Optimal. Leaf size=45 \[ \frac {x^{m+3} (A c+b B)}{m+3}+\frac {A b x^{m+2}}{m+2}+\frac {B c x^{m+4}}{m+4} \]
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Rubi [A] time = 0.02, antiderivative size = 45, 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 = {765} \begin {gather*} \frac {x^{m+3} (A c+b B)}{m+3}+\frac {A b x^{m+2}}{m+2}+\frac {B c x^{m+4}}{m+4} \end {gather*}
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin {align*} \int x^m (A+B x) \left (b x+c x^2\right ) \, dx &=\int \left (A b x^{1+m}+(b B+A c) x^{2+m}+B c x^{3+m}\right ) \, dx\\ &=\frac {A b x^{2+m}}{2+m}+\frac {(b B+A c) x^{3+m}}{3+m}+\frac {B c x^{4+m}}{4+m}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 57, normalized size = 1.27 \begin {gather*} \frac {x^{m+2} (A (m+4) (b (m+3)+c (m+2) x)+B (m+2) x (b (m+4)+c (m+3) x))}{(m+2) (m+3) (m+4)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.05, size = 0, normalized size = 0.00 \begin {gather*} \int x^m (A+B x) \left (b x+c x^2\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.42, size = 94, normalized size = 2.09 \begin {gather*} \frac {{\left ({\left (B c m^{2} + 5 \, B c m + 6 \, B c\right )} x^{4} + {\left ({\left (B b + A c\right )} m^{2} + 8 \, B b + 8 \, A c + 6 \, {\left (B b + A c\right )} m\right )} x^{3} + {\left (A b m^{2} + 7 \, A b m + 12 \, A b\right )} x^{2}\right )} x^{m}}{m^{3} + 9 \, m^{2} + 26 \, m + 24} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 149, normalized size = 3.31 \begin {gather*} \frac {B c m^{2} x^{4} x^{m} + B b m^{2} x^{3} x^{m} + A c m^{2} x^{3} x^{m} + 5 \, B c m x^{4} x^{m} + A b m^{2} x^{2} x^{m} + 6 \, B b m x^{3} x^{m} + 6 \, A c m x^{3} x^{m} + 6 \, B c x^{4} x^{m} + 7 \, A b m x^{2} x^{m} + 8 \, B b x^{3} x^{m} + 8 \, A c x^{3} x^{m} + 12 \, A b x^{2} x^{m}}{m^{3} + 9 \, m^{2} + 26 \, m + 24} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 98, normalized size = 2.18 \begin {gather*} \frac {\left (B c \,m^{2} x^{2}+A c \,m^{2} x +B b \,m^{2} x +5 B c m \,x^{2}+A b \,m^{2}+6 A c m x +6 B b m x +6 B c \,x^{2}+7 A b m +8 A c x +8 B b x +12 A b \right ) x^{m +2}}{\left (m +4\right ) \left (m +3\right ) \left (m +2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.86, size = 53, normalized size = 1.18 \begin {gather*} \frac {B c x^{m + 4}}{m + 4} + \frac {B b x^{m + 3}}{m + 3} + \frac {A c x^{m + 3}}{m + 3} + \frac {A b x^{m + 2}}{m + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.13, size = 97, normalized size = 2.16 \begin {gather*} x^m\,\left (\frac {x^3\,\left (A\,c+B\,b\right )\,\left (m^2+6\,m+8\right )}{m^3+9\,m^2+26\,m+24}+\frac {A\,b\,x^2\,\left (m^2+7\,m+12\right )}{m^3+9\,m^2+26\,m+24}+\frac {B\,c\,x^4\,\left (m^2+5\,m+6\right )}{m^3+9\,m^2+26\,m+24}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.81, size = 394, normalized size = 8.76 \begin {gather*} \begin {cases} - \frac {A b}{2 x^{2}} - \frac {A c}{x} - \frac {B b}{x} + B c \log {\relax (x )} & \text {for}\: m = -4 \\- \frac {A b}{x} + A c \log {\relax (x )} + B b \log {\relax (x )} + B c x & \text {for}\: m = -3 \\A b \log {\relax (x )} + A c x + B b x + \frac {B c x^{2}}{2} & \text {for}\: m = -2 \\\frac {A b m^{2} x^{2} x^{m}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac {7 A b m x^{2} x^{m}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac {12 A b x^{2} x^{m}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac {A c m^{2} x^{3} x^{m}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac {6 A c m x^{3} x^{m}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac {8 A c x^{3} x^{m}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac {B b m^{2} x^{3} x^{m}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac {6 B b m x^{3} x^{m}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac {8 B b x^{3} x^{m}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac {B c m^{2} x^{4} x^{m}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac {5 B c m x^{4} x^{m}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac {6 B c x^{4} x^{m}}{m^{3} + 9 m^{2} + 26 m + 24} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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