Integrand size = 20, antiderivative size = 266 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^7} \, dx=-\frac {\left (c d^2-b d e+a e^2\right )^3}{6 e^7 (d+e x)^6}+\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{5 e^7 (d+e x)^5}-\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{4 e^7 (d+e x)^4}+\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )}{3 e^7 (d+e x)^3}-\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{2 e^7 (d+e x)^2}+\frac {3 c^2 (2 c d-b e)}{e^7 (d+e x)}+\frac {c^3 \log (d+e x)}{e^7} \]
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Time = 0.16 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {712} \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^7} \, dx=-\frac {3 c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^7 (d+e x)^2}+\frac {(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{3 e^7 (d+e x)^3}-\frac {3 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{4 e^7 (d+e x)^4}+\frac {3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{5 e^7 (d+e x)^5}-\frac {\left (a e^2-b d e+c d^2\right )^3}{6 e^7 (d+e x)^6}+\frac {3 c^2 (2 c d-b e)}{e^7 (d+e x)}+\frac {c^3 \log (d+e x)}{e^7} \]
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Rule 712
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (c d^2-b d e+a e^2\right )^3}{e^6 (d+e x)^7}+\frac {3 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2}{e^6 (d+e x)^6}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right )}{e^6 (d+e x)^5}+\frac {(2 c d-b e) \left (-10 c^2 d^2-b^2 e^2+2 c e (5 b d-3 a e)\right )}{e^6 (d+e x)^4}+\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\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 {\left (c d^2-b d e+a e^2\right )^3}{6 e^7 (d+e x)^6}+\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{5 e^7 (d+e x)^5}-\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{4 e^7 (d+e x)^4}+\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )}{3 e^7 (d+e x)^3}-\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\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 = 0.10 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.45 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {c^3 d \left (147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right )-e^3 \left (10 a^3 e^3+6 a^2 b e^2 (d+6 e x)+3 a b^2 e \left (d^2+6 d e x+15 e^2 x^2\right )+b^3 \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )\right )-3 c e^2 \left (a^2 e^2 \left (d^2+6 d e x+15 e^2 x^2\right )+2 a b e \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+2 b^2 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )\right )-6 c^2 e \left (a e \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )+5 b \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )\right )+60 c^3 (d+e x)^6 \log (d+e x)}{60 e^7 (d+e x)^6} \]
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Time = 4.80 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.65
method | result | size |
