Optimal. Leaf size=55 \[ \frac{1}{3} a^2 A x^3+\frac{1}{5} b x^5 (2 a B+A b)+\frac{1}{4} a x^4 (a B+2 A b)+\frac{1}{6} b^2 B x^6 \]
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Rubi [A] time = 0.0287897, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {76} \[ \frac{1}{3} a^2 A x^3+\frac{1}{5} b x^5 (2 a B+A b)+\frac{1}{4} a x^4 (a B+2 A b)+\frac{1}{6} b^2 B x^6 \]
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin{align*} \int x^2 (a+b x)^2 (A+B x) \, dx &=\int \left (a^2 A x^2+a (2 A b+a B) x^3+b (A b+2 a B) x^4+b^2 B x^5\right ) \, dx\\ &=\frac{1}{3} a^2 A x^3+\frac{1}{4} a (2 A b+a B) x^4+\frac{1}{5} b (A b+2 a B) x^5+\frac{1}{6} b^2 B x^6\\ \end{align*}
Mathematica [A] time = 0.0105954, size = 50, normalized size = 0.91 \[ \frac{1}{60} x^3 \left (5 a^2 (4 A+3 B x)+6 a b x (5 A+4 B x)+2 b^2 x^2 (6 A+5 B x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 52, normalized size = 1. \begin{align*}{\frac{{b}^{2}B{x}^{6}}{6}}+{\frac{ \left ({b}^{2}A+2\,abB \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,abA+{a}^{2}B \right ){x}^{4}}{4}}+{\frac{{a}^{2}A{x}^{3}}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04979, size = 69, normalized size = 1.25 \begin{align*} \frac{1}{6} \, B b^{2} x^{6} + \frac{1}{3} \, A a^{2} x^{3} + \frac{1}{5} \,{\left (2 \, B a b + A b^{2}\right )} x^{5} + \frac{1}{4} \,{\left (B a^{2} + 2 \, A a b\right )} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75352, size = 128, normalized size = 2.33 \begin{align*} \frac{1}{6} x^{6} b^{2} B + \frac{2}{5} x^{5} b a B + \frac{1}{5} x^{5} b^{2} A + \frac{1}{4} x^{4} a^{2} B + \frac{1}{2} x^{4} b a A + \frac{1}{3} x^{3} a^{2} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.078262, size = 54, normalized size = 0.98 \begin{align*} \frac{A a^{2} x^{3}}{3} + \frac{B b^{2} x^{6}}{6} + x^{5} \left (\frac{A b^{2}}{5} + \frac{2 B a b}{5}\right ) + x^{4} \left (\frac{A a b}{2} + \frac{B a^{2}}{4}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21344, size = 72, normalized size = 1.31 \begin{align*} \frac{1}{6} \, B b^{2} x^{6} + \frac{2}{5} \, B a b x^{5} + \frac{1}{5} \, A b^{2} x^{5} + \frac{1}{4} \, B a^{2} x^{4} + \frac{1}{2} \, A a b x^{4} + \frac{1}{3} \, A a^{2} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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