Optimal. Leaf size=17 \[ \frac{1}{4} d \left (a+b x+c x^2\right )^4 \]
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Rubi [A] time = 0.0054493, 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} \[ \frac{1}{4} d \left (a+b x+c x^2\right )^4 \]
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 )^3 \, dx &=\frac{1}{4} d \left (a+b x+c x^2\right )^4\\ \end{align*}
Mathematica [B] time = 0.0176819, size = 52, normalized size = 3.06 \[ \frac{1}{4} d x (b+c x) \left (6 a^2 x (b+c x)+4 a^3+4 a x^2 (b+c x)^2+x^3 (b+c x)^3\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.039, size = 231, normalized size = 13.6 \begin{align*}{\frac{{c}^{4}d{x}^{8}}{4}}+bd{c}^{3}{x}^{7}+{\frac{ \left ( 3\,{b}^{2}d{c}^{2}+2\,cd \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{6}}{6}}+{\frac{ \left ( bd \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +2\,cd \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( bd \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) +2\,cd \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+{a}^{2}c \right ) \right ){x}^{4}}{4}}+{\frac{ \left ( bd \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+{a}^{2}c \right ) +6\,cdb{a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,cd{a}^{3}+3\,{b}^{2}d{a}^{2} \right ){x}^{2}}{2}}+bd{a}^{3}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.23692, size = 20, normalized size = 1.18 \begin{align*} \frac{1}{4} \,{\left (c x^{2} + b x + a\right )}^{4} d \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.65394, size = 306, normalized size = 18. \begin{align*} \frac{1}{4} x^{8} d c^{4} + x^{7} d c^{3} b + \frac{3}{2} x^{6} d c^{2} b^{2} + x^{6} d c^{3} a + x^{5} d c b^{3} + 3 x^{5} d c^{2} b a + \frac{1}{4} x^{4} d b^{4} + 3 x^{4} d c b^{2} a + \frac{3}{2} x^{4} d c^{2} a^{2} + x^{3} d b^{3} a + 3 x^{3} d c b a^{2} + \frac{3}{2} x^{2} d b^{2} a^{2} + x^{2} d c a^{3} + x d b a^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.128285, size = 144, normalized size = 8.47 \begin{align*} a^{3} b d x + b c^{3} d x^{7} + \frac{c^{4} d x^{8}}{4} + x^{6} \left (a c^{3} d + \frac{3 b^{2} c^{2} d}{2}\right ) + x^{5} \left (3 a b c^{2} d + b^{3} c d\right ) + x^{4} \left (\frac{3 a^{2} c^{2} d}{2} + 3 a b^{2} c d + \frac{b^{4} d}{4}\right ) + x^{3} \left (3 a^{2} b c d + a b^{3} d\right ) + x^{2} \left (a^{3} c d + \frac{3 a^{2} b^{2} d}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.22418, size = 189, normalized size = 11.12 \begin{align*} \frac{1}{4} \, c^{4} d x^{8} + b c^{3} d x^{7} + \frac{3}{2} \, b^{2} c^{2} d x^{6} + a c^{3} d x^{6} + b^{3} c d x^{5} + 3 \, a b c^{2} d x^{5} + \frac{1}{4} \, b^{4} d x^{4} + 3 \, a b^{2} c d x^{4} + \frac{3}{2} \, a^{2} c^{2} d x^{4} + a b^{3} d x^{3} + 3 \, a^{2} b c d x^{3} + \frac{3}{2} \, a^{2} b^{2} d x^{2} + a^{3} c d x^{2} + a^{3} b d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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