Integrand size = 19, antiderivative size = 137 \[ \int \frac {\left (b x+c x^2\right )^2}{(d+e x)^7} \, dx=-\frac {d^2 (c d-b e)^2}{6 e^5 (d+e x)^6}+\frac {2 d (c d-b e) (2 c d-b e)}{5 e^5 (d+e x)^5}-\frac {6 c^2 d^2-6 b c d e+b^2 e^2}{4 e^5 (d+e x)^4}+\frac {2 c (2 c d-b e)}{3 e^5 (d+e x)^3}-\frac {c^2}{2 e^5 (d+e x)^2} \]
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Time = 0.06 (sec) , antiderivative size = 137, 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} \[ \int \frac {\left (b x+c x^2\right )^2}{(d+e x)^7} \, dx=-\frac {b^2 e^2-6 b c d e+6 c^2 d^2}{4 e^5 (d+e x)^4}-\frac {d^2 (c d-b e)^2}{6 e^5 (d+e x)^6}+\frac {2 c (2 c d-b e)}{3 e^5 (d+e x)^3}+\frac {2 d (c d-b e) (2 c d-b e)}{5 e^5 (d+e x)^5}-\frac {c^2}{2 e^5 (d+e x)^2} \]
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Rule 712
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d^2 (c d-b e)^2}{e^4 (d+e x)^7}+\frac {2 d (c d-b e) (-2 c d+b e)}{e^4 (d+e x)^6}+\frac {6 c^2 d^2-6 b c d e+b^2 e^2}{e^4 (d+e x)^5}-\frac {2 c (2 c d-b e)}{e^4 (d+e x)^4}+\frac {c^2}{e^4 (d+e x)^3}\right ) \, dx \\ & = -\frac {d^2 (c d-b e)^2}{6 e^5 (d+e x)^6}+\frac {2 d (c d-b e) (2 c d-b e)}{5 e^5 (d+e x)^5}-\frac {6 c^2 d^2-6 b c d e+b^2 e^2}{4 e^5 (d+e x)^4}+\frac {2 c (2 c d-b e)}{3 e^5 (d+e x)^3}-\frac {c^2}{2 e^5 (d+e x)^2} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.85 \[ \int \frac {\left (b x+c x^2\right )^2}{(d+e x)^7} \, dx=-\frac {b^2 e^2 \left (d^2+6 d e x+15 e^2 x^2\right )+2 b c e \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+2 c^2 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )}{60 e^5 (d+e x)^6} \]
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Time = 2.00 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.92
method | result | size |
risch | \(\frac {-\frac {c^{2} x^{4}}{2 e}-\frac {2 c \left (b e +c d \right ) x^{3}}{3 e^{2}}-\frac {\left (b^{2} e^{2}+2 b c d e +2 c^{2} d^{2}\right ) x^{2}}{4 e^{3}}-\frac {d \left (b^{2} e^{2}+2 b c d e +2 c^{2} d^{2}\right ) x}{10 e^{4}}-\frac {d^{2} \left (b^{2} e^{2}+2 b c d e +2 c^{2} d^{2}\right )}{60 e^{5}}}{\left (e x +d \right )^{6}}\) | \(126\) |
gosper | \(-\frac {30 c^{2} x^{4} e^{4}+40 x^{3} b c \,e^{4}+40 x^{3} c^{2} d \,e^{3}+15 x^{2} b^{2} e^{4}+30 x^{2} b c d \,e^{3}+30 x^{2} c^{2} d^{2} e^{2}+6 x \,b^{2} d \,e^{3}+12 x b c \,d^{2} e^{2}+12 x \,c^{2} d^{3} e +b^{2} d^{2} e^{2}+2 d^{3} e b c +2 c^{2} d^{4}}{60 e^{5} \left (e x +d \right )^{6}}\) | \(140\) |
norman | \(\frac {-\frac {c^{2} x^{4}}{2 e}-\frac {2 \left (b c \,e^{2}+d e \,c^{2}\right ) x^{3}}{3 e^{3}}-\frac {\left (e^{3} b^{2}+2 b c d \,e^{2}+2 d^{2} e \,c^{2}\right ) x^{2}}{4 e^{4}}-\frac {d \left (e^{3} b^{2}+2 b c d \,e^{2}+2 d^{2} e \,c^{2}\right ) x}{10 e^{5}}-\frac {d^{2} \left (e^{3} b^{2}+2 b c d \,e^{2}+2 d^{2} e \,c^{2}\right )}{60 e^{6}}}{\left (e x +d \right )^{6}}\) | \(140\) |
default | \(-\frac {d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}{6 e^{5} \left (e x +d \right )^{6}}-\frac {2 c \left (b e -2 c d \right )}{3 e^{5} \left (e x +d \right )^{3}}-\frac {b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}}{4 e^{5} \left (e x +d \right )^{4}}+\frac {2 d \left (b^{2} e^{2}-3 b c d e +2 c^{2} d^{2}\right )}{5 e^{5} \left (e x +d \right )^{5}}-\frac {c^{2}}{2 e^{5} \left (e x +d \right )^{2}}\) | \(143\) |
