Integrand size = 26, antiderivative size = 155 \[ \int \frac {1}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {1}{5 (b d-a e) (a+b x)^5}+\frac {e}{4 (b d-a e)^2 (a+b x)^4}-\frac {e^2}{3 (b d-a e)^3 (a+b x)^3}+\frac {e^3}{2 (b d-a e)^4 (a+b x)^2}-\frac {e^4}{(b d-a e)^5 (a+b x)}-\frac {e^5 \log (a+b x)}{(b d-a e)^6}+\frac {e^5 \log (d+e x)}{(b d-a e)^6} \]
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Time = 0.08 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 46} \[ \int \frac {1}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {e^5 \log (a+b x)}{(b d-a e)^6}+\frac {e^5 \log (d+e x)}{(b d-a e)^6}-\frac {e^4}{(a+b x) (b d-a e)^5}+\frac {e^3}{2 (a+b x)^2 (b d-a e)^4}-\frac {e^2}{3 (a+b x)^3 (b d-a e)^3}+\frac {e}{4 (a+b x)^4 (b d-a e)^2}-\frac {1}{5 (a+b x)^5 (b d-a e)} \]
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Rule 27
Rule 46
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(a+b x)^6 (d+e x)} \, dx \\ & = \int \left (\frac {b}{(b d-a e) (a+b x)^6}-\frac {b e}{(b d-a e)^2 (a+b x)^5}+\frac {b e^2}{(b d-a e)^3 (a+b x)^4}-\frac {b e^3}{(b d-a e)^4 (a+b x)^3}+\frac {b e^4}{(b d-a e)^5 (a+b x)^2}-\frac {b e^5}{(b d-a e)^6 (a+b x)}+\frac {e^6}{(b d-a e)^6 (d+e x)}\right ) \, dx \\ & = -\frac {1}{5 (b d-a e) (a+b x)^5}+\frac {e}{4 (b d-a e)^2 (a+b x)^4}-\frac {e^2}{3 (b d-a e)^3 (a+b x)^3}+\frac {e^3}{2 (b d-a e)^4 (a+b x)^2}-\frac {e^4}{(b d-a e)^5 (a+b x)}-\frac {e^5 \log (a+b x)}{(b d-a e)^6}+\frac {e^5 \log (d+e x)}{(b d-a e)^6} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.98 \[ \int \frac {1}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {-12 (b d-a e)^5+15 e (b d-a e)^4 (a+b x)+20 e^2 (-b d+a e)^3 (a+b x)^2+30 e^3 (b d-a e)^2 (a+b x)^3+60 e^4 (-b d+a e) (a+b x)^4-60 e^5 (a+b x)^5 \log (a+b x)+60 e^5 (a+b x)^5 \log (d+e x)}{60 (b d-a e)^6 (a+b x)^5} \]
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Time = 2.54 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.95
method | result | size |
default | \(\frac {1}{5 \left (a e -b d \right ) \left (b x +a \right )^{5}}+\frac {e}{4 \left (a e -b d \right )^{2} \left (b x +a \right )^{4}}-\frac {e^{5} \ln \left (b x +a \right )}{\left (a e -b d \right )^{6}}+\frac {e^{2}}{3 \left (a e -b d \right )^{3} \left (b x +a \right )^{3}}+\frac {e^{3}}{2 \left (a e -b d \right )^{4} \left (b x +a \right )^{2}}+\frac {e^{4}}{\left (a e -b d \right )^{5} \left (b x +a \right )}+\frac {e^{5} \ln \left (e x +d \right )}{\left (a e -b d \right )^{6}}\) | \(147\) |
parallelrisch | \(-\frac {-60 x^{4} a \,b^{9} e^{5}+60 x^{4} b^{10} d \,e^{4}-270 x^{3} a^{2} b^{8} e^{5}-30 x^{3} b^{10} d^{2} e^{3}-470 x^{2} a^{3} b^{7} e^{5}+20 x^{2} b^{10} d^{3} e^{2}-385 x \,a^{4} b^{6} e^{5}-15 x \,b^{10} d^{4} e +60 \ln \left (b x +a \right ) x^{5} b^{10} e^{5}-60 \ln \left (e x +d \right ) x^{5} b^{10} e^{5}+60 \ln \left (b x +a \right ) a^{5} b^{5} e^{5}-60 \ln \left (e x +d \right ) a^{5} b^{5} e^{5}+600 \ln \left (b x +a \right ) x^{3} a^{2} b^{8} e^{5}-600 \ln \left (e x +d \right ) x^{3} a^{2} b^{8} e^{5}+600 \ln \left (b x +a \right ) x^{2} a^{3} b^{7} e^{5}-600 \ln \left (e x +d \right ) x^{2} a^{3} b^{7} e^{5}+300 \ln \left (b x +a \right ) x \,a^{4} b^{6} e^{5}-300 \ln \left (e x +d \right ) x \,a^{4} b^{6} e^{5}+300 x^{3} a \,b^{9} d \,e^{4}+600 x^{2} a^{2} b^{8} d \,e^{4}-150 x^{2} a \,b^{9} d^{2} e^{3}+600 x \,a^{3} b^{7} d \,e^{4}-300 x \,a^{2} b^{8} d^{2} e^{3}+100 x a \,b^{9} d^{3} e^{2}+300 \ln \left (b x +a \right ) x^{4} a \,b^{9} e^{5}+12 b^{10} d^{5}-137 a^{5} b^{5} e^{5}+300 a^{4} b^{6} d \,e^{4}-300 a^{3} b^{7} d^{2} e^{3}+200 a^{2} b^{8} d^{3} e^{2}-75 a \,b^{9} d^{4} e -300 \ln \left (e x +d \right ) x^{4} a \,b^{9} e^{5}}{60 \left (a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}\right ) \left (b x +a \right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2} b^{5}}\) | \(586\) |
risch | \(\frac {\frac {b^{4} e^{4} x^{4}}{a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}}+\frac {\left (9 a e -b d \right ) b^{3} e^{3} x^{3}}{2 a^{5} e^{5}-10 a^{4} b d \,e^{4}+20 a^{3} b^{2} d^{2} e^{3}-20 a^{2} b^{3} d^{3} e^{2}+10 a \,b^{4} d^{4} e -2 b^{5} d^{5}}+\frac {b^{2} e^{2} \left (47 a^{2} e^{2}-13 a b d e +2 b^{2} d^{2}\right ) x^{2}}{6 a^{5} e^{5}-30 a^{4} b d \,e^{4}+60 a^{3} b^{2} d^{2} e^{3}-60 a^{2} b^{3} d^{3} e^{2}+30 a \,b^{4} d^{4} e -6 b^{5} d^{5}}+\frac {\left (77 a^{3} e^{3}-43 a^{2} b d \,e^{2}+17 a \,b^{2} d^{2} e -3 b^{3} d^{3}\right ) e b x}{12 a^{5} e^{5}-60 a^{4} b d \,e^{4}+120 a^{3} b^{2} d^{2} e^{3}-120 a^{2} b^{3} d^{3} e^{2}+60 a \,b^{4} d^{4} e -12 b^{5} d^{5}}+\frac {137 e^{4} a^{4}-163 b \,e^{3} d \,a^{3}+137 b^{2} e^{2} d^{2} a^{2}-63 a \,b^{3} d^{3} e +12 b^{4} d^{4}}{60 a^{5} e^{5}-300 a^{4} b d \,e^{4}+600 a^{3} b^{2} d^{2} e^{3}-600 a^{2} b^{3} d^{3} e^{2}+300 a \,b^{4} d^{4} e -60 b^{5} d^{5}}}{\left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2} \left (b x +a \right )}+\frac {e^{5} \ln \left (-e x -d \right )}{a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}}-\frac {e^{5} \ln \left (b x +a \right )}{a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}}\) | \(700\) |
norman | \(\frac {\frac {b^{4} e^{4} x^{4}}{a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}}+\frac {137 a^{4} b^{5} e^{4}-163 a^{3} b^{6} d \,e^{3}+137 a^{2} b^{7} d^{2} e^{2}-63 a \,b^{8} d^{3} e +12 b^{9} d^{4}}{60 b^{5} \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}+\frac {\left (9 e^{4} a \,b^{5}-d \,e^{3} b^{6}\right ) x^{3}}{2 b^{2} \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}+\frac {\left (47 a^{2} b^{5} e^{4}-13 a \,b^{6} d \,e^{3}+2 b^{7} d^{2} e^{2}\right ) x^{2}}{6 b^{3} \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}+\frac {\left (77 a^{3} b^{5} e^{4}-43 a^{2} b^{6} d \,e^{3}+17 a \,b^{7} d^{2} e^{2}-3 b^{8} d^{3} e \right ) x}{12 b^{4} \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}}{\left (b x +a \right )^{5}}+\frac {e^{5} \ln \left (e x +d \right )}{a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}}-\frac {e^{5} \ln \left (b x +a \right )}{a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}}\) | \(710\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 923 vs. \(2 (147) = 294\).
