-\frac {\sqrt {a} \sqrt {b} \left (1+\frac {b x^2}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 c (b c-a d) \left (a+b x^2\right )^{3/4}}+\frac {\sqrt [4]{a} (5 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 (b c-a d)^2 x}+\frac {\sqrt [4]{a} (5 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 (b c-a d)^2 x} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.22 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.15 \[ \int \frac {1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^2} \, dx=\frac {x \left (\frac {b d x^2 \left (1+\frac {b x^2}{a}\right )^{3/4} \operatorname {AppellF1}\left (\frac {3}{2},\frac {3}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{-b c+a d}+\frac {c \left (36 a c \left (-2 b c+2 a d+b d x^2\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {3}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )-6 d x^2 \left (a+b x^2\right ) \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {3}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+3 b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {7}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}{(b c-a d) \left (c+d x^2\right ) \left (-6 a c \operatorname {AppellF1}\left (\frac {1}{2},\frac {3}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+x^2 \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {3}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+3 b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {7}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}\right )}{12 c^2 \left (a+b x^2\right )^{3/4}} \]
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\[\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {3}{4}} \left (d \,x^{2}+c \right )^{2}}d x\]
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Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^2} \, dx=\int \frac {1}{\left (a + b x^{2}\right )^{\frac {3}{4}} \left (c + d x^{2}\right )^{2}}\, dx \]
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\[ \int \frac {1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^2} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} {\left (d x^{2} + c\right )}^{2}} \,d x } \]
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\[ \int \frac {1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^2} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} {\left (d x^{2} + c\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^2} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{3/4}\,{\left (d\,x^2+c\right )}^2} \,d x \]
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