Integrand size = 26, antiderivative size = 171 \[ \int (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {(b d-a e)^6 (a+b x)^7}{7 b^7}+\frac {3 e (b d-a e)^5 (a+b x)^8}{4 b^7}+\frac {5 e^2 (b d-a e)^4 (a+b x)^9}{3 b^7}+\frac {2 e^3 (b d-a e)^3 (a+b x)^{10}}{b^7}+\frac {15 e^4 (b d-a e)^2 (a+b x)^{11}}{11 b^7}+\frac {e^5 (b d-a e) (a+b x)^{12}}{2 b^7}+\frac {e^6 (a+b x)^{13}}{13 b^7} \]
[Out]
Time = 0.24 (sec) , antiderivative size = 171, 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} \[ \int (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {e^5 (a+b x)^{12} (b d-a e)}{2 b^7}+\frac {15 e^4 (a+b x)^{11} (b d-a e)^2}{11 b^7}+\frac {2 e^3 (a+b x)^{10} (b d-a e)^3}{b^7}+\frac {5 e^2 (a+b x)^9 (b d-a e)^4}{3 b^7}+\frac {3 e (a+b x)^8 (b d-a e)^5}{4 b^7}+\frac {(a+b x)^7 (b d-a e)^6}{7 b^7}+\frac {e^6 (a+b x)^{13}}{13 b^7} \]
[In]
[Out]
Rule 27
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int (a+b x)^6 (d+e x)^6 \, dx \\ & = \int \left (\frac {(b d-a e)^6 (a+b x)^6}{b^6}+\frac {6 e (b d-a e)^5 (a+b x)^7}{b^6}+\frac {15 e^2 (b d-a e)^4 (a+b x)^8}{b^6}+\frac {20 e^3 (b d-a e)^3 (a+b x)^9}{b^6}+\frac {15 e^4 (b d-a e)^2 (a+b x)^{10}}{b^6}+\frac {6 e^5 (b d-a e) (a+b x)^{11}}{b^6}+\frac {e^6 (a+b x)^{12}}{b^6}\right ) \, dx \\ & = \frac {(b d-a e)^6 (a+b x)^7}{7 b^7}+\frac {3 e (b d-a e)^5 (a+b x)^8}{4 b^7}+\frac {5 e^2 (b d-a e)^4 (a+b x)^9}{3 b^7}+\frac {2 e^3 (b d-a e)^3 (a+b x)^{10}}{b^7}+\frac {15 e^4 (b d-a e)^2 (a+b x)^{11}}{11 b^7}+\frac {e^5 (b d-a e) (a+b x)^{12}}{2 b^7}+\frac {e^6 (a+b x)^{13}}{13 b^7} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(573\) vs. \(2(171)=342\).
Time = 0.05 (sec) , antiderivative size = 573, normalized size of antiderivative = 3.35 \[ \int (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=a^6 d^6 x+3 a^5 d^5 (b d+a e) x^2+a^4 d^4 \left (5 b^2 d^2+12 a b d e+5 a^2 e^2\right ) x^3+\frac {5}{2} a^3 d^3 \left (2 b^3 d^3+9 a b^2 d^2 e+9 a^2 b d e^2+2 a^3 e^3\right ) x^4+3 a^2 d^2 \left (b^4 d^4+8 a b^3 d^3 e+15 a^2 b^2 d^2 e^2+8 a^3 b d e^3+a^4 e^4\right ) x^5+a d \left (b^5 d^5+15 a b^4 d^4 e+50 a^2 b^3 d^3 e^2+50 a^3 b^2 d^2 e^3+15 a^4 b d e^4+a^5 e^5\right ) x^6+\frac {1}{7} \left (b^6 d^6+36 a b^5 d^5 e+225 a^2 b^4 d^4 e^2+400 a^3 b^3 d^3 e^3+225 a^4 b^2 d^2 e^4+36 a^5 b d e^5+a^6 e^6\right ) x^7+\frac {3}{4} b e \left (b^5 d^5+15 a b^4 d^4 e+50 a^2 b^3 d^3 e^2+50 a^3 b^2 d^2 e^3+15 a^4 b d e^4+a^5 e^5\right ) x^8+\frac {5}{3} b^2 e^2 \left (b^4 d^4+8 a b^3 d^3 e+15 a^2 b^2 d^2 e^2+8 a^3 b d e^3+a^4 e^4\right ) x^9+b^3 e^3 \left (2 b^3 d^3+9 a b^2 d^2 e+9 a^2 b d e^2+2 a^3 e^3\right ) x^{10}+\frac {3}{11} b^4 e^4 \left (5 b^2 d^2+12 a b d e+5 a^2 e^2\right ) x^{11}+\frac {1}{2} b^5 e^5 (b d+a e) x^{12}+\frac {1}{13} b^6 e^6 x^{13} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(603\) vs. \(2(159)=318\).
