Integrand size = 29, antiderivative size = 14 \[ \int \frac {1}{\left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right )^3} \, dx=-\frac {1}{8 b (a+b x)^8} \]
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Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2083, 32} \[ \int \frac {1}{\left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right )^3} \, dx=-\frac {1}{8 b (a+b x)^8} \]
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Rule 32
Rule 2083
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(a+b x)^9} \, dx \\ & = -\frac {1}{8 b (a+b x)^8} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right )^3} \, dx=-\frac {1}{8 b (a+b x)^8} \]
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Time = 0.11 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93
method | result | size |
default | \(-\frac {1}{8 b \left (b x +a \right )^{8}}\) | \(13\) |
norman | \(-\frac {1}{8 b \left (b x +a \right )^{8}}\) | \(13\) |
risch | \(-\frac {1}{8 b \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{3} \left (b x +a \right )^{2}}\) | \(31\) |
gosper | \(-\frac {1}{8 \left (b^{2} x^{2}+2 a b x +a^{2}\right ) \left (b^{3} x^{3}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}\right )^{2} b}\) | \(53\) |
parallelrisch | \(-\frac {1}{8 \left (b^{2} x^{2}+2 a b x +a^{2}\right ) \left (b^{3} x^{3}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}\right )^{2} b}\) | \(53\) |
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Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (12) = 24\).
Time = 0.24 (sec) , antiderivative size = 90, normalized size of antiderivative = 6.43 \[ \int \frac {1}{\left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right )^3} \, dx=-\frac {1}{8 \, {\left (b^{9} x^{8} + 8 \, a b^{8} x^{7} + 28 \, a^{2} b^{7} x^{6} + 56 \, a^{3} b^{6} x^{5} + 70 \, a^{4} b^{5} x^{4} + 56 \, a^{5} b^{4} x^{3} + 28 \, a^{6} b^{3} x^{2} + 8 \, a^{7} b^{2} x + a^{8} b\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (12) = 24\).
Time = 0.25 (sec) , antiderivative size = 97, normalized size of antiderivative = 6.93 \[ \int \frac {1}{\left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right )^3} \, dx=- \frac {1}{8 a^{8} b + 64 a^{7} b^{2} x + 224 a^{6} b^{3} x^{2} + 448 a^{5} b^{4} x^{3} + 560 a^{4} b^{5} x^{4} + 448 a^{3} b^{6} x^{5} + 224 a^{2} b^{7} x^{6} + 64 a b^{8} x^{7} + 8 b^{9} x^{8}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (12) = 24\).
Time = 0.20 (sec) , antiderivative size = 90, normalized size of antiderivative = 6.43 \[ \int \frac {1}{\left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right )^3} \, dx=-\frac {1}{8 \, {\left (b^{9} x^{8} + 8 \, a b^{8} x^{7} + 28 \, a^{2} b^{7} x^{6} + 56 \, a^{3} b^{6} x^{5} + 70 \, a^{4} b^{5} x^{4} + 56 \, a^{5} b^{4} x^{3} + 28 \, a^{6} b^{3} x^{2} + 8 \, a^{7} b^{2} x + a^{8} b\right )}} \]
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none
Time = 0.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right )^3} \, dx=-\frac {1}{8 \, {\left (b x + a\right )}^{8} b} \]
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Time = 9.02 (sec) , antiderivative size = 92, normalized size of antiderivative = 6.57 \[ \int \frac {1}{\left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right )^3} \, dx=-\frac {1}{8\,a^8\,b+64\,a^7\,b^2\,x+224\,a^6\,b^3\,x^2+448\,a^5\,b^4\,x^3+560\,a^4\,b^5\,x^4+448\,a^3\,b^6\,x^5+224\,a^2\,b^7\,x^6+64\,a\,b^8\,x^7+8\,b^9\,x^8} \]
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