Integrand size = 19, antiderivative size = 230 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^8} \, dx=-\frac {d^3 (c d-b e)^3}{7 e^7 (d+e x)^7}+\frac {d^2 (c d-b e)^2 (2 c d-b e)}{2 e^7 (d+e x)^6}-\frac {3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{5 e^7 (d+e x)^5}+\frac {(2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right )}{4 e^7 (d+e x)^4}-\frac {c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{e^7 (d+e x)^3}+\frac {3 c^2 (2 c d-b e)}{2 e^7 (d+e x)^2}-\frac {c^3}{e^7 (d+e x)} \]
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Time = 0.11 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {712} \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^8} \, dx=-\frac {c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^7 (d+e x)^3}+\frac {(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{4 e^7 (d+e x)^4}-\frac {3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{5 e^7 (d+e x)^5}+\frac {3 c^2 (2 c d-b e)}{2 e^7 (d+e x)^2}-\frac {d^3 (c d-b e)^3}{7 e^7 (d+e x)^7}+\frac {d^2 (c d-b e)^2 (2 c d-b e)}{2 e^7 (d+e x)^6}-\frac {c^3}{e^7 (d+e x)} \]
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Rule 712
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d^3 (c d-b e)^3}{e^6 (d+e x)^8}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{e^6 (d+e x)^7}+\frac {3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{e^6 (d+e x)^6}+\frac {(2 c d-b e) \left (-10 c^2 d^2+10 b c d e-b^2 e^2\right )}{e^6 (d+e x)^5}+\frac {3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{e^6 (d+e x)^4}-\frac {3 c^2 (2 c d-b e)}{e^6 (d+e x)^3}+\frac {c^3}{e^6 (d+e x)^2}\right ) \, dx \\ & = -\frac {d^3 (c d-b e)^3}{7 e^7 (d+e x)^7}+\frac {d^2 (c d-b e)^2 (2 c d-b e)}{2 e^7 (d+e x)^6}-\frac {3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{5 e^7 (d+e x)^5}+\frac {(2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right )}{4 e^7 (d+e x)^4}-\frac {c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{e^7 (d+e x)^3}+\frac {3 c^2 (2 c d-b e)}{2 e^7 (d+e x)^2}-\frac {c^3}{e^7 (d+e x)} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.96 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^8} \, dx=-\frac {b^3 e^3 \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+4 b^2 c e^2 \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )+10 b c^2 e \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )+20 c^3 \left (d^6+7 d^5 e x+21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+21 d e^5 x^5+7 e^6 x^6\right )}{140 e^7 (d+e x)^7} \]
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Time = 2.02 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.09
method | result | size |
risch | \(\frac {-\frac {c^{3} x^{6}}{e}-\frac {3 c^{2} \left (b e +2 c d \right ) x^{5}}{2 e^{2}}-\frac {c \left (2 b^{2} e^{2}+5 b c d e +10 c^{2} d^{2}\right ) x^{4}}{2 e^{3}}-\frac {\left (b^{3} e^{3}+4 b^{2} d \,e^{2} c +10 b \,c^{2} d^{2} e +20 c^{3} d^{3}\right ) x^{3}}{4 e^{4}}-\frac {3 d \left (b^{3} e^{3}+4 b^{2} d \,e^{2} c +10 b \,c^{2} d^{2} e +20 c^{3} d^{3}\right ) x^{2}}{20 e^{5}}-\frac {d^{2} \left (b^{3} e^{3}+4 b^{2} d \,e^{2} c +10 b \,c^{2} d^{2} e +20 c^{3} d^{3}\right ) x}{20 e^{6}}-\frac {d^{3} \left (b^{3} e^{3}+4 b^{2} d \,e^{2} c +10 b \,c^{2} d^{2} e +20 c^{3} d^{3}\right )}{140 e^{7}}}{\left (e x +d \right )^{7}}\) | \(250\) |
