Integrand size = 26, antiderivative size = 155 \[ \int \frac {(d+e x)^6}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {e^6 x}{b^6}-\frac {(b d-a e)^6}{5 b^7 (a+b x)^5}-\frac {3 e (b d-a e)^5}{2 b^7 (a+b x)^4}-\frac {5 e^2 (b d-a e)^4}{b^7 (a+b x)^3}-\frac {10 e^3 (b d-a e)^3}{b^7 (a+b x)^2}-\frac {15 e^4 (b d-a e)^2}{b^7 (a+b x)}+\frac {6 e^5 (b d-a e) \log (a+b x)}{b^7} \]
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Time = 0.12 (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, 45} \[ \int \frac {(d+e x)^6}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {6 e^5 (b d-a e) \log (a+b x)}{b^7}-\frac {15 e^4 (b d-a e)^2}{b^7 (a+b x)}-\frac {10 e^3 (b d-a e)^3}{b^7 (a+b x)^2}-\frac {5 e^2 (b d-a e)^4}{b^7 (a+b x)^3}-\frac {3 e (b d-a e)^5}{2 b^7 (a+b x)^4}-\frac {(b d-a e)^6}{5 b^7 (a+b x)^5}+\frac {e^6 x}{b^6} \]
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Rule 27
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \frac {(d+e x)^6}{(a+b x)^6} \, dx \\ & = \int \left (\frac {e^6}{b^6}+\frac {(b d-a e)^6}{b^6 (a+b x)^6}+\frac {6 e (b d-a e)^5}{b^6 (a+b x)^5}+\frac {15 e^2 (b d-a e)^4}{b^6 (a+b x)^4}+\frac {20 e^3 (b d-a e)^3}{b^6 (a+b x)^3}+\frac {15 e^4 (b d-a e)^2}{b^6 (a+b x)^2}+\frac {6 e^5 (b d-a e)}{b^6 (a+b x)}\right ) \, dx \\ & = \frac {e^6 x}{b^6}-\frac {(b d-a e)^6}{5 b^7 (a+b x)^5}-\frac {3 e (b d-a e)^5}{2 b^7 (a+b x)^4}-\frac {5 e^2 (b d-a e)^4}{b^7 (a+b x)^3}-\frac {10 e^3 (b d-a e)^3}{b^7 (a+b x)^2}-\frac {15 e^4 (b d-a e)^2}{b^7 (a+b x)}+\frac {6 e^5 (b d-a e) \log (a+b x)}{b^7} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.94 \[ \int \frac {(d+e x)^6}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {87 a^6 e^6+a^5 b e^5 (-137 d+375 e x)+5 a^4 b^2 e^4 \left (6 d^2-125 d e x+120 e^2 x^2\right )+10 a^3 b^3 e^3 \left (d^3+15 d^2 e x-110 d e^2 x^2+40 e^3 x^3\right )+5 a^2 b^4 e^2 \left (d^4+10 d^3 e x+60 d^2 e^2 x^2-180 d e^3 x^3+10 e^4 x^4\right )+a b^5 e \left (3 d^5+25 d^4 e x+100 d^3 e^2 x^2+300 d^2 e^3 x^3-300 d e^4 x^4-50 e^5 x^5\right )+b^6 \left (2 d^6+15 d^5 e x+50 d^4 e^2 x^2+100 d^3 e^3 x^3+150 d^2 e^4 x^4-10 e^6 x^6\right )+60 e^5 (-b d+a e) (a+b x)^5 \log (a+b x)}{10 b^7 (a+b x)^5} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(348\) vs. \(2(151)=302\).