Time = 0.35 (sec) , antiderivative size = 923, normalized size of antiderivative = 5.95 \[ \int \frac {1}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {12 \, b^{5} d^{5} - 75 \, a b^{4} d^{4} e + 200 \, a^{2} b^{3} d^{3} e^{2} - 300 \, a^{3} b^{2} d^{2} e^{3} + 300 \, a^{4} b d e^{4} - 137 \, a^{5} e^{5} + 60 \, {\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} - 30 \, {\left (b^{5} d^{2} e^{3} - 10 \, a b^{4} d e^{4} + 9 \, a^{2} b^{3} e^{5}\right )} x^{3} + 10 \, {\left (2 \, b^{5} d^{3} e^{2} - 15 \, a b^{4} d^{2} e^{3} + 60 \, a^{2} b^{3} d e^{4} - 47 \, a^{3} b^{2} e^{5}\right )} x^{2} - 5 \, {\left (3 \, b^{5} d^{4} e - 20 \, a b^{4} d^{3} e^{2} + 60 \, a^{2} b^{3} d^{2} e^{3} - 120 \, a^{3} b^{2} d e^{4} + 77 \, a^{4} b e^{5}\right )} x + 60 \, {\left (b^{5} e^{5} x^{5} + 5 \, a b^{4} e^{5} x^{4} + 10 \, a^{2} b^{3} e^{5} x^{3} + 10 \, a^{3} b^{2} e^{5} x^{2} + 5 \, a^{4} b e^{5} x + a^{5} e^{5}\right )} \log \left (b x + a\right ) - 60 \, {\left (b^{5} e^{5} x^{5} + 5 \, a b^{4} e^{5} x^{4} + 10 \, a^{2} b^{3} e^{5} x^{3} + 10 \, a^{3} b^{2} e^{5} x^{2} + 5 \, a^{4} b e^{5} x + a^{5} e^{5}\right )} \log \left (e x + d\right )}{60 \, {\left (a^{5} b^{6} d^{6} - 6 \, a^{6} b^{5} d^{5} e + 15 \, a^{7} b^{4} d^{4} e^{2} - 20 \, a^{8} b^{3} d^{3} e^{3} + 15 \, a^{9} b^{2} d^{2} e^{4} - 6 \, a^{10} b d e^{5} + a^{11} e^{6} + {\left (b^{11} d^{6} - 6 \, a b^{10} d^{5} e + 15 \, a^{2} b^{9} d^{4} e^{2} - 20 \, a^{3} b^{8} d^{3} e^{3} + 15 \, a^{4} b^{7} d^{2} e^{4} - 6 \, a^{5} b^{6} d e^{5} + a^{6} b^{5} e^{6}\right )} x^{5} + 5 \, {\left (a b^{10} d^{6} - 6 \, a^{2} b^{9} d^{5} e + 15 \, a^{3} b^{8} d^{4} e^{2} - 20 \, a^{4} b^{7} d^{3} e^{3} + 15 \, a^{5} b^{6} d^{2} e^{4} - 6 \, a^{6} b^{5} d e^{5} + a^{7} b^{4} e^{6}\right )} x^{4} + 10 \, {\left (a^{2} b^{9} d^{6} - 6 \, a^{3} b^{8} d^{5} e + 15 \, a^{4} b^{7} d^{4} e^{2} - 20 \, a^{5} b^{6} d^{3} e^{3} + 15 \, a^{6} b^{5} d^{2} e^{4} - 6 \, a^{7} b^{4} d e^{5} + a^{8} b^{3} e^{6}\right )} x^{3} + 10 \, {\left (a^{3} b^{8} d^{6} - 6 \, a^{4} b^{7} d^{5} e + 15 \, a^{5} b^{6} d^{4} e^{2} - 20 \, a^{6} b^{5} d^{3} e^{3} + 15 \, a^{7} b^{4} d^{2} e^{4} - 6 \, a^{8} b^{3} d e^{5} + a^{9} b^{2} e^{6}\right )} x^{2} + 5 \, {\left (a^{4} b^{7} d^{6} - 6 \, a^{5} b^{6} d^{5} e + 15 \, a^{6} b^{5} d^{4} e^{2} - 20 \, a^{7} b^{4} d^{3} e^{3} + 15 \, a^{8} b^{3} d^{2} e^{4} - 6 \, a^{9} b^{2} d e^{5} + a^{10} b e^{6}\right )} x\right )}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 1081 vs. \(2 (129) = 258\).
