Integrand size = 21, antiderivative size = 119 \[ \int \frac {x^{9/2}}{\left (b x^2+c x^4\right )^{3/2}} \, dx=-\frac {x^{3/2}}{c \sqrt {b x^2+c x^4}}+\frac {x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{b} c^{5/4} \sqrt {b x^2+c x^4}} \]
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Time = 0.09 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2047, 2057, 335, 226} \[ \int \frac {x^{9/2}}{\left (b x^2+c x^4\right )^{3/2}} \, dx=\frac {x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{b} c^{5/4} \sqrt {b x^2+c x^4}}-\frac {x^{3/2}}{c \sqrt {b x^2+c x^4}} \]
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Rule 226
Rule 335
Rule 2047
Rule 2057
Rubi steps \begin{align*} \text {integral}& = -\frac {x^{3/2}}{c \sqrt {b x^2+c x^4}}+\frac {\int \frac {\sqrt {x}}{\sqrt {b x^2+c x^4}} \, dx}{2 c} \\ & = -\frac {x^{3/2}}{c \sqrt {b x^2+c x^4}}+\frac {\left (x \sqrt {b+c x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x^2}} \, dx}{2 c \sqrt {b x^2+c x^4}} \\ & = -\frac {x^{3/2}}{c \sqrt {b x^2+c x^4}}+\frac {\left (x \sqrt {b+c x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b+c x^4}} \, dx,x,\sqrt {x}\right )}{c \sqrt {b x^2+c x^4}} \\ & = -\frac {x^{3/2}}{c \sqrt {b x^2+c x^4}}+\frac {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 )}{2 \sqrt [4]{b} c^{5/4} \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 = 10.02 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.50 \[ \int \frac {x^{9/2}}{\left (b x^2+c x^4\right )^{3/2}} \, dx=\frac {x^{3/2} \left (-1+\sqrt {1+\frac {c x^2}{b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {c x^2}{b}\right )\right )}{c \sqrt {x^2 \left (b+c x^2\right )}} \]
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Time = 0.07 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.01
method | result | size |
default | \(\frac {x^{\frac {5}{2}} \left (c \,x^{2}+b \right ) \left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {x c}{\sqrt {-b c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-b c}-2 c x \right )}{2 \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} c^{2}}\) | \(120\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.50 \[ \int \frac {x^{9/2}}{\left (b x^2+c x^4\right )^{3/2}} \, dx=\frac {{\left (c x^{3} + b x\right )} \sqrt {c} {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right ) - \sqrt {c x^{4} + b x^{2}} c \sqrt {x}}{c^{3} x^{3} + b c^{2} x} \]
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\[ \int \frac {x^{9/2}}{\left (b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {x^{\frac {9}{2}}}{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {x^{9/2}}{\left (b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {x^{\frac {9}{2}}}{{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x^{9/2}}{\left (b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {x^{\frac {9}{2}}}{{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^{9/2}}{\left (b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {x^{9/2}}{{\left (c\,x^4+b\,x^2\right )}^{3/2}} \,d x \]
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