Time = 2.29 (sec) , antiderivative size = 604, normalized size of antiderivative = 3.53
method | result | size |
norman | \(\frac {b^{6} e^{6} x^{13}}{13}+\left (\frac {1}{2} a \,b^{5} e^{6}+\frac {1}{2} b^{6} d \,e^{5}\right ) x^{12}+\left (\frac {15}{11} a^{2} b^{4} e^{6}+\frac {36}{11} a \,b^{5} d \,e^{5}+\frac {15}{11} b^{6} d^{2} e^{4}\right ) x^{11}+\left (2 a^{3} b^{3} e^{6}+9 a^{2} b^{4} d \,e^{5}+9 a \,b^{5} d^{2} e^{4}+2 b^{6} d^{3} e^{3}\right ) x^{10}+\left (\frac {5}{3} a^{4} b^{2} e^{6}+\frac {40}{3} a^{3} b^{3} d \,e^{5}+25 a^{2} b^{4} d^{2} e^{4}+\frac {40}{3} a \,b^{5} d^{3} e^{3}+\frac {5}{3} b^{6} d^{4} e^{2}\right ) x^{9}+\left (\frac {3}{4} a^{5} b \,e^{6}+\frac {45}{4} a^{4} b^{2} d \,e^{5}+\frac {75}{2} a^{3} b^{3} d^{2} e^{4}+\frac {75}{2} a^{2} b^{4} d^{3} e^{3}+\frac {45}{4} a \,b^{5} d^{4} e^{2}+\frac {3}{4} b^{6} d^{5} e \right ) x^{8}+\left (\frac {1}{7} a^{6} e^{6}+\frac {36}{7} a^{5} b d \,e^{5}+\frac {225}{7} a^{4} b^{2} d^{2} e^{4}+\frac {400}{7} a^{3} b^{3} d^{3} e^{3}+\frac {225}{7} a^{2} b^{4} d^{4} e^{2}+\frac {36}{7} a \,b^{5} d^{5} e +\frac {1}{7} b^{6} d^{6}\right ) x^{7}+\left (a^{6} d \,e^{5}+15 a^{5} b \,d^{2} e^{4}+50 a^{4} b^{2} d^{3} e^{3}+50 a^{3} b^{3} d^{4} e^{2}+15 a^{2} b^{4} d^{5} e +a \,b^{5} d^{6}\right ) x^{6}+\left (3 a^{6} d^{2} e^{4}+24 a^{5} b \,d^{3} e^{3}+45 a^{4} b^{2} d^{4} e^{2}+24 a^{3} b^{3} d^{5} e +3 a^{2} b^{4} d^{6}\right ) x^{5}+\left (5 a^{6} d^{3} e^{3}+\frac {45}{2} a^{5} b \,d^{4} e^{2}+\frac {45}{2} a^{4} b^{2} d^{5} e +5 a^{3} b^{3} d^{6}\right ) x^{4}+\left (5 a^{6} d^{4} e^{2}+12 a^{5} b \,d^{5} e +5 a^{4} b^{2} d^{6}\right ) x^{3}+\left (3 a^{6} d^{5} e +3 a^{5} b \,d^{6}\right ) x^{2}+a^{6} d^{6} x\) | \(604\) |
default | \(\frac {b^{6} e^{6} x^{13}}{13}+\frac {\left (6 a \,b^{5} e^{6}+6 b^{6} d \,e^{5}\right ) x^{12}}{12}+\frac {\left (15 a^{2} b^{4} e^{6}+36 a \,b^{5} d \,e^{5}+15 b^{6} d^{2} e^{4}\right ) x^{11}}{11}+\frac {\left (20 a^{3} b^{3} e^{6}+90 a^{2} b^{4} d \,e^{5}+90 a \,b^{5} d^{2} e^{4}+20 b^{6} d^{3} e^{3}\right ) x^{10}}{10}+\frac {\left (15 a^{4} b^{2} e^{6}+120 a^{3} b^{3} d \,e^{5}+225 a^{2} b^{4} d^{2} e^{4}+120 