Integrand size = 11, antiderivative size = 127 \[ \int \frac {1}{\left (a+b x^4\right )^{5/2}} \, dx=\frac {x}{6 a \left (a+b x^4\right )^{3/2}}+\frac {5 x}{12 a^2 \sqrt {a+b x^4}}+\frac {5 \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{24 a^{9/4} \sqrt [4]{b} \sqrt {a+b x^4}} \]
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Time = 0.02 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {205, 226} \[ \int \frac {1}{\left (a+b x^4\right )^{5/2}} \, dx=\frac {5 \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{24 a^{9/4} \sqrt [4]{b} \sqrt {a+b x^4}}+\frac {5 x}{12 a^2 \sqrt {a+b x^4}}+\frac {x}{6 a \left (a+b x^4\right )^{3/2}} \]
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Rule 205
Rule 226
Rubi steps \begin{align*} \text {integral}& = \frac {x}{6 a \left (a+b x^4\right )^{3/2}}+\frac {5 \int \frac {1}{\left (a+b x^4\right )^{3/2}} \, dx}{6 a} \\ & = \frac {x}{6 a \left (a+b x^4\right )^{3/2}}+\frac {5 x}{12 a^2 \sqrt {a+b x^4}}+\frac {5 \int \frac {1}{\sqrt {a+b x^4}} \, dx}{12 a^2} \\ & = \frac {x}{6 a \left (a+b x^4\right )^{3/2}}+\frac {5 x}{12 a^2 \sqrt {a+b x^4}}+\frac {5 \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{24 a^{9/4} \sqrt [4]{b} \sqrt {a+b x^4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.04 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.57 \[ \int \frac {1}{\left (a+b x^4\right )^{5/2}} \, dx=\frac {7 a x+5 b x^5+5 x \left (a+b x^4\right ) \sqrt {1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {b x^4}{a}\right )}{12 a^2 \left (a+b x^4\right )^{3/2}} \]
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Result contains complex when optimal does not.
Time = 4.08 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.97
method | result | size |
default | \(\frac {x \sqrt {b \,x^{4}+a}}{6 a \,b^{2} \left (x^{4}+\frac {a}{b}\right )^{2}}+\frac {5 x}{12 a^{2} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {5 \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{12 a^{2} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\) | \(123\) |
elliptic | \(\frac {x \sqrt {b \,x^{4}+a}}{6 a \,b^{2} \left (x^{4}+\frac {a}{b}\right )^{2}}+\frac {5 x}{12 a^{2} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {5 \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{12 a^{2} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\) | \(123\) |
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none
Time = 0.10 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\left (a+b x^4\right )^{5/2}} \, dx=-\frac {5 \, {\left (b^{2} x^{8} + 2 \, a b x^{4} + a^{2}\right )} \sqrt {a} \left (-\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - {\left (5 \, b^{2} x^{5} + 7 \, a b x\right )} \sqrt {b x^{4} + a}}{12 \, {\left (a^{2} b^{3} x^{8} + 2 \, a^{3} b^{2} x^{4} + a^{4} b\right )}} \]
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Result contains complex when optimal does not.
Time = 0.54 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.28 \[ \int \frac {1}{\left (a+b x^4\right )^{5/2}} \, dx=\frac {x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {5}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right )} \]
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\[ \int \frac {1}{\left (a+b x^4\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {1}{\left (a+b x^4\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {5}{2}}} \,d x } \]
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Time = 5.51 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.29 \[ \int \frac {1}{\left (a+b x^4\right )^{5/2}} \, dx=\frac {x\,{\left (\frac {b\,x^4}{a}+1\right )}^{5/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {5}{2};\ \frac {5}{4};\ -\frac {b\,x^4}{a}\right )}{{\left (b\,x^4+a\right )}^{5/2}} \]
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