Integrand size = 25, antiderivative size = 17 \[ \int \left (2 c x+3 d x^2\right ) \left (c x^2+d x^3\right )^7 \, dx=\frac {1}{8} \left (c x^2+d x^3\right )^8 \]
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Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {1602} \[ \int \left (2 c x+3 d x^2\right ) \left (c x^2+d x^3\right )^7 \, dx=\frac {1}{8} \left (c x^2+d x^3\right )^8 \]
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Rule 1602
Rubi steps \begin{align*} \text {integral}& = \frac {1}{8} \left (c x^2+d x^3\right )^8 \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(98\) vs. \(2(17)=34\).
Time = 0.01 (sec) , antiderivative size = 98, normalized size of antiderivative = 5.76 \[ \int \left (2 c x+3 d x^2\right ) \left (c x^2+d x^3\right )^7 \, dx=\frac {c^8 x^{16}}{8}+c^7 d x^{17}+\frac {7}{2} c^6 d^2 x^{18}+7 c^5 d^3 x^{19}+\frac {35}{4} c^4 d^4 x^{20}+7 c^3 d^5 x^{21}+\frac {7}{2} c^2 d^6 x^{22}+c d^7 x^{23}+\frac {d^8 x^{24}}{8} \]
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Time = 0.72 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76
method | result | size |
gosper | \(\frac {x^{16} \left (d x +c \right )^{8}}{8}\) | \(13\) |
default | \(\frac {\left (x^{3} d +c \,x^{2}\right )^{8}}{8}\) | \(16\) |
norman | \(\frac {7}{2} x^{22} c^{2} d^{6}+c \,d^{7} x^{23}+\frac {1}{8} d^{8} x^{24}+\frac {1}{8} x^{16} c^{8}+c^{7} d \,x^{17}+\frac {7}{2} x^{18} c^{6} d^{2}+7 c^{5} d^{3} x^{19}+\frac {35}{4} x^{20} c^{4} d^{4}+7 c^{3} d^{5} x^{21}\) | \(89\) |
risch | \(\frac {7}{2} x^{22} c^{2} d^{6}+c \,d^{7} x^{23}+\frac {1}{8} d^{8} x^{24}+\frac {1}{8} x^{16} c^{8}+c^{7} d \,x^{17}+\frac {7}{2} x^{18} c^{6} d^{2}+7 c^{5} d^{3} x^{19}+\frac {35}{4} x^{20} c^{4} d^{4}+7 c^{3} d^{5} x^{21}\) | \(89\) |
parallelrisch | \(\frac {7}{2} x^{22} c^{2} d^{6}+c \,d^{7} x^{23}+\frac {1}{8} d^{8} x^{24}+\frac {1}{8} x^{16} c^{8}+c^{7} d \,x^{17}+\frac {7}{2} x^{18} c^{6} d^{2}+7 c^{5} d^{3} x^{19}+\frac {35}{4} x^{20} c^{4} d^{4}+7 c^{3} d^{5} x^{21}\) | \(89\) |
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Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (15) = 30\).
Time = 0.23 (sec) , antiderivative size = 88, normalized size of antiderivative = 5.18 \[ \int \left (2 c x+3 d x^2\right ) \left (c x^2+d x^3\right )^7 \, dx=\frac {1}{8} \, d^{8} x^{24} + c d^{7} x^{23} + \frac {7}{2} \, c^{2} d^{6} x^{22} + 7 \, c^{3} d^{5} x^{21} + \frac {35}{4} \, c^{4} d^{4} x^{20} + 7 \, c^{5} d^{3} x^{19} + \frac {7}{2} \, c^{6} d^{2} x^{18} + c^{7} d x^{17} + \frac {1}{8} \, c^{8} x^{16} \]
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Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (12) = 24\).
Time = 0.04 (sec) , antiderivative size = 97, normalized size of antiderivative = 5.71 \[ \int \left (2 c x+3 d x^2\right ) \left (c x^2+d x^3\right )^7 \, dx=\frac {c^{8} x^{16}}{8} + c^{7} d x^{17} + \frac {7 c^{6} d^{2} x^{18}}{2} + 7 c^{5} d^{3} x^{19} + \frac {35 c^{4} d^{4} x^{20}}{4} + 7 c^{3} d^{5} x^{21} + \frac {7 c^{2} d^{6} x^{22}}{2} + c d^{7} x^{23} + \frac {d^{8} x^{24}}{8} \]
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none
Time = 0.20 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \left (2 c x+3 d x^2\right ) \left (c x^2+d x^3\right )^7 \, dx=\frac {1}{8} \, {\left (d x^{3} + c x^{2}\right )}^{8} \]
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Time = 0.38 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \left (2 c x+3 d x^2\right ) \left (c x^2+d x^3\right )^7 \, dx=\frac {1}{8} \, {\left (d x^{3} + c x^{2}\right )}^{8} \]
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Time = 9.33 (sec) , antiderivative size = 88, normalized size of antiderivative = 5.18 \[ \int \left (2 c x+3 d x^2\right ) \left (c x^2+d x^3\right )^7 \, dx=\frac {c^8\,x^{16}}{8}+c^7\,d\,x^{17}+\frac {7\,c^6\,d^2\,x^{18}}{2}+7\,c^5\,d^3\,x^{19}+\frac {35\,c^4\,d^4\,x^{20}}{4}+7\,c^3\,d^5\,x^{21}+\frac {7\,c^2\,d^6\,x^{22}}{2}+c\,d^7\,x^{23}+\frac {d^8\,x^{24}}{8} \]
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