Integrand size = 17, antiderivative size = 55 \[ \int (d+e x) \left (b x+c x^2\right )^2 \, dx=\frac {1}{3} b^2 d x^3+\frac {1}{4} b (2 c d+b e) x^4+\frac {1}{5} c (c d+2 b e) x^5+\frac {1}{6} c^2 e x^6 \]
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Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {645} \[ \int (d+e x) \left (b x+c x^2\right )^2 \, dx=\frac {1}{3} b^2 d x^3+\frac {1}{5} c x^5 (2 b e+c d)+\frac {1}{4} b x^4 (b e+2 c d)+\frac {1}{6} c^2 e x^6 \]
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Rule 645
Rubi steps \begin{align*} \text {integral}& = \int \left (b^2 d x^2+b (2 c d+b e) x^3+c (c d+2 b e) x^4+c^2 e x^5\right ) \, dx \\ & = \frac {1}{3} b^2 d x^3+\frac {1}{4} b (2 c d+b e) x^4+\frac {1}{5} c (c d+2 b e) x^5+\frac {1}{6} c^2 e x^6 \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.91 \[ \int (d+e x) \left (b x+c x^2\right )^2 \, dx=\frac {1}{60} x^3 \left (5 b^2 (4 d+3 e x)+6 b c x (5 d+4 e x)+2 c^2 x^2 (6 d+5 e x)\right ) \]
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Time = 2.08 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.95
method | result | size |
gosper | \(\frac {x^{3} \left (10 c^{2} e \,x^{3}+24 x^{2} b c e +12 c^{2} d \,x^{2}+15 b^{2} e x +30 b c d x +20 b^{2} d \right )}{60}\) | \(52\) |
default | \(\frac {c^{2} e \,x^{6}}{6}+\frac {\left (2 b c e +c^{2} d \right ) x^{5}}{5}+\frac {\left (b^{2} e +2 b c d \right ) x^{4}}{4}+\frac {d \,x^{3} b^{2}}{3}\) | \(52\) |
norman | \(\frac {c^{2} e \,x^{6}}{6}+\left (\frac {2}{5} b c e +\frac {1}{5} c^{2} d \right ) x^{5}+\left (\frac {1}{4} b^{2} e +\frac {1}{2} b c d \right ) x^{4}+\frac {d \,x^{3} b^{2}}{3}\) | \(52\) |
risch | \(\frac {1}{6} c^{2} e \,x^{6}+\frac {2}{5} x^{5} b c e +\frac {1}{5} c^{2} d \,x^{5}+\frac {1}{4} b^{2} e \,x^{4}+\frac {1}{2} c b d \,x^{4}+\frac {1}{3} d \,x^{3} b^{2}\) | \(54\) |
parallelrisch | \(\frac {1}{6} c^{2} e \,x^{6}+\frac {2}{5} x^{5} b c e +\frac {1}{5} c^{2} d \,x^{5}+\frac {1}{4} b^{2} e \,x^{4}+\frac {1}{2} c b d \,x^{4}+\frac {1}{3} d \,x^{3} b^{2}\) | \(54\) |
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Time = 0.26 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int (d+e x) \left (b x+c x^2\right )^2 \, dx=\frac {1}{6} \, c^{2} e x^{6} + \frac {1}{3} \, b^{2} d x^{3} + \frac {1}{5} \, {\left (c^{2} d + 2 \, b c e\right )} x^{5} + \frac {1}{4} \, {\left (2 \, b c d + b^{2} e\right )} x^{4} \]
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Time = 0.02 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.98 \[ \int (d+e x) \left (b x+c x^2\right )^2 \, dx=\frac {b^{2} d x^{3}}{3} + \frac {c^{2} e x^{6}}{6} + x^{5} \cdot \left (\frac {2 b c e}{5} + \frac {c^{2} d}{5}\right ) + x^{4} \left (\frac {b^{2} e}{4} + \frac {b c d}{2}\right ) \]
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Time = 0.22 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int (d+e x) \left (b x+c x^2\right )^2 \, dx=\frac {1}{6} \, c^{2} e x^{6} + \frac {1}{3} \, b^{2} d x^{3} + \frac {1}{5} \, {\left (c^{2} d + 2 \, b c e\right )} x^{5} + \frac {1}{4} \, {\left (2 \, b c d + b^{2} e\right )} x^{4} \]
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Time = 0.27 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.96 \[ \int (d+e x) \left (b x+c x^2\right )^2 \, dx=\frac {1}{6} \, c^{2} e x^{6} + \frac {1}{5} \, c^{2} d x^{5} + \frac {2}{5} \, b c e x^{5} + \frac {1}{2} \, b c d x^{4} + \frac {1}{4} \, b^{2} e x^{4} + \frac {1}{3} \, b^{2} d x^{3} \]
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Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int (d+e x) \left (b x+c x^2\right )^2 \, dx=x^4\,\left (\frac {e\,b^2}{4}+\frac {c\,d\,b}{2}\right )+x^5\,\left (\frac {d\,c^2}{5}+\frac {2\,b\,e\,c}{5}\right )+\frac {b^2\,d\,x^3}{3}+\frac {c^2\,e\,x^6}{6} \]
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