Time = 2.24 (sec) , antiderivative size = 349, normalized size of antiderivative = 2.25
method | result | size |
default | \(\frac {e^{6} x}{b^{6}}-\frac {5 e^{2} \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{b^{7} \left (b x +a \right )^{3}}-\frac {a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}}{5 b^{7} \left (b x +a \right )^{5}}-\frac {6 e^{5} \left (a e -b d \right ) \ln \left (b x +a \right )}{b^{7}}+\frac {3 e \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}{2 b^{7} \left (b x +a \right )^{4}}+\frac {10 e^{3} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{b^{7} \left (b x +a \right )^{2}}-\frac {15 e^{4} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{b^{7} \left (b x +a \right )}\) | \(349\) |
norman | \(\frac {\frac {e^{6} x^{6}}{b}-\frac {137 a^{6} e^{6}-137 a^{5} b d \,e^{5}+30 a^{4} b^{2} d^{2} e^{4}+10 a^{3} b^{3} d^{3} e^{3}+5 a^{2} b^{4} d^{4} e^{2}+3 a \,b^{5} d^{5} e +2 b^{6} d^{6}}{10 b^{7}}-\frac {5 \left (6 a^{2} e^{6}-6 a b d \,e^{5}+3 b^{2} d^{2} e^{4}\right ) x^{4}}{b^{3}}-\frac {10 \left (9 e^{6} a^{3}-9 a^{2} b d \,e^{5}+3 a \,b^{2} d^{2} e^{4}+b^{3} d^{3} e^{3}\right ) x^{3}}{b^{4}}-\frac {5 \left (22 a^{4} e^{6}-22 a^{3} b d \,e^{5}+6 a^{2} b^{2} d^{2} e^{4}+2 a \,b^{3} d^{3} e^{3}+d^{4} e^{2} b^{4}\right ) x^{2}}{b^{5}}-\frac {\left (125 a^{5} e^{6}-125 a^{4} b d \,e^{5}+30 a^{3} b^{2} d^{2} e^{4}+10 a^{2} b^{3} d^{3} e^{3}+5 a \,b^{4} d^{4} e^{2}+3 d^{5} e \,b^{5}\right ) x}{2 b^{6}}}{\left (b x +a \right )^{5}}-\frac {6 e^{5} \left (a e -b d \right ) \ln \left (b x +a \right )}{b^{7}}\) | \(349\) |
risch | \(\frac {e^{6} x}{b^{6}}+\frac {\left (-15 a^{2} b^{3} e^{6}+30 a \,b^{4} d \,e^{5}-15 d^{2} e^{4} b^{5}\right ) x^{4}-10 b^{2} e^{3} \left (5 a^{3} e^{3}-9 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{3}-5 b \,e^{2} \left (13 e^{4} a^{4}-22 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}+2 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x^{2}-\frac {e \left (77 a^{5} e^{5}-125 a^{4} b d \,e^{4}+30 a^{3} b^{2} d^{2} e^{3}+10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e +3 b^{5} d^{5}\right ) x}{2}-\frac {87 a^{6} e^{6}-137 a^{5} b d \,e^{5}+30 a^{4} b^{2} d^{2} e^{4}+10 a^{3} b^{3} d^{3} e^{3}+5 a^{2} b^{4} d^{4} e^{2}+3 a \,b^{5} d^{5} e +2 b^{6} d^{6}}{10 b}}{b^{6} \left (b x +a \right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2}}-\frac {6 e^{6} \ln \left (b x +a \right ) a}{b^{7}}+\frac {6 e^{5} \ln \left (b x +a \right ) d}{b^{6}}\) | \(366\) |
parallelrisch | \(-\frac {625 x \,a^{5} b \,e^{6}+15 x \,b^{6} d^{5} e +300 x^{4} a^{2} b^{4} e^{6}+150 x^{4} b^{6} d^{2} e^{4}+900 x^{3} a^{3} b^{3} e^{6}+100 x^{3} b^{6} d^{3} e^{3}+1100 x^{2} a^{4} b^{2} e^{6}+137 a^{6} e^{6}+2 b^{6} d^{6}+3 a \,b^{5} d^{5} e +30 a^{4} b^{2} d^{2} e^{4}+10 a^{3} b^{3} d^{3} e^{3}+5 a^{2} b^{4} d^{4} e^{2}-137 a^{5} b d \,e^{5}-600 \ln \left (b x +a \right ) x^{2} a^{3} b^{3} d \,e^{5}+60 \ln \left (b x +a \right ) a^{6} e^{6}-300 \ln \left (b x +a \right ) x^{4} a \,b^{5} d \,e^{5}-300 \ln \left (b x +a \right ) x \,a^{4} b^{2} d \,e^{5}-600 \ln \left (b x +a \right ) x^{3} a^{2} b^{4} d \,e^{5}-10 x^{6} b^{6} e^{6}+60 \ln \left (b x +a \right ) x^{5} a \,b^{5} e^{6}-60 \ln \left (b x +a \right ) x^{5} b^{6} d \,e^{5}+300 \ln \left (b x +a \right ) x^{4} a^{2} b^{4} e^{6}+600 \ln \left (b x +a \right ) x^{3} a^{3} b^{3} e^{6}+600 \ln \left (b x +a \right ) x^{2} a^{4} b^{2} e^{6}+300 \ln \left (b x +a \right ) x \,a^{5} b \,e^{6}-60 \ln \left (b x +a \right ) a^{5} b d \,e^{5}+50 x^{2} b^{6} d^{4} e^{2}-300 x^{4} a \,b^{5} d \,e^{5}-900 x^{3} a^{2} b^{4} d \,e^{5}+300 x^{3} a \,b^{5} d^{2} e^{4}-1100 x^{2} a^{3} b^{3} d \,e^{5}+300 x^{2} a^{2} b^{4} d^{2} e^{4}+100 x^{2} a \,b^{5} d^{3} e^{3}-625 x \,a^{4} b^{2} d \,e^{5}+150 x \,a^{3} b^{3} d^{2} e^{4}+50 x \,a^{2} b^{4} d^{3} e^{3}+25 x a \,b^{5} d^{4} e^{2}}{10 b^{7} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2} \left (b x +a \right )}\) | \(593\) |
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Leaf count of result is larger than twice the leaf count of optimal. 542 vs. \(2 (151) = 302\).
