Optimal. Leaf size=27 \[ \frac {\left (a+b x^{m+1}\right )^{n+1}}{b (m+1) (n+1)} \]
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Rubi [A] time = 0.01, antiderivative size = 27, 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} \[ \frac {\left (a+b x^{m+1}\right )^{n+1}}{b (m+1) (n+1)} \]
Antiderivative was successfully verified.
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Rule 261
Rubi steps
\begin {align*} \int x^m \left (a+b x^{1+m}\right )^n \, dx &=\frac {\left (a+b x^{1+m}\right )^{1+n}}{b (1+m) (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 1.00 \[ \frac {\left (a+b x^{m+1}\right )^{n+1}}{b (m+1) (n+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 35, normalized size = 1.30 \[ \frac {{\left (b x^{m + 1} + a\right )} {\left (b x^{m + 1} + a\right )}^{n}}{b m + {\left (b m + b\right )} n + b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 26, normalized size = 0.96 \[ \frac {{\left (b x^{m + 1} + a\right )}^{n + 1}}{{\left (b m + b\right )} {\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 60, normalized size = 2.22 \[ \frac {x \,{\mathrm e}^{m \ln \relax (x )} {\mathrm e}^{n \ln \left (b x \,{\mathrm e}^{m \ln \relax (x )}+a \right )}}{m n +m +n +1}+\frac {a \,{\mathrm e}^{n \ln \left (b x \,{\mathrm e}^{m \ln \relax (x )}+a \right )}}{\left (m n +m +n +1\right ) b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 27, normalized size = 1.00 \[ \frac {{\left (b x^{m + 1} + a\right )}^{n + 1}}{b {\left (m + 1\right )} {\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.49, size = 27, normalized size = 1.00 \[ \frac {{\left (a+b\,x^{m+1}\right )}^{n+1}}{b\,\left (m+1\right )\,\left (n+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 134.23, size = 100, normalized size = 3.70 \[ \begin {cases} \frac {\log {\relax (x )}}{a} & \text {for}\: b = 0 \wedge m = -1 \wedge n = -1 \\\frac {a^{n} x x^{m}}{m + 1} & \text {for}\: b = 0 \\\left (a + b\right )^{n} \log {\relax (x )} & \text {for}\: m = -1 \\\frac {\log {\left (\frac {a}{b} + x x^{m} \right )}}{b m + b} & \text {for}\: n = -1 \\\frac {a \left (a + b x x^{m}\right )^{n}}{b m n + b m + b n + b} + \frac {b x x^{m} \left (a + b x x^{m}\right )^{n}}{b m n + b m + b n + b} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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