Integrand size = 35, antiderivative size = 111 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=-\frac {\left (c d^2-a e^2\right )^3 (d+e x)^6}{6 e^4}+\frac {3 c d \left (c d^2-a e^2\right )^2 (d+e x)^7}{7 e^4}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^8}{8 e^4}+\frac {c^3 d^3 (d+e x)^9}{9 e^4} \]
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Time = 0.16 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 45} \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=-\frac {3 c^2 d^2 (d+e x)^8 \left (c d^2-a e^2\right )}{8 e^4}+\frac {3 c d (d+e x)^7 \left (c d^2-a e^2\right )^2}{7 e^4}-\frac {(d+e x)^6 \left (c d^2-a e^2\right )^3}{6 e^4}+\frac {c^3 d^3 (d+e x)^9}{9 e^4} \]
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Rule 45
Rule 640
Rubi steps \begin{align*} \text {integral}& = \int (a e+c d x)^3 (d+e x)^5 \, dx \\ & = \int \left (\frac {\left (-c d^2+a e^2\right )^3 (d+e x)^5}{e^3}+\frac {3 c d \left (c d^2-a e^2\right )^2 (d+e x)^6}{e^3}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^7}{e^3}+\frac {c^3 d^3 (d+e x)^8}{e^3}\right ) \, dx \\ & = -\frac {\left (c d^2-a e^2\right )^3 (d+e x)^6}{6 e^4}+\frac {3 c d \left (c d^2-a e^2\right )^2 (d+e x)^7}{7 e^4}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^8}{8 e^4}+\frac {c^3 d^3 (d+e x)^9}{9 e^4} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(255\) vs. \(2(111)=222\).
Time = 0.05 (sec) , antiderivative size = 255, normalized size of antiderivative = 2.30 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=\frac {1}{504} x \left (84 a^3 e^3 \left (6 d^5+15 d^4 e x+20 d^3 e^2 x^2+15 d^2 e^3 x^3+6 d e^4 x^4+e^5 x^5\right )+36 a^2 c d e^2 x \left (21 d^5+70 d^4 e x+105 d^3 e^2 x^2+84 d^2 e^3 x^3+35 d e^4 x^4+6 e^5 x^5\right )+9 a c^2 d^2 e x^2 \left (56 d^5+210 d^4 e x+336 d^3 e^2 x^2+280 d^2 e^3 x^3+120 d e^4 x^4+21 e^5 x^5\right )+c^3 d^3 x^3 \left (126 d^5+504 d^4 e x+840 d^3 e^2 x^2+720 d^2 e^3 x^3+315 d e^4 x^4+56 e^5 x^5\right )\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(298\) vs. \(2(103)=206\).
Time = 2.41 (sec) , antiderivative size = 299, normalized size of antiderivative = 2.69
method | result | size |
norman | \(\frac {c^{3} d^{3} e^{5} x^{9}}{9}+\left (\frac {3}{8} a \,c^{2} d^{2} e^{6}+\frac {5}{8} c^{3} d^{4} e^{4}\right ) x^{8}+\left (\frac {3}{7} d \,e^{7} a^{2} c +\frac {15}{7} a \,c^{2} d^{3} e^{5}+\frac {10}{7} c^{3} d^{5} e^{3}\right ) x^{7}+\left (\frac {1}{6} a^{3} e^{8}+\frac {5}{2} a^{2} c \,d^{2} e^{6}+5 a \,c^{2} d^{4} e^{4}+\frac {5}{3} c^{3} d^{6} e^{2}\right ) x^{6}+\left (a^{3} d \,e^{7}+6 a^{2} c \,d^{3} e^{5}+6 a \,c^{2} d^{5} e^{3}+c^{3} d^{7} e \right ) x^{5}+\left (\frac {5}{2} a^{3} e^{6} d^{2}+\frac {15}{2} d^{4} e^{4} a^{2} c +\frac {15}{4} a \,c^{2} d^{6} e^{2}+\frac {1}{4} c^{3} d^{8}\right ) x^{4}+\left (\frac {10}{3} a^{3} e^{5} d^{3}+5 d^{5} e^{3} a^{2} c +a \,c^{2} d^{7} e \right ) x^{3}+\left (\frac {5}{2} a^{3} e^{4} d^{4}+\frac {3}{2} d^{6} e^{2} a^{2} c \right ) x^{2}+a^{3} e^{3} d^{5} x\) | \(299\) |
