Optimal. Leaf size=50 \[ \frac{1}{8} (2 x+3) \sqrt{4 x^2+12 x+9} (2 d-3 e)+\frac{1}{12} e \left (4 x^2+12 x+9\right )^{3/2} \]
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Rubi [A] time = 0.0120345, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {640, 609} \[ \frac{1}{8} (2 x+3) \sqrt{4 x^2+12 x+9} (2 d-3 e)+\frac{1}{12} e \left (4 x^2+12 x+9\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 640
Rule 609
Rubi steps
\begin{align*} \int (d+e x) \sqrt{9+12 x+4 x^2} \, dx &=\frac{1}{12} e \left (9+12 x+4 x^2\right )^{3/2}+\frac{1}{2} (2 d-3 e) \int \sqrt{9+12 x+4 x^2} \, dx\\ &=\frac{1}{8} (2 d-3 e) (3+2 x) \sqrt{9+12 x+4 x^2}+\frac{1}{12} e \left (9+12 x+4 x^2\right )^{3/2}\\ \end{align*}
Mathematica [A] time = 0.0115699, size = 38, normalized size = 0.76 \[ \frac{x \sqrt{(2 x+3)^2} (6 d (x+3)+e x (4 x+9))}{6 (2 x+3)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.079, size = 38, normalized size = 0.8 \begin{align*}{\frac{x \left ( 4\,e{x}^{2}+6\,dx+9\,ex+18\,d \right ) }{18+12\,x}\sqrt{ \left ( 3+2\,x \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.59159, size = 105, normalized size = 2.1 \begin{align*} \frac{1}{12} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{3}{2}} e + \frac{1}{2} \, \sqrt{4 \, x^{2} + 12 \, x + 9} d x - \frac{3}{4} \, \sqrt{4 \, x^{2} + 12 \, x + 9} e x + \frac{3}{4} \, \sqrt{4 \, x^{2} + 12 \, x + 9} d - \frac{9}{8} \, \sqrt{4 \, x^{2} + 12 \, x + 9} e \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55504, size = 55, normalized size = 1.1 \begin{align*} \frac{2}{3} \, e x^{3} + \frac{1}{2} \,{\left (2 \, d + 3 \, e\right )} x^{2} + 3 \, d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d + e x\right ) \sqrt{\left (2 x + 3\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18821, size = 86, normalized size = 1.72 \begin{align*} \frac{2}{3} \, x^{3} e \mathrm{sgn}\left (2 \, x + 3\right ) + d x^{2} \mathrm{sgn}\left (2 \, x + 3\right ) + \frac{3}{2} \, x^{2} e \mathrm{sgn}\left (2 \, x + 3\right ) + 3 \, d x \mathrm{sgn}\left (2 \, x + 3\right ) + \frac{9}{8} \,{\left (2 \, d - e\right )} \mathrm{sgn}\left (2 \, x + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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