Integrand size = 18, antiderivative size = 69 \[ \int \frac {a+b x+c x^2}{(d+e x)^6} \, dx=-\frac {c d^2-b d e+a e^2}{5 e^3 (d+e x)^5}+\frac {2 c d-b e}{4 e^3 (d+e x)^4}-\frac {c}{3 e^3 (d+e x)^3} \]
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Time = 0.03 (sec) , antiderivative size = 69, 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} \[ \int \frac {a+b x+c x^2}{(d+e x)^6} \, dx=-\frac {a e^2-b d e+c d^2}{5 e^3 (d+e x)^5}+\frac {2 c d-b e}{4 e^3 (d+e x)^4}-\frac {c}{3 e^3 (d+e x)^3} \]
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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)^6}+\frac {-2 c d+b e}{e^2 (d+e x)^5}+\frac {c}{e^2 (d+e x)^4}\right ) \, dx \\ & = -\frac {c d^2-b d e+a e^2}{5 e^3 (d+e x)^5}+\frac {2 c d-b e}{4 e^3 (d+e x)^4}-\frac {c}{3 e^3 (d+e x)^3} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.74 \[ \int \frac {a+b x+c x^2}{(d+e x)^6} \, dx=-\frac {2 c \left (d^2+5 d e x+10 e^2 x^2\right )+3 e (4 a e+b (d+5 e x))}{60 e^3 (d+e x)^5} \]
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Time = 2.53 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.77
method | result | size |
gosper | \(-\frac {20 c \,x^{2} e^{2}+15 x b \,e^{2}+10 x c d e +12 e^{2} a +3 b d e +2 c \,d^{2}}{60 e^{3} \left (e x +d \right )^{5}}\) | \(53\) |
risch | \(\frac {-\frac {c \,x^{2}}{3 e}-\frac {\left (3 b e +2 c d \right ) x}{12 e^{2}}-\frac {12 e^{2} a +3 b d e +2 c \,d^{2}}{60 e^{3}}}{\left (e x +d \right )^{5}}\) | \(57\) |
parallelrisch | \(\frac {-20 c \,x^{2} e^{4}-15 b \,e^{4} x -10 c d \,e^{3} x -12 e^{4} a -3 b d \,e^{3}-2 d^{2} e^{2} c}{60 e^{5} \left (e x +d \right )^{5}}\) | \(60\) |
default | \(-\frac {e^{2} a -b d e +c \,d^{2}}{5 e^{3} \left (e x +d \right )^{5}}-\frac {c}{3 e^{3} \left (e x +d \right )^{3}}-\frac {b e -2 c d}{4 e^{3} \left (e x +d \right )^{4}}\) | \(63\) |
norman | \(\frac {-\frac {c \,x^{2}}{3 e}-\frac {\left (3 e^{3} b +2 d \,e^{2} c \right ) x}{12 e^{4}}-\frac {12 e^{4} a +3 b d \,e^{3}+2 d^{2} e^{2} c}{60 e^{5}}}{\left (e x +d \right )^{5}}\) | \(67\) |
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Time = 0.29 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.46 \[ \int \frac {a+b x+c x^2}{(d+e x)^6} \, dx=-\frac {20 \, c e^{2} x^{2} + 2 \, c d^{2} + 3 \, b d e + 12 \, a e^{2} + 5 \, {\left (2 \, c d e + 3 \, b e^{2}\right )} x}{60 \, {\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \]
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Time = 0.91 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.55 \[ \int \frac {a+b x+c x^2}{(d+e x)^6} \, dx=\frac {- 12 a e^{2} - 3 b d e - 2 c d^{2} - 20 c e^{2} x^{2} + x \left (- 15 b e^{2} - 10 c d e\right )}{60 d^{5} e^{3} + 300 d^{4} e^{4} x + 600 d^{3} e^{5} x^{2} + 600 d^{2} e^{6} x^{3} + 300 d e^{7} x^{4} + 60 e^{8} x^{5}} \]
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Time = 0.22 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.46 \[ \int \frac {a+b x+c x^2}{(d+e x)^6} \, dx=-\frac {20 \, c e^{2} x^{2} + 2 \, c d^{2} + 3 \, b d e + 12 \, a e^{2} + 5 \, {\left (2 \, c d e + 3 \, b e^{2}\right )} x}{60 \, {\left (e^{8} x^{5} + 5 \, d e^{7} x^{4} + 10 \, d^{2} e^{6} x^{3} + 10 \, d^{3} e^{5} x^{2} + 5 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \]
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Time = 0.26 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.75 \[ \int \frac {a+b x+c x^2}{(d+e x)^6} \, dx=-\frac {20 \, c e^{2} x^{2} + 10 \, c d e x + 15 \, b e^{2} x + 2 \, c d^{2} + 3 \, b d e + 12 \, a e^{2}}{60 \, {\left (e x + d\right )}^{5} e^{3}} \]
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Time = 0.07 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.46 \[ \int \frac {a+b x+c x^2}{(d+e x)^6} \, dx=-\frac {\frac {2\,c\,d^2+3\,b\,d\,e+12\,a\,e^2}{60\,e^3}+\frac {x\,\left (3\,b\,e+2\,c\,d\right )}{12\,e^2}+\frac {c\,x^2}{3\,e}}{d^5+5\,d^4\,e\,x+10\,d^3\,e^2\,x^2+10\,d^2\,e^3\,x^3+5\,d\,e^4\,x^4+e^5\,x^5} \]
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