a \,b^{5} d^{3} e^{3}+15 b^{6} d^{4} e^{2}\right ) x^{9}}{9}+\frac {\left (6 a^{5} b \,e^{6}+90 a^{4} b^{2} d \,e^{5}+300 a^{3} b^{3} d^{2} e^{4}+300 a^{2} b^{4} d^{3} e^{3}+90 a \,b^{5} d^{4} e^{2}+6 b^{6} d^{5} e \right ) x^{8}}{8}+\frac {\left (a^{6} e^{6}+36 a^{5} b d \,e^{5}+225 a^{4} b^{2} d^{2} e^{4}+400 a^{3} b^{3} d^{3} e^{3}+225 a^{2} b^{4} d^{4} e^{2}+36 a \,b^{5} d^{5} e +b^{6} d^{6}\right ) x^{7}}{7}+\frac {\left (6 a^{6} d \,e^{5}+90 a^{5} b \,d^{2} e^{4}+300 a^{4} b^{2} d^{3} e^{3}+300 a^{3} b^{3} d^{4} e^{2}+90 a^{2} b^{4} d^{5} e +6 a \,b^{5} d^{6}\right ) x^{6}}{6}+\frac {\left (15 a^{6} d^{2} e^{4}+120 a^{5} b \,d^{3} e^{3}+225 a^{4} b^{2} d^{4} e^{2}+120 a^{3} b^{3} d^{5} e +15 a^{2} b^{4} d^{6}\right ) x^{5}}{5}+\frac {\left (20 a^{6} d^{3} e^{3}+90 a^{5} b \,d^{4} e^{2}+90 a^{4} b^{2} d^{5} e +20 a^{3} b^{3} d^{6}\right ) x^{4}}{4}+\frac {\left (15 a^{6} d^{4} e^{2}+36 a^{5} b \,d^{5} e +15 a^{4} b^{2} d^{6}\right ) x^{3}}{3}+\frac {\left (6 a^{6} d^{5} e +6 a^{5} b \,d^{6}\right ) x^{2}}{2}+a^{6} d^{6} x\) | \(615\) |
risch | \(\frac {1}{2} x^{12} a \,b^{5} e^{6}+\frac {1}{13} b^{6} e^{6} x^{13}+24 a^{5} b \,d^{3} e^{3} x^{5}+45 a^{4} b^{2} d^{4} e^{2} x^{5}+24 a^{3} b^{3} d^{5} e \,x^{5}+12 a^{5} b \,d^{5} e \,x^{3}+a^{6} d^{6} x +\frac {1}{7} d^{6} x^{7} b^{6}+\frac {1}{2} x^{12} b^{6} d \,e^{5}+5 a^{4} b^{2} d^{6} x^{3}+3 a^{5} b \,d^{6} x^{2}+a \,b^{5} d^{6} x^{6}+3 a^{2} b^{4} d^{6} x^{5}+5 a^{3} b^{3} d^{6} x^{4}+9 a^{2} b^{4} d \,e^{5} x^{10}+\frac {36}{11} x^{11} a \,b^{5} d \,e^{5}+\frac {40}{3} x^{9} a^{3} b^{3} d \,e^{5}+25 x^{9} a^{2} b^{4} d^{2} e^{4}+\frac {40}{3} x^{9} a \,b^{5} d^{3} e^{3}+\frac {45}{4} x^{8} a^{4} b^{2} d \,e^{5}+\frac {75}{2} x^{8} a^{3} b^{3} d^{2} e^{4}+\frac {15}{11} x^{11} b^{6} d^{2} e^{4}+\frac {5}{3} x^{9} a^{4} b^{2} e^{6}+\frac {5}{3} x^{9} b^{6} d^{4} e^{2}+\frac {1}{7} x^{7} a^{6} e^{6}+\frac {3}{4} x^{8} a^{5} b \,e^{6}+\frac {3}{4} x^{8} b^{6} d^{5} e +\frac {15}{11} x^{11} a^{2} b^{4} e^{6}+\frac {75}{2} x^{8} a^{2} b^{4} d^{3} e^{3}+\frac {45}{4} x^{8} a \,b^{5} d^{4} e^{2}+\frac {36}{7} x^{7} a^{5} b d \,e^{5}+\frac {225}{7} x^{7} a^{4} b^{2} d^{2} e^{4}+\frac {400}{7} x^{7} a^{3} b^{3} d^{3} e^{3}+\frac {225}{7} x^{7} a^{2} b^{4} d^{4} e^{2}+\frac {36}{7} x^{7} a \,b^{5} d^{5} e +\frac {45}{2} x^{4} a^{5} b \,d^{4} e^{2}+\frac {45}{2} x^{4} a^{4} b^{2} d^{5} e +9 a \,b^{5} d^{2} e^{4} x^{10}+15 a^{5} b \,d^{2} e^{4} x^{6}+50 a^{4} b^{2} d^{3} e^{3} x^{6}+50 a^{3} b^{3} d^{4} e^{2} x^{6}+15 a^{2} b^{4} d^{5} e \,x^{6}+5 x^{4} a^{6} d^{3} e^{3}+2 a^{3} b^{3} e^{6} x^{10}+2 b^{6} d^{3} e^{3} x^{10}+a^{6} d \,e^{5} x^{6}+3 a^{6} d^{2} e^{4} x^{5}+5 a^{6} d^{4} e^{2} x^{3}+3 a^{6} d^{5} e \,x^{2}\) | \(690\) |
parallelrisch | \(\frac {1}{2} x^{12} a \,b^{5} e^{6}+\frac {1}{13} b^{6} e^{6} x^{13}+24 a^{5} b \,d^{3} e^{3} x^{5}+45 a^{4} b^{2} d^{4} e^{2} x^{5}+24 a^{3} b^{3} d^{5} e \,x^{5}+12 a^{5} b \,d^{5} e \,x^{3}+a^{6} d^{6} x +\frac {1}{7} d^{6} x^{7} b^{6}+\frac {1}{2} x^{12} b^{6} d \,e^{5}+5 a^{4} b^{2} d^{6} x^{3}+3 a^{5} b \,d^{6} x^{2}+a \,b^{5} d^{6} x^{6}+3 a^{2} b^{4} d^{6} x^{5}+5 a^{3} b^{3} d^{6} x^{4}+9 a^{2} b^{4} d \,e^{5} x^{10}+\frac {36}{11} x^{11} a \,b^{5} d \,e^{5}+\frac {40}{3} x^{9} a^{3} b^{3} d \,e^{5}+25 x^{9} a^{2} b^{4} d^{2} e^{4}+\frac {40}{3} x^{9} a \,b^{5} d^{3} e^{3}+\frac {45}{4} x^{8} a^{4} b^{2} d \,e^{5}+\frac {75}{2} x^{8} a^{3} b^{3} d^{2} e^{4}+\frac {15}{11} x^{11} b^{6} d^{2} e^{4}+\frac {5}{3} x^{9} a^{4} b^{2} e^{6}+\frac {5}{3} x^{9} b^{6} d^{4} e^{2}+\frac {1}{7} x^{7} a^{6} e^{6}+\frac {3}{4} x^{8} a^{5} b \,e^{6}+\frac {3}{4} x^{8} b^{6} d^{5} e +\frac {15}{11} x^{11} a^{2} b^{4} e^{6}+\frac {75}{2} x^{8} a^{2} b^{4} d^{3} e^{3}+\frac {45}{4} x^{8} a \,b^{5} d^{4} e^{2}+\frac {36}{7} x^{7} a^{5} b d \,e^{5}+\frac {225}{7} x^{7} a^{4} b^{2} d^{2} e^{4}+\frac {400}{7} x^{7} a^{3} b^{3} d^{3} e^{3}+\frac {225}{7} x^{7} a^{2} b^{4} d^{4} e^{2}+\frac {36}{7} x^{7} a \,b^{5} d^{5} e +\frac {45}{2} x^{4} a^{5} b \,d^{4} e^{2}+\frac {45}{2} x^{4} a^{4} b^{2} d^{5} e +9 a \,b^{5} d^{2} e^{4} x^{10}+15 a^{5} b \,d^{2} e^{4} x^{6}+50 a^{4} b^{2} d^{3} e^{3} x^{6}+50 a^{3} b^{3} d^{4} e^{2} x^{6}+15 a^{2} b^{4} d^{5} e \,x^{6}+5 x^{4} a^{6} d^{3} e^{3}+2 a^{3} b^{3} e^{6} x^{10}+2 b^{6} d^{3} e^{3} x^{10}+a^{6} d \,e^{5} x^{6}+3 a^{6} d^{2} e^{4} x^{5}+5 a^{6} d^{4} e^{2} x^{3}+3 a^{6} d^{5} e \,x^{2}\) | \(690\) |
