Optimal. Leaf size=17 \[ \frac{1}{3} d \left (a+b x+c x^2\right )^3 \]
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Rubi [A] time = 0.0055592, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {629} \[ \frac{1}{3} d \left (a+b x+c x^2\right )^3 \]
Antiderivative was successfully verified.
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Rule 629
Rubi steps
\begin{align*} \int (b d+2 c d x) \left (a+b x+c x^2\right )^2 \, dx &=\frac{1}{3} d \left (a+b x+c x^2\right )^3\\ \end{align*}
Mathematica [B] time = 0.0099525, size = 37, normalized size = 2.18 \[ \frac{1}{3} d x (b+c x) \left (3 a^2+3 a x (b+c x)+x^2 (b+c x)^2\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.039, size = 95, normalized size = 5.6 \begin{align*}{\frac{{c}^{3}d{x}^{6}}{3}}+bd{c}^{2}{x}^{5}+{\frac{ \left ( 2\,{b}^{2}dc+2\,cd \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( bd \left ( 2\,ac+{b}^{2} \right ) +4\,cabd \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,cd{a}^{2}+2\,a{b}^{2}d \right ){x}^{2}}{2}}+bd{a}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16972, size = 20, normalized size = 1.18 \begin{align*} \frac{1}{3} \,{\left (c x^{2} + b x + a\right )}^{3} d \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.84776, size = 174, normalized size = 10.24 \begin{align*} \frac{1}{3} x^{6} d c^{3} + x^{5} d c^{2} b + x^{4} d c b^{2} + x^{4} d c^{2} a + \frac{1}{3} x^{3} d b^{3} + 2 x^{3} d c b a + x^{2} d b^{2} a + x^{2} d c a^{2} + x d b a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.085105, size = 80, normalized size = 4.71 \begin{align*} a^{2} b d x + b c^{2} d x^{5} + \frac{c^{3} d x^{6}}{3} + x^{4} \left (a c^{2} d + b^{2} c d\right ) + x^{3} \left (2 a b c d + \frac{b^{3} d}{3}\right ) + x^{2} \left (a^{2} c d + a b^{2} d\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.23593, size = 108, normalized size = 6.35 \begin{align*} \frac{1}{3} \, c^{3} d x^{6} + b c^{2} d x^{5} + b^{2} c d x^{4} + a c^{2} d x^{4} + \frac{1}{3} \, b^{3} d x^{3} + 2 \, a b c d x^{3} + a b^{2} d x^{2} + a^{2} c d x^{2} + a^{2} b d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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