Integrand size = 31, antiderivative size = 39 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {1}{3} \left (a-\frac {c d^2}{e^2}\right ) (d+e x)^3+\frac {c d (d+e x)^4}{4 e^2} \]
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Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {640, 45} \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {1}{3} (d+e x)^3 \left (a-\frac {c d^2}{e^2}\right )+\frac {c d (d+e x)^4}{4 e^2} \]
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Rule 45
Rule 640
Rubi steps \begin{align*} \text {integral}& = \int (a e+c d x) (d+e x)^2 \, dx \\ & = \int \left (\frac {\left (-c d^2+a e^2\right ) (d+e x)^2}{e}+\frac {c d (d+e x)^3}{e}\right ) \, dx \\ & = \frac {1}{3} \left (a-\frac {c d^2}{e^2}\right ) (d+e x)^3+\frac {c d (d+e x)^4}{4 e^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.31 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {1}{12} x \left (4 a e \left (3 d^2+3 d e x+e^2 x^2\right )+c d x \left (6 d^2+8 d e x+3 e^2 x^2\right )\right ) \]
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Time = 0.21 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.38
method | result | size |
norman | \(\frac {c d \,e^{2} x^{4}}{4}+\left (\frac {1}{3} a \,e^{3}+\frac {2}{3} d^{2} e c \right ) x^{3}+\left (a d \,e^{2}+\frac {1}{2} d^{3} c \right ) x^{2}+x a e \,d^{2}\) | \(54\) |
gosper | \(\frac {x \left (3 c d \,e^{2} x^{3}+4 a \,e^{3} x^{2}+8 c \,d^{2} e \,x^{2}+12 a d \,e^{2} x +6 c \,d^{3} x +12 a e \,d^{2}\right )}{12}\) | \(56\) |
risch | \(\frac {1}{4} c d \,e^{2} x^{4}+\frac {1}{3} x^{3} a \,e^{3}+\frac {2}{3} x^{3} d^{2} e c +a d \,e^{2} x^{2}+\frac {1}{2} c \,d^{3} x^{2}+x a e \,d^{2}\) | \(56\) |
parallelrisch | \(\frac {1}{4} c d \,e^{2} x^{4}+\frac {1}{3} x^{3} a \,e^{3}+\frac {2}{3} x^{3} d^{2} e c +a d \,e^{2} x^{2}+\frac {1}{2} c \,d^{3} x^{2}+x a e \,d^{2}\) | \(56\) |
default | \(\frac {c d \,e^{2} x^{4}}{4}+\frac {\left (d^{2} e c +e \left (e^{2} a +c \,d^{2}\right )\right ) x^{3}}{3}+\frac {\left (d \left (e^{2} a +c \,d^{2}\right )+a d \,e^{2}\right ) x^{2}}{2}+x a e \,d^{2}\) | \(69\) |
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Time = 0.31 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.38 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {1}{4} \, c d e^{2} x^{4} + a d^{2} e x + \frac {1}{3} \, {\left (2 \, c d^{2} e + a e^{3}\right )} x^{3} + \frac {1}{2} \, {\left (c d^{3} + 2 \, a d e^{2}\right )} x^{2} \]
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Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.44 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=a d^{2} e x + \frac {c d e^{2} x^{4}}{4} + x^{3} \left (\frac {a e^{3}}{3} + \frac {2 c d^{2} e}{3}\right ) + x^{2} \left (a d e^{2} + \frac {c d^{3}}{2}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.38 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {1}{4} \, c d e^{2} x^{4} + a d^{2} e x + \frac {1}{3} \, {\left (2 \, c d^{2} e + a e^{3}\right )} x^{3} + \frac {1}{2} \, {\left (c d^{3} + 2 \, a d e^{2}\right )} x^{2} \]
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Time = 0.28 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.41 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {1}{4} \, c d e^{2} x^{4} + \frac {2}{3} \, c d^{2} e x^{3} + \frac {1}{3} \, a e^{3} x^{3} + \frac {1}{2} \, c d^{3} x^{2} + a d e^{2} x^{2} + a d^{2} e x \]
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Time = 0.05 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.36 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=x^2\,\left (\frac {c\,d^3}{2}+a\,d\,e^2\right )+x^3\,\left (\frac {2\,c\,d^2\,e}{3}+\frac {a\,e^3}{3}\right )+a\,d^2\,e\,x+\frac {c\,d\,e^2\,x^4}{4} \]
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