Optimal. Leaf size=25 \[ \frac {a x^{m+1}}{m+1}+\frac {b x^{m+2}}{m+2} \]
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Rubi [A] time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {43} \begin {gather*} \frac {a x^{m+1}}{m+1}+\frac {b x^{m+2}}{m+2} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int x^m (a+b x) \, dx &=\int \left (a x^m+b x^{1+m}\right ) \, dx\\ &=\frac {a x^{1+m}}{1+m}+\frac {b x^{2+m}}{2+m}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 22, normalized size = 0.88 \begin {gather*} x^{m+1} \left (\frac {a}{m+1}+\frac {b x}{m+2}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.01, size = 0, normalized size = 0.00 \begin {gather*} \int x^m (a+b x) \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.96, size = 33, normalized size = 1.32 \begin {gather*} \frac {{\left ({\left (b m + b\right )} x^{2} + {\left (a m + 2 \, a\right )} x\right )} x^{m}}{m^{2} + 3 \, m + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.01, size = 43, normalized size = 1.72 \begin {gather*} \frac {b m x^{2} x^{m} + a m x x^{m} + b x^{2} x^{m} + 2 \, a x x^{m}}{m^{2} + 3 \, m + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 31, normalized size = 1.24 \begin {gather*} \frac {\left (b m x +a m +b x +2 a \right ) x^{m +1}}{\left (m +2\right ) \left (m +1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 25, normalized size = 1.00 \begin {gather*} \frac {b x^{m + 2}}{m + 2} + \frac {a x^{m + 1}}{m + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.31, size = 30, normalized size = 1.20 \begin {gather*} \frac {x^{m+1}\,\left (2\,a+a\,m+b\,x+b\,m\,x\right )}{m^2+3\,m+2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.30, size = 87, normalized size = 3.48 \begin {gather*} \begin {cases} - \frac {a}{x} + b \log {\relax (x )} & \text {for}\: m = -2 \\a \log {\relax (x )} + b x & \text {for}\: m = -1 \\\frac {a m x x^{m}}{m^{2} + 3 m + 2} + \frac {2 a x x^{m}}{m^{2} + 3 m + 2} + \frac {b m x^{2} x^{m}}{m^{2} + 3 m + 2} + \frac {b x^{2} x^{m}}{m^{2} + 3 m + 2} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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