Optimal. Leaf size=45 \[ \frac{x^{m+2} (a B+A b)}{m+2}+\frac{a A x^{m+1}}{m+1}+\frac{b B x^{m+3}}{m+3} \]
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Rubi [A] time = 0.0181551, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {76} \[ \frac{x^{m+2} (a B+A b)}{m+2}+\frac{a A x^{m+1}}{m+1}+\frac{b B x^{m+3}}{m+3} \]
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin{align*} \int x^m (a+b x) (A+B x) \, dx &=\int \left (a A x^m+(A b+a B) x^{1+m}+b B x^{2+m}\right ) \, dx\\ &=\frac{a A x^{1+m}}{1+m}+\frac{(A b+a B) x^{2+m}}{2+m}+\frac{b B x^{3+m}}{3+m}\\ \end{align*}
Mathematica [A] time = 0.0326054, size = 57, normalized size = 1.27 \[ \frac{x^{m+1} (a (m+3) (A (m+2)+B (m+1) x)+b (m+1) x (A (m+3)+B (m+2) x))}{(m+1) (m+2) (m+3)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.003, size = 98, normalized size = 2.2 \begin{align*}{\frac{{x}^{1+m} \left ( Bb{m}^{2}{x}^{2}+Ab{m}^{2}x+Ba{m}^{2}x+3\,Bbm{x}^{2}+Aa{m}^{2}+4\,Abmx+4\,Bamx+2\,bB{x}^{2}+5\,Aam+3\,Abx+3\,Bax+6\,Aa \right ) }{ \left ( 3+m \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 [B] time = 2.28358, size = 208, normalized size = 4.62 \begin{align*} \frac{{\left ({\left (B b m^{2} + 3 \, B b m + 2 \, B b\right )} x^{3} +{\left ({\left (B a + A b\right )} m^{2} + 3 \, B a + 3 \, A b + 4 \,{\left (B a + A b\right )} m\right )} x^{2} +{\left (A a m^{2} + 5 \, A a m + 6 \, A a\right )} x\right )} x^{m}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.665872, size = 389, normalized size = 8.64 \begin{align*} \begin{cases} - \frac{A a}{2 x^{2}} - \frac{A b}{x} - \frac{B a}{x} + B b \log{\left (x \right )} & \text{for}\: m = -3 \\- \frac{A a}{x} + A b \log{\left (x \right )} + B a \log{\left (x \right )} + B b x & \text{for}\: m = -2 \\A a \log{\left (x \right )} + A b x + B a x + \frac{B b x^{2}}{2} & \text{for}\: m = -1 \\\frac{A a m^{2} x x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{5 A a m x x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{6 A a x x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{A b m^{2} x^{2} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{4 A b m x^{2} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{3 A b x^{2} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{B a m^{2} x^{2} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{4 B a m x^{2} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{3 B a x^{2} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{B b m^{2} x^{3} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{3 B b m x^{3} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} + \frac{2 B b x^{3} x^{m}}{m^{3} + 6 m^{2} + 11 m + 6} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.175, size = 193, normalized size = 4.29 \begin{align*} \frac{B b m^{2} x^{3} x^{m} + B a m^{2} x^{2} x^{m} + A b m^{2} x^{2} x^{m} + 3 \, B b m x^{3} x^{m} + A a m^{2} x x^{m} + 4 \, B a m x^{2} x^{m} + 4 \, A b m x^{2} x^{m} + 2 \, B b x^{3} x^{m} + 5 \, A a m x x^{m} + 3 \, B a x^{2} x^{m} + 3 \, A b x^{2} x^{m} + 6 \, A a x x^{m}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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