Optimal. Leaf size=16 \[ \frac{1}{3} \left (a+b x+c x^2\right )^3 \]
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Rubi [A] time = 0.0044276, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {629} \[ \frac{1}{3} \left (a+b x+c x^2\right )^3 \]
Antiderivative was successfully verified.
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Rule 629
Rubi steps
\begin{align*} \int (b+2 c x) \left (a+b x+c x^2\right )^2 \, dx &=\frac{1}{3} \left (a+b x+c x^2\right )^3\\ \end{align*}
Mathematica [B] time = 0.0097431, size = 36, normalized size = 2.25 \[ \frac{1}{3} x (b+c x) \left (3 a^2+3 a x (b+c x)+x^2 (b+c x)^2\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0., size = 86, normalized size = 5.4 \begin{align*}{\frac{{c}^{3}{x}^{6}}{3}}+b{c}^{2}{x}^{5}+{\frac{ \left ( 2\,{b}^{2}c+2\,c \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( b \left ( 2\,ac+{b}^{2} \right ) +4\,abc \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,c{a}^{2}+2\,{b}^{2}a \right ){x}^{2}}{2}}+b{a}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04444, size = 19, normalized size = 1.19 \begin{align*} \frac{1}{3} \,{\left (c x^{2} + b x + a\right )}^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.28025, size = 150, normalized size = 9.38 \begin{align*} \frac{1}{3} x^{6} c^{3} + x^{5} c^{2} b + x^{4} c b^{2} + x^{4} c^{2} a + \frac{1}{3} x^{3} b^{3} + 2 x^{3} c b a + x^{2} b^{2} a + x^{2} c a^{2} + x b a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.076576, size = 65, normalized size = 4.06 \begin{align*} a^{2} b x + b c^{2} x^{5} + \frac{c^{3} x^{6}}{3} + x^{4} \left (a c^{2} + b^{2} c\right ) + x^{3} \left (2 a b c + \frac{b^{3}}{3}\right ) + x^{2} \left (a^{2} c + a b^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17353, size = 96, normalized size = 6. \begin{align*} \frac{1}{3} \, c^{3} x^{6} + b c^{2} x^{5} + b^{2} c x^{4} + a c^{2} x^{4} + \frac{1}{3} \, b^{3} x^{3} + 2 \, a b c x^{3} + a b^{2} x^{2} + a^{2} c x^{2} + a^{2} b x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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