Integrand size = 33, antiderivative size = 39 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {1}{4} \left (a-\frac {c d^2}{e^2}\right ) (d+e x)^4+\frac {c d (d+e x)^5}{5 e^2} \]
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Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {640, 45} \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {1}{4} (d+e x)^4 \left (a-\frac {c d^2}{e^2}\right )+\frac {c d (d+e x)^5}{5 e^2} \]
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Rule 45
Rule 640
Rubi steps \begin{align*} \text {integral}& = \int (a e+c d x) (d+e x)^3 \, dx \\ & = \int \left (\frac {\left (-c d^2+a e^2\right ) (d+e x)^3}{e}+\frac {c d (d+e x)^4}{e}\right ) \, dx \\ & = \frac {1}{4} \left (a-\frac {c d^2}{e^2}\right ) (d+e x)^4+\frac {c d (d+e x)^5}{5 e^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.87 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {1}{20} x \left (5 a e \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+c d x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(75\) vs. \(2(35)=70\).
Time = 2.18 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.95
method | result | size |
norman | \(\frac {d \,e^{3} c \,x^{5}}{5}+\left (\frac {1}{4} e^{4} a +\frac {3}{4} d^{2} e^{2} c \right ) x^{4}+\left (a d \,e^{3}+c \,d^{3} e \right ) x^{3}+\left (\frac {3}{2} a \,d^{2} e^{2}+\frac {1}{2} c \,d^{4}\right ) x^{2}+a \,d^{3} e x\) | \(76\) |
risch | \(\frac {1}{5} d \,e^{3} c \,x^{5}+\frac {1}{4} x^{4} e^{4} a +\frac {3}{4} x^{4} d^{2} e^{2} c +x^{3} a d \,e^{3}+c \,d^{3} e \,x^{3}+\frac {3}{2} x^{2} a \,d^{2} e^{2}+\frac {1}{2} x^{2} c \,d^{4}+a \,d^{3} e x\) | \(79\) |
parallelrisch | \(\frac {1}{5} d \,e^{3} c \,x^{5}+\frac {1}{4} x^{4} e^{4} a +\frac {3}{4} x^{4} d^{2} e^{2} c +x^{3} a d \,e^{3}+c \,d^{3} e \,x^{3}+\frac {3}{2} x^{2} a \,d^{2} e^{2}+\frac {1}{2} x^{2} c \,d^{4}+a \,d^{3} e x\) | \(79\) |
gosper | \(\frac {x \left (4 d \,e^{3} c \,x^{4}+5 a \,e^{4} x^{3}+15 x^{3} d^{2} e^{2} c +20 a d \,e^{3} x^{2}+20 c \,d^{3} e \,x^{2}+30 a \,d^{2} e^{2} x +10 x c \,d^{4}+20 a \,d^{3} e \right )}{20}\) | \(80\) |
default | \(\frac {d \,e^{3} c \,x^{5}}{5}+\frac {\left (2 d^{2} e^{2} c +e^{2} \left (e^{2} a +c \,d^{2}\right )\right ) x^{4}}{4}+\frac {\left (c \,d^{3} e +2 d e \left (e^{2} a +c \,d^{2}\right )+a d \,e^{3}\right ) x^{3}}{3}+\frac {\left (d^{2} \left (e^{2} a +c \,d^{2}\right )+2 a \,d^{2} e^{2}\right ) x^{2}}{2}+a \,d^{3} e x\) | \(112\) |
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Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (35) = 70\).
Time = 0.27 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.92 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {1}{5} \, c d e^{3} x^{5} + a d^{3} e x + \frac {1}{4} \, {\left (3 \, c d^{2} e^{2} + a e^{4}\right )} x^{4} + {\left (c d^{3} e + a d e^{3}\right )} x^{3} + \frac {1}{2} \, {\left (c d^{4} + 3 \, a d^{2} e^{2}\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (34) = 68\).
Time = 0.03 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.05 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=a d^{3} e x + \frac {c d e^{3} x^{5}}{5} + x^{4} \left (\frac {a e^{4}}{4} + \frac {3 c d^{2} e^{2}}{4}\right ) + x^{3} \left (a d e^{3} + c d^{3} e\right ) + x^{2} \cdot \left (\frac {3 a d^{2} e^{2}}{2} + \frac {c d^{4}}{2}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (35) = 70\).
Time = 0.19 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.92 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {1}{5} \, c d e^{3} x^{5} + a d^{3} e x + \frac {1}{4} \, {\left (3 \, c d^{2} e^{2} + a e^{4}\right )} x^{4} + {\left (c d^{3} e + a d e^{3}\right )} x^{3} + \frac {1}{2} \, {\left (c d^{4} + 3 \, a d^{2} e^{2}\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (35) = 70\).
Time = 0.28 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.00 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {1}{5} \, c d e^{3} x^{5} + \frac {3}{4} \, c d^{2} e^{2} x^{4} + \frac {1}{4} \, a e^{4} x^{4} + c d^{3} e x^{3} + a d e^{3} x^{3} + \frac {1}{2} \, c d^{4} x^{2} + \frac {3}{2} \, a d^{2} e^{2} x^{2} + a d^{3} e x \]
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Time = 9.60 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.92 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=x^2\,\left (\frac {c\,d^4}{2}+\frac {3\,a\,d^2\,e^2}{2}\right )+x^4\,\left (\frac {3\,c\,d^2\,e^2}{4}+\frac {a\,e^4}{4}\right )+x^3\,\left (c\,d^3\,e+a\,d\,e^3\right )+a\,d^3\,e\,x+\frac {c\,d\,e^3\,x^5}{5} \]
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