Integrand size = 21, antiderivative size = 309 \[ \int \frac {\left (a+b x^2\right )^{3/4}}{\left (c+d x^2\right )^2} \, dx=-\frac {b x}{2 c d \sqrt [4]{a+b x^2}}+\frac {x \left (a+b x^2\right )^{3/4}}{2 c \left (c+d x^2\right )}+\frac {\sqrt {a} \sqrt {b} \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 c d \sqrt [4]{a+b x^2}}+\frac {\sqrt [4]{a} (b c+2 a d) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{4 c d^{3/2} \sqrt {-b c+a d} x}-\frac {\sqrt [4]{a} (b c+2 a d) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{4 c d^{3/2} \sqrt {-b c+a d} x} \]
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Time = 0.15 (sec) , antiderivative size = 309, 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 = {423, 544, 235, 233, 202, 408, 504, 1232} \[ \int \frac {\left (a+b x^2\right )^{3/4}}{\left (c+d x^2\right )^2} \, dx=\frac {\sqrt [4]{a} \sqrt {-\frac {b x^2}{a}} (2 a d+b c) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}},\arcsin \left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right ),-1\right )}{4 c d^{3/2} x \sqrt {a d-b c}}-\frac {\sqrt [4]{a} \sqrt {-\frac {b x^2}{a}} (2 a d+b c) \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}},\arcsin \left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right ),-1\right )}{4 c d^{3/2} x \sqrt {a d-b c}}+\frac {\sqrt {a} \sqrt {b} \sqrt [4]{\frac {b x^2}{a}+1} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{2 c d \sqrt [4]{a+b x^2}}-\frac {b x}{2 c d \sqrt [4]{a+b x^2}}+\frac {x \left (a+b x^2\right )^{3/4}}{2 c \left (c+d x^2\right )} \]
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Rule 202
Rule 233
Rule 235
Rule 408
Rule 423
Rule 504
Rule 544
Rule 1232
Rubi steps \begin{align*} \text {integral}& = \frac {x \left (a+b x^2\right )^{3/4}}{2 c \left (c+d x^2\right )}-\frac {\int \frac {-a+\frac {b x^2}{2}}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )} \, dx}{2 c} \\ & = \frac {x \left (a+b x^2\right )^{3/4}}{2 c \left (c+d x^2\right )}-\frac {b \int \frac {1}{\sqrt [4]{a+b x^2}} \, dx}{4 c d}+\frac {(b c+2 a d) \int \frac {1}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )} \, dx}{4 c d} \\ & = \frac {x \left (a+b x^2\right )^{3/4}}{2 c \left (c+d x^2\right )}+\frac {\left ((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 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 d \sqrt [4]{a+b x^2}} \\ & = -\frac {b x}{2 c d \sqrt [4]{a+b x^2}}+\frac {x \left (a+b x^2\right )^{3/4}}{2 c \left (c+d x^2\right )}-\frac {\left ((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 d^{3/2} x}+\frac {\left ((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 d^{3/2} 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 d \sqrt [4]{a+b x^2}} \\ & = -\frac {b x}{2 c d \sqrt [4]{a+b x^2}}+\frac {x \left (a+b x^2\right )^{3/4}}{2 c \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 d \sqrt [4]{a+b x^2}}+\frac {\sqrt [4]{a} (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^{3/2} \sqrt {-b c+a d} x}-\frac {\sqrt [4]{a} (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^{3/2} \sqrt {-b c+a d} x} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.16 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+b x^2\right )^{3/4}}{\left (c+d x^2\right )^2} \, dx=\frac {x \left (-\frac {b x^2 \sqrt [4]{1+\frac {b x^2}{a}} \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{c^2}+\frac {6 \left (\frac {a+b x^2}{c}-\frac {6 a^2 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{-6 a c \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+x^2 \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )}\right )}{c+d x^2}\right )}{12 \sqrt [4]{a+b x^2}} \]
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\[\int \frac {\left (b \,x^{2}+a \right )^{\frac {3}{4}}}{\left (d \,x^{2}+c \right )^{2}}d x\]
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Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/4}}{\left (c+d x^2\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (a+b x^2\right )^{3/4}}{\left (c+d x^2\right )^2} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {3}{4}}}{\left (c + d x^{2}\right )^{2}}\, dx \]
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\[ \int \frac {\left (a+b x^2\right )^{3/4}}{\left (c+d x^2\right )^2} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}}}{{\left (d x^{2} + c\right )}^{2}} \,d x } \]
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\[ \int \frac {\left (a+b x^2\right )^{3/4}}{\left (c+d x^2\right )^2} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}}}{{\left (d x^{2} + c\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/4}}{\left (c+d x^2\right )^2} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/4}}{{\left (d\,x^2+c\right )}^2} \,d x \]
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