Integrand size = 18, antiderivative size = 60 \[ \int (d+e x) \left (9+12 x+4 x^2\right )^p \, dx=\frac {(2 d-3 e) (3+2 x) \left (9+12 x+4 x^2\right )^p}{4 (1+2 p)}+\frac {e \left (9+12 x+4 x^2\right )^{1+p}}{8 (1+p)} \]
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Time = 0.01 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {654, 623} \[ \int (d+e x) \left (9+12 x+4 x^2\right )^p \, dx=\frac {(2 x+3) (2 d-3 e) \left (4 x^2+12 x+9\right )^p}{4 (2 p+1)}+\frac {e \left (4 x^2+12 x+9\right )^{p+1}}{8 (p+1)} \]
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Rule 623
Rule 654
Rubi steps \begin{align*} \text {integral}& = \frac {e \left (9+12 x+4 x^2\right )^{1+p}}{8 (1+p)}+\frac {1}{2} (2 d-3 e) \int \left (9+12 x+4 x^2\right )^p \, dx \\ & = \frac {(2 d-3 e) (3+2 x) \left (9+12 x+4 x^2\right )^p}{4 (1+2 p)}+\frac {e \left (9+12 x+4 x^2\right )^{1+p}}{8 (1+p)} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.80 \[ \int (d+e x) \left (9+12 x+4 x^2\right )^p \, dx=\frac {(3+2 x) \left ((3+2 x)^2\right )^p (4 d (1+p)+e (-3+(2+4 p) x))}{8 (1+p) (1+2 p)} \]
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Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 2.19 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.72
method | result | size |
meijerg | \(\frac {e 3^{2 p} x^{2} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (2,-2 p ;3;-\frac {2 x}{3}\right )}{2}+d 3^{2 p} x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (1,-2 p ;2;-\frac {2 x}{3}\right )\) | \(43\) |
gosper | \(\frac {\left (4 x^{2}+12 x +9\right )^{p} \left (4 e p x +4 d p +2 e x +4 d -3 e \right ) \left (2 x +3\right )}{16 p^{2}+24 p +8}\) | \(52\) |
risch | \(\frac {\left (8 e p \,x^{2}+8 d p x +12 e p x +4 e \,x^{2}+12 d p +8 d x +12 d -9 e \right ) \left (\left (2 x +3\right )^{2}\right )^{p}}{8 \left (1+p \right ) \left (1+2 p \right )}\) | \(62\) |
norman | \(\frac {3 \left (4 d p +4 d -3 e \right ) {\mathrm e}^{p \ln \left (4 x^{2}+12 x +9\right )}}{8 \left (2 p^{2}+3 p +1\right )}+\frac {e \,x^{2} {\mathrm e}^{p \ln \left (4 x^{2}+12 x +9\right )}}{2+2 p}+\frac {\left (2 d p +3 p e +2 d \right ) x \,{\mathrm e}^{p \ln \left (4 x^{2}+12 x +9\right )}}{4 p^{2}+6 p +2}\) | \(107\) |
parallelrisch | \(\frac {8 x^{2} \left (4 x^{2}+12 x +9\right )^{p} e p +4 x^{2} \left (4 x^{2}+12 x +9\right )^{p} e +8 x \left (4 x^{2}+12 x +9\right )^{p} d p +12 x \left (4 x^{2}+12 x +9\right )^{p} e p +8 x \left (4 x^{2}+12 x +9\right )^{p} d +12 \left (4 x^{2}+12 x +9\right )^{p} d p +12 \left (4 x^{2}+12 x +9\right )^{p} d -9 e \left (4 x^{2}+12 x +9\right )^{p}}{16 p^{2}+24 p +8}\) | \(149\) |
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Time = 0.31 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.07 \[ \int (d+e x) \left (9+12 x+4 x^2\right )^p \, dx=\frac {{\left (4 \, {\left (2 \, e p + e\right )} x^{2} + 12 \, d p + 4 \, {\left ({\left (2 \, d + 3 \, e\right )} p + 2 \, d\right )} x + 12 \, d - 9 \, e\right )} {\left (4 \, x^{2} + 12 \, x + 9\right )}^{p}}{8 \, {\left (2 \, p^{2} + 3 \, p + 1\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 321 vs. \(2 (49) = 98\).
