Integrand size = 19, antiderivative size = 228 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^7} \, dx=-\frac {d^3 (c d-b e)^3}{6 e^7 (d+e x)^6}+\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{5 e^7 (d+e x)^5}-\frac {3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{4 e^7 (d+e x)^4}+\frac {(2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right )}{3 e^7 (d+e x)^3}-\frac {3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{2 e^7 (d+e x)^2}+\frac {3 c^2 (2 c d-b e)}{e^7 (d+e x)}+\frac {c^3 \log (d+e x)}{e^7} \]
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Time = 0.12 (sec) , antiderivative size = 228, 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} \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^7} \, dx=-\frac {3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{2 e^7 (d+e x)^2}+\frac {(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{3 e^7 (d+e x)^3}-\frac {3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{4 e^7 (d+e x)^4}+\frac {3 c^2 (2 c d-b e)}{e^7 (d+e x)}-\frac {d^3 (c d-b e)^3}{6 e^7 (d+e x)^6}+\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{5 e^7 (d+e x)^5}+\frac {c^3 \log (d+e x)}{e^7} \]
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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)^7}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{e^6 (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 )}{e^6 (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 )}{e^6 (d+e x)^4}+\frac {3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{e^6 (d+e x)^3}-\frac {3 c^2 (2 c d-b e)}{e^6 (d+e x)^2}+\frac {c^3}{e^6 (d+e x)}\right ) \, dx \\ & = -\frac {d^3 (c d-b e)^3}{6 e^7 (d+e x)^6}+\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{5 e^7 (d+e x)^5}-\frac {3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{4 e^7 (d+e x)^4}+\frac {(2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right )}{3 e^7 (d+e x)^3}-\frac {3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{2 e^7 (d+e x)^2}+\frac {3 c^2 (2 c d-b e)}{e^7 (d+e x)}+\frac {c^3 \log (d+e x)}{e^7} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.01 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {-b^3 e^3 \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )-6 b^2 c e^2 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )-30 b c^2 e \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )+c^3 d \left (147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right )+60 c^3 (d+e x)^6 \log (d+e x)}{60 e^7 (d+e x)^6} \]
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Time = 2.03 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.11
method | result | size |
risch | \(\frac {-\frac {3 c^{2} \left (b e -2 c d \right ) x^{5}}{e^{2}}-\frac {3 c \left (b^{2} e^{2}+5 b c d e -15 c^{2} d^{2}\right ) x^{4}}{2 e^{3}}-\frac {\left (b^{3} e^{3}+6 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -110 c^{3} d^{3}\right ) x^{3}}{3 e^{4}}-\frac {d \left (b^{3} e^{3}+6 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -125 c^{3} d^{3}\right ) x^{2}}{4 e^{5}}-\frac {d^{2} \left (b^{3} e^{3}+6 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -137 c^{3} d^{3}\right ) x}{10 e^{6}}-\frac {d^{3} \left (b^{3} e^{3}+6 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -147 c^{3} d^{3}\right )}{60 e^{7}}}{\left (e x +d \right )^{6}}+\frac {c^{3} \ln \left (e x +d \right )}{e^{7}}\) | \(252\) |
