Optimal. Leaf size=27 \[ \frac{a x^{m+1}}{m+1}+\frac{b x^{m+n+1}}{m+n+1} \]
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Rubi [A] time = 0.0079004, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {14} \[ \frac{a x^{m+1}}{m+1}+\frac{b x^{m+n+1}}{m+n+1} \]
Antiderivative was successfully verified.
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Rule 14
Rubi steps
\begin{align*} \int x^m \left (a+b x^n\right ) \, dx &=\int \left (a x^m+b x^{m+n}\right ) \, dx\\ &=\frac{a x^{1+m}}{1+m}+\frac{b x^{1+m+n}}{1+m+n}\\ \end{align*}
Mathematica [A] time = 0.0189485, size = 27, normalized size = 1. \[ \frac{a x^{m+1}}{m+1}+\frac{b x^{m+n+1}}{m+n+1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 34, normalized size = 1.3 \begin{align*}{\frac{ax{{\rm e}^{m\ln \left ( x \right ) }}}{1+m}}+{\frac{bx{{\rm e}^{m\ln \left ( x \right ) }}{{\rm e}^{n\ln \left ( x \right ) }}}{m+n+1}} \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 = 1.06185, size = 101, normalized size = 3.74 \begin{align*} \frac{{\left (b m + b\right )} x x^{m} x^{n} +{\left (a m + a n + a\right )} x x^{m}}{m^{2} +{\left (m + 1\right )} n + 2 \, m + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.78093, size = 175, normalized size = 6.48 \begin{align*} \begin{cases} \left (a + b\right ) \log{\left (x \right )} & \text{for}\: m = -1 \wedge n = 0 \\a \log{\left (x \right )} + \frac{b x^{n}}{n} & \text{for}\: m = -1 \\\frac{a m x x^{m}}{m^{2} + m} + \frac{b m^{2} \log{\left (x \right )}}{m^{2} + m} + \frac{b m \log{\left (x \right )}}{m^{2} + m} & \text{for}\: n = - m - 1 \\\frac{a m x x^{m}}{m^{2} + m n + 2 m + n + 1} + \frac{a n x x^{m}}{m^{2} + m n + 2 m + n + 1} + \frac{a x x^{m}}{m^{2} + m n + 2 m + n + 1} + \frac{b m x x^{m} x^{n}}{m^{2} + m n + 2 m + n + 1} + \frac{b x x^{m} x^{n}}{m^{2} + m n + 2 m + n + 1} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.11245, size = 74, normalized size = 2.74 \begin{align*} \frac{b m x x^{m} x^{n} + a m x x^{m} + a n x x^{m} + b x x^{m} x^{n} + a x x^{m}}{m^{2} + m n + 2 \, m + n + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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