Integrand size = 31, antiderivative size = 159 \[ \int \frac {a+b x}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {b}{4 (b d-a e)^2 (a+b x)^4}+\frac {2 b e}{3 (b d-a e)^3 (a+b x)^3}-\frac {3 b e^2}{2 (b d-a e)^4 (a+b x)^2}+\frac {4 b e^3}{(b d-a e)^5 (a+b x)}+\frac {e^4}{(b d-a e)^5 (d+e x)}+\frac {5 b e^4 \log (a+b x)}{(b d-a e)^6}-\frac {5 b e^4 \log (d+e x)}{(b d-a e)^6} \]
[Out]
Time = 0.09 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 46} \[ \int \frac {a+b x}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {e^4}{(d+e x) (b d-a e)^5}+\frac {5 b e^4 \log (a+b x)}{(b d-a e)^6}-\frac {5 b e^4 \log (d+e x)}{(b d-a e)^6}+\frac {4 b e^3}{(a+b x) (b d-a e)^5}-\frac {3 b e^2}{2 (a+b x)^2 (b d-a e)^4}+\frac {2 b e}{3 (a+b x)^3 (b d-a e)^3}-\frac {b}{4 (a+b x)^4 (b d-a e)^2} \]
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Rule 27
Rule 46
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(a+b x)^5 (d+e x)^2} \, dx \\ & = \int \left (\frac {b^2}{(b d-a e)^2 (a+b x)^5}-\frac {2 b^2 e}{(b d-a e)^3 (a+b x)^4}+\frac {3 b^2 e^2}{(b d-a e)^4 (a+b x)^3}-\frac {4 b^2 e^3}{(b d-a e)^5 (a+b x)^2}+\frac {5 b^2 e^4}{(b d-a e)^6 (a+b x)}-\frac {e^5}{(b d-a e)^5 (d+e x)^2}-\frac {5 b e^5}{(b d-a e)^6 (d+e x)}\right ) \, dx \\ & = -\frac {b}{4 (b d-a e)^2 (a+b x)^4}+\frac {2 b e}{3 (b d-a e)^3 (a+b x)^3}-\frac {3 b e^2}{2 (b d-a e)^4 (a+b x)^2}+\frac {4 b e^3}{(b d-a e)^5 (a+b x)}+\frac {e^4}{(b d-a e)^5 (d+e x)}+\frac {5 b e^4 \log (a+b x)}{(b d-a e)^6}-\frac {5 b e^4 \log (d+e x)}{(b d-a e)^6} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.91 \[ \int \frac {a+b x}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {-\frac {3 b (b d-a e)^4}{(a+b x)^4}+\frac {8 b e (b d-a e)^3}{(a+b x)^3}-\frac {18 b e^2 (b d-a e)^2}{(a+b x)^2}+\frac {48 b e^3 (b d-a e)}{a+b x}+\frac {12 e^4 (b d-a e)}{d+e x}+60 b e^4 \log (a+b x)-60 b e^4 \log (d+e x)}{12 (b d-a e)^6} \]
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Time = 0.35 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.97
method | result | size |
default | \(-\frac {b}{4 \left (a e -b d \right )^{2} \left (b x +a \right )^{4}}+\frac {5 b \,e^{4} \ln \left (b x +a \right )}{\left (a e -b d \right )^{6}}-\frac {4 b \,e^{3}}{\left (a e -b d \right )^{5} \left (b x +a \right )}-\frac {3 b \,e^{2}}{2 \left (a e -b d \right )^{4} \left (b x +a \right )^{2}}-\frac {2 b e}{3 \left (a e -b d \right )^{3} \left (b x +a \right )^{3}}-\frac {e^{4}}{\left (a e -b d \right )^{5} \left (e x +d \right )}-\frac {5 b \,e^{4} \ln \left (e x +d \right )}{\left (a e -b d \right )^{6}}\) | \(155\) |
