Integrand size = 21, antiderivative size = 279 \[ \int \frac {\left (a+b x^2\right )^{5/4}}{\left (c+d x^2\right )^2} \, dx=-\frac {(b c-a d) x \sqrt [4]{a+b x^2}}{2 c d \left (c+d x^2\right )}+\frac {\sqrt {a} \sqrt {b} (3 b c+a d) \left (1+\frac {b x^2}{a}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{2 c d^2 \left (a+b x^2\right )^{3/4}}-\frac {\sqrt [4]{a} (3 b c+2 a d) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{4 c d^2 x}-\frac {\sqrt [4]{a} (3 b c+2 a d) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{4 c d^2 x} \]
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Time = 0.19 (sec) , antiderivative size = 279, 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 = {424, 544, 239, 237, 410, 109, 418, 1232} \[ \int \frac {\left (a+b x^2\right )^{5/4}}{\left (c+d x^2\right )^2} \, dx=-\frac {\sqrt [4]{a} \sqrt {-\frac {b x^2}{a}} (2 a d+3 b c) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}},\arcsin \left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right ),-1\right )}{4 c d^2 x}-\frac {\sqrt [4]{a} \sqrt {-\frac {b x^2}{a}} (2 a d+3 b c) \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}},\arcsin \left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right ),-1\right )}{4 c d^2 x}+\frac {\sqrt {a} \sqrt {b} \left (\frac {b x^2}{a}+1\right )^{3/4} (a d+3 b c) \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{2 c d^2 \left (a+b x^2\right )^{3/4}}-\frac {x \sqrt [4]{a+b x^2} (b c-a d)}{2 c d \left (c+d x^2\right )} \]
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Rule 109
Rule 237
Rule 239
Rule 410
Rule 418
Rule 424
Rule 544
Rule 1232
Rubi steps \begin{align*} \text {integral}& = -\frac {(b c-a d) x \sqrt [4]{a+b x^2}}{2 c d \left (c+d x^2\right )}+\frac {\int \frac {a (b c+a d)+\frac {1}{2} b (3 b c+a d) x^2}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )} \, dx}{2 c d} \\ & = -\frac {(b c-a d) x \sqrt [4]{a+b x^2}}{2 c d \left (c+d x^2\right )}+\frac {(b (3 b c+a d)) \int \frac {1}{\left (a+b x^2\right )^{3/4}} \, dx}{4 c d^2}-\frac {((b c-a d) (3 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 d^2} \\ & = -\frac {(b c-a d) x \sqrt [4]{a+b x^2}}{2 c d \left (c+d x^2\right )}-\frac {\left ((b c-a d) (3 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 d^2 x}+\frac {\left (b (3 b c+a d) \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 d^2 \left (a+b x^2\right )^{3/4}} \\ & = -\frac {(b c-a d) x \sqrt [4]{a+b x^2}}{2 c d \left (c+d x^2\right )}+\frac {\sqrt {a} \sqrt {b} (3 b c+a d) \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 d^2 \left (a+b x^2\right )^{3/4}}+\frac {\left ((b c-a d) (3 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 d^2 x} \\ & = -\frac {(b c-a d) x \sqrt [4]{a+b x^2}}{2 c d \left (c+d x^2\right )}+\frac {\sqrt {a} \sqrt {b} (3 b c+a d) \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 d^2 \left (a+b x^2\right )^{3/4}}-\frac {\left ((3 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 d^2 x}-\frac {\left ((3 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 d^2 x} \\ & = -\frac {(b c-a d) x \sqrt [4]{a+b x^2}}{2 c d \left (c+d x^2\right )}+\frac {\sqrt {a} \sqrt {b} (3 b c+a d) \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 d^2 \left (a+b x^2\right )^{3/4}}-\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 d^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 d^2 x} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.36 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.22 \[ \int \frac {\left (a+b x^2\right )^{5/4}}{\left (c+d x^2\right )^2} \, dx=\frac {x \left (b (3 b c+a d) x^2 \left (1+\frac {b x^2}{a}\right )^{3/4} \operatorname {AppellF1}\left (\frac {3}{2},\frac {3}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+\frac {6 c \left (-6 a c \left (2 a^2 d-b^2 c x^2+a b d x^2\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {3}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+(-b c+a d) x^2 \left (a+b x^2\right ) \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {3}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+3 b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {7}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}{\left (c+d x^2\right ) \left (-6 a c \operatorname {AppellF1}\left (\frac {1}{2},\frac {3}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+x^2 \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {3}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+3 b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {7}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}\right )}{12 c^2 d \left (a+b x^2\right )^{3/4}} \]
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\[\int \frac {\left (b \,x^{2}+a \right )^{\frac {5}{4}}}{\left (d \,x^{2}+c \right )^{2}}d x\]
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Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/4}}{\left (c+d x^2\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (a+b x^2\right )^{5/4}}{\left (c+d x^2\right )^2} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {5}{4}}}{\left (c + d x^{2}\right )^{2}}\, dx \]
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\[ \int \frac {\left (a+b x^2\right )^{5/4}}{\left (c+d x^2\right )^2} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{4}}}{{\left (d x^{2} + c\right )}^{2}} \,d x } \]
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\[ \int \frac {\left (a+b x^2\right )^{5/4}}{\left (c+d x^2\right )^2} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{4}}}{{\left (d x^{2} + c\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/4}}{\left (c+d x^2\right )^2} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/4}}{{\left (d\,x^2+c\right )}^2} \,d x \]
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