risch | \(\frac {1}{9} c^{3} d^{3} e^{5} x^{9}+\frac {3}{8} x^{8} a \,c^{2} d^{2} e^{6}+\frac {5}{8} x^{8} c^{3} d^{4} e^{4}+\frac {3}{7} x^{7} d \,e^{7} a^{2} c +\frac {15}{7} x^{7} a \,c^{2} d^{3} e^{5}+\frac {10}{7} x^{7} c^{3} d^{5} e^{3}+\frac {1}{6} x^{6} a^{3} e^{8}+\frac {5}{2} x^{6} a^{2} c \,d^{2} e^{6}+5 x^{6} a \,c^{2} d^{4} e^{4}+\frac {5}{3} x^{6} c^{3} d^{6} e^{2}+a^{3} d \,e^{7} x^{5}+6 a^{2} c \,d^{3} e^{5} x^{5}+6 a \,c^{2} d^{5} e^{3} x^{5}+c^{3} d^{7} e \,x^{5}+\frac {5}{2} x^{4} a^{3} e^{6} d^{2}+\frac {15}{2} x^{4} d^{4} e^{4} a^{2} c +\frac {15}{4} x^{4} a \,c^{2} d^{6} e^{2}+\frac {1}{4} x^{4} c^{3} d^{8}+\frac {10}{3} x^{3} a^{3} e^{5} d^{3}+5 x^{3} d^{5} e^{3} a^{2} c +x^{3} a \,c^{2} d^{7} e +\frac {5}{2} x^{2} a^{3} e^{4} d^{4}+\frac {3}{2} x^{2} d^{6} e^{2} a^{2} c +a^{3} e^{3} d^{5} x\) | \(330\) |
parallelrisch | \(\frac {1}{9} c^{3} d^{3} e^{5} x^{9}+\frac {3}{8} x^{8} a \,c^{2} d^{2} e^{6}+\frac {5}{8} x^{8} c^{3} d^{4} e^{4}+\frac {3}{7} x^{7} d \,e^{7} a^{2} c +\frac {15}{7} x^{7} a \,c^{2} d^{3} e^{5}+\frac {10}{7} x^{7} c^{3} d^{5} e^{3}+\frac {1}{6} x^{6} a^{3} e^{8}+\frac {5}{2} x^{6} a^{2} c \,d^{2} e^{6}+5 x^{6} a \,c^{2} d^{4} e^{4}+\frac {5}{3} x^{6} c^{3} d^{6} e^{2}+a^{3} d \,e^{7} x^{5}+6 a^{2} c \,d^{3} e^{5} x^{5}+6 a \,c^{2} d^{5} e^{3} x^{5}+c^{3} d^{7} e \,x^{5}+\frac {5}{2} x^{4} a^{3} e^{6} d^{2}+\frac {15}{2} x^{4} d^{4} e^{4} a^{2} c +\frac {15}{4} x^{4} a \,c^{2} d^{6} e^{2}+\frac {1}{4} x^{4} c^{3} d^{8}+\frac {10}{3} x^{3} a^{3} e^{5} d^{3}+5 x^{3} d^{5} e^{3} a^{2} c +x^{3} a \,c^{2} d^{7} e +\frac {5}{2} x^{2} a^{3} e^{4} d^{4}+\frac {3}{2} x^{2} d^{6} e^{2} a^{2} c +a^{3} e^{3} d^{5} x\) | \(330\) |
gosper | \(\frac {x \left (56 c^{3} d^{3} e^{5} x^{8}+189 x^{7} a \,c^{2} d^{2} e^{6}+315 x^{7} c^{3} d^{4} e^{4}+216 x^{6} d \,e^{7} a^{2} c +1080 x^{6} a \,c^{2} d^{3} e^{5}+720 x^{6} c^{3} d^{5} e^{3}+84 x^{5} a^{3} e^{8}+1260 x^{5} a^{2} c \,d^{2} e^{6}+2520 x^{5} a \,c^{2} d^{4} e^{4}+840 x^{5} c^{3} d^{6} e^{2}+504 a^{3} d \,e^{7} x^{4}+3024 a^{2} c \,d^{3} e^{5} x^{4}+3024 a \,c^{2} d^{5} e^{3} x^{4}+504 c^{3} d^{7} e \,x^{4}+1260 x^{3} a^{3} e^{6} d^{2}+3780 x^{3} d^{4} e^{4} a^{2} c +1890 x^{3} a \,c^{2} d^{6} e^{2}+126 x^{3} c^{3} d^{8}+1680 x^{2} a^{3} e^{5} d^{3}+2520 x^{2} d^{5} e^{3} a^{2} c +504 x^{2} a \,c^{2} d^{7} e +1260 x \,a^{3} e^{4} d^{4}+756 x \,d^{6} e^{2} a^{2} c +504 a^{3} e^{3} d^{5}\right )}{504}\) | \(332\) |
