Integrand size = 13, antiderivative size = 16 \[ \int \frac {x^3}{\left (a+c x^4\right )^2} \, dx=-\frac {1}{4 c \left (a+c x^4\right )} \]
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Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {267} \[ \int \frac {x^3}{\left (a+c x^4\right )^2} \, dx=-\frac {1}{4 c \left (a+c x^4\right )} \]
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Rule 267
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{4 c \left (a+c x^4\right )} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {x^3}{\left (a+c x^4\right )^2} \, dx=-\frac {1}{4 c \left (a+c x^4\right )} \]
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Time = 4.05 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94
method | result | size |
gosper | \(-\frac {1}{4 c \left (x^{4} c +a \right )}\) | \(15\) |
derivativedivides | \(-\frac {1}{4 c \left (x^{4} c +a \right )}\) | \(15\) |
default | \(-\frac {1}{4 c \left (x^{4} c +a \right )}\) | \(15\) |
norman | \(-\frac {1}{4 c \left (x^{4} c +a \right )}\) | \(15\) |
risch | \(-\frac {1}{4 c \left (x^{4} c +a \right )}\) | \(15\) |
parallelrisch | \(-\frac {1}{4 c \left (x^{4} c +a \right )}\) | \(15\) |
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none
Time = 0.27 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \frac {x^3}{\left (a+c x^4\right )^2} \, dx=-\frac {1}{4 \, {\left (c^{2} x^{4} + a c\right )}} \]
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Time = 0.10 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \frac {x^3}{\left (a+c x^4\right )^2} \, dx=- \frac {1}{4 a c + 4 c^{2} x^{4}} \]
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none
Time = 0.21 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {x^3}{\left (a+c x^4\right )^2} \, dx=-\frac {1}{4 \, {\left (c x^{4} + a\right )} c} \]
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none
Time = 0.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {x^3}{\left (a+c x^4\right )^2} \, dx=-\frac {1}{4 \, {\left (c x^{4} + a\right )} c} \]
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Time = 5.66 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {x^3}{\left (a+c x^4\right )^2} \, dx=-\frac {1}{4\,c\,\left (c\,x^4+a\right )} \]
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