Integrand size = 21, antiderivative size = 16 \[ \int x^7 \left (b+d x^2\right )^7 \left (b+3 d x^2\right ) \, dx=\frac {1}{8} x^8 \left (b+d x^2\right )^8 \]
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Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {457, 75} \[ \int x^7 \left (b+d x^2\right )^7 \left (b+3 d x^2\right ) \, dx=\frac {1}{8} x^8 \left (b+d x^2\right )^8 \]
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Rule 75
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int x^3 (b+d x)^7 (b+3 d x) \, dx,x,x^2\right ) \\ & = \frac {1}{8} x^8 \left (b+d x^2\right )^8 \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(98\) vs. \(2(16)=32\).
Time = 0.00 (sec) , antiderivative size = 98, normalized size of antiderivative = 6.12 \[ \int x^7 \left (b+d x^2\right )^7 \left (b+3 d x^2\right ) \, dx=\frac {b^8 x^8}{8}+b^7 d x^{10}+\frac {7}{2} b^6 d^2 x^{12}+7 b^5 d^3 x^{14}+\frac {35}{4} b^4 d^4 x^{16}+7 b^3 d^5 x^{18}+\frac {7}{2} b^2 d^6 x^{20}+b d^7 x^{22}+\frac {d^8 x^{24}}{8} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(88\) vs. \(2(14)=28\).
Time = 0.77 (sec) , antiderivative size = 89, normalized size of antiderivative = 5.56
method | result | size |
gosper | \(\frac {1}{8} d^{8} x^{24}+x^{22} b \,d^{7}+7 x^{18} b^{3} d^{5}+\frac {7}{2} x^{20} b^{2} d^{6}+7 x^{14} b^{5} d^{3}+\frac {35}{4} x^{16} b^{4} d^{4}+\frac {1}{8} x^{8} b^{8}+x^{10} b^{7} d +\frac {7}{2} x^{12} b^{6} d^{2}\) | \(89\) |
default | \(\frac {1}{8} d^{8} x^{24}+x^{22} b \,d^{7}+7 x^{18} b^{3} d^{5}+\frac {7}{2} x^{20} b^{2} d^{6}+7 x^{14} b^{5} d^{3}+\frac {35}{4} x^{16} b^{4} d^{4}+\frac {1}{8} x^{8} b^{8}+x^{10} b^{7} d +\frac {7}{2} x^{12} b^{6} d^{2}\) | \(89\) |
norman | \(\frac {1}{8} d^{8} x^{24}+x^{22} b \,d^{7}+7 x^{18} b^{3} d^{5}+\frac {7}{2} x^{20} b^{2} d^{6}+7 x^{14} b^{5} d^{3}+\frac {35}{4} x^{16} b^{4} d^{4}+\frac {1}{8} x^{8} b^{8}+x^{10} b^{7} d +\frac {7}{2} x^{12} b^{6} d^{2}\) | \(89\) |
risch | \(\frac {1}{8} d^{8} x^{24}+x^{22} b \,d^{7}+7 x^{18} b^{3} d^{5}+\frac {7}{2} x^{20} b^{2} d^{6}+7 x^{14} b^{5} d^{3}+\frac {35}{4} x^{16} b^{4} d^{4}+\frac {1}{8} x^{8} b^{8}+x^{10} b^{7} d +\frac {7}{2} x^{12} b^{6} d^{2}\) | \(89\) |
parallelrisch | \(\frac {1}{8} d^{8} x^{24}+x^{22} b \,d^{7}+7 x^{18} b^{3} d^{5}+\frac {7}{2} x^{20} b^{2} d^{6}+7 x^{14} b^{5} d^{3}+\frac {35}{4} x^{16} b^{4} d^{4}+\frac {1}{8} x^{8} b^{8}+x^{10} b^{7} d +\frac {7}{2} x^{12} b^{6} d^{2}\) | \(89\) |
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Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (14) = 28\).
Time = 0.25 (sec) , antiderivative size = 88, normalized size of antiderivative = 5.50 \[ \int x^7 \left (b+d x^2\right )^7 \left (b+3 d x^2\right ) \, dx=\frac {1}{8} \, d^{8} x^{24} + b d^{7} x^{22} + \frac {7}{2} \, b^{2} d^{6} x^{20} + 7 \, b^{3} d^{5} x^{18} + \frac {35}{4} \, b^{4} d^{4} x^{16} + 7 \, b^{5} d^{3} x^{14} + \frac {7}{2} \, b^{6} d^{2} x^{12} + b^{7} d x^{10} + \frac {1}{8} \, b^{8} x^{8} \]
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Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (12) = 24\).
Time = 0.03 (sec) , antiderivative size = 97, normalized size of antiderivative = 6.06 \[ \int x^7 \left (b+d x^2\right )^7 \left (b+3 d x^2\right ) \, dx=\frac {b^{8} x^{8}}{8} + b^{7} d x^{10} + \frac {7 b^{6} d^{2} x^{12}}{2} + 7 b^{5} d^{3} x^{14} + \frac {35 b^{4} d^{4} x^{16}}{4} + 7 b^{3} d^{5} x^{18} + \frac {7 b^{2} d^{6} x^{20}}{2} + b d^{7} x^{22} + \frac {d^{8} x^{24}}{8} \]
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Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (14) = 28\).
Time = 0.20 (sec) , antiderivative size = 88, normalized size of antiderivative = 5.50 \[ \int x^7 \left (b+d x^2\right )^7 \left (b+3 d x^2\right ) \, dx=\frac {1}{8} \, d^{8} x^{24} + b d^{7} x^{22} + \frac {7}{2} \, b^{2} d^{6} x^{20} + 7 \, b^{3} d^{5} x^{18} + \frac {35}{4} \, b^{4} d^{4} x^{16} + 7 \, b^{5} d^{3} x^{14} + \frac {7}{2} \, b^{6} d^{2} x^{12} + b^{7} d x^{10} + \frac {1}{8} \, b^{8} x^{8} \]
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none
Time = 0.33 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81 \[ \int x^7 \left (b+d x^2\right )^7 \left (b+3 d x^2\right ) \, dx=\frac {1}{8} \, {\left (d x^{3} + b x\right )}^{8} \]
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Time = 0.02 (sec) , antiderivative size = 88, normalized size of antiderivative = 5.50 \[ \int x^7 \left (b+d x^2\right )^7 \left (b+3 d x^2\right ) \, dx=\frac {b^8\,x^8}{8}+b^7\,d\,x^{10}+\frac {7\,b^6\,d^2\,x^{12}}{2}+7\,b^5\,d^3\,x^{14}+\frac {35\,b^4\,d^4\,x^{16}}{4}+7\,b^3\,d^5\,x^{18}+\frac {7\,b^2\,d^6\,x^{20}}{2}+b\,d^7\,x^{22}+\frac {d^8\,x^{24}}{8} \]
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