Time = 4.75 (sec) , antiderivative size = 1081, normalized size of antiderivative = 6.97 \[ \int \frac {1}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {e^{5} \log {\left (x + \frac {- \frac {a^{7} e^{12}}{\left (a e - b d\right )^{6}} + \frac {7 a^{6} b d e^{11}}{\left (a e - b d\right )^{6}} - \frac {21 a^{5} b^{2} d^{2} e^{10}}{\left (a e - b d\right )^{6}} + \frac {35 a^{4} b^{3} d^{3} e^{9}}{\left (a e - b d\right )^{6}} - \frac {35 a^{3} b^{4} d^{4} e^{8}}{\left (a e - b d\right )^{6}} + \frac {21 a^{2} b^{5} d^{5} e^{7}}{\left (a e - b d\right )^{6}} - \frac {7 a b^{6} d^{6} e^{6}}{\left (a e - b d\right )^{6}} + a e^{6} + \frac {b^{7} d^{7} e^{5}}{\left (a e - b d\right )^{6}} + b d e^{5}}{2 b e^{6}} \right )}}{\left (a e - b d\right )^{6}} - \frac {e^{5} \log {\left (x + \frac {\frac {a^{7} e^{12}}{\left (a e - b d\right )^{6}} - \frac {7 a^{6} b d e^{11}}{\left (a e - b d\right )^{6}} + \frac {21 a^{5} b^{2} d^{2} e^{10}}{\left (a e - b d\right )^{6}} - \frac {35 a^{4} b^{3} d^{3} e^{9}}{\left (a e - b d\right )^{6}} + \frac {35 a^{3} b^{4} d^{4} e^{8}}{\left (a e - b d\right )^{6}} - \frac {21 a^{2} b^{5} d^{5} e^{7}}{\left (a e - b d\right )^{6}} + \frac {7 a b^{6} d^{6} e^{6}}{\left (a e - b d\right )^{6}} + a e^{6} - \frac {b^{7} d^{7} e^{5}}{\left (a e - b d\right )^{6}} + b d e^{5}}{2 b e^{6}} \right )}}{\left (a e - b d\right )^{6}} + \frac {137 a^{4} e^{4} - 163 a^{3} b d e^{3} + 137 a^{2} b^{2} d^{2} e^{2} - 63 a b^{3} d^{3} e + 12 b^{4} d^{4} + 60 b^{4} e^{4} x^{4} + x^{3} \cdot \left (270 a b^{3} e^{4} - 30 b^{4} d e^{3}\right ) + x^{2} \cdot \left (470 a^{2} b^{2} e^{4} - 130 a b^{3} d e^{3} + 20 b^{4} d^{2} e^{2}\right ) + x \left (385 a^{3} b e^{4} - 215 a^{2} b^{2} d e^{3} + 85 a b^{3} d^{2} e^{2} - 15 b^{4} d^{3} e\right )}{60 a^{10} e^{5} - 300 a^{9} b d e^{4} + 600 a^{8} b^{2} d^{2} e^{3} - 600 a^{7} b^{3} d^{3} e^{2} + 300 a^{6} b^{4} d^{4} e - 60 a^{5} b^{5} d^{5} + x^{5} \cdot \left (60 a^{5} b^{5} e^{5} - 300 a^{4} b^{6} d e^{4} + 600 a^{3} b^{7} d^{2} e^{3} - 600 a^{2} b^{8} d^{3} e^{2} + 300 a b^{9} d^{4} e - 60 b^{10} d^{5}\right ) + x^{4} \cdot \left (300 a^{6} b^{4} e^{5} - 1500 a^{5} b^{5} d e^{4} + 3000 a^{4} b^{6} d^{2} e^{3} - 3000 a^{3} b^{7} d^{3} e^{2} + 1500 a^{2} b^{8} d^{4} e - 300 a b^{9} d^{5}\right ) + x^{3} \cdot \left (600 a^{7} b^{3} e^{5} - 3000 a^{6} b^{4} d e^{4} + 6000 a^{5} b^{5} d^{2} e^{3} - 6000 a^{4} b^{6} d^{3} e^{2} + 3000 a^{3} b^{7} d^{4} e - 600 a^{2} b^{8} d^{5}\right ) + x^{2} \cdot \left (600 a^{8} b^{2} e^{5} - 3000 a^{7} b^{3} d e^{4} + 6000 a^{6} b^{4} d^{2} e^{3} - 6000 a^{5} b^{5} d^{3} e^{2} + 3000 a^{4} b^{6} d^{4} e - 600 a^{3} b^{7} d^{5}\right ) + x \left (300 a^{9} b e^{5} - 1500 a^{8} b^{2} d e^{4} + 3000 a^{7} b^{3} d^{2} e^{3} - 3000 a^{6} b^{4} d^{3} e^{2} + 1500 a^{5} b^{5} d^{4} e - 300 a^{4} b^{6} d^{5}\right )} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 805 vs. \(2 (147) = 294\).
Time = 0.25 (sec) , antiderivative size = 805, normalized size of antiderivative = 5.19 \[ \int \frac {1}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {e^{5} \log \left (b x + a\right )}{b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}} + \frac {e^{5} \log \left (e x + d\right )}{b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}} - \frac {60 \, b^{4} e^{4} x^{4} + 12 \, b^{4} d^{4} - 63 \, a b^{3} d^{3} e + 137 \, a^{2} b^{2} d^{2} e^{2} - 163 \, a^{3} b d e^{3} + 137 \, a^{4} e^{4} - 30 \, {\left (b^{4} d e^{3} - 9 \, a b^{3} e^{4}\right )} x^{3} + 10 \, {\left (2 \, b^{4} d^{2} e^{2} - 13 \, a b^{3} d e^{3} + 47 \, a^{2} b^{2} e^{4}\right )} x^{2} - 5 \, {\left (3 \, b^{4} d^{3} e - 17 \, a b^{3} d^{2} e^{2} + 43 \, a^{2} b^{2} d e^{3} - 77 \, a^{3} b e^{4}\right )} x}{60 \, {\left (a^{5} b^{5} d^{5} - 5 \, a^{6} b^{4} d^{4} e + 10 \, a^{7} b^{3} d^{3} e^{2} - 10 \, a^{8} b^{2} d^{2} e^{3} + 5 \, a^{9} b d e^{4} - a^{10} e^{5} + {\left (b^{10} d^{5} - 5 \, a b^{9} d^{4} e + 10 \, a^{2} b^{8} d^{3} e^{2} - 10 \, a^{3} b^{7} d^{2} e^{3} + 5 \, a^{4} b^{6} d e^{4} - a^{5} b^{5} e^{5}\right )} x^{5} + 5 \, {\left (a b^{9} d^{5} - 5 \, a^{2} b^{8} d^{4} e + 10 \, a^{3} b^{7} d^{3} e^{2} - 10 \, a^{4} b^{6} d^{2} e^{3} + 5 \, a^{5} b^{5} d e^{4} - a^{6} b^{4} e^{5}\right )} x^{4} + 10 \, {\left (a^{2} b^{8} d^{5} - 5 \, a^{3} b^{7} d^{4} e + 10 \, a^{4} b^{6} d^{3} e^{2} - 10 \, a^{5} b^{5} d^{2} e^{3} + 5 \, a^{6} b^{4} d e^{4} - a^{7} b^{3} e^{5}\right )} x^{3} + 10 \, {\left (a^{3} b^{7} d^{5} - 5 \, a^{4} b^{6} d^{4} e + 10 \, a^{5} b^{5} d^{3} e^{2} - 10 \, a^{6} b^{4} d^{2} e^{3} + 5 \, a^{7} b^{3} d e^{4} - a^{8} b^{2} e^{5}\right )} x^{2} + 5 \, {\left (a^{4} b^{6} d^{5} - 5 \, a^{5} b^{5} d^{4} e + 10 \, a^{6} b^{4} d^{3} e^{2} - 10 \, a^{7} b^{3} d^{2} e^{3} + 5 \, a^{8} b^{2} d e^{4} - a^{9} b e^{5}\right )} x\right )}} \]
[In]
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Leaf count of result is larger than twice the leaf count of optimal. 445 vs. \(2 (147) = 294\).
