Integrand size = 7, antiderivative size = 14 \[ \int (c+d x)^7 \, dx=\frac {(c+d x)^8}{8 d} \]
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Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {32} \[ \int (c+d x)^7 \, dx=\frac {(c+d x)^8}{8 d} \]
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Rule 32
Rubi steps \begin{align*} \text {integral}& = \frac {(c+d x)^8}{8 d} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int (c+d x)^7 \, dx=\frac {(c+d x)^8}{8 d} \]
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Time = 0.19 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93
method | result | size |
default | \(\frac {\left (d x +c \right )^{8}}{8 d}\) | \(13\) |
gosper | \(\frac {1}{8} d^{7} x^{8}+c \,d^{6} x^{7}+\frac {7}{2} c^{2} d^{5} x^{6}+7 c^{3} d^{4} x^{5}+\frac {35}{4} c^{4} d^{3} x^{4}+7 c^{5} d^{2} x^{3}+\frac {7}{2} c^{6} d \,x^{2}+c^{7} x\) | \(76\) |
norman | \(\frac {1}{8} d^{7} x^{8}+c \,d^{6} x^{7}+\frac {7}{2} c^{2} d^{5} x^{6}+7 c^{3} d^{4} x^{5}+\frac {35}{4} c^{4} d^{3} x^{4}+7 c^{5} d^{2} x^{3}+\frac {7}{2} c^{6} d \,x^{2}+c^{7} x\) | \(76\) |
parallelrisch | \(\frac {1}{8} d^{7} x^{8}+c \,d^{6} x^{7}+\frac {7}{2} c^{2} d^{5} x^{6}+7 c^{3} d^{4} x^{5}+\frac {35}{4} c^{4} d^{3} x^{4}+7 c^{5} d^{2} x^{3}+\frac {7}{2} c^{6} d \,x^{2}+c^{7} x\) | \(76\) |
risch | \(\frac {d^{7} x^{8}}{8}+c \,d^{6} x^{7}+\frac {7 c^{2} d^{5} x^{6}}{2}+7 c^{3} d^{4} x^{5}+\frac {35 c^{4} d^{3} x^{4}}{4}+7 c^{5} d^{2} x^{3}+\frac {7 c^{6} d \,x^{2}}{2}+c^{7} x +\frac {c^{8}}{8 d}\) | \(84\) |
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Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (12) = 24\).
Time = 0.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 5.36 \[ \int (c+d x)^7 \, dx=\frac {1}{8} \, d^{7} x^{8} + c d^{6} x^{7} + \frac {7}{2} \, c^{2} d^{5} x^{6} + 7 \, c^{3} d^{4} x^{5} + \frac {35}{4} \, c^{4} d^{3} x^{4} + 7 \, c^{5} d^{2} x^{3} + \frac {7}{2} \, c^{6} d x^{2} + c^{7} x \]
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Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (8) = 16\).
Time = 0.03 (sec) , antiderivative size = 83, normalized size of antiderivative = 5.93 \[ \int (c+d x)^7 \, dx=c^{7} x + \frac {7 c^{6} d x^{2}}{2} + 7 c^{5} d^{2} x^{3} + \frac {35 c^{4} d^{3} x^{4}}{4} + 7 c^{3} d^{4} x^{5} + \frac {7 c^{2} d^{5} x^{6}}{2} + c d^{6} x^{7} + \frac {d^{7} x^{8}}{8} \]
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none
Time = 0.20 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int (c+d x)^7 \, dx=\frac {{\left (d x + c\right )}^{8}}{8 \, d} \]
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none
Time = 0.29 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int (c+d x)^7 \, dx=\frac {{\left (d x + c\right )}^{8}}{8 \, d} \]
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Time = 0.06 (sec) , antiderivative size = 75, normalized size of antiderivative = 5.36 \[ \int (c+d x)^7 \, dx=c^7\,x+\frac {7\,c^6\,d\,x^2}{2}+7\,c^5\,d^2\,x^3+\frac {35\,c^4\,d^3\,x^4}{4}+7\,c^3\,d^4\,x^5+\frac {7\,c^2\,d^5\,x^6}{2}+c\,d^6\,x^7+\frac {d^7\,x^8}{8} \]
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