Integrand size = 22, antiderivative size = 38 \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {(b d-a e) (a+b x)^3}{3 b^2}+\frac {e (a+b x)^4}{4 b^2} \]
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Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {27, 45} \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {(a+b x)^3 (b d-a e)}{3 b^2}+\frac {e (a+b x)^4}{4 b^2} \]
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Rule 27
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int (a+b x)^2 (d+e x) \, dx \\ & = \int \left (\frac {(b d-a e) (a+b x)^2}{b}+\frac {e (a+b x)^3}{b}\right ) \, dx \\ & = \frac {(b d-a e) (a+b x)^3}{3 b^2}+\frac {e (a+b x)^4}{4 b^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.21 \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {1}{12} x \left (6 a^2 (2 d+e x)+4 a b x (3 d+2 e x)+b^2 x^2 (4 d+3 e x)\right ) \]
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Time = 0.18 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.26
method | result | size |
norman | \(\frac {b^{2} e \,x^{4}}{4}+\left (\frac {2}{3} a e b +\frac {1}{3} b^{2} d \right ) x^{3}+\left (\frac {1}{2} a^{2} e +a b d \right ) x^{2}+a^{2} d x\) | \(48\) |
default | \(\frac {b^{2} e \,x^{4}}{4}+\frac {\left (2 a e b +b^{2} d \right ) x^{3}}{3}+\frac {\left (a^{2} e +2 a b d \right ) x^{2}}{2}+a^{2} d x\) | \(49\) |
gosper | \(\frac {x \left (3 b^{2} e \,x^{3}+8 a b e \,x^{2}+4 x^{2} b^{2} d +6 a^{2} e x +12 a x b d +12 a^{2} d \right )}{12}\) | \(50\) |
risch | \(\frac {1}{4} b^{2} e \,x^{4}+\frac {2}{3} a b e \,x^{3}+\frac {1}{3} d \,x^{3} b^{2}+\frac {1}{2} a^{2} e \,x^{2}+x^{2} a b d +a^{2} d x\) | \(50\) |
parallelrisch | \(\frac {1}{4} b^{2} e \,x^{4}+\frac {2}{3} a b e \,x^{3}+\frac {1}{3} d \,x^{3} b^{2}+\frac {1}{2} a^{2} e \,x^{2}+x^{2} a b d +a^{2} d x\) | \(50\) |
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Time = 0.24 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.26 \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {1}{4} \, b^{2} e x^{4} + a^{2} d x + \frac {1}{3} \, {\left (b^{2} d + 2 \, a b e\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a b d + a^{2} e\right )} x^{2} \]
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Time = 0.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.29 \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx=a^{2} d x + \frac {b^{2} e x^{4}}{4} + x^{3} \cdot \left (\frac {2 a b e}{3} + \frac {b^{2} d}{3}\right ) + x^{2} \left (\frac {a^{2} e}{2} + a b d\right ) \]
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none
Time = 0.19 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.26 \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {1}{4} \, b^{2} e x^{4} + a^{2} d x + \frac {1}{3} \, {\left (b^{2} d + 2 \, a b e\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a b d + a^{2} e\right )} x^{2} \]
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Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.29 \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {1}{4} \, b^{2} e x^{4} + \frac {1}{3} \, b^{2} d x^{3} + \frac {2}{3} \, a b e x^{3} + a b d x^{2} + \frac {1}{2} \, a^{2} e x^{2} + a^{2} d x \]
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Time = 0.05 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.24 \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx=x^2\,\left (\frac {e\,a^2}{2}+b\,d\,a\right )+x^3\,\left (\frac {d\,b^2}{3}+\frac {2\,a\,e\,b}{3}\right )+\frac {b^2\,e\,x^4}{4}+a^2\,d\,x \]
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