Integrand size = 15, antiderivative size = 262 \[ \int \frac {1}{\left (a+\frac {b}{x^4}\right )^{5/2} x^4} \, dx=-\frac {1}{6 a \left (a+\frac {b}{x^4}\right )^{3/2} x^3}-\frac {1}{4 a^2 \sqrt {a+\frac {b}{x^4}} x^3}+\frac {\sqrt {a+\frac {b}{x^4}}}{4 a^2 \sqrt {b} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) x}-\frac {\sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{4 a^{7/4} b^{3/4} \sqrt {a+\frac {b}{x^4}}}+\frac {\sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) \operatorname {EllipticF}\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{8 a^{7/4} b^{3/4} \sqrt {a+\frac {b}{x^4}}} \]
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Time = 0.11 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {342, 296, 311, 226, 1210} \[ \int \frac {1}{\left (a+\frac {b}{x^4}\right )^{5/2} x^4} \, dx=\frac {\sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) \operatorname {EllipticF}\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{8 a^{7/4} b^{3/4} \sqrt {a+\frac {b}{x^4}}}-\frac {\sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{4 a^{7/4} b^{3/4} \sqrt {a+\frac {b}{x^4}}}-\frac {1}{4 a^2 x^3 \sqrt {a+\frac {b}{x^4}}}+\frac {\sqrt {a+\frac {b}{x^4}}}{4 a^2 \sqrt {b} x \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )}-\frac {1}{6 a x^3 \left (a+\frac {b}{x^4}\right )^{3/2}} \]
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Rule 226
Rule 296
Rule 311
Rule 342
Rule 1210
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {x^2}{\left (a+b x^4\right )^{5/2}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {1}{6 a \left (a+\frac {b}{x^4}\right )^{3/2} x^3}-\frac {\text {Subst}\left (\int \frac {x^2}{\left (a+b x^4\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{2 a} \\ & = -\frac {1}{6 a \left (a+\frac {b}{x^4}\right )^{3/2} x^3}-\frac {1}{4 a^2 \sqrt {a+\frac {b}{x^4}} x^3}+\frac {\text {Subst}\left (\int \frac {x^2}{\sqrt {a+b x^4}} \, dx,x,\frac {1}{x}\right )}{4 a^2} \\ & = -\frac {1}{6 a \left (a+\frac {b}{x^4}\right )^{3/2} x^3}-\frac {1}{4 a^2 \sqrt {a+\frac {b}{x^4}} x^3}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {a+b x^4}} \, dx,x,\frac {1}{x}\right )}{4 a^{3/2} \sqrt {b}}-\frac {\text {Subst}\left (\int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx,x,\frac {1}{x}\right )}{4 a^{3/2} \sqrt {b}} \\ & = -\frac {1}{6 a \left (a+\frac {b}{x^4}\right )^{3/2} x^3}-\frac {1}{4 a^2 \sqrt {a+\frac {b}{x^4}} x^3}+\frac {\sqrt {a+\frac {b}{x^4}}}{4 a^2 \sqrt {b} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) x}-\frac {\sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{4 a^{7/4} b^{3/4} \sqrt {a+\frac {b}{x^4}}}+\frac {\sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) F\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{8 a^{7/4} b^{3/4} \sqrt {a+\frac {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 = 77, normalized size of antiderivative = 0.29 \[ \int \frac {1}{\left (a+\frac {b}{x^4}\right )^{5/2} x^4} \, dx=\frac {-b x+x \left (b+a x^4\right ) \sqrt {1+\frac {a x^4}{b}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {5}{2},\frac {7}{4},-\frac {a x^4}{b}\right )}{3 a b \sqrt {a+\frac {b}{x^4}} \left (b+a x^4\right )} \]
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Result contains complex when optimal does not.
Time = 0.43 (sec) , antiderivative size = 484, normalized size of antiderivative = 1.85
method | result | size |
default | \(\frac {3 \sqrt {b}\, a^{\frac {7}{2}} \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, x^{11}+4 b^{\frac {3}{2}} a^{\frac {5}{2}} \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, x^{7}-3 i \sqrt {\frac {-i \sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}}\, \sqrt {\frac {i \sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}}\, F\left (x \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}, i\right ) a^{3} b \,x^{8}+3 i \sqrt {\frac {-i \sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}}\, \sqrt {\frac {i \sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}}\, E\left (x \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}, i\right ) a^{3} b \,x^{8}+b^{\frac {5}{2}} a^{\frac {3}{2}} \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, x^{3}-6 i \sqrt {\frac {-i \sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}}\, \sqrt {\frac {i \sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}}\, F\left (x \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}, i\right ) a^{2} b^{2} x^{4}+6 i \sqrt {\frac {-i \sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}}\, \sqrt {\frac {i \sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}}\, E\left (x \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}, i\right ) a^{2} b^{2} x^{4}-3 i \sqrt {\frac {-i \sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}}\, \sqrt {\frac {i \sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}}\, F\left (x \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}, i\right ) a \,b^{3}+3 i \sqrt {\frac {-i \sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}}\, \sqrt {\frac {i \sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}}\, E\left (x \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}, i\right ) a \,b^{3}}{12 a^{\frac {5}{2}} \left (\frac {a \,x^{4}+b}{x^{4}}\right )^{\frac {5}{2}} x^{10} b^{\frac {3}{2}} \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}}\) | \(484\) |
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none
Time = 0.08 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.57 \[ \int \frac {1}{\left (a+\frac {b}{x^4}\right )^{5/2} x^4} \, dx=\frac {3 \, {\left (a^{2} x^{8} + 2 \, a b x^{4} + b^{2}\right )} \sqrt {b} \left (-\frac {a}{b}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )\,|\,-1) - 3 \, {\left (a^{2} x^{8} + 2 \, a b x^{4} + b^{2}\right )} \sqrt {b} \left (-\frac {a}{b}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )\,|\,-1) + {\left (3 \, a^{2} x^{9} + a b x^{5}\right )} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{12 \, {\left (a^{4} b x^{8} + 2 \, a^{3} b^{2} x^{4} + a^{2} b^{3}\right )}} \]
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Result contains complex when optimal does not.
Time = 0.96 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.15 \[ \int \frac {1}{\left (a+\frac {b}{x^4}\right )^{5/2} x^4} \, dx=- \frac {\Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {5}{2} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{4}}} \right )}}{4 a^{\frac {5}{2}} x^{3} \Gamma \left (\frac {7}{4}\right )} \]
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\[ \int \frac {1}{\left (a+\frac {b}{x^4}\right )^{5/2} x^4} \, dx=\int { \frac {1}{{\left (a + \frac {b}{x^{4}}\right )}^{\frac {5}{2}} x^{4}} \,d x } \]
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\[ \int \frac {1}{\left (a+\frac {b}{x^4}\right )^{5/2} x^4} \, dx=\int { \frac {1}{{\left (a + \frac {b}{x^{4}}\right )}^{\frac {5}{2}} x^{4}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (a+\frac {b}{x^4}\right )^{5/2} x^4} \, dx=\int \frac {1}{x^4\,{\left (a+\frac {b}{x^4}\right )}^{5/2}} \,d x \]
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