Optimal. Leaf size=28 \[ \frac {x^n}{b n}-\frac {a \log \left (a+b x^n\right )}{b^2 n} \]
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Rubi [A] time = 0.02, antiderivative size = 28, 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} \[ \frac {x^n}{b n}-\frac {a \log \left (a+b x^n\right )}{b^2 n} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^{-1+2 n}}{a+b x^n} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x}{a+b x} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{b}-\frac {a}{b (a+b x)}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {x^n}{b n}-\frac {a \log \left (a+b x^n\right )}{b^2 n}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 26, normalized size = 0.93 \[ \frac {\frac {x^n}{b}-\frac {a \log \left (a+b x^n\right )}{b^2}}{n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 24, normalized size = 0.86 \[ \frac {b x^{n} - a \log \left (b x^{n} + a\right )}{b^{2} n} \]
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 \[ \int \frac {x^{2 \, n - 1}}{b x^{n} + a}\,{d x} \]
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 \[ -\frac {a \ln \left (b \,{\mathrm e}^{n \ln \relax (x )}+a \right )}{b^{2} n}+\frac {{\mathrm e}^{n \ln \relax (x )}}{b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 32, normalized size = 1.14 \[ \frac {x^{n}}{b n} - \frac {a \log \left (\frac {b x^{n} + a}{b}\right )}{b^{2} n} \]
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 \[ \int \frac {x^{2\,n-1}}{a+b\,x^n} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 12.23, size = 41, normalized size = 1.46 \[ \begin {cases} \frac {\log {\relax (x )}}{a} & \text {for}\: b = 0 \wedge n = 0 \\\frac {\log {\relax (x )}}{a + b} & \text {for}\: n = 0 \\\frac {x^{2 n}}{2 a n} & \text {for}\: b = 0 \\- \frac {a \log {\left (\frac {a}{b} + x^{n} \right )}}{b^{2} n} + \frac {x^{n}}{b n} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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