Optimal. Leaf size=25 \[ -\frac {a x^{-2 n}}{2 n}-\frac {b x^{-n}}{n} \]
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Rubi [A]
time = 0.00, antiderivative size = 25, 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}
\begin {gather*} -\frac {a x^{-2 n}}{2 n}-\frac {b x^{-n}}{n} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rubi steps
\begin {align*} \int x^{-1-2 n} \left (a+b x^n\right ) \, dx &=\int \left (a x^{-1-2 n}+b x^{-1-n}\right ) \, dx\\ &=-\frac {a x^{-2 n}}{2 n}-\frac {b x^{-n}}{n}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 22, normalized size = 0.88 \begin {gather*} \frac {x^{-2 n} \left (-a-2 b x^n\right )}{2 n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 24, normalized size = 0.96
method | result | size |
risch | \(-\frac {b \,x^{-n}}{n}-\frac {a \,x^{-2 n}}{2 n}\) | \(24\) |
norman | \(\left (-\frac {a}{2 n}-\frac {b \,{\mathrm e}^{n \ln \left (x \right )}}{n}\right ) {\mathrm e}^{-2 n \ln \left (x \right )}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 25, normalized size = 1.00 \begin {gather*} -\frac {a}{2 \, n x^{2 \, n}} - \frac {b}{n x^{n}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 20, normalized size = 0.80 \begin {gather*} -\frac {2 \, b x^{n} + a}{2 \, n x^{2 \, n}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.62, size = 24, normalized size = 0.96 \begin {gather*} \begin {cases} - \frac {a x^{- 2 n}}{2 n} - \frac {b x^{- n}}{n} & \text {for}\: n \neq 0 \\\left (a + b\right ) \log {\left (x \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.18, size = 20, normalized size = 0.80 \begin {gather*} -\frac {2 \, b x^{n} + a}{2 \, n x^{2 \, n}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.21, size = 20, normalized size = 0.80 \begin {gather*} -\frac {a+2\,b\,x^n}{2\,n\,x^{2\,n}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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