Time = 0.81 (sec) , antiderivative size = 321, normalized size of antiderivative = 5.35 \[ \int (d+e x) \left (9+12 x+4 x^2\right )^p \, dx=\begin {cases} - \frac {2 d}{8 x + 12} + \frac {2 e x \log {\left (x + \frac {3}{2} \right )}}{8 x + 12} + \frac {3 e \log {\left (x + \frac {3}{2} \right )}}{8 x + 12} + \frac {3 e}{8 x + 12} & \text {for}\: p = -1 \\\frac {e \sqrt {4 x^{2} + 12 x + 9}}{4} + \frac {\left (d - \frac {3 e}{2}\right ) \left (x + \frac {3}{2}\right ) \log {\left (x + \frac {3}{2} \right )}}{2 \sqrt {\left (x + \frac {3}{2}\right )^{2}}} & \text {for}\: p = - \frac {1}{2} \\\frac {8 d p x \left (4 x^{2} + 12 x + 9\right )^{p}}{16 p^{2} + 24 p + 8} + \frac {12 d p \left (4 x^{2} + 12 x + 9\right )^{p}}{16 p^{2} + 24 p + 8} + \frac {8 d x \left (4 x^{2} + 12 x + 9\right )^{p}}{16 p^{2} + 24 p + 8} + \frac {12 d \left (4 x^{2} + 12 x + 9\right )^{p}}{16 p^{2} + 24 p + 8} + \frac {8 e p x^{2} \left (4 x^{2} + 12 x + 9\right )^{p}}{16 p^{2} + 24 p + 8} + \frac {12 e p x \left (4 x^{2} + 12 x + 9\right )^{p}}{16 p^{2} + 24 p + 8} + \frac {4 e x^{2} \left (4 x^{2} + 12 x + 9\right )^{p}}{16 p^{2} + 24 p + 8} - \frac {9 e \left (4 x^{2} + 12 x + 9\right )^{p}}{16 p^{2} + 24 p + 8} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.08 \[ \int (d+e x) \left (9+12 x+4 x^2\right )^p \, dx=\frac {{\left (4 \, {\left (2 \, p + 1\right )} x^{2} + 12 \, p x - 9\right )} e {\left (2 \, x + 3\right )}^{2 \, p}}{8 \, {\left (2 \, p^{2} + 3 \, p + 1\right )}} + \frac {d {\left (2 \, x + 3\right )}^{2 \, p} {\left (2 \, x + 3\right )}}{2 \, {\left (2 \, p + 1\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (56) = 112\).
Time = 0.28 (sec) , antiderivative size = 148, normalized size of antiderivative = 2.47 \[ \int (d+e x) \left (9+12 x+4 x^2\right )^p \, dx=\frac {8 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{p} e p x^{2} + 8 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{p} d p x + 12 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{p} e p x + 4 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{p} e x^{2} + 12 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{p} d p + 8 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{p} d x + 12 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{p} d - 9 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}^{p} e}{8 \, {\left (2 \, p^{2} + 3 \, p + 1\right )}} \]
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Time = 0.11 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.45 \[ \int (d+e x) \left (9+12 x+4 x^2\right )^p \, dx=\left (\frac {12\,d-9\,e+12\,d\,p}{16\,p^2+24\,p+8}+\frac {x\,\left (8\,d+8\,d\,p+12\,e\,p\right )}{16\,p^2+24\,p+8}+\frac {4\,e\,x^2\,\left (2\,p+1\right )}{16\,p^2+24\,p+8}\right )\,{\left (4\,x^2+12\,x+9\right )}^p \]
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