Optimal. Leaf size=44 \[ \frac{1}{2} x^2 (a \text{c1}+2 b \text{b1})+a \text{b1} x+\frac{1}{3} x^3 (2 b \text{c1}+\text{b1} c)+\frac{1}{4} c \text{c1} x^4 \]
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Rubi [A] time = 0.0365853, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {631} \[ \frac{1}{2} x^2 (a \text{c1}+2 b \text{b1})+a \text{b1} x+\frac{1}{3} x^3 (2 b \text{c1}+\text{b1} c)+\frac{1}{4} c \text{c1} x^4 \]
Antiderivative was successfully verified.
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Rule 631
Rubi steps
\begin{align*} \int (\text{b1}+\text{c1} x) \left (a+2 b x+c x^2\right ) \, dx &=\int \left (a \text{b1}+(2 b \text{b1}+a \text{c1}) x+(\text{b1} c+2 b \text{c1}) x^2+c \text{c1} x^3\right ) \, dx\\ &=a \text{b1} x+\frac{1}{2} (2 b \text{b1}+a \text{c1}) x^2+\frac{1}{3} (\text{b1} c+2 b \text{c1}) x^3+\frac{1}{4} c \text{c1} x^4\\ \end{align*}
Mathematica [A] time = 0.0116867, size = 41, normalized size = 0.93 \[ \frac{1}{12} x (6 a (2 \text{b1}+\text{c1} x)+x (4 b (3 \text{b1}+2 \text{c1} x)+c x (4 \text{b1}+3 \text{c1} x))) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 39, normalized size = 0.9 \begin{align*} a{\it b1}\,x+{\frac{ \left ( a{\it c1}+2\,b{\it b1} \right ){x}^{2}}{2}}+{\frac{ \left ( 2\,b{\it c1}+{\it b1}\,c \right ){x}^{3}}{3}}+{\frac{c{\it c1}\,{x}^{4}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.925517, size = 51, normalized size = 1.16 \begin{align*} \frac{1}{4} \, c c_{1} x^{4} + \frac{1}{3} \,{\left (b_{1} c + 2 \, b c_{1}\right )} x^{3} + a b_{1} x + \frac{1}{2} \,{\left (2 \, b b_{1} + a c_{1}\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47087, size = 107, normalized size = 2.43 \begin{align*} \frac{1}{4} x^{4} c_{1} c + \frac{1}{3} x^{3} c b_{1} + \frac{2}{3} x^{3} c_{1} b + x^{2} b_{1} b + \frac{1}{2} x^{2} c_{1} a + x b_{1} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.058321, size = 39, normalized size = 0.89 \begin{align*} a b_{1} x + \frac{c c_{1} x^{4}}{4} + x^{3} \left (\frac{2 b c_{1}}{3} + \frac{b_{1} c}{3}\right ) + x^{2} \left (\frac{a c_{1}}{2} + b b_{1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0673, size = 53, normalized size = 1.2 \begin{align*} \frac{1}{4} \, c c_{1} x^{4} + \frac{1}{3} \, b_{1} c x^{3} + \frac{2}{3} \, b c_{1} x^{3} + b b_{1} x^{2} + \frac{1}{2} \, a c_{1} x^{2} + a b_{1} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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