risch | \(\frac {-\frac {3 c^{2} \left (b e -2 c d \right ) x^{5}}{e^{2}}-\frac {3 c \left (a c \,e^{2}+b^{2} e^{2}+5 b c d e -15 c^{2} d^{2}\right ) x^{4}}{2 e^{3}}-\frac {\left (6 a b c \,e^{3}+6 c^{2} a d \,e^{2}+b^{3} e^{3}+6 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -110 c^{3} d^{3}\right ) x^{3}}{3 e^{4}}-\frac {\left (3 e^{4} a^{2} c +3 a \,b^{2} e^{4}+6 a b c d \,e^{3}+6 d^{2} e^{2} c^{2} a +b^{3} d \,e^{3}+6 b^{2} c \,d^{2} e^{2}+30 d^{3} e b \,c^{2}-125 d^{4} c^{3}\right ) x^{2}}{4 e^{5}}-\frac {\left (6 a^{2} b \,e^{5}+3 d \,e^{4} a^{2} c +3 a \,b^{2} d \,e^{4}+6 a b c \,d^{2} e^{3}+6 d^{3} e^{2} c^{2} a +b^{3} d^{2} e^{3}+6 b^{2} c \,d^{3} e^{2}+30 b \,c^{2} d^{4} e -137 d^{5} c^{3}\right ) x}{10 e^{6}}-\frac {10 e^{6} a^{3}+6 a^{2} b d \,e^{5}+3 d^{2} e^{4} a^{2} c +3 a \,b^{2} d^{2} e^{4}+6 a b c \,d^{3} e^{3}+6 d^{4} e^{2} c^{2} a +b^{3} d^{3} e^{3}+6 b^{2} c \,d^{4} e^{2}+30 b \,c^{2} d^{5} e -147 c^{3} d^{6}}{60 e^{7}}}{\left (e x +d \right )^{6}}+\frac {c^{3} \ln \left (e x +d \right )}{e^{7}}\) | \(438\) |
norman | \(\frac {-\frac {10 e^{6} a^{3}+6 a^{2} b d \,e^{5}+3 d^{2} e^{4} a^{2} c +3 a \,b^{2} d^{2} e^{4}+6 a b c \,d^{3} e^{3}+6 d^{4} e^{2} c^{2} a +b^{3} d^{3} e^{3}+6 b^{2} c \,d^{4} e^{2}+30 b \,c^{2} d^{5} e -147 c^{3} d^{6}}{60 e^{7}}-\frac {3 \left (b \,c^{2} e -2 c^{3} d \right ) x^{5}}{e^{2}}-\frac {3 \left (a \,c^{2} e^{2}+b^{2} e^{2} c +5 d e b \,c^{2}-15 c^{3} d^{2}\right ) x^{4}}{2 e^{3}}-\frac {\left (6 a b c \,e^{3}+6 c^{2} a d \,e^{2}+b^{3} e^{3}+6 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -110 c^{3} d^{3}\right ) x^{3}}{3 e^{4}}-\frac {\left (3 e^{4} a^{2} c +3 a \,b^{2} e^{4}+6 a b c d \,e^{3}+6 d^{2} e^{2} c^{2} a +b^{3} d \,e^{3}+6 b^{2} c \,d^{2} e^{2}+30 d^{3} e b \,c^{2}-125 d^{4} c^{3}\right ) x^{2}}{4 e^{5}}-\frac {\left (6 a^{2} b \,e^{5}+3 d \,e^{4} a^{2} c +3 a \,b^{2} d \,e^{4}+6 a b c \,d^{2} e^{3}+6 d^{3} e^{2} c^{2} a +b^{3} d^{2} e^{3}+6 b^{2} c \,d^{3} e^{2}+30 b \,c^{2} d^{4} e -137 d^{5} c^{3}\right ) x}{10 e^{6}}}{\left (e x +d \right )^{6}}+\frac {c^{3} \ln \left (e x +d \right )}{e^{7}}\) | \(444\) |
default | \(-\frac {3 a^{2} b \,e^{5}-6 d \,e^{4} a^{2} c -6 a \,b^{2} d \,e^{4}+18 a b c \,d^{2} e^{3}-12 d^{3} e^{2} c^{2} a +3 b^{3} d^{2} e^{3}-12 b^{2} c \,d^{3} e^{2}+15 b \,c^{2} d^{4} e -6 d^{5} c^{3}}{5 e^{7} \left (e x +d \right )^{5}}-\frac {3 c^{2} \left (b e -2 c d \right )}{e^{7} \left (e x +d \right )}-\frac {6 a b c \,e^{3}-12 c^{2} a d \,e^{2}+b^{3} e^{3}-12 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -20 c^{3} d^{3}}{3 e^{7} \left (e x +d \right )^{3}}-\frac {3 e^{4} a^{2} c +3 a \,b^{2} e^{4}-18 a b c d \,e^{3}+18 d^{2} e^{2} c^{2} a -3 b^{3} d \,e^{3}+18 b^{2} c \,d^{2} e^{2}-30 d^{3} e b \,c^{2}+15 d^{4} c^{3}}{4 e^{7} \left (e x +d \right )^{4}}-\frac {3 c \left (a c \,e^{2}+b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right )}{2 e^{7} \left (e x +d \right )^{2}}+\frac {c^{3} \ln \left (e x +d \right )}{e^{7}}-\frac {e^{6} a^{3}-3 a^{2} b d \,e^{5}+3 d^{2} e^{4} a^{2} c +3 a \,b^{2} d^{2} e^{4}-6 a b c \,d^{3} e^{3}+3 d^{4} e^{2} c^{2} a -b^{3} d^{3} e^{3}+3 b^{2} c \,d^{4} e^{2}-3 b \,c^{2} d^{5} e +c^{3} d^{6}}{6 e^{7} \left (e x +d \right )^{6}}\) | \(459\) |
