Integrand size = 15, antiderivative size = 54 \[ \int \frac {a+c x^2}{(d+e x)^4} \, dx=\frac {-c d^2-a e^2}{3 e^3 (d+e x)^3}+\frac {c d}{e^3 (d+e x)^2}-\frac {c}{e^3 (d+e x)} \]
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Time = 0.02 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.96, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {711} \[ \int \frac {a+c x^2}{(d+e x)^4} \, dx=-\frac {a e^2+c d^2}{3 e^3 (d+e x)^3}-\frac {c}{e^3 (d+e x)}+\frac {c d}{e^3 (d+e x)^2} \]
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Rule 711
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c d^2+a e^2}{e^2 (d+e x)^4}-\frac {2 c d}{e^2 (d+e x)^3}+\frac {c}{e^2 (d+e x)^2}\right ) \, dx \\ & = -\frac {c d^2+a e^2}{3 e^3 (d+e x)^3}+\frac {c d}{e^3 (d+e x)^2}-\frac {c}{e^3 (d+e x)} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.72 \[ \int \frac {a+c x^2}{(d+e x)^4} \, dx=-\frac {a e^2+c \left (d^2+3 d e x+3 e^2 x^2\right )}{3 e^3 (d+e x)^3} \]
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Time = 2.06 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.72
method | result | size |
gosper | \(-\frac {3 c \,x^{2} e^{2}+3 x c d e +e^{2} a +c \,d^{2}}{3 \left (e x +d \right )^{3} e^{3}}\) | \(39\) |
parallelrisch | \(\frac {-3 c \,x^{2} e^{2}-3 x c d e -e^{2} a -c \,d^{2}}{3 e^{3} \left (e x +d \right )^{3}}\) | \(41\) |
norman | \(\frac {-\frac {c \,x^{2}}{e}-\frac {c d x}{e^{2}}-\frac {e^{2} a +c \,d^{2}}{3 e^{3}}}{\left (e x +d \right )^{3}}\) | \(43\) |
risch | \(\frac {-\frac {c \,x^{2}}{e}-\frac {c d x}{e^{2}}-\frac {e^{2} a +c \,d^{2}}{3 e^{3}}}{\left (e x +d \right )^{3}}\) | \(43\) |
default | \(-\frac {c}{e^{3} \left (e x +d \right )}-\frac {e^{2} a +c \,d^{2}}{3 e^{3} \left (e x +d \right )^{3}}+\frac {c d}{e^{3} \left (e x +d \right )^{2}}\) | \(51\) |
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Time = 0.27 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.17 \[ \int \frac {a+c x^2}{(d+e x)^4} \, dx=-\frac {3 \, c e^{2} x^{2} + 3 \, c d e x + c d^{2} + a e^{2}}{3 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \]
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Time = 0.23 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.22 \[ \int \frac {a+c x^2}{(d+e x)^4} \, dx=\frac {- a e^{2} - c d^{2} - 3 c d e x - 3 c e^{2} x^{2}}{3 d^{3} e^{3} + 9 d^{2} e^{4} x + 9 d e^{5} x^{2} + 3 e^{6} x^{3}} \]
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Time = 0.20 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.17 \[ \int \frac {a+c x^2}{(d+e x)^4} \, dx=-\frac {3 \, c e^{2} x^{2} + 3 \, c d e x + c d^{2} + a e^{2}}{3 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \]
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Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.70 \[ \int \frac {a+c x^2}{(d+e x)^4} \, dx=-\frac {3 \, c e^{2} x^{2} + 3 \, c d e x + c d^{2} + a e^{2}}{3 \, {\left (e x + d\right )}^{3} e^{3}} \]
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Time = 0.04 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.17 \[ \int \frac {a+c x^2}{(d+e x)^4} \, dx=-\frac {\frac {c\,d^2+a\,e^2}{3\,e^3}+\frac {c\,x^2}{e}+\frac {c\,d\,x}{e^2}}{d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3} \]
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