Integrand size = 15, antiderivative size = 83 \[ \int \frac {x^4}{\left (a+b x^4\right )^{3/4}} \, dx=\frac {x \sqrt [4]{a+b x^4}}{2 b}+\frac {\sqrt {a} \left (1+\frac {a}{b x^4}\right )^{3/4} x^3 \operatorname {EllipticF}\left (\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{2 \sqrt {b} \left (a+b x^4\right )^{3/4}} \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {327, 243, 342, 281, 237} \[ \int \frac {x^4}{\left (a+b x^4\right )^{3/4}} \, dx=\frac {x \sqrt [4]{a+b x^4}}{2 b}+\frac {\sqrt {a} x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{2 \sqrt {b} \left (a+b x^4\right )^{3/4}} \]
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Rule 237
Rule 243
Rule 281
Rule 327
Rule 342
Rubi steps \begin{align*} \text {integral}& = \frac {x \sqrt [4]{a+b x^4}}{2 b}-\frac {a \int \frac {1}{\left (a+b x^4\right )^{3/4}} \, dx}{2 b} \\ & = \frac {x \sqrt [4]{a+b x^4}}{2 b}-\frac {\left (a \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}{2 b \left (a+b x^4\right )^{3/4}} \\ & = \frac {x \sqrt [4]{a+b x^4}}{2 b}+\frac {\left (a \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 )}{2 b \left (a+b x^4\right )^{3/4}} \\ & = \frac {x \sqrt [4]{a+b x^4}}{2 b}+\frac {\left (a \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 )}{4 b \left (a+b x^4\right )^{3/4}} \\ & = \frac {x \sqrt [4]{a+b x^4}}{2 b}+\frac {\sqrt {a} \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 )}{2 \sqrt {b} \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 = 7.52 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.75 \[ \int \frac {x^4}{\left (a+b x^4\right )^{3/4}} \, dx=\frac {x \left (a+b x^4-a \left (1+\frac {b x^4}{a}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{4},\frac {5}{4},-\frac {b x^4}{a}\right )\right )}{2 b \left (a+b x^4\right )^{3/4}} \]
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\[\int \frac {x^{4}}{\left (b \,x^{4}+a \right )^{\frac {3}{4}}}d x\]
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\[ \int \frac {x^4}{\left (a+b x^4\right )^{3/4}} \, dx=\int { \frac {x^{4}}{{\left (b x^{4} + a\right )}^{\frac {3}{4}}} \,d x } \]
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Result contains complex when optimal does not.
Time = 0.46 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.45 \[ \int \frac {x^4}{\left (a+b x^4\right )^{3/4}} \, dx=\frac {x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{4}} \Gamma \left (\frac {9}{4}\right )} \]
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\[ \int \frac {x^4}{\left (a+b x^4\right )^{3/4}} \, dx=\int { \frac {x^{4}}{{\left (b x^{4} + a\right )}^{\frac {3}{4}}} \,d x } \]
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\[ \int \frac {x^4}{\left (a+b x^4\right )^{3/4}} \, dx=\int { \frac {x^{4}}{{\left (b x^{4} + a\right )}^{\frac {3}{4}}} \,d x } \]
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Timed out. \[ \int \frac {x^4}{\left (a+b x^4\right )^{3/4}} \, dx=\int \frac {x^4}{{\left (b\,x^4+a\right )}^{3/4}} \,d x \]
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