Time = 0.29 (sec) , antiderivative size = 542, normalized size of antiderivative = 3.50 \[ \int \frac {(d+e x)^6}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {10 \, b^{6} e^{6} x^{6} + 50 \, a b^{5} e^{6} x^{5} - 2 \, b^{6} d^{6} - 3 \, a b^{5} d^{5} e - 5 \, a^{2} b^{4} d^{4} e^{2} - 10 \, a^{3} b^{3} d^{3} e^{3} - 30 \, a^{4} b^{2} d^{2} e^{4} + 137 \, a^{5} b d e^{5} - 87 \, a^{6} e^{6} - 50 \, {\left (3 \, b^{6} d^{2} e^{4} - 6 \, a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} - 100 \, {\left (b^{6} d^{3} e^{3} + 3 \, a b^{5} d^{2} e^{4} - 9 \, a^{2} b^{4} d e^{5} + 4 \, a^{3} b^{3} e^{6}\right )} x^{3} - 50 \, {\left (b^{6} d^{4} e^{2} + 2 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 22 \, a^{3} b^{3} d e^{5} + 12 \, a^{4} b^{2} e^{6}\right )} x^{2} - 5 \, {\left (3 \, b^{6} d^{5} e + 5 \, a b^{5} d^{4} e^{2} + 10 \, a^{2} b^{4} d^{3} e^{3} + 30 \, a^{3} b^{3} d^{2} e^{4} - 125 \, a^{4} b^{2} d e^{5} + 75 \, a^{5} b e^{6}\right )} x + 60 \, {\left (a^{5} b d e^{5} - a^{6} e^{6} + {\left (b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} + 5 \, {\left (a b^{5} d e^{5} - a^{2} b^{4} e^{6}\right )} x^{4} + 10 \, {\left (a^{2} b^{4} d e^{5} - a^{3} b^{3} e^{6}\right )} x^{3} + 10 \, {\left (a^{3} b^{3} d e^{5} - a^{4} b^{2} e^{6}\right )} x^{2} + 5 \, {\left (a^{4} b^{2} d e^{5} - a^{5} b e^{6}\right )} x\right )} \log \left (b x + a\right )}{10 \, {\left (b^{12} x^{5} + 5 \, a b^{11} x^{4} + 10 \, a^{2} b^{10} x^{3} + 10 \, a^{3} b^{9} x^{2} + 5 \, a^{4} b^{8} x + a^{5} b^{7}\right )}} \]
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Timed out. \[ \int \frac {(d+e x)^6}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 397 vs. \(2 (151) = 302\).
