Integrand size = 11, antiderivative size = 83 \[ \int \frac {1}{\left (a+b x^4\right )^{7/4}} \, dx=\frac {x}{3 a \left (a+b x^4\right )^{3/4}}-\frac {2 \sqrt {b} \left (1+\frac {a}{b x^4}\right )^{3/4} x^3 \operatorname {EllipticF}\left (\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{3 a^{3/2} \left (a+b x^4\right )^{3/4}} \]
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Time = 0.02 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {205, 243, 342, 281, 237} \[ \int \frac {1}{\left (a+b x^4\right )^{7/4}} \, dx=\frac {x}{3 a \left (a+b x^4\right )^{3/4}}-\frac {2 \sqrt {b} x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{3 a^{3/2} \left (a+b x^4\right )^{3/4}} \]
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Rule 205
Rule 237
Rule 243
Rule 281
Rule 342
Rubi steps \begin{align*} \text {integral}& = \frac {x}{3 a \left (a+b x^4\right )^{3/4}}+\frac {2 \int \frac {1}{\left (a+b x^4\right )^{3/4}} \, dx}{3 a} \\ & = \frac {x}{3 a \left (a+b x^4\right )^{3/4}}+\frac {\left (2 \left (1+\frac {a}{b x^4}\right )^{3/4} x^3\right ) \int \frac {1}{\left (1+\frac {a}{b x^4}\right )^{3/4} x^3} \, dx}{3 a \left (a+b x^4\right )^{3/4}} \\ & = \frac {x}{3 a \left (a+b x^4\right )^{3/4}}-\frac {\left (2 \left (1+\frac {a}{b x^4}\right )^{3/4} x^3\right ) \text {Subst}\left (\int \frac {x}{\left (1+\frac {a x^4}{b}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )}{3 a \left (a+b x^4\right )^{3/4}} \\ & = \frac {x}{3 a \left (a+b x^4\right )^{3/4}}-\frac {\left (\left (1+\frac {a}{b x^4}\right )^{3/4} x^3\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {a x^2}{b}\right )^{3/4}} \, dx,x,\frac {1}{x^2}\right )}{3 a \left (a+b x^4\right )^{3/4}} \\ & = \frac {x}{3 a \left (a+b x^4\right )^{3/4}}-\frac {2 \sqrt {b} \left (1+\frac {a}{b x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{3 a^{3/2} \left (a+b x^4\right )^{3/4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 8.40 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\left (a+b x^4\right )^{7/4}} \, dx=\frac {x+2 x \left (1+\frac {b x^4}{a}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{4},\frac {5}{4},-\frac {b x^4}{a}\right )}{3 a \left (a+b x^4\right )^{3/4}} \]
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\[\int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {7}{4}}}d x\]
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\[ \int \frac {1}{\left (a+b x^4\right )^{7/4}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {7}{4}}} \,d x } \]
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Result contains complex when optimal does not.
Time = 0.53 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.43 \[ \int \frac {1}{\left (a+b x^4\right )^{7/4}} \, dx=\frac {x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {7}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {7}{4}} \Gamma \left (\frac {5}{4}\right )} \]
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\[ \int \frac {1}{\left (a+b x^4\right )^{7/4}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {7}{4}}} \,d x } \]
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\[ \int \frac {1}{\left (a+b x^4\right )^{7/4}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{\frac {7}{4}}} \,d x } \]
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Time = 5.65 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.45 \[ \int \frac {1}{\left (a+b x^4\right )^{7/4}} \, dx=\frac {x\,{\left (\frac {b\,x^4}{a}+1\right )}^{7/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {7}{4};\ \frac {5}{4};\ -\frac {b\,x^4}{a}\right )}{{\left (b\,x^4+a\right )}^{7/4}} \]
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