Optimal. Leaf size=56 \[ -\frac {8 \log \left (b x^n+2\right )}{b^4 n}+\frac {4 x^n}{b^3 n}-\frac {x^{2 n}}{b^2 n}+\frac {x^{3 n}}{3 b n} \]
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Rubi [A] time = 0.03, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {266, 43} \[ \frac {4 x^n}{b^3 n}-\frac {x^{2 n}}{b^2 n}-\frac {8 \log \left (b x^n+2\right )}{b^4 n}+\frac {x^{3 n}}{3 b n} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^{-1+4 n}}{2+b x^n} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^3}{2+b x} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {4}{b^3}-\frac {2 x}{b^2}+\frac {x^2}{b}-\frac {8}{b^3 (2+b x)}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {4 x^n}{b^3 n}-\frac {x^{2 n}}{b^2 n}+\frac {x^{3 n}}{3 b n}-\frac {8 \log \left (2+b x^n\right )}{b^4 n}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 43, normalized size = 0.77 \[ \frac {b x^n \left (b^2 x^{2 n}-3 b x^n+12\right )-24 \log \left (b x^n+2\right )}{3 b^4 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 44, normalized size = 0.79 \[ \frac {b^{3} x^{3 \, n} - 3 \, b^{2} x^{2 \, n} + 12 \, b x^{n} - 24 \, \log \left (b x^{n} + 2\right )}{3 \, b^{4} n} \]
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 \[ \int \frac {x^{4 \, n - 1}}{b x^{n} + 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 63, normalized size = 1.12 \[ \frac {{\mathrm e}^{3 n \ln \relax (x )}}{3 b n}-\frac {{\mathrm e}^{2 n \ln \relax (x )}}{b^{2} n}+\frac {4 \,{\mathrm e}^{n \ln \relax (x )}}{b^{3} n}-\frac {8 \ln \left (b \,{\mathrm e}^{n \ln \relax (x )}+2\right )}{b^{4} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 52, normalized size = 0.93 \[ \frac {b^{2} x^{3 \, n} - 3 \, b x^{2 \, n} + 12 \, x^{n}}{3 \, b^{3} n} - \frac {8 \, \log \left (\frac {b x^{n} + 2}{b}\right )}{b^{4} n} \]
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 \[ \int \frac {x^{4\,n-1}}{b\,x^n+2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 36.36, size = 63, normalized size = 1.12 \[ \begin {cases} \frac {\log {\relax (x )}}{2} & \text {for}\: b = 0 \wedge n = 0 \\\frac {\log {\relax (x )}}{b + 2} & \text {for}\: n = 0 \\\frac {x^{4 n}}{8 n} & \text {for}\: b = 0 \\\frac {x^{3 n}}{3 b n} - \frac {x^{2 n}}{b^{2} n} + \frac {4 x^{n}}{b^{3} n} - \frac {8 \log {\left (x^{n} + \frac {2}{b} \right )}}{b^{4} n} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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