Optimal. Leaf size=18 \[ x \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-b x^4\right ) \]
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Rubi [A] time = 0.0042675, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {245} \[ x \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-b x^4\right ) \]
Antiderivative was successfully verified.
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Rule 245
Rubi steps
\begin{align*} \int \left (1+b x^4\right )^p \, dx &=x \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-b x^4\right )\\ \end{align*}
Mathematica [A] time = 0.0016625, size = 18, normalized size = 1. \[ x \, _2F_1\left (\frac{1}{4},-p;\frac{5}{4};-b x^4\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 17, normalized size = 0.9 \begin{align*} x{\mbox{$_2$F$_1$}({\frac{1}{4}},-p;\,{\frac{5}{4}};\,-b{x}^{4})} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{4} + 1\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b x^{4} + 1\right )}^{p}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 9.93296, size = 29, normalized size = 1.61 \begin{align*} \frac{x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, - p \\ \frac{5}{4} \end{matrix}\middle |{b x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{4} + 1\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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