Optimal. Leaf size=46 \[ \frac {b^2 \log \left (b+c x^n\right )}{c^3 n}-\frac {b x^n}{c^2 n}+\frac {x^{2 n}}{2 c n} \]
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Rubi [A] time = 0.04, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1584, 266, 43} \[ \frac {b^2 \log \left (b+c x^n\right )}{c^3 n}-\frac {b x^n}{c^2 n}+\frac {x^{2 n}}{2 c n} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^{-1+4 n}}{b x^n+c x^{2 n}} \, dx &=\int \frac {x^{-1+3 n}}{b+c x^n} \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{b+c x} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {b}{c^2}+\frac {x}{c}+\frac {b^2}{c^2 (b+c x)}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac {b x^n}{c^2 n}+\frac {x^{2 n}}{2 c n}+\frac {b^2 \log \left (b+c x^n\right )}{c^3 n}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 38, normalized size = 0.83 \[ \frac {2 b^2 \log \left (b+c x^n\right )+c x^n \left (c x^n-2 b\right )}{2 c^3 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.06, size = 38, normalized size = 0.83 \[ \frac {c^{2} x^{2 \, n} - 2 \, b c x^{n} + 2 \, b^{2} \log \left (c x^{n} + b\right )}{2 \, c^{3} n} \]
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 \[ \int \frac {x^{4 \, n - 1}}{c x^{2 \, n} + b x^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 62, normalized size = 1.35 \[ \left (-\frac {b \,{\mathrm e}^{2 n \ln \relax (x )}}{c^{2} n}+\frac {{\mathrm e}^{3 n \ln \relax (x )}}{2 c n}\right ) {\mathrm e}^{-n \ln \relax (x )}+\frac {b^{2} \ln \left (c \,{\mathrm e}^{n \ln \relax (x )}+b \right )}{c^{3} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.93, size = 45, normalized size = 0.98 \[ \frac {b^{2} \log \left (\frac {c x^{n} + b}{c}\right )}{c^{3} n} + \frac {c x^{2 \, n} - 2 \, b x^{n}}{2 \, c^{2} n} \]
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 \[ \int \frac {x^{4\,n-1}}{b\,x^n+c\,x^{2\,n}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 19.83, size = 42, normalized size = 0.91 \[ \frac {b^{2} \left (\begin {cases} \frac {x^{n}}{b} & \text {for}\: c = 0 \\\frac {\log {\left (b + c x^{n} \right )}}{c} & \text {otherwise} \end {cases}\right )}{c^{2} n} - \frac {b x^{n}}{c^{2} n} + \frac {x^{2 n}}{2 c n} \]
Verification of antiderivative is not currently implemented for this CAS.
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