Optimal. Leaf size=33 \[ \frac{1}{3} x^3 (b e+c d)+\frac{1}{2} b d x^2+\frac{1}{4} c e x^4 \]
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Rubi [A] time = 0.0219701, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {631} \[ \frac{1}{3} x^3 (b e+c d)+\frac{1}{2} b d x^2+\frac{1}{4} c e x^4 \]
Antiderivative was successfully verified.
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Rule 631
Rubi steps
\begin{align*} \int (d+e x) \left (b x+c x^2\right ) \, dx &=\int \left (b d x+(c d+b e) x^2+c e x^3\right ) \, dx\\ &=\frac{1}{2} b d x^2+\frac{1}{3} (c d+b e) x^3+\frac{1}{4} c e x^4\\ \end{align*}
Mathematica [A] time = 0.0050811, size = 29, normalized size = 0.88 \[ \frac{1}{12} x^2 (b (6 d+4 e x)+c x (4 d+3 e x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 28, normalized size = 0.9 \begin{align*}{\frac{bd{x}^{2}}{2}}+{\frac{ \left ( be+cd \right ){x}^{3}}{3}}+{\frac{ce{x}^{4}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11657, size = 36, normalized size = 1.09 \begin{align*} \frac{1}{4} \, c e x^{4} + \frac{1}{2} \, b d x^{2} + \frac{1}{3} \,{\left (c d + b e\right )} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43521, size = 74, normalized size = 2.24 \begin{align*} \frac{1}{4} x^{4} e c + \frac{1}{3} x^{3} d c + \frac{1}{3} x^{3} e b + \frac{1}{2} x^{2} d b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.282943, size = 29, normalized size = 0.88 \begin{align*} \frac{b d x^{2}}{2} + \frac{c e x^{4}}{4} + x^{3} \left (\frac{b e}{3} + \frac{c d}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.3321, size = 42, normalized size = 1.27 \begin{align*} \frac{1}{4} \, c x^{4} e + \frac{1}{3} \, c d x^{3} + \frac{1}{3} \, b x^{3} e + \frac{1}{2} \, b d x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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