default | \(\frac {c^{3} d^{3} e^{5} x^{9}}{9}+\frac {\left (2 c^{3} d^{4} e^{4}+3 e^{4} \left (e^{2} a +c \,d^{2}\right ) c^{2} d^{2}\right ) x^{8}}{8}+\frac {\left (c^{3} d^{5} e^{3}+6 d^{3} e^{3} \left (e^{2} a +c \,d^{2}\right ) c^{2}+e^{2} \left (a \,d^{3} e^{3} c^{2}+2 \left (e^{2} a +c \,d^{2}\right )^{2} c d e +c d e \left (2 a c \,d^{2} e^{2}+\left (e^{2} a +c \,d^{2}\right )^{2}\right )\right )\right ) x^{7}}{7}+\frac {\left (3 d^{4} \left (e^{2} a +c \,d^{2}\right ) c^{2} e^{2}+2 d e \left (a \,d^{3} e^{3} c^{2}+2 \left (e^{2} a +c \,d^{2}\right )^{2} c d e +c d e \left (2 a c \,d^{2} e^{2}+\left (e^{2} a +c \,d^{2}\right )^{2}\right )\right )+e^{2} \left (4 a \,d^{2} e^{2} c \left (e^{2} a +c \,d^{2}\right )+\left (e^{2} a +c \,d^{2}\right ) \left (2 a c \,d^{2} e^{2}+\left (e^{2} a +c \,d^{2}\right )^{2}\right )\right )\right ) x^{6}}{6}+\frac {\left (d^{2} \left (a \,d^{3} e^{3} c^{2}+2 \left (e^{2} a +c \,d^{2}\right )^{2} c d e +c d e \left (2 a c \,d^{2} e^{2}+\left (e^{2} a +c \,d^{2}\right )^{2}\right )\right )+2 d e \left (4 a \,d^{2} e^{2} c \left (e^{2} a +c \,d^{2}\right )+\left (e^{2} a +c \,d^{2}\right ) \left (2 a c \,d^{2} e^{2}+\left (e^{2} a +c \,d^{2}\right )^{2}\right )\right )+e^{2} \left (a d e \left (2 a c \,d^{2} e^{2}+\left (e^{2} a +c \,d^{2}\right )^{2}\right )+2 \left (e^{2} a +c \,d^{2}\right )^{2} a d e +c \,d^{3} e^{3} a^{2}\right )\right ) x^{5}}{5}+\frac {\left (d^{2} \left (4 a \,d^{2} e^{2} c \left (e^{2} a +c \,d^{2}\right )+\left (e^{2} a +c \,d^{2}\right ) \left (2 a c \,d^{2} e^{2}+\left (e^{2} a +c \,d^{2}\right )^{2}\right )\right )+2 d e \left (a d e \left (2 a c \,d^{2} e^{2}+\left (e^{2} a +c \,d^{2}\right )^{2}\right )+2 \left (e^{2} a +c \,d^{2}\right )^{2} a d e +c \,d^{3} e^{3} a^{2}\right )+3 e^{4} a^{2} d^{2} \left (e^{2} a +c \,d^{2}\right )\right ) x^{4}}{4}+\frac {\left (d^{2} \left (a d e \left (2 a c \,d^{2} e^{2}+\left (e^{2} a +c \,d^{2}\right )^{2}\right )+2 \left (e^{2} a +c \,d^{2}\right )^{2} a d e +c \,d^{3} e^{3} a^{2}\right )+6 d^{3} e^{3} a^{2} \left (e^{2} a +c \,d^{2}\right )+a^{3} e^{5} d^{3}\right ) x^{3}}{3}+\frac {\left (3 d^{4} a^{2} e^{2} \left (e^{2} a +c \,d^{2}\right )+2 a^{3} e^{4} d^{4}\right ) x^{2}}{2}+a^{3} e^{3} d^{5} x\) | \(801\) |
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Leaf count of result is larger than twice the leaf count of optimal. 303 vs. \(2 (103) = 206\).
