Optimal. Leaf size=23 \[ \frac {\left (a+b x^k\right )^{n+1}}{b k (n+1)} \]
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Rubi [A] time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {261} \[ \frac {\left (a+b x^k\right )^{n+1}}{b k (n+1)} \]
Antiderivative was successfully verified.
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Rule 261
Rubi steps
\begin {align*} \int x^{-1+k} \left (a+b x^k\right )^n \, dx &=\frac {\left (a+b x^k\right )^{1+n}}{b k (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 23, normalized size = 1.00 \[ \frac {\left (a+b x^k\right )^{n+1}}{b k (n+1)} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{-1+k} \left (a+b x^k\right )^n \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.80, size = 27, normalized size = 1.17 \[ \frac {{\left (b x^{k} + a\right )} {\left (b x^{k} + a\right )}^{n}}{b k n + b k} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.98, size = 23, normalized size = 1.00 \[ \frac {{\left (b x^{k} + a\right )}^{n + 1}}{b k {\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 29, normalized size = 1.26
method | result | size |
risch | \(\frac {\left (a +b \,x^{k}\right ) \left (a +b \,x^{k}\right )^{n}}{b \left (1+n \right ) k}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 23, normalized size = 1.00 \[ \frac {{\left (b x^{k} + a\right )}^{n + 1}}{b k {\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.73, size = 23, normalized size = 1.00 \[ \frac {{\left (a+b\,x^k\right )}^{n+1}}{b\,k\,\left (n+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 60.37, size = 75, normalized size = 3.26 \[ \begin {cases} \frac {\log {\relax (x )}}{a} & \text {for}\: b = 0 \wedge k = 0 \wedge n = -1 \\\frac {a^{n} x^{k}}{k} & \text {for}\: b = 0 \\\left (a + b\right )^{n} \log {\relax (x )} & \text {for}\: k = 0 \\\frac {\log {\left (\frac {a}{b} + x^{k} \right )}}{b k} & \text {for}\: n = -1 \\\frac {a \left (a + b x^{k}\right )^{n}}{b k n + b k} + \frac {b x^{k} \left (a + b x^{k}\right )^{n}}{b k n + b k} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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