Integrand size = 16, antiderivative size = 18 \[ \int \frac {a+b x^2+c x^4}{x^4} \, dx=-\frac {a}{3 x^3}-\frac {b}{x}+c x \]
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Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {14} \[ \int \frac {a+b x^2+c x^4}{x^4} \, dx=-\frac {a}{3 x^3}-\frac {b}{x}+c x \]
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Rule 14
Rubi steps \begin{align*} \text {integral}& = \int \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right ) \, dx \\ & = -\frac {a}{3 x^3}-\frac {b}{x}+c x \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {a+b x^2+c x^4}{x^4} \, dx=-\frac {a}{3 x^3}-\frac {b}{x}+c x \]
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Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94
method | result | size |
default | \(-\frac {a}{3 x^{3}}-\frac {b}{x}+c x\) | \(17\) |
risch | \(c x +\frac {-b \,x^{2}-\frac {a}{3}}{x^{3}}\) | \(19\) |
gosper | \(-\frac {-3 c \,x^{4}+3 b \,x^{2}+a}{3 x^{3}}\) | \(20\) |
norman | \(\frac {c \,x^{4}-b \,x^{2}-\frac {1}{3} a}{x^{3}}\) | \(20\) |
parallelrisch | \(\frac {3 c \,x^{4}-3 b \,x^{2}-a}{3 x^{3}}\) | \(22\) |
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none
Time = 0.23 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.17 \[ \int \frac {a+b x^2+c x^4}{x^4} \, dx=\frac {3 \, c x^{4} - 3 \, b x^{2} - a}{3 \, x^{3}} \]
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Time = 0.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {a+b x^2+c x^4}{x^4} \, dx=c x + \frac {- a - 3 b x^{2}}{3 x^{3}} \]
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none
Time = 0.21 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {a+b x^2+c x^4}{x^4} \, dx=c x - \frac {3 \, b x^{2} + a}{3 \, x^{3}} \]
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none
Time = 0.29 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {a+b x^2+c x^4}{x^4} \, dx=c x - \frac {3 \, b x^{2} + a}{3 \, x^{3}} \]
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Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {a+b x^2+c x^4}{x^4} \, dx=c\,x-\frac {b\,x^2+\frac {a}{3}}{x^3} \]
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