Integrand size = 15, antiderivative size = 99 \[ \int x^2 \left (a+b x^4\right )^{3/4} \, dx=\frac {a x^3}{4 \sqrt [4]{a+b x^4}}+\frac {1}{6} x^3 \left (a+b x^4\right )^{3/4}+\frac {a^{3/2} \sqrt [4]{1+\frac {a}{b x^4}} x E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{4 \sqrt {b} \sqrt [4]{a+b x^4}} \]
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Time = 0.03 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {285, 316, 287, 342, 281, 202} \[ \int x^2 \left (a+b x^4\right )^{3/4} \, dx=\frac {a^{3/2} x \sqrt [4]{\frac {a}{b x^4}+1} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{4 \sqrt {b} \sqrt [4]{a+b x^4}}+\frac {1}{6} x^3 \left (a+b x^4\right )^{3/4}+\frac {a x^3}{4 \sqrt [4]{a+b x^4}} \]
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Rule 202
Rule 281
Rule 285
Rule 287
Rule 316
Rule 342
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} x^3 \left (a+b x^4\right )^{3/4}+\frac {1}{2} a \int \frac {x^2}{\sqrt [4]{a+b x^4}} \, dx \\ & = \frac {a x^3}{4 \sqrt [4]{a+b x^4}}+\frac {1}{6} x^3 \left (a+b x^4\right )^{3/4}-\frac {1}{4} a^2 \int \frac {x^2}{\left (a+b x^4\right )^{5/4}} \, dx \\ & = \frac {a x^3}{4 \sqrt [4]{a+b x^4}}+\frac {1}{6} x^3 \left (a+b x^4\right )^{3/4}-\frac {\left (a^2 \sqrt [4]{1+\frac {a}{b x^4}} x\right ) \int \frac {1}{\left (1+\frac {a}{b x^4}\right )^{5/4} x^3} \, dx}{4 b \sqrt [4]{a+b x^4}} \\ & = \frac {a x^3}{4 \sqrt [4]{a+b x^4}}+\frac {1}{6} x^3 \left (a+b x^4\right )^{3/4}+\frac {\left (a^2 \sqrt [4]{1+\frac {a}{b x^4}} x\right ) \text {Subst}\left (\int \frac {x}{\left (1+\frac {a x^4}{b}\right )^{5/4}} \, dx,x,\frac {1}{x}\right )}{4 b \sqrt [4]{a+b x^4}} \\ & = \frac {a x^3}{4 \sqrt [4]{a+b x^4}}+\frac {1}{6} x^3 \left (a+b x^4\right )^{3/4}+\frac {\left (a^2 \sqrt [4]{1+\frac {a}{b x^4}} x\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {a x^2}{b}\right )^{5/4}} \, dx,x,\frac {1}{x^2}\right )}{8 b \sqrt [4]{a+b x^4}} \\ & = \frac {a x^3}{4 \sqrt [4]{a+b x^4}}+\frac {1}{6} x^3 \left (a+b x^4\right )^{3/4}+\frac {a^{3/2} \sqrt [4]{1+\frac {a}{b x^4}} x E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{4 \sqrt {b} \sqrt [4]{a+b x^4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 8.73 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.52 \[ \int x^2 \left (a+b x^4\right )^{3/4} \, dx=\frac {x^3 \left (a+b x^4\right )^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {3}{4},\frac {7}{4},-\frac {b x^4}{a}\right )}{3 \left (1+\frac {b x^4}{a}\right )^{3/4}} \]
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\[\int x^{2} \left (b \,x^{4}+a \right )^{\frac {3}{4}}d x\]
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\[ \int x^2 \left (a+b x^4\right )^{3/4} \, dx=\int { {\left (b x^{4} + a\right )}^{\frac {3}{4}} x^{2} \,d x } \]
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Result contains complex when optimal does not.
Time = 0.55 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.39 \[ \int x^2 \left (a+b x^4\right )^{3/4} \, dx=\frac {a^{\frac {3}{4}} x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} \]
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\[ \int x^2 \left (a+b x^4\right )^{3/4} \, dx=\int { {\left (b x^{4} + a\right )}^{\frac {3}{4}} x^{2} \,d x } \]
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\[ \int x^2 \left (a+b x^4\right )^{3/4} \, dx=\int { {\left (b x^{4} + a\right )}^{\frac {3}{4}} x^{2} \,d x } \]
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Timed out. \[ \int x^2 \left (a+b x^4\right )^{3/4} \, dx=\int x^2\,{\left (b\,x^4+a\right )}^{3/4} \,d x \]
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