Optimal. Leaf size=31 \[ \frac{1}{28} (2 x+3)^7 (2 d-3 e)+\frac{1}{32} e (2 x+3)^8 \]
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Rubi [A] time = 0.0115803, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {27, 43} \[ \frac{1}{28} (2 x+3)^7 (2 d-3 e)+\frac{1}{32} e (2 x+3)^8 \]
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 )^3 \, dx &=\int (3+2 x)^6 (d+e x) \, dx\\ &=\int \left (\frac{1}{2} (2 d-3 e) (3+2 x)^6+\frac{1}{2} e (3+2 x)^7\right ) \, dx\\ &=\frac{1}{28} (2 d-3 e) (3+2 x)^7+\frac{1}{32} e (3+2 x)^8\\ \end{align*}
Mathematica [B] time = 0.0150449, size = 81, normalized size = 2.61 \[ \frac{64}{7} x^7 (d+9 e)+24 x^6 (4 d+15 e)+432 x^5 (d+2 e)+135 x^4 (8 d+9 e)+324 x^3 (5 d+3 e)+\frac{729}{2} x^2 (4 d+e)+729 d x+8 e x^8 \]
Antiderivative was successfully verified.
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Maple [B] time = 0.04, size = 84, normalized size = 2.7 \begin{align*} 8\,e{x}^{8}+{\frac{ \left ( 64\,d+576\,e \right ){x}^{7}}{7}}+{\frac{ \left ( 576\,d+2160\,e \right ){x}^{6}}{6}}+{\frac{ \left ( 2160\,d+4320\,e \right ){x}^{5}}{5}}+{\frac{ \left ( 4320\,d+4860\,e \right ){x}^{4}}{4}}+{\frac{ \left ( 4860\,d+2916\,e \right ){x}^{3}}{3}}+{\frac{ \left ( 2916\,d+729\,e \right ){x}^{2}}{2}}+729\,dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.09033, size = 104, normalized size = 3.35 \begin{align*} 8 \, e x^{8} + \frac{64}{7} \,{\left (d + 9 \, e\right )} x^{7} + 24 \,{\left (4 \, d + 15 \, e\right )} x^{6} + 432 \,{\left (d + 2 \, e\right )} x^{5} + 135 \,{\left (8 \, d + 9 \, e\right )} x^{4} + 324 \,{\left (5 \, d + 3 \, e\right )} x^{3} + \frac{729}{2} \,{\left (4 \, d + e\right )} x^{2} + 729 \, d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.34382, size = 231, normalized size = 7.45 \begin{align*} 8 x^{8} e + \frac{576}{7} x^{7} e + \frac{64}{7} x^{7} d + 360 x^{6} e + 96 x^{6} d + 864 x^{5} e + 432 x^{5} d + 1215 x^{4} e + 1080 x^{4} d + 972 x^{3} e + 1620 x^{3} d + \frac{729}{2} x^{2} e + 1458 x^{2} d + 729 x d \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.089094, size = 76, normalized size = 2.45 \begin{align*} 729 d x + 8 e x^{8} + x^{7} \left (\frac{64 d}{7} + \frac{576 e}{7}\right ) + x^{6} \left (96 d + 360 e\right ) + x^{5} \left (432 d + 864 e\right ) + x^{4} \left (1080 d + 1215 e\right ) + x^{3} \left (1620 d + 972 e\right ) + x^{2} \left (1458 d + \frac{729 e}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13701, size = 122, normalized size = 3.94 \begin{align*} 8 \, x^{8} e + \frac{64}{7} \, d x^{7} + \frac{576}{7} \, x^{7} e + 96 \, d x^{6} + 360 \, x^{6} e + 432 \, d x^{5} + 864 \, x^{5} e + 1080 \, d x^{4} + 1215 \, x^{4} e + 1620 \, d x^{3} + 972 \, x^{3} e + 1458 \, d x^{2} + \frac{729}{2} \, x^{2} e + 729 \, d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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