risch | \(\frac {-\frac {5 b^{4} e^{4} x^{4}}{e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}}-\frac {5 b^{3} \left (7 a e +b d \right ) e^{3} x^{3}}{2 \left (e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}\right )}-\frac {5 b^{2} e^{2} \left (26 e^{2} a^{2}+11 a b d e -b^{2} d^{2}\right ) x^{2}}{6 \left (e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}\right )}-\frac {5 \left (25 a^{3} e^{3}+29 a^{2} b d \,e^{2}-7 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) b e x}{12 \left (e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}\right )}-\frac {12 e^{4} a^{4}+77 b d \,e^{3} a^{3}-43 b^{2} d^{2} e^{2} a^{2}+17 b^{3} d^{3} e a -3 b^{4} d^{4}}{12 \left (e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}\right )}}{\left (b x +a \right )^{2} \left (e x +d \right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )}+\frac {5 b \,e^{4} \ln \left (-b x -a \right )}{e^{6} a^{6}-6 b d \,e^{5} a^{5}+15 b^{2} d^{2} e^{4} a^{4}-20 b^{3} d^{3} e^{3} a^{3}+15 b^{4} d^{4} e^{2} a^{2}-6 b^{5} d^{5} e a +b^{6} d^{6}}-\frac {5 b \,e^{4} \ln \left (e x +d \right )}{e^{6} a^{6}-6 b d \,e^{5} a^{5}+15 b^{2} d^{2} e^{4} a^{4}-20 b^{3} d^{3} e^{3} a^{3}+15 b^{4} d^{4} e^{2} a^{2}-6 b^{5} d^{5} e a +b^{6} d^{6}}\) | \(709\) |
parallelrisch | \(\frac {-360 \ln \left (e x +d \right ) x^{3} a^{2} b^{7} e^{6}+240 \ln \left (b x +a \right ) x^{2} a^{3} b^{6} e^{6}-240 \ln \left (e x +d \right ) x^{2} a^{3} b^{6} e^{6}+60 \ln \left (b x +a \right ) x \,a^{4} b^{5} e^{6}-60 \ln \left (e x +d \right ) x \,a^{4} b^{5} e^{6}+60 \ln \left (b x +a \right ) a^{4} b^{5} d \,e^{5}-60 \ln \left (e x +d \right ) a^{4} b^{5} d \,e^{5}+360 \ln \left (b x +a \right ) x^{3} a^{2} b^{7} e^{6}-12 a^{5} b^{4} e^{6}-3 b^{9} d^{5} e +180 x^{3} a \,b^{8} d \,e^{5}+150 x^{2} a^{2} b^{7} d \,e^{5}+120 x^{2} a \,b^{8} d^{2} e^{4}-20 x \,a^{3} b^{6} d \,e^{5}+180 x \,a^{2} b^{7} d^{2} e^{4}-40 x a \,b^{8} d^{3} e^{3}+240 \ln \left (b x +a \right ) x^{4} a \,b^{8} e^{6}+60 \ln \left (b x +a \right ) x^{4} b^{9} d \,e^{5}-240 \ln \left (e x +d \right ) x^{4} a \,b^{8} e^{6}-60 \ln \left (e x +d \right ) x^{4} b^{9} d \,e^{5}-60 x^{4} a \,b^{8} e^{6}+60 x^{4} b^{9} d \,e^{5}-240 \ln \left (e x +d \right ) x^{3} a \,b^{8} d \,e^{5}+360 \ln \left (b x +a \right ) x^{2} a^{2} b^{7} d \,e^{5}-360 \ln \left (e x +d \right ) x^{2} a^{2} b^{7} d \,e^{5}+240 \ln \left (b x +a \right ) x \,a^{3} b^{6} d \,e^{5}-240 \ln \left (e x +d \right ) x \,a^{3} b^{6} d \,e^{5}+20 a \,b^{8} d^{4} e^{2}+240 \ln \left (b x +a \right ) x^{3} a \,b^{8} d \,e^{5}-210 x^{3} a^{2} b^{7} e^{6}+30 x^{3} b^{9} d^{2} e^{4}-260 x^{2} a^{3} b^{6} e^{6}-10 x^{2} b^{9} d^{3} e^{3}-125 x \,a^{4} b^{5} e^{6}+5 x \,b^{9} d^{4} e^{2}+60 \ln \left (b x +a \right ) x^{5} b^{9} e^{6}-60 \ln \left (e x +d \right ) x^{5} b^{9} e^{6}-65 a^{4} b^{5} d \,e^{5}+120 a^{3} b^{6} d^{2} e^{4}-60 a^{2} b^{7} d^{3} e^{3}}{12 \left (e^{6} a^{6}-6 b d \,e^{5} a^{5}+15 b^{2} d^{2} e^{4} a^{4}-20 b^{3} d^{3} e^{3} a^{3}+15 b^{4} d^{4} e^{2} a^{2}-6 b^{5} d^{5} e a +b^{6} d^{6}\right ) \left (b x +a \right )^{2} \left (b^{2} x^{2}+2 a b x +a^{2}\right ) \left (e x +d \right ) b^{4} e}\) | \(757\) |
norman | \(\frac {-\frac {5 b^{5} e^{4} x^{5}}{e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}}+\frac {a \left (-12 a^{4} b^{5} e^{5}-77 a^{3} b^{6} d \,e^{4}+43 a^{2} b^{7} d^{2} e^{3}-17 a \,b^{8} d^{3} e^{2}+3 b^{9} d^{4} e \right )}{12 e \,b^{5} \left (e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}\right )}+\frac {5 \left (-77 a^{3} b^{6} e^{5}-51 a^{2} b^{7} d \,e^{4}+9 a \,b^{8} d^{2} e^{3}-b^{9} d^{3} e^{2}\right ) x^{2}}{12 b^{4} e \left (e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}\right )}+\frac {5 \left (-9 e^{5} a \,b^{6}-d \,e^{4} b^{7}\right ) x^{4}}{2 e \,b^{2} \left (e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}\right )}+\frac {5 \left (-47 a^{2} b^{6} e^{5}-14 a \,b^{7} d \,e^{4}+b^{8} d^{2} e^{3}\right ) x^{3}}{6 e \,b^{3} \left (e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}\right )}+\frac {\left (-137 a^{4} b^{6} e^{5}-222 a^{3} b^{7} d \,e^{4}+78 a^{2} b^{8} d^{2} e^{3}-22 a \,b^{9} d^{3} e^{2}+3 b^{10} d^{4} e \right ) x}{12 e \,b^{5} \left (e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}\right )}}{\left (b x +a \right )^{5} \left (e x +d \right )}+\frac {5 b \,e^{4} \ln \left (b x +a \right )}{e^{6} a^{6}-6 b d \,e^{5} a^{5}+15 b^{2} d^{2} e^{4} a^{4}-20 b^{3} d^{3} e^{3} a^{3}+15 b^{4} d^{4} e^{2} a^{2}-6 b^{5} d^{5} e a +b^{6} d^{6}}-\frac {5 b \,e^{4} \ln \left (e x +d \right )}{e^{6} a^{6}-6 b d \,e^{5} a^{5}+15 b^{2} d^{2} e^{4} a^{4}-20 b^{3} d^{3} e^{3} a^{3}+15 b^{4} d^{4} e^{2} a^{2}-6 b^{5} d^{5} e a +b^{6} d^{6}}\) | \(874\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 1083 vs. \(2 (153) = 306\).