norman | \(\frac {-\frac {c^{3} x^{6}}{e}-\frac {3 \left (b \,c^{2} e +2 c^{3} d \right ) x^{5}}{2 e^{2}}-\frac {\left (2 b^{2} e^{2} c +5 d e b \,c^{2}+10 c^{3} d^{2}\right ) x^{4}}{2 e^{3}}-\frac {\left (b^{3} e^{3}+4 b^{2} d \,e^{2} c +10 b \,c^{2} d^{2} e +20 c^{3} d^{3}\right ) x^{3}}{4 e^{4}}-\frac {3 d \left (b^{3} e^{3}+4 b^{2} d \,e^{2} c +10 b \,c^{2} d^{2} e +20 c^{3} d^{3}\right ) x^{2}}{20 e^{5}}-\frac {d^{2} \left (b^{3} e^{3}+4 b^{2} d \,e^{2} c +10 b \,c^{2} d^{2} e +20 c^{3} d^{3}\right ) x}{20 e^{6}}-\frac {d^{3} \left (b^{3} e^{3}+4 b^{2} d \,e^{2} c +10 b \,c^{2} d^{2} e +20 c^{3} d^{3}\right )}{140 e^{7}}}{\left (e x +d \right )^{7}}\) | \(254\) |
default | \(-\frac {d^{2} \left (b^{3} e^{3}-4 b^{2} d \,e^{2} c +5 b \,c^{2} d^{2} e -2 c^{3} d^{3}\right )}{2 e^{7} \left (e x +d \right )^{6}}-\frac {c^{3}}{e^{7} \left (e x +d \right )}-\frac {c \left (b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right )}{e^{7} \left (e x +d \right )^{3}}-\frac {b^{3} e^{3}-12 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -20 c^{3} d^{3}}{4 e^{7} \left (e x +d \right )^{4}}+\frac {d^{3} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}{7 e^{7} \left (e x +d \right )^{7}}+\frac {3 d \left (b^{3} e^{3}-6 b^{2} d \,e^{2} c +10 b \,c^{2} d^{2} e -5 c^{3} d^{3}\right )}{5 e^{7} \left (e x +d \right )^{5}}-\frac {3 c^{2} \left (b e -2 c d \right )}{2 e^{7} \left (e x +d \right )^{2}}\) | \(274\) |
gosper | \(-\frac {140 x^{6} c^{3} e^{6}+210 x^{5} b \,c^{2} e^{6}+420 x^{5} c^{3} d \,e^{5}+140 x^{4} b^{2} c \,e^{6}+350 x^{4} b \,c^{2} d \,e^{5}+700 x^{4} c^{3} d^{2} e^{4}+35 x^{3} b^{3} e^{6}+140 x^{3} b^{2} c d \,e^{5}+350 x^{3} b \,c^{2} d^{2} e^{4}+700 x^{3} c^{3} d^{3} e^{3}+21 x^{2} b^{3} d \,e^{5}+84 x^{2} b^{2} c \,d^{2} e^{4}+210 x^{2} b \,c^{2} d^{3} e^{3}+420 x^{2} c^{3} d^{4} e^{2}+7 x \,b^{3} d^{2} e^{4}+28 x \,b^{2} c \,d^{3} e^{3}+70 x b \,c^{2} d^{4} e^{2}+140 x \,c^{3} d^{5} e +b^{3} d^{3} e^{3}+4 b^{2} c \,d^{4} e^{2}+10 b \,c^{2} d^{5} e +20 c^{3} d^{6}}{140 e^{7} \left (e x +d \right )^{7}}\) | \(285\) |
parallelrisch | \(\frac {-140 x^{6} c^{3} e^{6}-210 x^{5} b \,c^{2} e^{6}-420 x^{5} c^{3} d \,e^{5}-140 x^{4} b^{2} c \,e^{6}-350 x^{4} b \,c^{2} d \,e^{5}-700 x^{4} c^{3} d^{2} e^{4}-35 x^{3} b^{3} e^{6}-140 x^{3} b^{2} c d \,e^{5}-350 x^{3} b \,c^{2} d^{2} e^{4}-700 x^{3} c^{3} d^{3} e^{3}-21 x^{2} b^{3} d \,e^{5}-84 x^{2} b^{2} c \,d^{2} e^{4}-210 x^{2} b \,c^{2} d^{3} e^{3}-420 x^{2} c^{3} d^{4} e^{2}-7 x \,b^{3} d^{2} e^{4}-28 x \,b^{2} c \,d^{3} e^{3}-70 x b \,c^{2} d^{4} e^{2}-140 x \,c^{3} d^{5} e -b^{3} d^{3} e^{3}-4 b^{2} c \,d^{4} e^{2}-10 b \,c^{2} d^{5} e -20 c^{3} d^{6}}{140 e^{7} \left (e x +d \right )^{7}}\) | \(286\) |
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Time = 0.26 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.45 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^8} \, dx=-\frac {140 \, c^{3} e^{6} x^{6} + 20 \, c^{3} d^{6} + 10 \, b c^{2} d^{5} e + 4 \, b^{2} c d^{4} e^{2} + b^{3} d^{3} e^{3} + 210 \, {\left (2 \, c^{3} d e^{5} + b c^{2} e^{6}\right )} x^{5} + 70 \, {\left (10 \, c^{3} d^{2} e^{4} + 5 \, b c^{2} d e^{5} + 2 \, b^{2} c e^{6}\right )} x^{4} + 35 \, {\left (20 \, c^{3} d^{3} e^{3} + 10 \, b c^{2} d^{2} e^{4} + 4 \, b^{2} c d e^{5} + b^{3} e^{6}\right )} x^{3} + 21 \, {\left (20 \, c^{3} d^{4} e^{2} + 10 \, b c^{2} d^{3} e^{3} + 4 \, b^{2} c d^{2} e^{4} + b^{3} d e^{5}\right )} x^{2} + 7 \, {\left (20 \, c^{3} d^{5} e + 10 \, b c^{2} d^{4} e^{2} + 4 \, b^{2} c d^{3} e^{3} + b^{3} d^{2} e^{4}\right )} x}{140 \, {\left (e^{14} x^{7} + 7 \, d e^{13} x^{6} + 21 \, d^{2} e^{12} x^{5} + 35 \, d^{3} e^{11} x^{4} + 35 \, d^{4} e^{10} x^{3} + 21 \, d^{5} e^{9} x^{2} + 7 \, d^{6} e^{8} x + d^{7} e^{7}\right )}} \]
