Integrand size = 17, antiderivative size = 62 \[ \int \frac {b x+c x^2}{(d+e x)^5} \, dx=-\frac {d (c d-b e)}{4 e^3 (d+e x)^4}+\frac {2 c d-b e}{3 e^3 (d+e x)^3}-\frac {c}{2 e^3 (d+e x)^2} \]
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Time = 0.03 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {712} \[ \int \frac {b x+c x^2}{(d+e x)^5} \, dx=\frac {2 c d-b e}{3 e^3 (d+e x)^3}-\frac {d (c d-b e)}{4 e^3 (d+e x)^4}-\frac {c}{2 e^3 (d+e x)^2} \]
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Rule 712
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d (c d-b e)}{e^2 (d+e x)^5}+\frac {-2 c d+b e}{e^2 (d+e x)^4}+\frac {c}{e^2 (d+e x)^3}\right ) \, dx \\ & = -\frac {d (c d-b e)}{4 e^3 (d+e x)^4}+\frac {2 c d-b e}{3 e^3 (d+e x)^3}-\frac {c}{2 e^3 (d+e x)^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.69 \[ \int \frac {b x+c x^2}{(d+e x)^5} \, dx=-\frac {b e (d+4 e x)+c \left (d^2+4 d e x+6 e^2 x^2\right )}{12 e^3 (d+e x)^4} \]
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Time = 1.84 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.73
method | result | size |
gosper | \(-\frac {6 c \,x^{2} e^{2}+4 x b \,e^{2}+4 x c d e +b d e +c \,d^{2}}{12 e^{3} \left (e x +d \right )^{4}}\) | \(45\) |
risch | \(\frac {-\frac {c \,x^{2}}{2 e}-\frac {\left (b e +c d \right ) x}{3 e^{2}}-\frac {d \left (b e +c d \right )}{12 e^{3}}}{\left (e x +d \right )^{4}}\) | \(45\) |
norman | \(\frac {-\frac {c \,x^{2}}{2 e}-\frac {\left (e^{2} b +c d e \right ) x}{3 e^{3}}-\frac {d \left (e^{2} b +c d e \right )}{12 e^{4}}}{\left (e x +d \right )^{4}}\) | \(51\) |
parallelrisch | \(\frac {-6 c \,x^{2} e^{3}-4 b \,e^{3} x -4 c d \,e^{2} x -b d \,e^{2}-d^{2} e c}{12 e^{4} \left (e x +d \right )^{4}}\) | \(52\) |
default | \(-\frac {b e -2 c d}{3 e^{3} \left (e x +d \right )^{3}}+\frac {d \left (b e -c d \right )}{4 e^{3} \left (e x +d \right )^{4}}-\frac {c}{2 e^{3} \left (e x +d \right )^{2}}\) | \(56\) |
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Time = 0.26 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.29 \[ \int \frac {b x+c x^2}{(d+e x)^5} \, dx=-\frac {6 \, c e^{2} x^{2} + c d^{2} + b d e + 4 \, {\left (c d e + b e^{2}\right )} x}{12 \, {\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \]
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Time = 0.31 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.37 \[ \int \frac {b x+c x^2}{(d+e x)^5} \, dx=\frac {- b d e - c d^{2} - 6 c e^{2} x^{2} + x \left (- 4 b e^{2} - 4 c d e\right )}{12 d^{4} e^{3} + 48 d^{3} e^{4} x + 72 d^{2} e^{5} x^{2} + 48 d e^{6} x^{3} + 12 e^{7} x^{4}} \]
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none
Time = 0.20 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.29 \[ \int \frac {b x+c x^2}{(d+e x)^5} \, dx=-\frac {6 \, c e^{2} x^{2} + c d^{2} + b d e + 4 \, {\left (c d e + b e^{2}\right )} x}{12 \, {\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \]
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Time = 0.27 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.23 \[ \int \frac {b x+c x^2}{(d+e x)^5} \, dx=-\frac {\frac {6 \, c}{{\left (e x + d\right )}^{2} e^{2}} - \frac {8 \, c d}{{\left (e x + d\right )}^{3} e^{2}} + \frac {3 \, c d^{2}}{{\left (e x + d\right )}^{4} e^{2}} + \frac {4 \, b}{{\left (e x + d\right )}^{3} e} - \frac {3 \, b d}{{\left (e x + d\right )}^{4} e}}{12 \, e} \]
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Time = 9.96 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.26 \[ \int \frac {b x+c x^2}{(d+e x)^5} \, dx=-\frac {\frac {d\,\left (b\,e+c\,d\right )}{12\,e^3}+\frac {x\,\left (b\,e+c\,d\right )}{3\,e^2}+\frac {c\,x^2}{2\,e}}{d^4+4\,d^3\,e\,x+6\,d^2\,e^2\,x^2+4\,d\,e^3\,x^3+e^4\,x^4} \]
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