Time = 0.46 (sec) , antiderivative size = 1083, normalized size of antiderivative = 6.81 \[ \int \frac {a+b x}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {3 \, b^{5} d^{5} - 20 \, a b^{4} d^{4} e + 60 \, a^{2} b^{3} d^{3} e^{2} - 120 \, a^{3} b^{2} d^{2} e^{3} + 65 \, a^{4} b d e^{4} + 12 \, 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} + 6 \, a b^{4} d e^{4} - 7 \, a^{2} b^{3} e^{5}\right )} x^{3} + 10 \, {\left (b^{5} d^{3} e^{2} - 12 \, a b^{4} d^{2} e^{3} - 15 \, a^{2} b^{3} d e^{4} + 26 \, a^{3} b^{2} e^{5}\right )} x^{2} - 5 \, {\left (b^{5} d^{4} e - 8 \, a b^{4} d^{3} e^{2} + 36 \, a^{2} b^{3} d^{2} e^{3} - 4 \, a^{3} b^{2} d e^{4} - 25 \, a^{4} b e^{5}\right )} x - 60 \, {\left (b^{5} e^{5} x^{5} + a^{4} b d e^{4} + {\left (b^{5} d e^{4} + 4 \, a b^{4} e^{5}\right )} x^{4} + 2 \, {\left (2 \, a b^{4} d e^{4} + 3 \, a^{2} b^{3} e^{5}\right )} x^{3} + 2 \, {\left (3 \, a^{2} b^{3} d e^{4} + 2 \, a^{3} b^{2} e^{5}\right )} x^{2} + {\left (4 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x\right )} \log \left (b x + a\right ) + 60 \, {\left (b^{5} e^{5} x^{5} + a^{4} b d e^{4} + {\left (b^{5} d e^{4} + 4 \, a b^{4} e^{5}\right )} x^{4} + 2 \, {\left (2 \, a b^{4} d e^{4} + 3 \, a^{2} b^{3} e^{5}\right )} x^{3} + 2 \, {\left (3 \, a^{2} b^{3} d e^{4} + 2 \, a^{3} b^{2} e^{5}\right )} x^{2} + {\left (4 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x\right )} \log \left (e x + d\right )}{12 \, {\left (a^{4} b^{6} d^{7} - 6 \, a^{5} b^{5} d^{6} e + 15 \, a^{6} b^{4} d^{5} e^{2} - 20 \, a^{7} b^{3} d^{4} e^{3} + 15 \, a^{8} b^{2} d^{3} e^{4} - 6 \, a^{9} b d^{2} e^{5} + a^{10} d e^{6} + {\left (b^{10} d^{6} e - 6 \, a b^{9} d^{5} e^{2} + 15 \, a^{2} b^{8} d^{4} e^{3} - 20 \, a^{3} b^{7} d^{3} e^{4} + 15 \, a^{4} b^{6} d^{2} e^{5} - 6 \, a^{5} b^{5} d e^{6} + a^{6} b^{4} e^{7}\right )} x^{5} + {\left (b^{10} d^{7} - 2 \, a b^{9} d^{6} e - 9 \, a^{2} b^{8} d^{5} e^{2} + 40 \, a^{3} b^{7} d^{4} e^{3} - 65 \, a^{4} b^{6} d^{3} e^{4} + 54 \, a^{5} b^{5} d^{2} e^{5} - 23 \, a^{6} b^{4} d e^{6} + 4 \, a^{7} b^{3} e^{7}\right )} x^{4} + 2 \, {\left (2 \, a b^{9} d^{7} - 9 \, a^{2} b^{8} d^{6} e + 12 \, a^{3} b^{7} d^{5} e^{2} + 5 \, a^{4} b^{6} d^{4} e^{3} - 30 \, a^{5} b^{5} d^{3} e^{4} + 33 \, a^{6} b^{4} d^{2} e^{5} - 16 \, a^{7} b^{3} d e^{6} + 3 \, a^{8} b^{2} e^{7}\right )} x^{3} + 2 \, {\left (3 \, a^{2} b^{8} d^{7} - 16 \, a^{3} b^{7} d^{6} e + 33 \, a^{4} b^{6} d^{5} e^{2} - 30 \, a^{5} b^{5} d^{4} e^{3} + 5 \, a^{6} b^{4} d^{3} e^{4} + 12 \, a^{7} b^{3} d^{2} e^{5} - 9 \, a^{8} b^{2} d e^{6} + 2 \, a^{9} b e^{7}\right )} x^{2} + {\left (4 \, a^{3} b^{7} d^{7} - 23 \, a^{4} b^{6} d^{6} e + 54 \, a^{5} b^{5} d^{5} e^{2} - 65 \, a^{6} b^{4} d^{4} e^{3} + 40 \, a^{7} b^{3} d^{3} e^{4} - 9 \, a^{8} b^{2} d^{2} e^{5} - 2 \, a^{9} b d e^{6} + a^{10} e^{7}\right )} x\right )}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 1178 vs. \(2 (144) = 288\).
