Optimal. Leaf size=23 \[ \frac {\left (a+b x^k\right )^{1+n}}{b k (1+n)} \]
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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 = {267}
\begin {gather*} \frac {\left (a+b x^k\right )^{n+1}}{b k (n+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
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 = 0.03, size = 23, normalized size = 1.00 \begin {gather*} \frac {\left (a+b x^k\right )^{1+n}}{b k (1+n)} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 22.19, size = 112, normalized size = 4.87 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {\text {Log}\left [x\right ]}{a},b\text {==}0\text {\&\&}k\text {==}0\text {\&\&}n\text {==}-1\right \},\left \{\frac {a^n x^k}{k},b\text {==}0\right \},\left \{\text {Log}\left [x\right ] \left (a+b\right )^n,k\text {==}0\right \},\left \{\frac {\text {Log}\left [\frac {a}{b}+x^k\right ]}{b k},n\text {==}-1\right \}\right \},\frac {a {\left (a+b x^k\right )}^n}{b k+b k n}+\frac {b x^k {\left (a+b x^k\right )}^n}{b k+b k n}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.08, 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 = 0.27, size = 23, normalized size = 1.00 \begin {gather*} \frac {{\left (b x^{k} + a\right )}^{n + 1}}{b k {\left (n + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 27, normalized size = 1.17 \begin {gather*} \frac {{\left (b x^{k} + a\right )} {\left (b x^{k} + a\right )}^{n}}{b k n + b k} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 19.41, size = 75, normalized size = 3.26 \begin {gather*} \begin {cases} \frac {\log {\left (x \right )}}{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 {\left (x \right )} & \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 20, normalized size = 0.87 \begin {gather*} \frac {\left (a+b x^{k}\right )^{n+1}}{b k \left (n+1\right )} \end {gather*}
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 \begin {gather*} \frac {{\left (a+b\,x^k\right )}^{n+1}}{b\,k\,\left (n+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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