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 = 0.03, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {27, 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 27
Rule 76
Rubi steps
\begin {align*} \int x^2 (A+B x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.65, size = 53, normalized size = 0.96 \[ \frac {1}{6} x^{6} b^{2} B + \frac {2}{5} x^{5} b a B + \frac {1}{5} x^{5} b^{2} A + \frac {1}{4} x^{4} a^{2} B + \frac {1}{2} x^{4} b a A + \frac {1}{3} x^{3} a^{2} A \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 53, normalized size = 0.96 \[ \frac {1}{6} \, B b^{2} x^{6} + \frac {2}{5} \, B a b x^{5} + \frac {1}{5} \, A b^{2} x^{5} + \frac {1}{4} \, B a^{2} x^{4} + \frac {1}{2} \, A a b x^{4} + \frac {1}{3} \, A a^{2} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 52, normalized size = 0.95 \[ \frac {B \,b^{2} x^{6}}{6}+\frac {A \,a^{2} x^{3}}{3}+\frac {\left (A \,b^{2}+2 a b B \right ) x^{5}}{5}+\frac {\left (2 A a b +a^{2} B \right ) x^{4}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 51, normalized size = 0.93 \[ \frac {1}{6} \, B b^{2} x^{6} + \frac {1}{3} \, A a^{2} x^{3} + \frac {1}{5} \, {\left (2 \, B a b + A b^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B a^{2} + 2 \, A a b\right )} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 51, normalized size = 0.93 \[ x^4\,\left (\frac {B\,a^2}{4}+\frac {A\,b\,a}{2}\right )+x^5\,\left (\frac {A\,b^2}{5}+\frac {2\,B\,a\,b}{5}\right )+\frac {A\,a^2\,x^3}{3}+\frac {B\,b^2\,x^6}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 54, normalized size = 0.98 \[ \frac {A a^{2} x^{3}}{3} + \frac {B b^{2} x^{6}}{6} + x^{5} \left (\frac {A b^{2}}{5} + \frac {2 B a b}{5}\right ) + x^{4} \left (\frac {A a b}{2} + \frac {B a^{2}}{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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