Time = 3.51 (sec) , antiderivative size = 1178, normalized size of antiderivative = 7.41 \[ \int \frac {a+b x}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=- \frac {5 b e^{4} \log {\left (x + \frac {- \frac {5 a^{7} b e^{11}}{\left (a e - b d\right )^{6}} + \frac {35 a^{6} b^{2} d e^{10}}{\left (a e - b d\right )^{6}} - \frac {105 a^{5} b^{3} d^{2} e^{9}}{\left (a e - b d\right )^{6}} + \frac {175 a^{4} b^{4} d^{3} e^{8}}{\left (a e - b d\right )^{6}} - \frac {175 a^{3} b^{5} d^{4} e^{7}}{\left (a e - b d\right )^{6}} + \frac {105 a^{2} b^{6} d^{5} e^{6}}{\left (a e - b d\right )^{6}} - \frac {35 a b^{7} d^{6} e^{5}}{\left (a e - b d\right )^{6}} + 5 a b e^{5} + \frac {5 b^{8} d^{7} e^{4}}{\left (a e - b d\right )^{6}} + 5 b^{2} d e^{4}}{10 b^{2} e^{5}} \right )}}{\left (a e - b d\right )^{6}} + \frac {5 b e^{4} \log {\left (x + \frac {\frac {5 a^{7} b e^{11}}{\left (a e - b d\right )^{6}} - \frac {35 a^{6} b^{2} d e^{10}}{\left (a e - b d\right )^{6}} + \frac {105 a^{5} b^{3} d^{2} e^{9}}{\left (a e - b d\right )^{6}} - \frac {175 a^{4} b^{4} d^{3} e^{8}}{\left (a e - b d\right )^{6}} + \frac {175 a^{3} b^{5} d^{4} e^{7}}{\left (a e - b d\right )^{6}} - \frac {105 a^{2} b^{6} d^{5} e^{6}}{\left (a e - b d\right )^{6}} + \frac {35 a b^{7} d^{6} e^{5}}{\left (a e - b d\right )^{6}} + 5 a b e^{5} - \frac {5 b^{8} d^{7} e^{4}}{\left (a e - b d\right )^{6}} + 5 b^{2} d e^{4}}{10 b^{2} e^{5}} \right )}}{\left (a e - b d\right )^{6}} + \frac {- 12 a^{4} e^{4} - 77 a^{3} b d e^{3} + 43 a^{2} b^{2} d^{2} e^{2} - 17 a b^{3} d^{3} e + 3 b^{4} d^{4} - 60 b^{4} e^{4} x^{4} + x^{3} \left (- 210 a b^{3} e^{4} - 30 b^{4} d e^{3}\right ) + x^{2} \left (- 260 a^{2} b^{2} e^{4} - 110 a b^{3} d e^{3} + 10 b^{4} d^{2} e^{2}\right ) + x \left (- 125 a^{3} b e^{4} - 145 a^{2} b^{2} d e^{3} + 35 a b^{3} d^{2} e^{2} - 5 b^{4} d^{3} e\right )}{12 a^{9} d e^{5} - 60 a^{8} b d^{2} e^{4} + 120 a^{7} b^{2} d^{3} e^{3} - 120 a^{6} b^{3} d^{4} e^{2} + 60 a^{5} b^{4} d^{5} e - 12 a^{4} b^{5} d^{6} + x^{5} \cdot \left (12 a^{5} b^{4} e^{6} - 60 a^{4} b^{5} d e^{5} + 120 a^{3} b^{6} d^{2} e^{4} - 120 a^{2} b^{7} d^{3} e^{3} + 60 a b^{8} d^{4} e^{2} - 12 b^{9} d^{5} e\right ) + x^{4} \cdot \left (48 a^{6} b^{3} e^{6} - 228 a^{5} b^{4} d e^{5} + 420 a^{4} b^{5} d^{2} e^{4} - 360 a^{3} b^{6} d^{3} e^{3} + 120 a^{2} b^{7} d^{4} e^{2} + 12 a b^{8} d^{5} e - 12 b^{9} d^{6}\right ) + x^{3} \cdot \left (72 a^{7} b^{2} e^{6} - 312 a^{6} b^{3} d e^{5} + 480 a^{5} b^{4} d^{2} e^{4} - 240 a^{4} b^{5} d^{3} e^{3} - 120 a^{3} b^{6} d^{4} e^{2} + 168 a^{2} b^{7} d^{5} e - 48 a b^{8} d^{6}\right ) + x^{2} \cdot \left (48 a^{8} b e^{6} - 168 a^{7} b^{2} d e^{5} + 120 a^{6} b^{3} d^{2} e^{4} + 240 a^{5} b^{4} d^{3} e^{3} - 480 a^{4} b^{5} d^{4} e^{2} + 312 a^{3} b^{6} d^{5} e - 72 a^{2} b^{7} d^{6}\right ) + x \left (12 a^{9} e^{6} - 12 a^{8} b d e^{5} - 120 a^{7} b^{2} d^{2} e^{4} + 360 a^{6} b^{3} d^{3} e^{3} - 420 a^{5} b^{4} d^{4} e^{2} + 228 a^{4} b^{5} d^{5} e - 48 a^{3} b^{6} d^{6}\right )} \]
[In]
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Leaf count of result is larger than twice the leaf count of optimal. 858 vs. \(2 (153) = 306\).
