Optimal. Leaf size=28 \[ \frac {x^n}{c n}-\frac {b \log \left (b+c x^n\right )}{c^2 n} \]
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Rubi [A] time = 0.03, antiderivative size = 28, 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} \[ \frac {x^n}{c n}-\frac {b \log \left (b+c x^n\right )}{c^2 n} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^{-1+3 n}}{b x^n+c x^{2 n}} \, dx &=\int \frac {x^{-1+2 n}}{b+c x^n} \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {x}{b+c x} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{c}-\frac {b}{c (b+c x)}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {x^n}{c n}-\frac {b \log \left (b+c x^n\right )}{c^2 n}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 0.93 \[ \frac {\frac {x^n}{c}-\frac {b \log \left (b+c x^n\right )}{c^2}}{n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 24, normalized size = 0.86 \[ \frac {c x^{n} - b \log \left (c x^{n} + b\right )}{c^{2} n} \]
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 \[ \int \frac {x^{3 \, n - 1}}{c x^{2 \, n} + b x^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 33, normalized size = 1.18 \[ -\frac {b \ln \left (c \,{\mathrm e}^{n \ln \relax (x )}+b \right )}{c^{2} n}+\frac {{\mathrm e}^{n \ln \relax (x )}}{c n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.87, size = 32, normalized size = 1.14 \[ \frac {x^{n}}{c n} - \frac {b \log \left (\frac {c x^{n} + b}{c}\right )}{c^{2} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {x^{3\,n-1}}{b\,x^n+c\,x^{2\,n}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 13.89, size = 26, normalized size = 0.93 \[ - \frac {b \left (\begin {cases} \frac {x^{n}}{b} & \text {for}\: c = 0 \\\frac {\log {\left (b + c x^{n} \right )}}{c} & \text {otherwise} \end {cases}\right )}{c n} + \frac {x^{n}}{c n} \]
Verification of antiderivative is not currently implemented for this CAS.
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