Optimal. Leaf size=26 \[ a^2 \log (x)+\frac {1}{2} a b x^4+\frac {b^2 x^8}{8} \]
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Rubi [A] time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \[ a^2 \log (x)+\frac {1}{2} a b x^4+\frac {b^2 x^8}{8} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+b x^4\right )^2}{x} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {(a+b x)^2}{x} \, dx,x,x^4\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \left (2 a b+\frac {a^2}{x}+b^2 x\right ) \, dx,x,x^4\right )\\ &=\frac {1}{2} a b x^4+\frac {b^2 x^8}{8}+a^2 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 26, normalized size = 1.00 \[ a^2 \log (x)+\frac {1}{2} a b x^4+\frac {b^2 x^8}{8} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 22, normalized size = 0.85 \[ \frac {1}{8} \, b^{2} x^{8} + \frac {1}{2} \, a b x^{4} + a^{2} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 25, normalized size = 0.96 \[ \frac {1}{8} \, b^{2} x^{8} + \frac {1}{2} \, a b x^{4} + \frac {1}{4} \, a^{2} \log \left (x^{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 23, normalized size = 0.88 \[ \frac {b^{2} x^{8}}{8}+\frac {a b \,x^{4}}{2}+a^{2} \ln \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 25, normalized size = 0.96 \[ \frac {1}{8} \, b^{2} x^{8} + \frac {1}{2} \, a b x^{4} + \frac {1}{4} \, a^{2} \log \left (x^{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.96, size = 22, normalized size = 0.85 \[ a^2\,\ln \relax (x)+\frac {b^2\,x^8}{8}+\frac {a\,b\,x^4}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 22, normalized size = 0.85 \[ a^{2} \log {\relax (x )} + \frac {a b x^{4}}{2} + \frac {b^{2} x^{8}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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