norman | \(\frac {-\frac {d^{3} \left (b^{3} e^{3}+6 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -147 c^{3} d^{3}\right )}{60 e^{7}}-\frac {3 \left (b \,c^{2} e -2 c^{3} d \right ) x^{5}}{e^{2}}-\frac {3 \left (b^{2} e^{2} c +5 d e b \,c^{2}-15 c^{3} d^{2}\right ) x^{4}}{2 e^{3}}-\frac {\left (b^{3} e^{3}+6 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -110 c^{3} d^{3}\right ) x^{3}}{3 e^{4}}-\frac {d \left (b^{3} e^{3}+6 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -125 c^{3} d^{3}\right ) x^{2}}{4 e^{5}}-\frac {d^{2} \left (b^{3} e^{3}+6 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -137 c^{3} d^{3}\right ) x}{10 e^{6}}}{\left (e x +d \right )^{6}}+\frac {c^{3} \ln \left (e x +d \right )}{e^{7}}\) | \(256\) |
default | \(\frac {d^{3} \left (b^{3} e^{3}-3 b^{2} d \,e^{2} c +3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}{6 e^{7} \left (e x +d \right )^{6}}-\frac {3 c^{2} \left (b e -2 c d \right )}{e^{7} \left (e x +d \right )}-\frac {b^{3} e^{3}-12 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -20 c^{3} d^{3}}{3 e^{7} \left (e x +d \right )^{3}}+\frac {3 d \left (b^{3} e^{3}-6 b^{2} d \,e^{2} c +10 b \,c^{2} d^{2} e -5 c^{3} d^{3}\right )}{4 e^{7} \left (e x +d \right )^{4}}-\frac {3 d^{2} \left (b^{3} e^{3}-4 b^{2} d \,e^{2} c +5 b \,c^{2} d^{2} e -2 c^{3} d^{3}\right )}{5 e^{7} \left (e x +d \right )^{5}}-\frac {3 c \left (b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right )}{2 e^{7} \left (e x +d \right )^{2}}+\frac {c^{3} \ln \left (e x +d \right )}{e^{7}}\) | \(272\) |
parallelrisch | \(\frac {-b^{3} d^{3} e^{3}-180 x^{5} b \,c^{2} e^{6}-90 x^{4} b^{2} c \,e^{6}+1350 x^{4} c^{3} d^{2} e^{4}+2200 x^{3} c^{3} d^{3} e^{3}-15 x^{2} b^{3} d \,e^{5}+1875 x^{2} c^{3} d^{4} e^{2}-6 x \,b^{3} d^{2} e^{4}+822 x \,c^{3} d^{5} e +147 c^{3} d^{6}+360 \ln \left (e x +d \right ) x \,c^{3} d^{5} e +60 \ln \left (e x +d \right ) x^{6} c^{3} e^{6}-450 x^{4} b \,c^{2} d \,e^{5}-120 x^{3} b^{2} c d \,e^{5}+900 \ln \left (e x +d \right ) x^{2} c^{3} d^{4} e^{2}+1200 \ln \left (e x +d \right ) x^{3} c^{3} d^{3} e^{3}-6 b^{2} c \,d^{4} e^{2}-30 b \,c^{2} d^{5} e +360 x^{5} c^{3} d \,e^{5}+900 \ln \left (e x +d \right ) x^{4} c^{3} d^{2} e^{4}-600 x^{3} b \,c^{2} d^{2} e^{4}-90 x^{2} b^{2} c \,d^{2} e^{4}-450 x^{2} b \,c^{2} d^{3} e^{3}-36 x \,b^{2} c \,d^{3} e^{3}+360 \ln \left (e x +d \right ) x^{5} c^{3} d \,e^{5}-20 x^{3} b^{3} e^{6}+60 \ln \left (e x +d \right ) c^{3} d^{6}-180 x b \,c^{2} d^{4} e^{2}}{60 e^{7} \left (e x +d \right )^{6}}\) | \(400\) |
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Time = 0.25 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.79 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {147 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e - 6 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3} + 180 \, {\left (2 \, c^{3} d e^{5} - b c^{2} e^{6}\right )} x^{5} + 90 \, {\left (15 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} - b^{2} c e^{6}\right )} x^{4} + 20 \, {\left (110 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} - 6 \, b^{2} c d e^{5} - b^{3} e^{6}\right )} x^{3} + 15 \, {\left (125 \, c^{3} d^{4} e^{2} - 30 \, b c^{2} d^{3} e^{3} - 6 \, b^{2} c d^{2} e^{4} - b^{3} d e^{5}\right )} x^{2} + 6 \, {\left (137 \, c^{3} d^{5} e - 30 \, b c^{2} d^{4} e^{2} - 6 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4}\right )} x + 60 \, {\left (c^{3} e^{6} x^{6} + 6 \, c^{3} d e^{5} x^{5} + 15 \, c^{3} d^{2} e^{4} x^{4} + 20 \, c^{3} d^{3} e^{3} x^{3} + 15 \, c^{3} d^{4} e^{2} x^{2} + 6 \, c^{3} d^{5} e x + c^{3} d^{6}\right )} \log \left (e x + d\right )}{60 \, {\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} \]