parallelrisch | \(\frac {-30 x^{4} c^{2} e^{5}-40 b c \,e^{5} x^{3}-40 c^{2} d \,e^{4} x^{3}-15 b^{2} e^{5} x^{2}-30 b c d \,e^{4} x^{2}-30 c^{2} d^{2} e^{3} x^{2}-6 b^{2} d \,e^{4} x -12 b c \,d^{2} e^{3} x -12 c^{2} d^{3} e^{2} x -b^{2} d^{2} e^{3}-2 b c \,d^{3} e^{2}-2 c^{2} d^{4} e}{60 e^{6} \left (e x +d \right )^{6}}\) | \(146\) |
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Time = 0.26 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.39 \[ \int \frac {\left (b x+c x^2\right )^2}{(d+e x)^7} \, dx=-\frac {30 \, c^{2} e^{4} x^{4} + 2 \, c^{2} d^{4} + 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 40 \, {\left (c^{2} d e^{3} + b c e^{4}\right )} x^{3} + 15 \, {\left (2 \, c^{2} d^{2} e^{2} + 2 \, b c d e^{3} + b^{2} e^{4}\right )} x^{2} + 6 \, {\left (2 \, c^{2} d^{3} e + 2 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x}{60 \, {\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} e^{5}\right )}} \]
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Time = 21.43 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.51 \[ \int \frac {\left (b x+c x^2\right )^2}{(d+e x)^7} \, dx=\frac {- b^{2} d^{2} e^{2} - 2 b c d^{3} e - 2 c^{2} d^{4} - 30 c^{2} e^{4} x^{4} + x^{3} \left (- 40 b c e^{4} - 40 c^{2} d e^{3}\right ) + x^{2} \left (- 15 b^{2} e^{4} - 30 b c d e^{3} - 30 c^{2} d^{2} e^{2}\right ) + x \left (- 6 b^{2} d e^{3} - 12 b c d^{2} e^{2} - 12 c^{2} d^{3} e\right )}{60 d^{6} e^{5} + 360 d^{5} e^{6} x + 900 d^{4} e^{7} x^{2} + 1200 d^{3} e^{8} x^{3} + 900 d^{2} e^{9} x^{4} + 360 d e^{10} x^{5} + 60 e^{11} x^{6}} \]
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Time = 0.22 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.39 \[ \int \frac {\left (b x+c x^2\right )^2}{(d+e x)^7} \, dx=-\frac {30 \, c^{2} e^{4} x^{4} + 2 \, c^{2} d^{4} + 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 40 \, {\left (c^{2} d e^{3} + b c e^{4}\right )} x^{3} + 15 \, {\left (2 \, c^{2} d^{2} e^{2} + 2 \, b c d e^{3} + b^{2} e^{4}\right )} x^{2} + 6 \, {\left (2 \, c^{2} d^{3} e + 2 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x}{60 \, {\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} e^{5}\right )}} \]
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Time = 0.26 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.01 \[ \int \frac {\left (b x+c x^2\right )^2}{(d+e x)^7} \, dx=-\frac {30 \, c^{2} e^{4} x^{4} + 40 \, c^{2} d e^{3} x^{3} + 40 \, b c e^{4} x^{3} + 30 \, c^{2} d^{2} e^{2} x^{2} + 30 \, b c d e^{3} x^{2} + 15 \, b^{2} e^{4} x^{2} + 12 \, c^{2} d^{3} e x + 12 \, b c d^{2} e^{2} x + 6 \, b^{2} d e^{3} x + 2 \, c^{2} d^{4} + 2 \, b c d^{3} e + b^{2} d^{2} e^{2}}{60 \, {\left (e x + d\right )}^{6} e^{5}} \]
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Time = 0.09 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.32 \[ \int \frac {\left (b x+c x^2\right )^2}{(d+e x)^7} \, dx=-\frac {\frac {x^2\,\left (b^2\,e^2+2\,b\,c\,d\,e+2\,c^2\,d^2\right )}{4\,e^3}+\frac {c^2\,x^4}{2\,e}+\frac {d^2\,\left (b^2\,e^2+2\,b\,c\,d\,e+2\,c^2\,d^2\right )}{60\,e^5}+\frac {2\,c\,x^3\,\left (b\,e+c\,d\right )}{3\,e^2}+\frac {d\,x\,\left (b^2\,e^2+2\,b\,c\,d\,e+2\,c^2\,d^2\right )}{10\,e^4}}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6} \]
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