Optimal. Leaf size=16 \[ b \log (x)-\frac {a x^{-n}}{n} \]
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Rubi [A] time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {14} \[ b \log (x)-\frac {a x^{-n}}{n} \]
Antiderivative was successfully verified.
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Rule 14
Rubi steps
\begin {align*} \int x^{-1-n} \left (a+b x^n\right ) \, dx &=\int \left (\frac {b}{x}+a x^{-1-n}\right ) \, dx\\ &=-\frac {a x^{-n}}{n}+b \log (x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 16, normalized size = 1.00 \[ b \log (x)-\frac {a x^{-n}}{n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 21, normalized size = 1.31 \[ \frac {b n x^{n} \log \relax (x) - a}{n x^{n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 21, normalized size = 1.31 \[ \frac {b n x^{n} \log \relax (x) - a}{n x^{n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 25, normalized size = 1.56 \[ \left (b \,{\mathrm e}^{n \ln \relax (x )} \ln \relax (x )-\frac {a}{n}\right ) {\mathrm e}^{-n \ln \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 16, normalized size = 1.00 \[ b \log \relax (x) - \frac {a}{n x^{n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.41, size = 29, normalized size = 1.81 \[ \left \{\begin {array}{cl} \ln \relax (x)\,\left (a+b\right ) & \text {\ if\ \ }n=0\\ b\,\ln \relax (x)-\frac {a}{n\,x^n} & \text {\ if\ \ }n\neq 0 \end {array}\right . \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 12.86, size = 107, normalized size = 6.69 \[ \begin {cases} a x + b \log {\relax (x )} & \text {for}\: n = -1 \\\left (a + b\right ) \log {\relax (x )} & \text {for}\: n = 0 \\- \frac {a n}{n^{2} x^{n} + n x^{n}} - \frac {a}{n^{2} x^{n} + n x^{n}} + \frac {b n^{2} x^{n} \log {\relax (x )}}{n^{2} x^{n} + n x^{n}} + \frac {b n x^{n} \log {\relax (x )}}{n^{2} x^{n} + n x^{n}} + \frac {b n x^{n}}{n^{2} x^{n} + n x^{n}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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