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Timed out. \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^7} \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.45 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {147 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e - 6 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3} + 180 \, {\left (2 \, c^{3} d e^{5} - b c^{2} e^{6}\right )} x^{5} + 90 \, {\left (15 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} - b^{2} c e^{6}\right )} x^{4} + 20 \, {\left (110 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} - 6 \, b^{2} c d e^{5} - b^{3} e^{6}\right )} x^{3} + 15 \, {\left (125 \, c^{3} d^{4} e^{2} - 30 \, b c^{2} d^{3} e^{3} - 6 \, b^{2} c d^{2} e^{4} - b^{3} d e^{5}\right )} x^{2} + 6 \, {\left (137 \, c^{3} d^{5} e - 30 \, b c^{2} d^{4} e^{2} - 6 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4}\right )} x}{60 \, {\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} + \frac {c^{3} \log \left (e x + d\right )}{e^{7}} \]
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Time = 0.28 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.20 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {c^{3} \log \left ({\left | e x + d \right |}\right )}{e^{7}} + \frac {180 \, {\left (2 \, c^{3} d e^{4} - b c^{2} e^{5}\right )} x^{5} + 90 \, {\left (15 \, c^{3} d^{2} e^{3} - 5 \, b c^{2} d e^{4} - b^{2} c e^{5}\right )} x^{4} + 20 \, {\left (110 \, c^{3} d^{3} e^{2} - 30 \, b c^{2} d^{2} e^{3} - 6 \, b^{2} c d e^{4} - b^{3} e^{5}\right )} x^{3} + 15 \, {\left (125 \, c^{3} d^{4} e - 30 \, b c^{2} d^{3} e^{2} - 6 \, b^{2} c d^{2} e^{3} - b^{3} d e^{4}\right )} x^{2} + 6 \, {\left (137 \, c^{3} d^{5} - 30 \, b c^{2} d^{4} e - 6 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3}\right )} x + \frac {147 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e - 6 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}}{e}}{60 \, {\left (e x + d\right )}^{6} e^{6}} \]
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Time = 9.66 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.18 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {c^3\,\ln \left (d+e\,x\right )}{e^7}-\frac {x^5\,\left (3\,b\,c^2\,e^6-6\,c^3\,d\,e^5\right )+x^4\,\left (\frac {3\,b^2\,c\,e^6}{2}+\frac {15\,b\,c^2\,d\,e^5}{2}-\frac {45\,c^3\,d^2\,e^4}{2}\right )+x\,\left (\frac {b^3\,d^2\,e^4}{10}+\frac {3\,b^2\,c\,d^3\,e^3}{5}+3\,b\,c^2\,d^4\,e^2-\frac {137\,c^3\,d^5\,e}{10}\right )+x^2\,\left (\frac {b^3\,d\,e^5}{4}+\frac {3\,b^2\,c\,d^2\,e^4}{2}+\frac {15\,b\,c^2\,d^3\,e^3}{2}-\frac {125\,c^3\,d^4\,e^2}{4}\right )+x^3\,\left (\frac {b^3\,e^6}{3}+2\,b^2\,c\,d\,e^5+10\,b\,c^2\,d^2\,e^4-\frac {110\,c^3\,d^3\,e^3}{3}\right )-\frac {49\,c^3\,d^6}{20}+\frac {b^3\,d^3\,e^3}{60}+\frac {b^2\,c\,d^4\,e^2}{10}+\frac {b\,c^2\,d^5\,e}{2}}{e^7\,{\left (d+e\,x\right )}^6} \]
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