Integrand size = 18, antiderivative size = 67 \[ \int \frac {a+b x+c x^2}{(d+e x)^4} \, dx=-\frac {c d^2-b d e+a e^2}{3 e^3 (d+e x)^3}+\frac {2 c d-b e}{2 e^3 (d+e x)^2}-\frac {c}{e^3 (d+e x)} \]
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Time = 0.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {712} \[ \int \frac {a+b x+c x^2}{(d+e x)^4} \, dx=-\frac {a e^2-b d e+c d^2}{3 e^3 (d+e x)^3}+\frac {2 c d-b e}{2 e^3 (d+e x)^2}-\frac {c}{e^3 (d+e x)} \]
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Rule 712
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c d^2-b d e+a e^2}{e^2 (d+e x)^4}+\frac {-2 c d+b e}{e^2 (d+e x)^3}+\frac {c}{e^2 (d+e x)^2}\right ) \, dx \\ & = -\frac {c d^2-b d e+a e^2}{3 e^3 (d+e x)^3}+\frac {2 c d-b e}{2 e^3 (d+e x)^2}-\frac {c}{e^3 (d+e x)} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.75 \[ \int \frac {a+b x+c x^2}{(d+e x)^4} \, dx=-\frac {2 c \left (d^2+3 d e x+3 e^2 x^2\right )+e (2 a e+b (d+3 e x))}{6 e^3 (d+e x)^3} \]
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Time = 2.59 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.78
method | result | size |
gosper | \(-\frac {6 c \,x^{2} e^{2}+3 x b \,e^{2}+6 x c d e +2 e^{2} a +b d e +2 c \,d^{2}}{6 \left (e x +d \right )^{3} e^{3}}\) | \(52\) |
parallelrisch | \(\frac {-6 c \,x^{2} e^{2}-3 x b \,e^{2}-6 x c d e -2 e^{2} a -b d e -2 c \,d^{2}}{6 e^{3} \left (e x +d \right )^{3}}\) | \(53\) |
norman | \(\frac {-\frac {c \,x^{2}}{e}-\frac {\left (b e +2 c d \right ) x}{2 e^{2}}-\frac {2 e^{2} a +b d e +2 c \,d^{2}}{6 e^{3}}}{\left (e x +d \right )^{3}}\) | \(55\) |
risch | \(\frac {-\frac {c \,x^{2}}{e}-\frac {\left (b e +2 c d \right ) x}{2 e^{2}}-\frac {2 e^{2} a +b d e +2 c \,d^{2}}{6 e^{3}}}{\left (e x +d \right )^{3}}\) | \(55\) |
default | \(-\frac {c}{e^{3} \left (e x +d \right )}-\frac {e^{2} a -b d e +c \,d^{2}}{3 e^{3} \left (e x +d \right )^{3}}-\frac {b e -2 c d}{2 e^{3} \left (e x +d \right )^{2}}\) | \(63\) |
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none
Time = 0.29 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.15 \[ \int \frac {a+b x+c x^2}{(d+e x)^4} \, dx=-\frac {6 \, c e^{2} x^{2} + 2 \, c d^{2} + b d e + 2 \, a e^{2} + 3 \, {\left (2 \, c d e + b e^{2}\right )} x}{6 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \]
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Time = 0.41 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.22 \[ \int \frac {a+b x+c x^2}{(d+e x)^4} \, dx=\frac {- 2 a e^{2} - b d e - 2 c d^{2} - 6 c e^{2} x^{2} + x \left (- 3 b e^{2} - 6 c d e\right )}{6 d^{3} e^{3} + 18 d^{2} e^{4} x + 18 d e^{5} x^{2} + 6 e^{6} x^{3}} \]
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none
Time = 0.19 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.15 \[ \int \frac {a+b x+c x^2}{(d+e x)^4} \, dx=-\frac {6 \, c e^{2} x^{2} + 2 \, c d^{2} + b d e + 2 \, a e^{2} + 3 \, {\left (2 \, c d e + b e^{2}\right )} x}{6 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \]
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none
Time = 0.26 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.76 \[ \int \frac {a+b x+c x^2}{(d+e x)^4} \, dx=-\frac {6 \, c e^{2} x^{2} + 6 \, c d e x + 3 \, b e^{2} x + 2 \, c d^{2} + b d e + 2 \, a e^{2}}{6 \, {\left (e x + d\right )}^{3} e^{3}} \]
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Time = 0.04 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.13 \[ \int \frac {a+b x+c x^2}{(d+e x)^4} \, dx=-\frac {\frac {2\,c\,d^2+b\,d\,e+2\,a\,e^2}{6\,e^3}+\frac {x\,\left (b\,e+2\,c\,d\right )}{2\,e^2}+\frac {c\,x^2}{e}}{d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3} \]
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