Optimal. Leaf size=41 \[ -\frac {\left (a+b x^4\right )^{1+p} \, _2F_1\left (1,1+p;2+p;1+\frac {b x^4}{a}\right )}{4 a (1+p)} \]
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Rubi [A]
time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, 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 = {272, 67}
\begin {gather*} -\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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 67
Rule 272
Rubi steps
\begin {align*} \int \frac {\left (a+b x^4\right )^p}{x} \, dx &=\frac {1}{4} \text {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*}
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Mathematica [A]
time = 0.04, size = 41, normalized size = 1.00 \begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{4}+a \right )^{p}}{x}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.38, size = 15, normalized size = 0.37 \begin {gather*} {\rm integral}\left (\frac {{\left (b x^{4} + a\right )}^{p}}{x}, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 2.52, size = 39, normalized size = 0.95 \begin {gather*} - \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (b\,x^4+a\right )}^p}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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