Optimal. Leaf size=37 \[ \frac{a x^{m+2}}{m+2}+\frac{b x^{m+4}}{m+4}+\frac{c x^{m+6}}{m+6} \]
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Rubi [A] time = 0.0129212, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {14} \[ \frac{a x^{m+2}}{m+2}+\frac{b x^{m+4}}{m+4}+\frac{c x^{m+6}}{m+6} \]
Antiderivative was successfully verified.
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Rule 14
Rubi steps
\begin{align*} \int x^m \left (a x+b x^3+c x^5\right ) \, dx &=\int \left (a x^{1+m}+b x^{3+m}+c x^{5+m}\right ) \, dx\\ &=\frac{a x^{2+m}}{2+m}+\frac{b x^{4+m}}{4+m}+\frac{c x^{6+m}}{6+m}\\ \end{align*}
Mathematica [A] time = 0.0288071, size = 34, normalized size = 0.92 \[ x^{m+2} \left (\frac{a}{m+2}+\frac{b x^2}{m+4}+\frac{c x^4}{m+6}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.003, size = 77, normalized size = 2.1 \begin{align*}{\frac{{x}^{2+m} \left ( c{m}^{2}{x}^{4}+6\,cm{x}^{4}+b{m}^{2}{x}^{2}+8\,c{x}^{4}+8\,bm{x}^{2}+a{m}^{2}+12\,b{x}^{2}+10\,am+24\,a \right ) }{ \left ( 6+m \right ) \left ( 4+m \right ) \left ( 2+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 = 1.31314, size = 161, normalized size = 4.35 \begin{align*} \frac{{\left ({\left (c m^{2} + 6 \, c m + 8 \, c\right )} x^{6} +{\left (b m^{2} + 8 \, b m + 12 \, b\right )} x^{4} +{\left (a m^{2} + 10 \, a m + 24 \, a\right )} x^{2}\right )} x^{m}}{m^{3} + 12 \, m^{2} + 44 \, m + 48} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.09027, size = 280, normalized size = 7.57 \begin{align*} \begin{cases} - \frac{a}{4 x^{4}} - \frac{b}{2 x^{2}} + c \log{\left (x \right )} & \text{for}\: m = -6 \\- \frac{a}{2 x^{2}} + b \log{\left (x \right )} + \frac{c x^{2}}{2} & \text{for}\: m = -4 \\a \log{\left (x \right )} + \frac{b x^{2}}{2} + \frac{c x^{4}}{4} & \text{for}\: m = -2 \\\frac{a m^{2} x^{2} x^{m}}{m^{3} + 12 m^{2} + 44 m + 48} + \frac{10 a m x^{2} x^{m}}{m^{3} + 12 m^{2} + 44 m + 48} + \frac{24 a x^{2} x^{m}}{m^{3} + 12 m^{2} + 44 m + 48} + \frac{b m^{2} x^{4} x^{m}}{m^{3} + 12 m^{2} + 44 m + 48} + \frac{8 b m x^{4} x^{m}}{m^{3} + 12 m^{2} + 44 m + 48} + \frac{12 b x^{4} x^{m}}{m^{3} + 12 m^{2} + 44 m + 48} + \frac{c m^{2} x^{6} x^{m}}{m^{3} + 12 m^{2} + 44 m + 48} + \frac{6 c m x^{6} x^{m}}{m^{3} + 12 m^{2} + 44 m + 48} + \frac{8 c x^{6} x^{m}}{m^{3} + 12 m^{2} + 44 m + 48} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.11225, size = 144, normalized size = 3.89 \begin{align*} \frac{c m^{2} x^{6} x^{m} + 6 \, c m x^{6} x^{m} + b m^{2} x^{4} x^{m} + 8 \, c x^{6} x^{m} + 8 \, b m x^{4} x^{m} + a m^{2} x^{2} x^{m} + 12 \, b x^{4} x^{m} + 10 \, a m x^{2} x^{m} + 24 \, a x^{2} x^{m}}{m^{3} + 12 \, m^{2} + 44 \, m + 48} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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