Integrand size = 9, antiderivative size = 44 \[ \int \left (a+b x^4\right )^p \, dx=x \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},-p,\frac {5}{4},-\frac {b x^4}{a}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {252, 251} \[ \int \left (a+b x^4\right )^p \, dx=x \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},-p,\frac {5}{4},-\frac {b x^4}{a}\right ) \]
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Rule 251
Rule 252
Rubi steps \begin{align*} \text {integral}& = \left (\left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p}\right ) \int \left (1+\frac {b x^4}{a}\right )^p \, dx \\ & = x \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \, _2F_1\left (\frac {1}{4},-p;\frac {5}{4};-\frac {b x^4}{a}\right ) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^4\right )^p \, dx=x \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},-p,\frac {5}{4},-\frac {b x^4}{a}\right ) \]
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\[\int \left (b \,x^{4}+a \right )^{p}d x\]
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\[ \int \left (a+b x^4\right )^p \, dx=\int { {\left (b x^{4} + a\right )}^{p} \,d x } \]
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Result contains complex when optimal does not.
Time = 3.66 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.77 \[ \int \left (a+b x^4\right )^p \, dx=\frac {a^{p} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, - p \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} \]
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\[ \int \left (a+b x^4\right )^p \, dx=\int { {\left (b x^{4} + a\right )}^{p} \,d x } \]
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\[ \int \left (a+b x^4\right )^p \, dx=\int { {\left (b x^{4} + a\right )}^{p} \,d x } \]
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Time = 13.65 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.93 \[ \int \left (a+b x^4\right )^p \, dx=\frac {x\,{\left (b\,x^4+a\right )}^p\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},-p;\ \frac {5}{4};\ -\frac {b\,x^4}{a}\right )}{{\left (\frac {b\,x^4}{a}+1\right )}^p} \]
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