Integrand size = 51, antiderivative size = 14 \[ \int \frac {1}{a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5} \, dx=-\frac {1}{4 b (a+b x)^4} \]
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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} \[ \int \frac {1}{a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5} \, dx=-\frac {1}{4 b (a+b x)^4} \]
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Rule 32
Rule 2083
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(a+b x)^5} \, dx \\ & = -\frac {1}{4 b (a+b x)^4} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5} \, dx=-\frac {1}{4 b (a+b x)^4} \]
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Time = 0.06 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93
method | result | size |
default | \(-\frac {1}{4 b \left (b x +a \right )^{4}}\) | \(13\) |
norman | \(-\frac {1}{4 b \left (b x +a \right )^{4}}\) | \(13\) |
gosper | \(-\frac {1}{4 b \left (b^{4} x^{4}+4 a \,b^{3} x^{3}+6 a^{2} b^{2} x^{2}+4 a^{3} b x +a^{4}\right )}\) | \(46\) |
risch | \(-\frac {1}{4 b \left (b^{4} x^{4}+4 a \,b^{3} x^{3}+6 a^{2} b^{2} x^{2}+4 a^{3} b x +a^{4}\right )}\) | \(46\) |
parallelrisch | \(-\frac {1}{4 b \left (b^{4} x^{4}+4 a \,b^{3} x^{3}+6 a^{2} b^{2} x^{2}+4 a^{3} b x +a^{4}\right )}\) | \(46\) |
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Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (12) = 24\).
Time = 0.29 (sec) , antiderivative size = 46, normalized size of antiderivative = 3.29 \[ \int \frac {1}{a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5} \, dx=-\frac {1}{4 \, {\left (b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (12) = 24\).
Time = 0.13 (sec) , antiderivative size = 49, normalized size of antiderivative = 3.50 \[ \int \frac {1}{a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5} \, dx=- \frac {1}{4 a^{4} b + 16 a^{3} b^{2} x + 24 a^{2} b^{3} x^{2} + 16 a b^{4} x^{3} + 4 b^{5} x^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (12) = 24\).
Time = 0.20 (sec) , antiderivative size = 46, normalized size of antiderivative = 3.29 \[ \int \frac {1}{a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5} \, dx=-\frac {1}{4 \, {\left (b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b\right )}} \]
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none
Time = 0.29 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5} \, dx=-\frac {1}{4 \, {\left (b x + a\right )}^{4} b} \]
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Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 3.43 \[ \int \frac {1}{a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5} \, dx=-\frac {1}{4\,a^4\,b+16\,a^3\,b^2\,x+24\,a^2\,b^3\,x^2+16\,a\,b^4\,x^3+4\,b^5\,x^4} \]
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