Integrand size = 15, antiderivative size = 77 \[ \int \frac {x^5}{\left (a+b x^4\right )^{5/4}} \, dx=\frac {x^2}{b \sqrt [4]{a+b x^4}}-\frac {2 \sqrt {a} \sqrt [4]{1+\frac {b x^4}{a}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{b^{3/2} \sqrt [4]{a+b x^4}} \]
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Time = 0.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {281, 291, 203, 202} \[ \int \frac {x^5}{\left (a+b x^4\right )^{5/4}} \, dx=\frac {x^2}{b \sqrt [4]{a+b x^4}}-\frac {2 \sqrt {a} \sqrt [4]{\frac {b x^4}{a}+1} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{b^{3/2} \sqrt [4]{a+b x^4}} \]
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Rule 202
Rule 203
Rule 281
Rule 291
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x^2}{\left (a+b x^2\right )^{5/4}} \, dx,x,x^2\right ) \\ & = \frac {x^2}{b \sqrt [4]{a+b x^4}}-\frac {a \text {Subst}\left (\int \frac {1}{\left (a+b x^2\right )^{5/4}} \, dx,x,x^2\right )}{b} \\ & = \frac {x^2}{b \sqrt [4]{a+b x^4}}-\frac {\sqrt [4]{1+\frac {b x^4}{a}} \text {Subst}\left (\int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{5/4}} \, dx,x,x^2\right )}{b \sqrt [4]{a+b x^4}} \\ & = \frac {x^2}{b \sqrt [4]{a+b x^4}}-\frac {2 \sqrt {a} \sqrt [4]{1+\frac {b x^4}{a}} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{b^{3/2} \sqrt [4]{a+b x^4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 8.67 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.70 \[ \int \frac {x^5}{\left (a+b x^4\right )^{5/4}} \, dx=\frac {x^2 \left (-1+\sqrt [4]{1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {3}{2},-\frac {b x^4}{a}\right )\right )}{b \sqrt [4]{a+b x^4}} \]
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\[\int \frac {x^{5}}{\left (b \,x^{4}+a \right )^{\frac {5}{4}}}d x\]
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\[ \int \frac {x^5}{\left (a+b x^4\right )^{5/4}} \, dx=\int { \frac {x^{5}}{{\left (b x^{4} + a\right )}^{\frac {5}{4}}} \,d x } \]
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Result contains complex when optimal does not.
Time = 0.49 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.35 \[ \int \frac {x^5}{\left (a+b x^4\right )^{5/4}} \, dx=\frac {x^{6} {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{6 a^{\frac {5}{4}}} \]
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\[ \int \frac {x^5}{\left (a+b x^4\right )^{5/4}} \, dx=\int { \frac {x^{5}}{{\left (b x^{4} + a\right )}^{\frac {5}{4}}} \,d x } \]
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\[ \int \frac {x^5}{\left (a+b x^4\right )^{5/4}} \, dx=\int { \frac {x^{5}}{{\left (b x^{4} + a\right )}^{\frac {5}{4}}} \,d x } \]
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Timed out. \[ \int \frac {x^5}{\left (a+b x^4\right )^{5/4}} \, dx=\int \frac {x^5}{{\left (b\,x^4+a\right )}^{5/4}} \,d x \]
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