Optimal. Leaf size=41 \[ -\frac{\left (a+b x^4\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b x^4}{a}+1\right )}{4 a (p+1)} \]
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Rubi [A] time = 0.0199358, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 65} \[ -\frac{\left (a+b x^4\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b x^4}{a}+1\right )}{4 a (p+1)} \]
Antiderivative was successfully verified.
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Rule 266
Rule 65
Rubi steps
\begin{align*} \int \frac{\left (a+b x^4\right )^p}{x} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{(a+b x)^p}{x} \, dx,x,x^4\right )\\ &=-\frac{\left (a+b x^4\right )^{1+p} \, _2F_1\left (1,1+p;2+p;1+\frac{b x^4}{a}\right )}{4 a (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0111831, size = 41, normalized size = 1. \[ -\frac{\left (a+b x^4\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b x^4}{a}+1\right )}{4 a (p+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.027, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( b{x}^{4}+a \right ) ^{p}}{x}}\, dx \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 \frac{{\left (b x^{4} + a\right )}^{p}}{x}\,{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 (\frac{{\left (b x^{4} + a\right )}^{p}}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 15.6636, size = 39, normalized size = 0.95 \begin{align*} - \frac{b^{p} x^{4 p} \Gamma \left (- p\right ){{}_{2}F_{1}\left (\begin{matrix} - p, - p \\ 1 - p \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{4}}} \right )}}{4 \Gamma \left (1 - p\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 \frac{{\left (b x^{4} + a\right )}^{p}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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