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 = 0.0064292, antiderivative size = 25, normalized size of antiderivative = 1., 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} \[ \frac{a x^{m+1}}{m+1}+\frac{b x^{m+2}}{m+2} \]
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*}
Mathematica [A] time = 0.012999, size = 22, normalized size = 0.88 \[ x^{m+1} \left (\frac{a}{m+1}+\frac{b x}{m+2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 31, normalized size = 1.2 \begin{align*}{\frac{{x}^{1+m} \left ( bmx+am+bx+2\,a \right ) }{ \left ( 2+m \right ) \left ( 1+m \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57877, size = 72, normalized size = 2.88 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.262392, size = 87, normalized size = 3.48 \begin{align*} \begin{cases} - \frac{a}{x} + b \log{\left (x \right )} & \text{for}\: m = -2 \\a \log{\left (x \right )} + 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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06295, size = 58, normalized size = 2.32 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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