Integrand size = 21, antiderivative size = 302 \[ \int \frac {\left (a+b x^2\right )^{5/4}}{c+d x^2} \, dx=\frac {2 b x \sqrt [4]{a+b x^2}}{3 d}+\frac {2 a^{3/2} \sqrt {b} \left (1+\frac {b x^2}{a}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{3 d \left (a+b x^2\right )^{3/4}}-\frac {2 \sqrt {a} \sqrt {b} (b c-a d) \left (1+\frac {b x^2}{a}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{d^2 \left (a+b x^2\right )^{3/4}}+\frac {\sqrt [4]{a} (b c-a d) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{d^2 x}+\frac {\sqrt [4]{a} (b c-a d) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{d^2 x} \]
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Time = 0.16 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {411, 201, 239, 237, 410, 109, 418, 1232} \[ \int \frac {\left (a+b x^2\right )^{5/4}}{c+d x^2} \, dx=\frac {2 a^{3/2} \sqrt {b} \left (\frac {b x^2}{a}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{3 d \left (a+b x^2\right )^{3/4}}+\frac {\sqrt [4]{a} \sqrt {-\frac {b x^2}{a}} (b c-a d) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}},\arcsin \left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right ),-1\right )}{d^2 x}+\frac {\sqrt [4]{a} \sqrt {-\frac {b x^2}{a}} (b c-a d) \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {a d-b c}},\arcsin \left (\frac {\sqrt [4]{b x^2+a}}{\sqrt [4]{a}}\right ),-1\right )}{d^2 x}-\frac {2 \sqrt {a} \sqrt {b} \left (\frac {b x^2}{a}+1\right )^{3/4} (b c-a d) \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{d^2 \left (a+b x^2\right )^{3/4}}+\frac {2 b x \sqrt [4]{a+b x^2}}{3 d} \]
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Rule 109
Rule 201
Rule 237
Rule 239
Rule 410
Rule 411
Rule 418
Rule 1232
Rubi steps \begin{align*} \text {integral}& = \frac {b \int \sqrt [4]{a+b x^2} \, dx}{d}-\frac {(b c-a d) \int \frac {\sqrt [4]{a+b x^2}}{c+d x^2} \, dx}{d} \\ & = \frac {2 b x \sqrt [4]{a+b x^2}}{3 d}+\frac {(a b) \int \frac {1}{\left (a+b x^2\right )^{3/4}} \, dx}{3 d}-\frac {(b (b c-a d)) \int \frac {1}{\left (a+b x^2\right )^{3/4}} \, dx}{d^2}+\frac {(b c-a d)^2 \int \frac {1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )} \, dx}{d^2} \\ & = \frac {2 b x \sqrt [4]{a+b x^2}}{3 d}+\frac {\left ((b c-a d)^2 \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 )}{2 d^2 x}+\frac {\left (a 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}{3 d \left (a+b x^2\right )^{3/4}}-\frac {\left (b (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}{d^2 \left (a+b x^2\right )^{3/4}} \\ & = \frac {2 b x \sqrt [4]{a+b x^2}}{3 d}+\frac {2 a^{3/2} \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 )}{3 d \left (a+b x^2\right )^{3/4}}-\frac {2 \sqrt {a} \sqrt {b} (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 )}{d^2 \left (a+b x^2\right )^{3/4}}-\frac {\left (2 (b c-a d)^2 \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 )}{d^2 x} \\ & = \frac {2 b x \sqrt [4]{a+b x^2}}{3 d}+\frac {2 a^{3/2} \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 )}{3 d \left (a+b x^2\right )^{3/4}}-\frac {2 \sqrt {a} \sqrt {b} (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 )}{d^2 \left (a+b x^2\right )^{3/4}}+\frac {\left ((b c-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 )}{d^2 x}+\frac {\left ((b c-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 )}{d^2 x} \\ & = \frac {2 b x \sqrt [4]{a+b x^2}}{3 d}+\frac {2 a^{3/2} \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 )}{3 d \left (a+b x^2\right )^{3/4}}-\frac {2 \sqrt {a} \sqrt {b} (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 )}{d^2 \left (a+b x^2\right )^{3/4}}+\frac {\sqrt [4]{a} (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^2 x}+\frac {\sqrt [4]{a} (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^2 x} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.19 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.15 \[ \int \frac {\left (a+b x^2\right )^{5/4}}{c+d x^2} \, dx=\frac {x \left (\frac {b (-3 b c+4 a d) x^2 \left (1+\frac {b x^2}{a}\right )^{3/4} \operatorname {AppellF1}\left (\frac {3}{2},\frac {3}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{c}+\frac {6 \left (-3 a c \left (3 a^2 d+2 a b d x^2+2 b^2 x^2 \left (c+d x^2\right )\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {3}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b x^2 \left (a+b x^2\right ) \left (c+d x^2\right ) \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {3}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+3 b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {7}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}{\left (c+d x^2\right ) \left (-6 a c \operatorname {AppellF1}\left (\frac {1}{2},\frac {3}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+x^2 \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {3}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+3 b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {7}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}\right )}{9 d \left (a+b x^2\right )^{3/4}} \]
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\[\int \frac {\left (b \,x^{2}+a \right )^{\frac {5}{4}}}{d \,x^{2}+c}d x\]
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Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/4}}{c+d x^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (a+b x^2\right )^{5/4}}{c+d x^2} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {5}{4}}}{c + d x^{2}}\, dx \]
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\[ \int \frac {\left (a+b x^2\right )^{5/4}}{c+d x^2} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{4}}}{d x^{2} + c} \,d x } \]
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\[ \int \frac {\left (a+b x^2\right )^{5/4}}{c+d x^2} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{4}}}{d x^{2} + c} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/4}}{c+d x^2} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/4}}{d\,x^2+c} \,d x \]
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