Optimal. Leaf size=28 \[ \frac{1}{2} x^2 (a B+A b)+a A x+\frac{1}{3} b B x^3 \]
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Rubi [A] time = 0.0182654, antiderivative size = 28, 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 = {43} \[ \frac{1}{2} x^2 (a B+A b)+a A x+\frac{1}{3} b B x^3 \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int (a+b x) (A+B x) \, dx &=\int \left (a A+(A b+a B) x+b B x^2\right ) \, dx\\ &=a A x+\frac{1}{2} (A b+a B) x^2+\frac{1}{3} b B x^3\\ \end{align*}
Mathematica [A] time = 0.004805, size = 28, normalized size = 1. \[ \frac{1}{2} x^2 (a B+A b)+a A x+\frac{1}{3} b B x^3 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 25, normalized size = 0.9 \begin{align*} aAx+{\frac{ \left ( Ab+Ba \right ){x}^{2}}{2}}+{\frac{bB{x}^{3}}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07243, size = 32, normalized size = 1.14 \begin{align*} \frac{1}{3} \, B b x^{3} + A a x + \frac{1}{2} \,{\left (B a + A b\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52043, size = 66, normalized size = 2.36 \begin{align*} \frac{1}{3} x^{3} b B + \frac{1}{2} x^{2} a B + \frac{1}{2} x^{2} b A + x a A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.057172, size = 26, normalized size = 0.93 \begin{align*} A a x + \frac{B b x^{3}}{3} + x^{2} \left (\frac{A b}{2} + \frac{B a}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22534, size = 35, normalized size = 1.25 \begin{align*} \frac{1}{3} \, B b x^{3} + \frac{1}{2} \, B a x^{2} + \frac{1}{2} \, A b x^{2} + A a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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