Optimal. Leaf size=31 \[ \frac{1}{12} (2 x+3)^3 (2 d-3 e)+\frac{1}{16} e (2 x+3)^4 \]
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Rubi [A] time = 0.0206185, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {27, 43} \[ \frac{1}{12} (2 x+3)^3 (2 d-3 e)+\frac{1}{16} e (2 x+3)^4 \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int (d+e x) \left (9+12 x+4 x^2\right ) \, dx &=\int (3+2 x)^2 (d+e x) \, dx\\ &=\int \left (\frac{1}{2} (2 d-3 e) (3+2 x)^2+\frac{1}{2} e (3+2 x)^3\right ) \, dx\\ &=\frac{1}{12} (2 d-3 e) (3+2 x)^3+\frac{1}{16} e (3+2 x)^4\\ \end{align*}
Mathematica [A] time = 0.005751, size = 36, normalized size = 1.16 \[ \frac{4}{3} x^3 (d+3 e)+\frac{3}{2} x^2 (4 d+3 e)+9 d x+e x^4 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.038, size = 35, normalized size = 1.1 \begin{align*} e{x}^{4}+{\frac{ \left ( 4\,d+12\,e \right ){x}^{3}}{3}}+{\frac{ \left ( 12\,d+9\,e \right ){x}^{2}}{2}}+9\,dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11375, size = 43, normalized size = 1.39 \begin{align*} e x^{4} + \frac{4}{3} \,{\left (d + 3 \, e\right )} x^{3} + \frac{3}{2} \,{\left (4 \, d + 3 \, e\right )} x^{2} + 9 \, d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.36334, size = 80, normalized size = 2.58 \begin{align*} x^{4} e + 4 x^{3} e + \frac{4}{3} x^{3} d + \frac{9}{2} x^{2} e + 6 x^{2} d + 9 x d \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.077802, size = 32, normalized size = 1.03 \begin{align*} 9 d x + e x^{4} + x^{3} \left (\frac{4 d}{3} + 4 e\right ) + x^{2} \left (6 d + \frac{9 e}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11945, size = 50, normalized size = 1.61 \begin{align*} x^{4} e + \frac{4}{3} \, d x^{3} + 4 \, x^{3} e + 6 \, d x^{2} + \frac{9}{2} \, x^{2} e + 9 \, d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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