Time = 0.25 (sec) , antiderivative size = 303, normalized size of antiderivative = 2.73 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=\frac {1}{9} \, c^{3} d^{3} e^{5} x^{9} + a^{3} d^{5} e^{3} x + \frac {1}{8} \, {\left (5 \, c^{3} d^{4} e^{4} + 3 \, a c^{2} d^{2} e^{6}\right )} x^{8} + \frac {1}{7} \, {\left (10 \, c^{3} d^{5} e^{3} + 15 \, a c^{2} d^{3} e^{5} + 3 \, a^{2} c d e^{7}\right )} x^{7} + \frac {1}{6} \, {\left (10 \, c^{3} d^{6} e^{2} + 30 \, a c^{2} d^{4} e^{4} + 15 \, a^{2} c d^{2} e^{6} + a^{3} e^{8}\right )} x^{6} + {\left (c^{3} d^{7} e + 6 \, a c^{2} d^{5} e^{3} + 6 \, a^{2} c d^{3} e^{5} + a^{3} d e^{7}\right )} x^{5} + \frac {1}{4} \, {\left (c^{3} d^{8} + 15 \, a c^{2} d^{6} e^{2} + 30 \, a^{2} c d^{4} e^{4} + 10 \, a^{3} d^{2} e^{6}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, a c^{2} d^{7} e + 15 \, a^{2} c d^{5} e^{3} + 10 \, a^{3} d^{3} e^{5}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} c d^{6} e^{2} + 5 \, a^{3} d^{4} e^{4}\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 335 vs. \(2 (100) = 200\).
Time = 0.05 (sec) , antiderivative size = 335, normalized size of antiderivative = 3.02 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=a^{3} d^{5} e^{3} x + \frac {c^{3} d^{3} e^{5} x^{9}}{9} + x^{8} \cdot \left (\frac {3 a c^{2} d^{2} e^{6}}{8} + \frac {5 c^{3} d^{4} e^{4}}{8}\right ) + x^{7} \cdot \left (\frac {3 a^{2} c d e^{7}}{7} + \frac {15 a c^{2} d^{3} e^{5}}{7} + \frac {10 c^{3} d^{5} e^{3}}{7}\right ) + x^{6} \left (\frac {a^{3} e^{8}}{6} + \frac {5 a^{2} c d^{2} e^{6}}{2} + 5 a c^{2} d^{4} e^{4} + \frac {5 c^{3} d^{6} e^{2}}{3}\right ) + x^{5} \left (a^{3} d e^{7} + 6 a^{2} c d^{3} e^{5} + 6 a c^{2} d^{5} e^{3} + c^{3} d^{7} e\right ) + x^{4} \cdot \left (\frac {5 a^{3} d^{2} e^{6}}{2} + \frac {15 a^{2} c d^{4} e^{4}}{2} + \frac {15 a c^{2} d^{6} e^{2}}{4} + \frac {c^{3} d^{8}}{4}\right ) + x^{3} \cdot \left (\frac {10 a^{3} d^{3} e^{5}}{3} + 5 a^{2} c d^{5} e^{3} + a c^{2} d^{7} e\right ) + x^{2} \cdot \left (\frac {5 a^{3} d^{4} e^{4}}{2} + \frac {3 a^{2} c d^{6} e^{2}}{2}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 303 vs. \(2 (103) = 206\).
