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 = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {a x^{m+2}}{m+2}+\frac {b x^{m+4}}{m+4}+\frac {c x^{m+6}}{m+6} \end {gather*}
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*}
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Mathematica [A] time = 0.03, size = 34, normalized size = 0.92 \begin {gather*} x^{m+2} \left (\frac {a}{m+2}+\frac {b x^2}{m+4}+\frac {c x^4}{m+6}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.31, size = 0, normalized size = 0.00 \begin {gather*} \int x^m \left (a x+b x^3+c x^5\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.26, size = 71, normalized size = 1.92 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.48, size = 107, normalized size = 2.89 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 77, normalized size = 2.08 \begin {gather*} \frac {\left (c \,m^{2} x^{4}+6 c m \,x^{4}+b \,m^{2} x^{2}+8 c \,x^{4}+8 b m \,x^{2}+a \,m^{2}+12 b \,x^{2}+10 a m +24 a \right ) x^{m +2}}{\left (m +6\right ) \left (m +4\right ) \left (m +2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 37, normalized size = 1.00 \begin {gather*} \frac {c x^{m + 6}}{m + 6} + \frac {b x^{m + 4}}{m + 4} + \frac {a x^{m + 2}}{m + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.08, size = 89, normalized size = 2.41 \begin {gather*} x^m\,\left (\frac {a\,x^2\,\left (m^2+10\,m+24\right )}{m^3+12\,m^2+44\,m+48}+\frac {b\,x^4\,\left (m^2+8\,m+12\right )}{m^3+12\,m^2+44\,m+48}+\frac {c\,x^6\,\left (m^2+6\,m+8\right )}{m^3+12\,m^2+44\,m+48}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.18, size = 280, normalized size = 7.57 \begin {gather*} \begin {cases} - \frac {a}{4 x^{4}} - \frac {b}{2 x^{2}} + c \log {\relax (x )} & \text {for}\: m = -6 \\- \frac {a}{2 x^{2}} + b \log {\relax (x )} + \frac {c x^{2}}{2} & \text {for}\: m = -4 \\a \log {\relax (x )} + \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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