Time = 0.24 (sec) , antiderivative size = 858, normalized size of antiderivative = 5.40 \[ \int \frac {a+b x}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {5 \, b e^{4} \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 {5 \, b e^{4} \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} - 3 \, b^{4} d^{4} + 17 \, a b^{3} d^{3} e - 43 \, a^{2} b^{2} d^{2} e^{2} + 77 \, a^{3} b d e^{3} + 12 \, a^{4} e^{4} + 30 \, {\left (b^{4} d e^{3} + 7 \, a b^{3} e^{4}\right )} x^{3} - 10 \, {\left (b^{4} d^{2} e^{2} - 11 \, a b^{3} d e^{3} - 26 \, a^{2} b^{2} e^{4}\right )} x^{2} + 5 \, {\left (b^{4} d^{3} e - 7 \, a b^{3} d^{2} e^{2} + 29 \, a^{2} b^{2} d e^{3} + 25 \, a^{3} b e^{4}\right )} x}{12 \, {\left (a^{4} b^{5} d^{6} - 5 \, a^{5} b^{4} d^{5} e + 10 \, a^{6} b^{3} d^{4} e^{2} - 10 \, a^{7} b^{2} d^{3} e^{3} + 5 \, a^{8} b d^{2} e^{4} - a^{9} d e^{5} + {\left (b^{9} d^{5} e - 5 \, a b^{8} d^{4} e^{2} + 10 \, a^{2} b^{7} d^{3} e^{3} - 10 \, a^{3} b^{6} d^{2} e^{4} + 5 \, a^{4} b^{5} d e^{5} - a^{5} b^{4} e^{6}\right )} x^{5} + {\left (b^{9} d^{6} - a b^{8} d^{5} e - 10 \, a^{2} b^{7} d^{4} e^{2} + 30 \, a^{3} b^{6} d^{3} e^{3} - 35 \, a^{4} b^{5} d^{2} e^{4} + 19 \, a^{5} b^{4} d e^{5} - 4 \, a^{6} b^{3} e^{6}\right )} x^{4} + 2 \, {\left (2 \, a b^{8} d^{6} - 7 \, a^{2} b^{7} d^{5} e + 5 \, a^{3} b^{6} d^{4} e^{2} + 10 \, a^{4} b^{5} d^{3} e^{3} - 20 \, a^{5} b^{4} d^{2} e^{4} + 13 \, a^{6} b^{3} d e^{5} - 3 \, a^{7} b^{2} e^{6}\right )} x^{3} + 2 \, {\left (3 \, a^{2} b^{7} d^{6} - 13 \, a^{3} b^{6} d^{5} e + 20 \, a^{4} b^{5} d^{4} e^{2} - 10 \, a^{5} b^{4} d^{3} e^{3} - 5 \, a^{6} b^{3} d^{2} e^{4} + 7 \, a^{7} b^{2} d e^{5} - 2 \, a^{8} b e^{6}\right )} x^{2} + {\left (4 \, a^{3} b^{6} d^{6} - 19 \, a^{4} b^{5} d^{5} e + 35 \, a^{5} b^{4} d^{4} e^{2} - 30 \, a^{6} b^{3} d^{3} e^{3} + 10 \, a^{7} b^{2} d^{2} e^{4} + a^{8} b d e^{5} - a^{9} e^{6}\right )} x\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 373 vs. \(2 (153) = 306\).