Time = 0.23 (sec) , antiderivative size = 303, normalized size of antiderivative = 2.73 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=\frac {1}{9} \, c^{3} d^{3} e^{5} x^{9} + a^{3} d^{5} e^{3} x + \frac {1}{8} \, {\left (5 \, c^{3} d^{4} e^{4} + 3 \, a c^{2} d^{2} e^{6}\right )} x^{8} + \frac {1}{7} \, {\left (10 \, c^{3} d^{5} e^{3} + 15 \, a c^{2} d^{3} e^{5} + 3 \, a^{2} c d e^{7}\right )} x^{7} + \frac {1}{6} \, {\left (10 \, c^{3} d^{6} e^{2} + 30 \, a c^{2} d^{4} e^{4} + 15 \, a^{2} c d^{2} e^{6} + a^{3} e^{8}\right )} x^{6} + {\left (c^{3} d^{7} e + 6 \, a c^{2} d^{5} e^{3} + 6 \, a^{2} c d^{3} e^{5} + a^{3} d e^{7}\right )} x^{5} + \frac {1}{4} \, {\left (c^{3} d^{8} + 15 \, a c^{2} d^{6} e^{2} + 30 \, a^{2} c d^{4} e^{4} + 10 \, a^{3} d^{2} e^{6}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, a c^{2} d^{7} e + 15 \, a^{2} c d^{5} e^{3} + 10 \, a^{3} d^{3} e^{5}\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a^{2} c d^{6} e^{2} + 5 \, a^{3} d^{4} e^{4}\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 329 vs. \(2 (103) = 206\).
Time = 0.27 (sec) , antiderivative size = 329, normalized size of antiderivative = 2.96 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=\frac {1}{9} \, c^{3} d^{3} e^{5} x^{9} + \frac {5}{8} \, c^{3} d^{4} e^{4} x^{8} + \frac {3}{8} \, a c^{2} d^{2} e^{6} x^{8} + \frac {10}{7} \, c^{3} d^{5} e^{3} x^{7} + \frac {15}{7} \, a c^{2} d^{3} e^{5} x^{7} + \frac {3}{7} \, a^{2} c d e^{7} x^{7} + \frac {5}{3} \, c^{3} d^{6} e^{2} x^{6} + 5 \, a c^{2} d^{4} e^{4} x^{6} + \frac {5}{2} \, a^{2} c d^{2} e^{6} x^{6} + \frac {1}{6} \, a^{3} e^{8} x^{6} + c^{3} d^{7} e x^{5} + 6 \, a c^{2} d^{5} e^{3} x^{5} + 6 \, a^{2} c d^{3} e^{5} x^{5} + a^{3} d e^{7} x^{5} + \frac {1}{4} \, c^{3} d^{8} x^{4} + \frac {15}{4} \, a c^{2} d^{6} e^{2} x^{4} + \frac {15}{2} \, a^{2} c d^{4} e^{4} x^{4} + \frac {5}{2} \, a^{3} d^{2} e^{6} x^{4} + a c^{2} d^{7} e x^{3} + 5 \, a^{2} c d^{5} e^{3} x^{3} + \frac {10}{3} \, a^{3} d^{3} e^{5} x^{3} + \frac {3}{2} \, a^{2} c d^{6} e^{2} x^{2} + \frac {5}{2} \, a^{3} d^{4} e^{4} x^{2} + a^{3} d^{5} e^{3} x \]
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Time = 0.11 (sec) , antiderivative size = 295, normalized size of antiderivative = 2.66 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=x^4\,\left (\frac {5\,a^3\,d^2\,e^6}{2}+\frac {15\,a^2\,c\,d^4\,e^4}{2}+\frac {15\,a\,c^2\,d^6\,e^2}{4}+\frac {c^3\,d^8}{4}\right )+x^6\,\left (\frac {a^3\,e^8}{6}+\frac {5\,a^2\,c\,d^2\,e^6}{2}+5\,a\,c^2\,d^4\,e^4+\frac {5\,c^3\,d^6\,e^2}{3}\right )+x^5\,\left (a^3\,d\,e^7+6\,a^2\,c\,d^3\,e^5+6\,a\,c^2\,d^5\,e^3+c^3\,d^7\,e\right )+a^3\,d^5\,e^3\,x+\frac {c^3\,d^3\,e^5\,x^9}{9}+\frac {a\,d^3\,e\,x^3\,\left (10\,a^2\,e^4+15\,a\,c\,d^2\,e^2+3\,c^2\,d^4\right )}{3}+\frac {c\,d\,e^3\,x^7\,\left (3\,a^2\,e^4+15\,a\,c\,d^2\,e^2+10\,c^2\,d^4\right )}{7}+\frac {a^2\,d^4\,e^2\,x^2\,\left (3\,c\,d^2+5\,a\,e^2\right )}{2}+\frac {c^2\,d^2\,e^4\,x^8\,\left (5\,c\,d^2+3\,a\,e^2\right )}{8} \]
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