Time = 0.27 (sec) , antiderivative size = 445, normalized size of antiderivative = 2.87 \[ \int \frac {1}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {b e^{5} \log \left ({\left | b x + a \right |}\right )}{b^{7} d^{6} - 6 \, a b^{6} d^{5} e + 15 \, a^{2} b^{5} d^{4} e^{2} - 20 \, a^{3} b^{4} d^{3} e^{3} + 15 \, a^{4} b^{3} d^{2} e^{4} - 6 \, a^{5} b^{2} d e^{5} + a^{6} b e^{6}} + \frac {e^{6} \log \left ({\left | e x + d \right |}\right )}{b^{6} d^{6} e - 6 \, a b^{5} d^{5} e^{2} + 15 \, a^{2} b^{4} d^{4} e^{3} - 20 \, a^{3} b^{3} d^{3} e^{4} + 15 \, a^{4} b^{2} d^{2} e^{5} - 6 \, a^{5} b d e^{6} + a^{6} e^{7}} - \frac {12 \, b^{5} d^{5} - 75 \, a b^{4} d^{4} e + 200 \, a^{2} b^{3} d^{3} e^{2} - 300 \, a^{3} b^{2} d^{2} e^{3} + 300 \, a^{4} b d e^{4} - 137 \, a^{5} e^{5} + 60 \, {\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} - 30 \, {\left (b^{5} d^{2} e^{3} - 10 \, a b^{4} d e^{4} + 9 \, a^{2} b^{3} e^{5}\right )} x^{3} + 10 \, {\left (2 \, b^{5} d^{3} e^{2} - 15 \, a b^{4} d^{2} e^{3} + 60 \, a^{2} b^{3} d e^{4} - 47 \, a^{3} b^{2} e^{5}\right )} x^{2} - 5 \, {\left (3 \, b^{5} d^{4} e - 20 \, a b^{4} d^{3} e^{2} + 60 \, a^{2} b^{3} d^{2} e^{3} - 120 \, a^{3} b^{2} d e^{4} + 77 \, a^{4} b e^{5}\right )} x}{60 \, {\left (b d - a e\right )}^{6} {\left (b x + a\right )}^{5}} \]
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Time = 9.72 (sec) , antiderivative size = 721, normalized size of antiderivative = 4.65 \[ \int \frac {1}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\frac {137\,a^4\,e^4-163\,a^3\,b\,d\,e^3+137\,a^2\,b^2\,d^2\,e^2-63\,a\,b^3\,d^3\,e+12\,b^4\,d^4}{60\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}-\frac {e\,x\,\left (-77\,a^3\,b\,e^3+43\,a^2\,b^2\,d\,e^2-17\,a\,b^3\,d^2\,e+3\,b^4\,d^3\right )}{12\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}+\frac {b^4\,e^4\,x^4}{a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5}-\frac {e^3\,x^3\,\left (b^4\,d-9\,a\,b^3\,e\right )}{2\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}+\frac {e^2\,x^2\,\left (47\,a^2\,b^2\,e^2-13\,a\,b^3\,d\,e+2\,b^4\,d^2\right )}{6\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}}{a^5+5\,a^4\,b\,x+10\,a^3\,b^2\,x^2+10\,a^2\,b^3\,x^3+5\,a\,b^4\,x^4+b^5\,x^5}-\frac {2\,e^5\,\mathrm {atanh}\left (\frac {a^6\,e^6-4\,a^5\,b\,d\,e^5+5\,a^4\,b^2\,d^2\,e^4-5\,a^2\,b^4\,d^4\,e^2+4\,a\,b^5\,d^5\,e-b^6\,d^6}{{\left (a\,e-b\,d\right )}^6}+\frac {2\,b\,e\,x\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}{{\left (a\,e-b\,d\right )}^6}\right )}{{\left (a\,e-b\,d\right )}^6} \]
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