Optimal. Leaf size=50 \[ \frac{1}{16} (2 x+3) \left (4 x^2+12 x+9\right )^{3/2} (2 d-3 e)+\frac{1}{20} e \left (4 x^2+12 x+9\right )^{5/2} \]
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Rubi [A] time = 0.0121757, 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} \[ \frac{1}{16} (2 x+3) \left (4 x^2+12 x+9\right )^{3/2} (2 d-3 e)+\frac{1}{20} e \left (4 x^2+12 x+9\right )^{5/2} \]
Antiderivative was successfully verified.
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Rule 640
Rule 609
Rubi steps
\begin{align*} \int (d+e x) \left (9+12 x+4 x^2\right )^{3/2} \, dx &=\frac{1}{20} e \left (9+12 x+4 x^2\right )^{5/2}+\frac{1}{2} (2 d-3 e) \int \left (9+12 x+4 x^2\right )^{3/2} \, dx\\ &=\frac{1}{16} (2 d-3 e) (3+2 x) \left (9+12 x+4 x^2\right )^{3/2}+\frac{1}{20} e \left (9+12 x+4 x^2\right )^{5/2}\\ \end{align*}
Mathematica [A] time = 0.0218325, size = 57, normalized size = 1.14 \[ \frac{x \sqrt{(2 x+3)^2} \left (10 d \left (2 x^3+12 x^2+27 x+27\right )+e x \left (16 x^3+90 x^2+180 x+135\right )\right )}{20 x+30} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.08, size = 62, normalized size = 1.2 \begin{align*}{\frac{x \left ( 16\,e{x}^{4}+20\,d{x}^{3}+90\,{x}^{3}e+120\,d{x}^{2}+180\,e{x}^{2}+270\,dx+135\,ex+270\,d \right ) }{10\, \left ( 3+2\,x \right ) ^{3}} \left ( \left ( 3+2\,x \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.5557, size = 105, normalized size = 2.1 \begin{align*} \frac{1}{20} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{5}{2}} e + \frac{1}{4} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{3}{2}} d x - \frac{3}{8} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{3}{2}} e x + \frac{3}{8} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{3}{2}} d - \frac{9}{16} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{3}{2}} e \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58034, size = 107, normalized size = 2.14 \begin{align*} \frac{8}{5} \, e x^{5} +{\left (2 \, d + 9 \, e\right )} x^{4} + 6 \,{\left (2 \, d + 3 \, e\right )} x^{3} + \frac{27}{2} \,{\left (2 \, d + e\right )} x^{2} + 27 \, 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 ) \left (\left (2 x + 3\right )^{2}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19265, size = 155, normalized size = 3.1 \begin{align*} \frac{8}{5} \, x^{5} e \mathrm{sgn}\left (2 \, x + 3\right ) + 2 \, d x^{4} \mathrm{sgn}\left (2 \, x + 3\right ) + 9 \, x^{4} e \mathrm{sgn}\left (2 \, x + 3\right ) + 12 \, d x^{3} \mathrm{sgn}\left (2 \, x + 3\right ) + 18 \, x^{3} e \mathrm{sgn}\left (2 \, x + 3\right ) + 27 \, d x^{2} \mathrm{sgn}\left (2 \, x + 3\right ) + \frac{27}{2} \, x^{2} e \mathrm{sgn}\left (2 \, x + 3\right ) + 27 \, d x \mathrm{sgn}\left (2 \, x + 3\right ) + \frac{81}{80} \,{\left (10 \, d - 3 \, 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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