Optimal. Leaf size=42 \[ \frac{1}{2} x^2 (a e+b d)+a d x+\frac{1}{3} x^3 (b e+c d)+\frac{1}{4} c e x^4 \]
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Rubi [A] time = 0.028922, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {631} \[ \frac{1}{2} x^2 (a e+b d)+a d x+\frac{1}{3} x^3 (b e+c d)+\frac{1}{4} c e x^4 \]
Antiderivative was successfully verified.
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Rule 631
Rubi steps
\begin{align*} \int (d+e x) \left (a+b x+c x^2\right ) \, dx &=\int \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right ) \, dx\\ &=a d x+\frac{1}{2} (b d+a e) x^2+\frac{1}{3} (c d+b e) x^3+\frac{1}{4} c e x^4\\ \end{align*}
Mathematica [A] time = 0.0129499, size = 42, normalized size = 1. \[ \frac{1}{2} x^2 (a e+b d)+a d x+\frac{1}{3} x^3 (b e+c d)+\frac{1}{4} c e x^4 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 37, normalized size = 0.9 \begin{align*} adx+{\frac{ \left ( ae+bd \right ){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 = 0.967219, size = 49, normalized size = 1.17 \begin{align*} \frac{1}{4} \, c e x^{4} + \frac{1}{3} \,{\left (c d + b e\right )} x^{3} + a d x + \frac{1}{2} \,{\left (b d + a e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81696, size = 104, normalized size = 2.48 \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 + \frac{1}{2} x^{2} e a + x d a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.063283, size = 39, normalized size = 0.93 \begin{align*} a d x + \frac{c e x^{4}}{4} + x^{3} \left (\frac{b e}{3} + \frac{c d}{3}\right ) + x^{2} \left (\frac{a e}{2} + \frac{b d}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08335, size = 58, normalized size = 1.38 \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} + \frac{1}{2} \, a x^{2} e + a d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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