Integrand size = 21, antiderivative size = 244 \[ \int \frac {\left (a+b x^2\right )^{3/4}}{c+d x^2} \, dx=\frac {2 b x}{d \sqrt [4]{a+b x^2}}-\frac {2 \sqrt {a} \sqrt {b} \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{d \sqrt [4]{a+b x^2}}+\frac {\sqrt [4]{a} \sqrt {-b c+a d} \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{d^{3/2} x}-\frac {\sqrt [4]{a} \sqrt {-b c+a d} \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{d^{3/2} x} \]
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Time = 0.11 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {411, 235, 233, 202, 408, 504, 1232} \[ \int \frac {\left (a+b x^2\right )^{3/4}}{c+d x^2} \, dx=\frac {\sqrt [4]{a} \sqrt {-\frac {b x^2}{a}} \sqrt {a d-b c} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}},\arcsin \left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right ),-1\right )}{d^{3/2} x}-\frac {\sqrt [4]{a} \sqrt {-\frac {b x^2}{a}} \sqrt {a d-b c} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}},\arcsin \left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right ),-1\right )}{d^{3/2} x}-\frac {2 \sqrt {a} \sqrt {b} \sqrt [4]{\frac {b x^2}{a}+1} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{d \sqrt [4]{a+b x^2}}+\frac {2 b x}{d \sqrt [4]{a+b x^2}} \]
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Rule 202
Rule 233
Rule 235
Rule 408
Rule 411
Rule 504
Rule 1232
Rubi steps \begin{align*} \text {integral}& = \frac {b \int \frac {1}{\sqrt [4]{a+b x^2}} \, dx}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt [4]{a+b x^2} \left (c+d x^2\right )} \, dx}{d} \\ & = -\frac {\left (2 (b c-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 )}{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}{d \sqrt [4]{a+b x^2}} \\ & = \frac {2 b x}{d \sqrt [4]{a+b x^2}}+\frac {\left ((b c-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 )}{d^{3/2} x}-\frac {\left ((b c-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 )}{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}{d \sqrt [4]{a+b x^2}} \\ & = \frac {2 b x}{d \sqrt [4]{a+b x^2}}-\frac {2 \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 )}{d \sqrt [4]{a+b x^2}}+\frac {\sqrt [4]{a} \sqrt {-b c+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 )}{d^{3/2} x}-\frac {\sqrt [4]{a} \sqrt {-b c+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 )}{d^{3/2} x} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 9.24 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.66 \[ \int \frac {\left (a+b x^2\right )^{3/4}}{c+d x^2} \, dx=\frac {6 a c x \left (a+b x^2\right )^{3/4} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {3}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{\left (c+d x^2\right ) \left (6 a c \operatorname {AppellF1}\left (\frac {1}{2},-\frac {3}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+x^2 \left (-4 a d \operatorname {AppellF1}\left (\frac {3}{2},-\frac {3}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+3 b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )} \]
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\[\int \frac {\left (b \,x^{2}+a \right )^{\frac {3}{4}}}{d \,x^{2}+c}d x\]
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Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/4}}{c+d x^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (a+b x^2\right )^{3/4}}{c+d x^2} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {3}{4}}}{c + d x^{2}}\, dx \]
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\[ \int \frac {\left (a+b x^2\right )^{3/4}}{c+d x^2} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}}}{d x^{2} + c} \,d x } \]
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\[ \int \frac {\left (a+b x^2\right )^{3/4}}{c+d x^2} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}}}{d x^{2} + c} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/4}}{c+d x^2} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/4}}{d\,x^2+c} \,d x \]
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