Integrand size = 51, antiderivative size = 14 \[ \int \frac {1}{\left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^3} \, dx=-\frac {1}{14 b (a+b x)^{14}} \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.039, Rules used = {2083, 32} \[ \int \frac {1}{\left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^3} \, dx=-\frac {1}{14 b (a+b x)^{14}} \]
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Rule 32
Rule 2083
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(a+b x)^{15}} \, dx \\ & = -\frac {1}{14 b (a+b x)^{14}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^3} \, dx=-\frac {1}{14 b (a+b x)^{14}} \]
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Time = 0.24 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93
method | result | size |
default | \(-\frac {1}{14 b \left (b x +a \right )^{14}}\) | \(13\) |
norman | \(-\frac {1}{14 b \left (b x +a \right )^{14}}\) | \(13\) |
risch | \(-\frac {1}{14 b \left (b^{4} x^{4}+4 a \,b^{3} x^{3}+6 a^{2} b^{2} x^{2}+4 a^{3} b x +a^{4}\right )^{3} \left (b x +a \right )^{2}}\) | \(53\) |
gosper | \(-\frac {1}{14 \left (b^{4} x^{4}+4 a \,b^{3} x^{3}+6 a^{2} b^{2} x^{2}+4 a^{3} b x +a^{4}\right ) \left (b^{5} x^{5}+5 a \,b^{4} x^{4}+10 a^{2} b^{3} x^{3}+10 a^{3} b^{2} x^{2}+5 x \,a^{4} b +a^{5}\right )^{2} b}\) | \(97\) |
parallelrisch | \(-\frac {1}{14 \left (b^{4} x^{4}+4 a \,b^{3} x^{3}+6 a^{2} b^{2} x^{2}+4 a^{3} b x +a^{4}\right ) \left (b^{5} x^{5}+5 a \,b^{4} x^{4}+10 a^{2} b^{3} x^{3}+10 a^{3} b^{2} x^{2}+5 x \,a^{4} b +a^{5}\right )^{2} b}\) | \(97\) |
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Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (12) = 24\).
Time = 0.28 (sec) , antiderivative size = 156, normalized size of antiderivative = 11.14 \[ \int \frac {1}{\left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^3} \, dx=-\frac {1}{14 \, {\left (b^{15} x^{14} + 14 \, a b^{14} x^{13} + 91 \, a^{2} b^{13} x^{12} + 364 \, a^{3} b^{12} x^{11} + 1001 \, a^{4} b^{11} x^{10} + 2002 \, a^{5} b^{10} x^{9} + 3003 \, a^{6} b^{9} x^{8} + 3432 \, a^{7} b^{8} x^{7} + 3003 \, a^{8} b^{7} x^{6} + 2002 \, a^{9} b^{6} x^{5} + 1001 \, a^{10} b^{5} x^{4} + 364 \, a^{11} b^{4} x^{3} + 91 \, a^{12} b^{3} x^{2} + 14 \, a^{13} b^{2} x + a^{14} b\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (12) = 24\).
Time = 0.43 (sec) , antiderivative size = 168, normalized size of antiderivative = 12.00 \[ \int \frac {1}{\left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^3} \, dx=- \frac {1}{14 a^{14} b + 196 a^{13} b^{2} x + 1274 a^{12} b^{3} x^{2} + 5096 a^{11} b^{4} x^{3} + 14014 a^{10} b^{5} x^{4} + 28028 a^{9} b^{6} x^{5} + 42042 a^{8} b^{7} x^{6} + 48048 a^{7} b^{8} x^{7} + 42042 a^{6} b^{9} x^{8} + 28028 a^{5} b^{10} x^{9} + 14014 a^{4} b^{11} x^{10} + 5096 a^{3} b^{12} x^{11} + 1274 a^{2} b^{13} x^{12} + 196 a b^{14} x^{13} + 14 b^{15} x^{14}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (12) = 24\).
Time = 0.22 (sec) , antiderivative size = 156, normalized size of antiderivative = 11.14 \[ \int \frac {1}{\left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^3} \, dx=-\frac {1}{14 \, {\left (b^{15} x^{14} + 14 \, a b^{14} x^{13} + 91 \, a^{2} b^{13} x^{12} + 364 \, a^{3} b^{12} x^{11} + 1001 \, a^{4} b^{11} x^{10} + 2002 \, a^{5} b^{10} x^{9} + 3003 \, a^{6} b^{9} x^{8} + 3432 \, a^{7} b^{8} x^{7} + 3003 \, a^{8} b^{7} x^{6} + 2002 \, a^{9} b^{6} x^{5} + 1001 \, a^{10} b^{5} x^{4} + 364 \, a^{11} b^{4} x^{3} + 91 \, a^{12} b^{3} x^{2} + 14 \, a^{13} b^{2} x + a^{14} b\right )}} \]
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none
Time = 0.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^3} \, dx=-\frac {1}{14 \, {\left (b x + a\right )}^{14} b} \]
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Time = 11.83 (sec) , antiderivative size = 158, normalized size of antiderivative = 11.29 \[ \int \frac {1}{\left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right )^3} \, dx=-\frac {1}{14\,a^{14}\,b+196\,a^{13}\,b^2\,x+1274\,a^{12}\,b^3\,x^2+5096\,a^{11}\,b^4\,x^3+14014\,a^{10}\,b^5\,x^4+28028\,a^9\,b^6\,x^5+42042\,a^8\,b^7\,x^6+48048\,a^7\,b^8\,x^7+42042\,a^6\,b^9\,x^8+28028\,a^5\,b^{10}\,x^9+14014\,a^4\,b^{11}\,x^{10}+5096\,a^3\,b^{12}\,x^{11}+1274\,a^2\,b^{13}\,x^{12}+196\,a\,b^{14}\,x^{13}+14\,b^{15}\,x^{14}} \]
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