Optimal. Leaf size=50 \[ \frac{1}{24} (2 x+3) \left (4 x^2+12 x+9\right )^{5/2} (2 d-3 e)+\frac{1}{28} e \left (4 x^2+12 x+9\right )^{7/2} \]
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Rubi [A] time = 0.0127645, 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} \[ \frac{1}{24} (2 x+3) \left (4 x^2+12 x+9\right )^{5/2} (2 d-3 e)+\frac{1}{28} e \left (4 x^2+12 x+9\right )^{7/2} \]
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 )^{5/2} \, dx &=\frac{1}{28} e \left (9+12 x+4 x^2\right )^{7/2}+\frac{1}{2} (2 d-3 e) \int \left (9+12 x+4 x^2\right )^{5/2} \, dx\\ &=\frac{1}{24} (2 d-3 e) (3+2 x) \left (9+12 x+4 x^2\right )^{5/2}+\frac{1}{28} e \left (9+12 x+4 x^2\right )^{7/2}\\ \end{align*}
Mathematica [A] time = 0.0311574, size = 81, normalized size = 1.62 \[ \frac{x \sqrt{(2 x+3)^2} \left (14 d \left (16 x^5+144 x^4+540 x^3+1080 x^2+1215 x+729\right )+3 e x \left (64 x^5+560 x^4+2016 x^3+3780 x^2+3780 x+1701\right )\right )}{42 (2 x+3)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.081, size = 86, normalized size = 1.7 \begin{align*}{\frac{x \left ( 192\,e{x}^{6}+224\,{x}^{5}d+1680\,{x}^{5}e+2016\,d{x}^{4}+6048\,e{x}^{4}+7560\,d{x}^{3}+11340\,{x}^{3}e+15120\,d{x}^{2}+11340\,e{x}^{2}+17010\,dx+5103\,ex+10206\,d \right ) }{42\, \left ( 3+2\,x \right ) ^{5}} \left ( \left ( 3+2\,x \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.60213, size = 105, normalized size = 2.1 \begin{align*} \frac{1}{28} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{7}{2}} e + \frac{1}{6} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{5}{2}} d x - \frac{1}{4} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{5}{2}} e x + \frac{1}{4} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{5}{2}} d - \frac{3}{8} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{5}{2}} e \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47588, size = 176, normalized size = 3.52 \begin{align*} \frac{32}{7} \, e x^{7} + \frac{8}{3} \,{\left (2 \, d + 15 \, e\right )} x^{6} + 48 \,{\left (d + 3 \, e\right )} x^{5} + 90 \,{\left (2 \, d + 3 \, e\right )} x^{4} + 90 \,{\left (4 \, d + 3 \, e\right )} x^{3} + \frac{81}{2} \,{\left (10 \, d + 3 \, e\right )} x^{2} + 243 \, 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{5}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15301, size = 223, normalized size = 4.46 \begin{align*} \frac{32}{7} \, x^{7} e \mathrm{sgn}\left (2 \, x + 3\right ) + \frac{16}{3} \, d x^{6} \mathrm{sgn}\left (2 \, x + 3\right ) + 40 \, x^{6} e \mathrm{sgn}\left (2 \, x + 3\right ) + 48 \, d x^{5} \mathrm{sgn}\left (2 \, x + 3\right ) + 144 \, x^{5} e \mathrm{sgn}\left (2 \, x + 3\right ) + 180 \, d x^{4} \mathrm{sgn}\left (2 \, x + 3\right ) + 270 \, x^{4} e \mathrm{sgn}\left (2 \, x + 3\right ) + 360 \, d x^{3} \mathrm{sgn}\left (2 \, x + 3\right ) + 270 \, x^{3} e \mathrm{sgn}\left (2 \, x + 3\right ) + 405 \, d x^{2} \mathrm{sgn}\left (2 \, x + 3\right ) + \frac{243}{2} \, x^{2} e \mathrm{sgn}\left (2 \, x + 3\right ) + 243 \, d x \mathrm{sgn}\left (2 \, x + 3\right ) + \frac{243}{56} \,{\left (14 \, 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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