Time = 0.21 (sec) , antiderivative size = 397, normalized size of antiderivative = 2.56 \[ \int \frac {(d+e x)^6}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {e^{6} x}{b^{6}} - \frac {2 \, b^{6} d^{6} + 3 \, a b^{5} d^{5} e + 5 \, a^{2} b^{4} d^{4} e^{2} + 10 \, a^{3} b^{3} d^{3} e^{3} + 30 \, a^{4} b^{2} d^{2} e^{4} - 137 \, a^{5} b d e^{5} + 87 \, a^{6} e^{6} + 150 \, {\left (b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 100 \, {\left (b^{6} d^{3} e^{3} + 3 \, a b^{5} d^{2} e^{4} - 9 \, a^{2} b^{4} d e^{5} + 5 \, a^{3} b^{3} e^{6}\right )} x^{3} + 50 \, {\left (b^{6} d^{4} e^{2} + 2 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 22 \, a^{3} b^{3} d e^{5} + 13 \, a^{4} b^{2} e^{6}\right )} x^{2} + 5 \, {\left (3 \, b^{6} d^{5} e + 5 \, a b^{5} d^{4} e^{2} + 10 \, a^{2} b^{4} d^{3} e^{3} + 30 \, a^{3} b^{3} d^{2} e^{4} - 125 \, a^{4} b^{2} d e^{5} + 77 \, a^{5} b e^{6}\right )} x}{10 \, {\left (b^{12} x^{5} + 5 \, a b^{11} x^{4} + 10 \, a^{2} b^{10} x^{3} + 10 \, a^{3} b^{9} x^{2} + 5 \, a^{4} b^{8} x + a^{5} b^{7}\right )}} + \frac {6 \, {\left (b d e^{5} - a e^{6}\right )} \log \left (b x + a\right )}{b^{7}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 351 vs. \(2 (151) = 302\).
Time = 0.27 (sec) , antiderivative size = 351, normalized size of antiderivative = 2.26 \[ \int \frac {(d+e x)^6}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {e^{6} x}{b^{6}} + \frac {6 \, {\left (b d e^{5} - a e^{6}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{7}} - \frac {2 \, b^{6} d^{6} + 3 \, a b^{5} d^{5} e + 5 \, a^{2} b^{4} d^{4} e^{2} + 10 \, a^{3} b^{3} d^{3} e^{3} + 30 \, a^{4} b^{2} d^{2} e^{4} - 137 \, a^{5} b d e^{5} + 87 \, a^{6} e^{6} + 150 \, {\left (b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 100 \, {\left (b^{6} d^{3} e^{3} + 3 \, a b^{5} d^{2} e^{4} - 9 \, a^{2} b^{4} d e^{5} + 5 \, a^{3} b^{3} e^{6}\right )} x^{3} + 50 \, {\left (b^{6} d^{4} e^{2} + 2 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 22 \, a^{3} b^{3} d e^{5} + 13 \, a^{4} b^{2} e^{6}\right )} x^{2} + 5 \, {\left (3 \, b^{6} d^{5} e + 5 \, a b^{5} d^{4} e^{2} + 10 \, a^{2} b^{4} d^{3} e^{3} + 30 \, a^{3} b^{3} d^{2} e^{4} - 125 \, a^{4} b^{2} d e^{5} + 77 \, a^{5} b e^{6}\right )} x}{10 \, {\left (b x + a\right )}^{5} b^{7}} \]
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Time = 9.75 (sec) , antiderivative size = 399, normalized size of antiderivative = 2.57 \[ \int \frac {(d+e x)^6}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {e^6\,x}{b^6}-\frac {\ln \left (a+b\,x\right )\,\left (6\,a\,e^6-6\,b\,d\,e^5\right )}{b^7}-\frac {x^2\,\left (65\,a^4\,b\,e^6-110\,a^3\,b^2\,d\,e^5+30\,a^2\,b^3\,d^2\,e^4+10\,a\,b^4\,d^3\,e^3+5\,b^5\,d^4\,e^2\right )+x^4\,\left (15\,a^2\,b^3\,e^6-30\,a\,b^4\,d\,e^5+15\,b^5\,d^2\,e^4\right )+\frac {87\,a^6\,e^6-137\,a^5\,b\,d\,e^5+30\,a^4\,b^2\,d^2\,e^4+10\,a^3\,b^3\,d^3\,e^3+5\,a^2\,b^4\,d^4\,e^2+3\,a\,b^5\,d^5\,e+2\,b^6\,d^6}{10\,b}+x\,\left (\frac {77\,a^5\,e^6}{2}-\frac {125\,a^4\,b\,d\,e^5}{2}+15\,a^3\,b^2\,d^2\,e^4+5\,a^2\,b^3\,d^3\,e^3+\frac {5\,a\,b^4\,d^4\,e^2}{2}+\frac {3\,b^5\,d^5\,e}{2}\right )+x^3\,\left (50\,a^3\,b^2\,e^6-90\,a^2\,b^3\,d\,e^5+30\,a\,b^4\,d^2\,e^4+10\,b^5\,d^3\,e^3\right )}{a^5\,b^6+5\,a^4\,b^7\,x+10\,a^3\,b^8\,x^2+10\,a^2\,b^9\,x^3+5\,a\,b^{10}\,x^4+b^{11}\,x^5} \]
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