Time = 0.28 (sec) , antiderivative size = 373, normalized size of antiderivative = 2.35 \[ \int \frac {a+b x}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {e^{9}}{{\left (b^{5} d^{5} e^{5} - 5 \, a b^{4} d^{4} e^{6} + 10 \, a^{2} b^{3} d^{3} e^{7} - 10 \, a^{3} b^{2} d^{2} e^{8} + 5 \, a^{4} b d e^{9} - a^{5} e^{10}\right )} {\left (e x + d\right )}} + \frac {5 \, b e^{5} \log \left ({\left | b - \frac {b d}{e x + d} + \frac {a e}{e x + d} \right |}\right )}{b^{6} d^{6} e - 6 \, a b^{5} d^{5} e^{2} + 15 \, a^{2} b^{4} d^{4} e^{3} - 20 \, a^{3} b^{3} d^{3} e^{4} + 15 \, a^{4} b^{2} d^{2} e^{5} - 6 \, a^{5} b d e^{6} + a^{6} e^{7}} + \frac {77 \, b^{5} e^{4} - \frac {260 \, {\left (b^{5} d e^{5} - a b^{4} e^{6}\right )}}{{\left (e x + d\right )} e} + \frac {300 \, {\left (b^{5} d^{2} e^{6} - 2 \, a b^{4} d e^{7} + a^{2} b^{3} e^{8}\right )}}{{\left (e x + d\right )}^{2} e^{2}} - \frac {120 \, {\left (b^{5} d^{3} e^{7} - 3 \, a b^{4} d^{2} e^{8} + 3 \, a^{2} b^{3} d e^{9} - a^{3} b^{2} e^{10}\right )}}{{\left (e x + d\right )}^{3} e^{3}}}{12 \, {\left (b d - a e\right )}^{6} {\left (b - \frac {b d}{e x + d} + \frac {a e}{e x + d}\right )}^{4}} \]
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Time = 11.11 (sec) , antiderivative size = 763, normalized size of antiderivative = 4.80 \[ \int \frac {a+b x}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {10\,b\,e^4\,\mathrm {atanh}\left (\frac {a^6\,e^6-4\,a^5\,b\,d\,e^5+5\,a^4\,b^2\,d^2\,e^4-5\,a^2\,b^4\,d^4\,e^2+4\,a\,b^5\,d^5\,e-b^6\,d^6}{{\left (a\,e-b\,d\right )}^6}+\frac {2\,b\,e\,x\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}{{\left (a\,e-b\,d\right )}^6}\right )}{{\left (a\,e-b\,d\right )}^6}-\frac {\frac {12\,a^4\,e^4+77\,a^3\,b\,d\,e^3-43\,a^2\,b^2\,d^2\,e^2+17\,a\,b^3\,d^3\,e-3\,b^4\,d^4}{12\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}+\frac {5\,e\,x\,\left (25\,a^3\,b\,e^3+29\,a^2\,b^2\,d\,e^2-7\,a\,b^3\,d^2\,e+b^4\,d^3\right )}{12\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}+\frac {5\,b^4\,e^4\,x^4}{a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5}+\frac {5\,e^3\,x^3\,\left (d\,b^4+7\,a\,e\,b^3\right )}{2\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}+\frac {5\,e^2\,x^2\,\left (26\,a^2\,b^2\,e^2+11\,a\,b^3\,d\,e-b^4\,d^2\right )}{6\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )}}{x^4\,\left (d\,b^4+4\,a\,e\,b^3\right )+a^4\,d+x\,\left (e\,a^4+4\,b\,d\,a^3\right )+x^2\,\left (4\,e\,a^3\,b+6\,d\,a^2\,b^2\right )+x^3\,\left (6\,e\,a^2\,b^2+4\,d\,a\,b^3\right )+b^4\,e\,x^5} \]
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