Integrand size = 21, antiderivative size = 149 \[ \int \frac {x^{9/2}}{\sqrt {b x^2+c x^4}} \, dx=-\frac {10 b \sqrt {b x^2+c x^4}}{21 c^2 \sqrt {x}}+\frac {2 x^{3/2} \sqrt {b x^2+c x^4}}{7 c}+\frac {5 b^{7/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{21 c^{9/4} \sqrt {b x^2+c x^4}} \]
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Time = 0.12 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2049, 2057, 335, 226} \[ \int \frac {x^{9/2}}{\sqrt {b x^2+c x^4}} \, dx=\frac {5 b^{7/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{21 c^{9/4} \sqrt {b x^2+c x^4}}-\frac {10 b \sqrt {b x^2+c x^4}}{21 c^2 \sqrt {x}}+\frac {2 x^{3/2} \sqrt {b x^2+c x^4}}{7 c} \]
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Rule 226
Rule 335
Rule 2049
Rule 2057
Rubi steps \begin{align*} \text {integral}& = \frac {2 x^{3/2} \sqrt {b x^2+c x^4}}{7 c}-\frac {(5 b) \int \frac {x^{5/2}}{\sqrt {b x^2+c x^4}} \, dx}{7 c} \\ & = -\frac {10 b \sqrt {b x^2+c x^4}}{21 c^2 \sqrt {x}}+\frac {2 x^{3/2} \sqrt {b x^2+c x^4}}{7 c}+\frac {\left (5 b^2\right ) \int \frac {\sqrt {x}}{\sqrt {b x^2+c x^4}} \, dx}{21 c^2} \\ & = -\frac {10 b \sqrt {b x^2+c x^4}}{21 c^2 \sqrt {x}}+\frac {2 x^{3/2} \sqrt {b x^2+c x^4}}{7 c}+\frac {\left (5 b^2 x \sqrt {b+c x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x^2}} \, dx}{21 c^2 \sqrt {b x^2+c x^4}} \\ & = -\frac {10 b \sqrt {b x^2+c x^4}}{21 c^2 \sqrt {x}}+\frac {2 x^{3/2} \sqrt {b x^2+c x^4}}{7 c}+\frac {\left (10 b^2 x \sqrt {b+c x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{21 c^2 \sqrt {b x^2+c x^4}} \\ & = -\frac {10 b \sqrt {b x^2+c x^4}}{21 c^2 \sqrt {x}}+\frac {2 x^{3/2} \sqrt {b x^2+c x^4}}{7 c}+\frac {5 b^{7/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{21 c^{9/4} \sqrt {b x^2+c x^4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.03 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.58 \[ \int \frac {x^{9/2}}{\sqrt {b x^2+c x^4}} \, dx=\frac {2 x^{3/2} \left (-5 b^2-2 b c x^2+3 c^2 x^4+5 b^2 \sqrt {1+\frac {c x^2}{b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {c x^2}{b}\right )\right )}{21 c^2 \sqrt {x^2 \left (b+c x^2\right )}} \]
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Time = 0.21 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.92
method | result | size |
default | \(\frac {\sqrt {x}\, \left (5 b^{2} \sqrt {-b c}\, \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )+6 c^{3} x^{5}-4 b \,c^{2} x^{3}-10 b^{2} c x \right )}{21 \sqrt {c \,x^{4}+b \,x^{2}}\, c^{3}}\) | \(137\) |
risch | \(-\frac {2 \left (-3 c \,x^{2}+5 b \right ) x^{\frac {3}{2}} \left (c \,x^{2}+b \right )}{21 c^{2} \sqrt {x^{2} \left (c \,x^{2}+b \right )}}+\frac {5 b^{2} \sqrt {-b c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {x}\, \sqrt {x \left (c \,x^{2}+b \right )}}{21 c^{3} \sqrt {c \,x^{3}+b x}\, \sqrt {x^{2} \left (c \,x^{2}+b \right )}}\) | \(178\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.38 \[ \int \frac {x^{9/2}}{\sqrt {b x^2+c x^4}} \, dx=\frac {2 \, {\left (5 \, b^{2} \sqrt {c} x {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right ) + \sqrt {c x^{4} + b x^{2}} {\left (3 \, c^{2} x^{2} - 5 \, b c\right )} \sqrt {x}\right )}}{21 \, c^{3} x} \]
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\[ \int \frac {x^{9/2}}{\sqrt {b x^2+c x^4}} \, dx=\int \frac {x^{\frac {9}{2}}}{\sqrt {x^{2} \left (b + c x^{2}\right )}}\, dx \]
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\[ \int \frac {x^{9/2}}{\sqrt {b x^2+c x^4}} \, dx=\int { \frac {x^{\frac {9}{2}}}{\sqrt {c x^{4} + b x^{2}}} \,d x } \]
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\[ \int \frac {x^{9/2}}{\sqrt {b x^2+c x^4}} \, dx=\int { \frac {x^{\frac {9}{2}}}{\sqrt {c x^{4} + b x^{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^{9/2}}{\sqrt {b x^2+c x^4}} \, dx=\int \frac {x^{9/2}}{\sqrt {c\,x^4+b\,x^2}} \,d x \]
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