Integrand size = 15, antiderivative size = 257 \[ \int \frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{x^4} \, dx=-\frac {2 a \sqrt {a+\frac {b}{x^4}}}{15 x^3}-\frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{9 x^3}-\frac {4 a^2 \sqrt {a+\frac {b}{x^4}}}{15 \sqrt {b} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) x}+\frac {4 a^{9/4} \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+\frac {b}{x^4}}}-\frac {2 a^{9/4} \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) \operatorname {EllipticF}\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+\frac {b}{x^4}}} \]
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Time = 0.10 (sec) , antiderivative size = 257, 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, 285, 311, 226, 1210} \[ \int \frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{x^4} \, dx=-\frac {2 a^{9/4} \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) \operatorname {EllipticF}\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+\frac {b}{x^4}}}+\frac {4 a^{9/4} \sqrt {\frac {a+\frac {b}{x^4}}{\left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )^2}} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) E\left (2 \cot ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a+\frac {b}{x^4}}}-\frac {4 a^2 \sqrt {a+\frac {b}{x^4}}}{15 \sqrt {b} x \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right )}-\frac {2 a \sqrt {a+\frac {b}{x^4}}}{15 x^3}-\frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{9 x^3} \]
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Rule 226
Rule 285
Rule 311
Rule 342
Rule 1210
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int x^2 \left (a+b x^4\right )^{3/2} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{9 x^3}-\frac {1}{3} (2 a) \text {Subst}\left (\int x^2 \sqrt {a+b x^4} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {2 a \sqrt {a+\frac {b}{x^4}}}{15 x^3}-\frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{9 x^3}-\frac {1}{15} \left (4 a^2\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+b x^4}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {2 a \sqrt {a+\frac {b}{x^4}}}{15 x^3}-\frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{9 x^3}-\frac {\left (4 a^{5/2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^4}} \, dx,x,\frac {1}{x}\right )}{15 \sqrt {b}}+\frac {\left (4 a^{5/2}\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx,x,\frac {1}{x}\right )}{15 \sqrt {b}} \\ & = -\frac {2 a \sqrt {a+\frac {b}{x^4}}}{15 x^3}-\frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{9 x^3}-\frac {4 a^2 \sqrt {a+\frac {b}{x^4}}}{15 \sqrt {b} \left (\sqrt {a}+\frac {\sqrt {b}}{x^2}\right ) x}+\frac {4 a^{9/4} \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 )}{15 b^{3/4} \sqrt {a+\frac {b}{x^4}}}-\frac {2 a^{9/4} \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 )}{15 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 = 10.02 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.20 \[ \int \frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{x^4} \, dx=-\frac {b \sqrt {a+\frac {b}{x^4}} \operatorname {Hypergeometric2F1}\left (-\frac {9}{4},-\frac {3}{2},-\frac {5}{4},-\frac {a x^4}{b}\right )}{9 x^7 \sqrt {1+\frac {a x^4}{b}}} \]
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Result contains complex when optimal does not.
Time = 0.71 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.61
method | result | size |
risch | \(-\frac {\left (12 a^{2} x^{8}+11 a b \,x^{4}+5 b^{2}\right ) \sqrt {\frac {a \,x^{4}+b}{x^{4}}}}{45 x^{7} b}+\frac {4 i a^{\frac {5}{2}} \sqrt {1-\frac {i \sqrt {a}\, x^{2}}{\sqrt {b}}}\, \sqrt {1+\frac {i \sqrt {a}\, x^{2}}{\sqrt {b}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}, i\right )\right ) \sqrt {\frac {a \,x^{4}+b}{x^{4}}}\, x^{2}}{15 \sqrt {b}\, \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, \left (a \,x^{4}+b \right )}\) | \(156\) |
default | \(-\frac {\left (\frac {a \,x^{4}+b}{x^{4}}\right )^{\frac {3}{2}} \left (-12 i a^{\frac {5}{2}} \sqrt {\frac {-i \sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}}\, \sqrt {\frac {i \sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}}\, x^{9} b F\left (x \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}, i\right )+12 i a^{\frac {5}{2}} \sqrt {\frac {-i \sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}}\, \sqrt {\frac {i \sqrt {a}\, x^{2}+\sqrt {b}}{\sqrt {b}}}\, x^{9} b E\left (x \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}, i\right )+12 \sqrt {b}\, \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, a^{3} x^{12}+23 b^{\frac {3}{2}} \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, a^{2} x^{8}+16 b^{\frac {5}{2}} \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\, a \,x^{4}+5 b^{\frac {7}{2}} \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}\right )}{45 x^{3} \left (a \,x^{4}+b \right )^{2} b^{\frac {3}{2}} \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}}\) | \(251\) |
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none
Time = 0.09 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.42 \[ \int \frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{x^4} \, dx=-\frac {12 \, a^{2} \sqrt {b} x^{7} \left (-\frac {a}{b}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )\,|\,-1) - 12 \, a^{2} \sqrt {b} x^{7} \left (-\frac {a}{b}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )\,|\,-1) + {\left (12 \, a^{2} x^{8} + 11 \, a b x^{4} + 5 \, b^{2}\right )} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{45 \, b x^{7}} \]
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Result contains complex when optimal does not.
Time = 0.72 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.16 \[ \int \frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{x^4} \, dx=- \frac {a^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{4}}} \right )}}{4 x^{3} \Gamma \left (\frac {7}{4}\right )} \]
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\[ \int \frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{x^4} \, dx=\int { \frac {{\left (a + \frac {b}{x^{4}}\right )}^{\frac {3}{2}}}{x^{4}} \,d x } \]
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\[ \int \frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{x^4} \, dx=\int { \frac {{\left (a + \frac {b}{x^{4}}\right )}^{\frac {3}{2}}}{x^{4}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+\frac {b}{x^4}\right )^{3/2}}{x^4} \, dx=\int \frac {{\left (a+\frac {b}{x^4}\right )}^{3/2}}{x^4} \,d x \]
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