Integrand size = 21, antiderivative size = 314 \[ \int \frac {1}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^2} \, dx=-\frac {d x}{2 c (b c-a d) \sqrt [4]{a+b x^2} \left (c+d x^2\right )}+\frac {\sqrt {b} (4 b c+a d) \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 \sqrt {a} c (b c-a d)^2 \sqrt [4]{a+b x^2}}-\frac {\sqrt [4]{a} \sqrt {d} (7 b c-2 a d) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{4 c (-b c+a d)^{5/2} x}+\frac {\sqrt [4]{a} \sqrt {d} (7 b c-2 a d) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{4 c (-b c+a d)^{5/2} x} \]
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Time = 0.29 (sec) , antiderivative size = 314, 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 = {425, 541, 544, 235, 233, 202, 408, 504, 1232} \[ \int \frac {1}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^2} \, dx=-\frac {\sqrt [4]{a} \sqrt {d} \sqrt {-\frac {b x^2}{a}} (7 b c-2 a d) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}},\arcsin \left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right ),-1\right )}{4 c x (a d-b c)^{5/2}}+\frac {\sqrt [4]{a} \sqrt {d} \sqrt {-\frac {b x^2}{a}} (7 b c-2 a d) \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}},\arcsin \left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right ),-1\right )}{4 c x (a d-b c)^{5/2}}+\frac {\sqrt {b} \sqrt [4]{\frac {b x^2}{a}+1} (a d+4 b c) E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 \sqrt {a} c \sqrt [4]{a+b x^2} (b c-a d)^2}-\frac {d x}{2 c \sqrt [4]{a+b x^2} \left (c+d x^2\right ) (b c-a d)} \]
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Rule 202
Rule 233
Rule 235
Rule 408
Rule 425
Rule 504
Rule 541
Rule 544
Rule 1232
Rubi steps \begin{align*} \text {integral}& = -\frac {d x}{2 c (b c-a d) \sqrt [4]{a+b x^2} \left (c+d x^2\right )}+\frac {\int \frac {2 b c-a d-\frac {3}{2} b d x^2}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )} \, dx}{2 c (b c-a d)} \\ & = \frac {b (4 b c+a d) x}{2 a c (b c-a d)^2 \sqrt [4]{a+b x^2}}-\frac {d x}{2 c (b c-a d) \sqrt [4]{a+b x^2} \left (c+d x^2\right )}-\frac {\int \frac {\frac {1}{2} \left (2 b^2 c^2+4 a b c d-a^2 d^2\right )+\frac {1}{4} b d (4 b c+a d) x^2}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )} \, dx}{a c (b c-a d)^2} \\ & = \frac {b (4 b c+a d) x}{2 a c (b c-a d)^2 \sqrt [4]{a+b x^2}}-\frac {d x}{2 c (b c-a d) \sqrt [4]{a+b x^2} \left (c+d x^2\right )}-\frac {(d (7 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)^2}-\frac {(b (4 b c+a d)) \int \frac {1}{\sqrt [4]{a+b x^2}} \, dx}{4 a c (b c-a d)^2} \\ & = \frac {b (4 b c+a d) x}{2 a c (b c-a d)^2 \sqrt [4]{a+b x^2}}-\frac {d x}{2 c (b c-a d) \sqrt [4]{a+b x^2} \left (c+d x^2\right )}-\frac {\left (d (7 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)^2 x}-\frac {\left (b (4 b c+a d) \sqrt [4]{1+\frac {b x^2}{a}}\right ) \int \frac {1}{\sqrt [4]{1+\frac {b x^2}{a}}} \, dx}{4 a c (b c-a d)^2 \sqrt [4]{a+b x^2}} \\ & = -\frac {d x}{2 c (b c-a d) \sqrt [4]{a+b x^2} \left (c+d x^2\right )}+\frac {\left (\sqrt {d} (7 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 (b c-a d)^2 x}-\frac {\left (\sqrt {d} (7 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 (b c-a d)^2 x}+\frac {\left (b (4 b c+a d) \sqrt [4]{1+\frac {b x^2}{a}}\right ) \int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{5/4}} \, dx}{4 a c (b c-a d)^2 \sqrt [4]{a+b x^2}} \\ & = -\frac {d x}{2 c (b c-a d) \sqrt [4]{a+b x^2} \left (c+d x^2\right )}+\frac {\sqrt {b} (4 b c+a d) \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 \sqrt {a} c (b c-a d)^2 \sqrt [4]{a+b x^2}}-\frac {\sqrt [4]{a} \sqrt {d} (7 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)^{5/2} x}+\frac {\sqrt [4]{a} \sqrt {d} (7 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)^{5/2} x} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.33 (sec) , antiderivative size = 380, normalized size of antiderivative = 1.21 \[ \int \frac {1}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^2} \, dx=\frac {x \left (-b d (4 b c+a d) x^2 \sqrt [4]{1+\frac {b x^2}{a}} \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+\frac {c \left (36 a c \left (2 a^2 d^2+a b d \left (-4 c+d x^2\right )+2 b^2 c \left (c+2 d x^2\right )\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )-6 x^2 \left (a^2 d^2+a b d^2 x^2+4 b^2 c \left (c+d x^2\right )\right ) \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}{\left (c+d x^2\right ) \left (6 a c \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )-x^2 \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}\right )}{12 a c^2 (b c-a d)^2 \sqrt [4]{a+b x^2}} \]
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\[\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {5}{4}} \left (d \,x^{2}+c \right )^{2}}d x\]
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Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^2} \, dx=\int \frac {1}{\left (a + b x^{2}\right )^{\frac {5}{4}} \left (c + d x^{2}\right )^{2}}\, dx \]
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\[ \int \frac {1}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^2} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{4}} {\left (d x^{2} + c\right )}^{2}} \,d x } \]
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\[ \int \frac {1}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^2} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{4}} {\left (d x^{2} + c\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )^2} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{5/4}\,{\left (d\,x^2+c\right )}^2} \,d x \]
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