Optimal. Leaf size=31 \[ \frac{1}{20} (2 x+3)^5 (2 d-3 e)+\frac{1}{24} e (2 x+3)^6 \]
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Rubi [A] time = 0.0108392, 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} \[ \frac{1}{20} (2 x+3)^5 (2 d-3 e)+\frac{1}{24} e (2 x+3)^6 \]
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 )^2 \, dx &=\int (3+2 x)^4 (d+e x) \, dx\\ &=\int \left (\frac{1}{2} (2 d-3 e) (3+2 x)^4+\frac{1}{2} e (3+2 x)^5\right ) \, dx\\ &=\frac{1}{20} (2 d-3 e) (3+2 x)^5+\frac{1}{24} e (3+2 x)^6\\ \end{align*}
Mathematica [A] time = 0.0092227, size = 59, normalized size = 1.9 \[ \frac{16}{5} x^5 (d+6 e)+6 x^4 (4 d+9 e)+72 x^3 (d+e)+\frac{27}{2} x^2 (8 d+3 e)+81 d x+\frac{8 e x^6}{3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.039, size = 60, normalized size = 1.9 \begin{align*}{\frac{8\,e{x}^{6}}{3}}+{\frac{ \left ( 16\,d+96\,e \right ){x}^{5}}{5}}+{\frac{ \left ( 96\,d+216\,e \right ){x}^{4}}{4}}+{\frac{ \left ( 216\,d+216\,e \right ){x}^{3}}{3}}+{\frac{ \left ( 216\,d+81\,e \right ){x}^{2}}{2}}+81\,dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16255, size = 72, normalized size = 2.32 \begin{align*} \frac{8}{3} \, e x^{6} + \frac{16}{5} \,{\left (d + 6 \, e\right )} x^{5} + 6 \,{\left (4 \, d + 9 \, e\right )} x^{4} + 72 \,{\left (d + e\right )} x^{3} + \frac{27}{2} \,{\left (8 \, d + 3 \, e\right )} x^{2} + 81 \, d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.33961, size = 155, normalized size = 5. \begin{align*} \frac{8}{3} x^{6} e + \frac{96}{5} x^{5} e + \frac{16}{5} x^{5} d + 54 x^{4} e + 24 x^{4} d + 72 x^{3} e + 72 x^{3} d + \frac{81}{2} x^{2} e + 108 x^{2} d + 81 x d \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.077843, size = 58, normalized size = 1.87 \begin{align*} 81 d x + \frac{8 e x^{6}}{3} + x^{5} \left (\frac{16 d}{5} + \frac{96 e}{5}\right ) + x^{4} \left (24 d + 54 e\right ) + x^{3} \left (72 d + 72 e\right ) + x^{2} \left (108 d + \frac{81 e}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.18605, size = 86, normalized size = 2.77 \begin{align*} \frac{8}{3} \, x^{6} e + \frac{16}{5} \, d x^{5} + \frac{96}{5} \, x^{5} e + 24 \, d x^{4} + 54 \, x^{4} e + 72 \, d x^{3} + 72 \, x^{3} e + 108 \, d x^{2} + \frac{81}{2} \, x^{2} e + 81 \, d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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