parallelrisch | \(\frac {-90 a \,c^{2} e^{6} x^{4}+1350 c^{3} d^{2} e^{4} x^{4}-6 a b c \,d^{3} e^{3}-b^{3} d^{3} e^{3}-6 d^{4} e^{2} c^{2} a -3 d^{2} e^{4} a^{2} c +2200 x^{3} c^{3} d^{3} e^{3}+1875 x^{2} c^{3} d^{4} e^{2}+147 c^{3} d^{6}-90 x^{2} a b c d \,e^{5}-36 x a b c \,d^{2} e^{4}-18 x \,a^{2} c d \,e^{5}-36 x a \,c^{2} d^{3} e^{3}-6 b^{2} c \,d^{4} e^{2}-6 a^{2} b d \,e^{5}-3 a \,b^{2} d^{2} e^{4}-10 e^{6} a^{3}+60 \ln \left (e x +d \right ) c^{3} d^{6}-120 x^{3} a \,c^{2} d \,e^{5}-90 x^{2} a \,c^{2} d^{2} e^{4}+360 x^{5} c^{3} d \,e^{5}-45 x^{2} a^{2} c \,e^{6}-30 b \,c^{2} d^{5} e +822 x \,c^{3} d^{5} e -180 x^{5} b \,c^{2} e^{6}-90 x^{4} b^{2} c \,e^{6}-45 x^{2} a \,b^{2} e^{6}-15 x^{2} b^{3} d \,e^{5}-36 x \,a^{2} b \,e^{6}-6 x \,b^{3} d^{2} e^{4}-20 x^{3} b^{3} e^{6}+60 \ln \left (e x +d \right ) x^{6} c^{3} e^{6}+360 \ln \left (e x +d \right ) x^{5} c^{3} d \,e^{5}+900 \ln \left (e x +d \right ) x^{4} c^{3} d^{2} e^{4}-90 x^{2} b^{2} c \,d^{2} e^{4}-450 x^{2} b \,c^{2} d^{3} e^{3}-18 x a \,b^{2} d \,e^{5}-36 x \,b^{2} c \,d^{3} e^{3}-180 x b \,c^{2} d^{4} e^{2}-450 x^{4} b \,c^{2} d \,e^{5}-120 x^{3} a b c \,e^{6}-120 x^{3} b^{2} c d \,e^{5}-600 x^{3} b \,c^{2} d^{2} e^{4}+900 \ln \left (e x +d \right ) x^{2} c^{3} d^{4} e^{2}+1200 \ln \left (e x +d \right ) x^{3} c^{3} d^{3} e^{3}+360 \ln \left (e x +d \right ) x \,c^{3} d^{5} e}{60 e^{7} \left (e x +d \right )^{6}}\) | \(609\) |
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Leaf count of result is larger than twice the leaf count of optimal. 545 vs. \(2 (256) = 512\).
Time = 0.27 (sec) , antiderivative size = 545, normalized size of antiderivative = 2.05 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {147 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e - 6 \, a^{2} b d e^{5} - 10 \, a^{3} e^{6} - 6 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} - 3 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 180 \, {\left (2 \, c^{3} d e^{5} - b c^{2} e^{6}\right )} x^{5} + 90 \, {\left (15 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} - {\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + 20 \, {\left (110 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} - 6 \, {\left (b^{2} c + a c^{2}\right )} d e^{5} - {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 15 \, {\left (125 \, c^{3} d^{4} e^{2} - 30 \, b c^{2} d^{3} e^{3} - 6 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - {\left (b^{3} + 6 \, a b c\right )} d e^{5} - 3 \, {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 6 \, {\left (137 \, c^{3} d^{5} e - 30 \, b c^{2} d^{4} e^{2} - 6 \, a^{2} b e^{6} - 6 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} - 3 \, {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x + 60 \, {\left (c^{3} e^{6} x^{6} + 6 \, c^{3} d e^{5} x^{5} + 15 \, c^{3} d^{2} e^{4} x^{4} + 20 \, c^{3} d^{3} e^{3} x^{3} + 15 \, c^{3} d^{4} e^{2} x^{2} + 6 \, c^{3} d^{5} e x + c^{3} d^{6}\right )} \log \left (e x + d\right )}{60 \, {\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} \]
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Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^7} \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 469, normalized size of antiderivative = 1.76 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {147 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e - 6 \, a^{2} b d e^{5} - 10 \, a^{3} e^{6} - 6 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} - 3 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 180 \, {\left (2 \, c^{3} d e^{5} - b c^{2} e^{6}\right )} x^{5} + 90 \, {\left (15 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} - {\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + 20 \, {\left (110 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} - 6 \, {\left (b^{2} c + a c^{2}\right )} d e^{5} - {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 15 \, {\left (125 \, c^{3} d^{4} e^{2} - 30 \, b c^{2} d^{3} e^{3} - 6 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - {\left (b^{3} + 6 \, a b c\right )} d e^{5} - 3 \, {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 6 \, {\left (137 \, c^{3} d^{5} e - 30 \, b c^{2} d^{4} e^{2} - 6 \, a^{2} b e^{6} - 6 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} - 3 \, {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x}{60 \, {\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} + \frac {c^{3} \log \left (e x + d\right )}{e^{7}} \]
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Time = 0.27 (sec) , antiderivative size = 457, normalized size of antiderivative = 1.72 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {c^{3} \log \left ({\left | e x + d \right |}\right )}{e^{7}} + \frac {180 \, {\left (2 \, c^{3} d e^{4} - b c^{2} e^{5}\right )} x^{5} + 90 \, {\left (15 \, c^{3} d^{2} e^{3} - 5 \, b c^{2} d e^{4} - b^{2} c e^{5} - a c^{2} e^{5}\right )} x^{4} + 20 \, {\left (110 \, c^{3} d^{3} e^{2} - 30 \, b c^{2} d^{2} e^{3} - 6 \, b^{2} c d e^{4} - 6 \, a c^{2} d e^{4} - b^{3} e^{5} - 6 \, a b c e^{5}\right )} x^{3} + 15 \, {\left (125 \, c^{3} d^{4} e - 30 \, b c^{2} d^{3} e^{2} - 6 \, b^{2} c d^{2} e^{3} - 6 \, a c^{2} d^{2} e^{3} - b^{3} d e^{4} - 6 \, a b c d e^{4} - 3 \, a b^{2} e^{5} - 3 \, a^{2} c e^{5}\right )} x^{2} + 6 \, {\left (137 \, c^{3} d^{5} - 30 \, b c^{2} d^{4} e - 6 \, b^{2} c d^{3} e^{2} - 6 \, a c^{2} d^{3} e^{2} - b^{3} d^{2} e^{3} - 6 \, a b c d^{2} e^{3} - 3 \, a b^{2} d e^{4} - 3 \, a^{2} c d e^{4} - 6 \, a^{2} b e^{5}\right )} x + \frac {147 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e - 6 \, b^{2} c d^{4} e^{2} - 6 \, a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} - 6 \, a b c d^{3} e^{3} - 3 \, a b^{2} d^{2} e^{4} - 3 \, a^{2} c d^{2} e^{4} - 6 \, a^{2} b d e^{5} - 10 \, a^{3} e^{6}}{e}}{60 \, {\left (e x + d\right )}^{6} e^{6}} \]
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Time = 9.91 (sec) , antiderivative size = 497, normalized size of antiderivative = 1.87 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {c^3\,\ln \left (d+e\,x\right )}{e^7}-\frac {\frac {10\,a^3\,e^6+6\,a^2\,b\,d\,e^5+3\,a^2\,c\,d^2\,e^4+3\,a\,b^2\,d^2\,e^4+6\,a\,b\,c\,d^3\,e^3+6\,a\,c^2\,d^4\,e^2+b^3\,d^3\,e^3+6\,b^2\,c\,d^4\,e^2+30\,b\,c^2\,d^5\,e-147\,c^3\,d^6}{60\,e^7}+\frac {3\,x^4\,\left (b^2\,c\,e^2+5\,b\,c^2\,d\,e-15\,c^3\,d^2+a\,c^2\,e^2\right )}{2\,e^3}+\frac {x^3\,\left (b^3\,e^3+6\,b^2\,c\,d\,e^2+30\,b\,c^2\,d^2\,e+6\,a\,b\,c\,e^3-110\,c^3\,d^3+6\,a\,c^2\,d\,e^2\right )}{3\,e^4}+\frac {x^2\,\left (3\,a^2\,c\,e^4+3\,a\,b^2\,e^4+6\,a\,b\,c\,d\,e^3+6\,a\,c^2\,d^2\,e^2+b^3\,d\,e^3+6\,b^2\,c\,d^2\,e^2+30\,b\,c^2\,d^3\,e-125\,c^3\,d^4\right )}{4\,e^5}+\frac {x\,\left (6\,a^2\,b\,e^5+3\,a^2\,c\,d\,e^4+3\,a\,b^2\,d\,e^4+6\,a\,b\,c\,d^2\,e^3+6\,a\,c^2\,d^3\,e^2+b^3\,d^2\,e^3+6\,b^2\,c\,d^3\,e^2+30\,b\,c^2\,d^4\,e-137\,c^3\,d^5\right )}{10\,e^6}+\frac {3\,c^2\,x^5\,\left (b\,e-2\,c\,d\right )}{e^2}}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6} \]
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