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Timed out. \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^8} \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.45 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^8} \, dx=-\frac {140 \, c^{3} e^{6} x^{6} + 20 \, c^{3} d^{6} + 10 \, b c^{2} d^{5} e + 4 \, b^{2} c d^{4} e^{2} + b^{3} d^{3} e^{3} + 210 \, {\left (2 \, c^{3} d e^{5} + b c^{2} e^{6}\right )} x^{5} + 70 \, {\left (10 \, c^{3} d^{2} e^{4} + 5 \, b c^{2} d e^{5} + 2 \, b^{2} c e^{6}\right )} x^{4} + 35 \, {\left (20 \, c^{3} d^{3} e^{3} + 10 \, b c^{2} d^{2} e^{4} + 4 \, b^{2} c d e^{5} + b^{3} e^{6}\right )} x^{3} + 21 \, {\left (20 \, c^{3} d^{4} e^{2} + 10 \, b c^{2} d^{3} e^{3} + 4 \, b^{2} c d^{2} e^{4} + b^{3} d e^{5}\right )} x^{2} + 7 \, {\left (20 \, c^{3} d^{5} e + 10 \, b c^{2} d^{4} e^{2} + 4 \, b^{2} c d^{3} e^{3} + b^{3} d^{2} e^{4}\right )} x}{140 \, {\left (e^{14} x^{7} + 7 \, d e^{13} x^{6} + 21 \, d^{2} e^{12} x^{5} + 35 \, d^{3} e^{11} x^{4} + 35 \, d^{4} e^{10} x^{3} + 21 \, d^{5} e^{9} x^{2} + 7 \, d^{6} e^{8} x + d^{7} e^{7}\right )}} \]
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Time = 0.26 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.23 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^8} \, dx=-\frac {140 \, c^{3} e^{6} x^{6} + 420 \, c^{3} d e^{5} x^{5} + 210 \, b c^{2} e^{6} x^{5} + 700 \, c^{3} d^{2} e^{4} x^{4} + 350 \, b c^{2} d e^{5} x^{4} + 140 \, b^{2} c e^{6} x^{4} + 700 \, c^{3} d^{3} e^{3} x^{3} + 350 \, b c^{2} d^{2} e^{4} x^{3} + 140 \, b^{2} c d e^{5} x^{3} + 35 \, b^{3} e^{6} x^{3} + 420 \, c^{3} d^{4} e^{2} x^{2} + 210 \, b c^{2} d^{3} e^{3} x^{2} + 84 \, b^{2} c d^{2} e^{4} x^{2} + 21 \, b^{3} d e^{5} x^{2} + 140 \, c^{3} d^{5} e x + 70 \, b c^{2} d^{4} e^{2} x + 28 \, b^{2} c d^{3} e^{3} x + 7 \, b^{3} d^{2} e^{4} x + 20 \, c^{3} d^{6} + 10 \, b c^{2} d^{5} e + 4 \, b^{2} c d^{4} e^{2} + b^{3} d^{3} e^{3}}{140 \, {\left (e x + d\right )}^{7} e^{7}} \]
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Time = 9.59 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.37 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^8} \, dx=-\frac {\frac {d^3\,\left (b^3\,e^3+4\,b^2\,c\,d\,e^2+10\,b\,c^2\,d^2\,e+20\,c^3\,d^3\right )}{140\,e^7}+\frac {x^3\,\left (b^3\,e^3+4\,b^2\,c\,d\,e^2+10\,b\,c^2\,d^2\,e+20\,c^3\,d^3\right )}{4\,e^4}+\frac {c^3\,x^6}{e}+\frac {3\,c^2\,x^5\,\left (b\,e+2\,c\,d\right )}{2\,e^2}+\frac {c\,x^4\,\left (2\,b^2\,e^2+5\,b\,c\,d\,e+10\,c^2\,d^2\right )}{2\,e^3}+\frac {3\,d\,x^2\,\left (b^3\,e^3+4\,b^2\,c\,d\,e^2+10\,b\,c^2\,d^2\,e+20\,c^3\,d^3\right )}{20\,e^5}+\frac {d^2\,x\,\left (b^3\,e^3+4\,b^2\,c\,d\,e^2+10\,b\,c^2\,d^2\,e+20\,c^3\,d^3\right )}{20\,e^6}}{d^7+7\,d^6\,e\,x+21\,d^5\,e^2\,x^2+35\,d^4\,e^3\,x^3+35\,d^3\,e^4\,x^4+21\,d^2\,e^5\,x^5+7\,d\,e^6\,x^6+e^7\,x^7} \]
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