gosper | \(\frac {x \left (924 b^{6} e^{6} x^{12}+6006 x^{11} a \,b^{5} e^{6}+6006 x^{11} b^{6} d \,e^{5}+16380 x^{10} a^{2} b^{4} e^{6}+39312 x^{10} a \,b^{5} d \,e^{5}+16380 x^{10} b^{6} d^{2} e^{4}+24024 a^{3} b^{3} e^{6} x^{9}+108108 a^{2} b^{4} d \,e^{5} x^{9}+108108 a \,b^{5} d^{2} e^{4} x^{9}+24024 b^{6} d^{3} e^{3} x^{9}+20020 x^{8} a^{4} b^{2} e^{6}+160160 x^{8} a^{3} b^{3} d \,e^{5}+300300 x^{8} a^{2} b^{4} d^{2} e^{4}+160160 x^{8} a \,b^{5} d^{3} e^{3}+20020 x^{8} b^{6} d^{4} e^{2}+9009 x^{7} a^{5} b \,e^{6}+135135 x^{7} a^{4} b^{2} d \,e^{5}+450450 x^{7} a^{3} b^{3} d^{2} e^{4}+450450 x^{7} a^{2} b^{4} d^{3} e^{3}+135135 x^{7} a \,b^{5} d^{4} e^{2}+9009 x^{7} b^{6} d^{5} e +1716 x^{6} a^{6} e^{6}+61776 x^{6} a^{5} b d \,e^{5}+386100 x^{6} a^{4} b^{2} d^{2} e^{4}+686400 x^{6} a^{3} b^{3} d^{3} e^{3}+386100 x^{6} a^{2} b^{4} d^{4} e^{2}+61776 x^{6} a \,b^{5} d^{5} e +1716 x^{6} b^{6} d^{6}+12012 a^{6} d \,e^{5} x^{5}+180180 a^{5} b \,d^{2} e^{4} x^{5}+600600 a^{4} b^{2} d^{3} e^{3} x^{5}+600600 a^{3} b^{3} d^{4} e^{2} x^{5}+180180 a^{2} b^{4} d^{5} e \,x^{5}+12012 a \,b^{5} d^{6} x^{5}+36036 a^{6} d^{2} e^{4} x^{4}+288288 a^{5} b \,d^{3} e^{3} x^{4}+540540 a^{4} b^{2} d^{4} e^{2} x^{4}+288288 a^{3} b^{3} d^{5} e \,x^{4}+36036 a^{2} b^{4} d^{6} x^{4}+60060 x^{3} a^{6} d^{3} e^{3}+270270 x^{3} a^{5} b \,d^{4} e^{2}+270270 x^{3} a^{4} b^{2} d^{5} e +60060 x^{3} a^{3} b^{3} d^{6}+60060 a^{6} d^{4} e^{2} x^{2}+144144 a^{5} b \,d^{5} e \,x^{2}+60060 a^{4} b^{2} d^{6} x^{2}+36036 a^{6} d^{5} e x +36036 a^{5} b \,d^{6} x +12012 a^{6} d^{6}\right )}{12012}\) | \(691\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 599 vs. \(2 (159) = 318\).
Time = 0.30 (sec) , antiderivative size = 599, normalized size of antiderivative = 3.50 \[ \int (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{13} \, b^{6} e^{6} x^{13} + a^{6} d^{6} x + \frac {1}{2} \, {\left (b^{6} d e^{5} + a b^{5} e^{6}\right )} x^{12} + \frac {3}{11} \, {\left (5 \, b^{6} d^{2} e^{4} + 12 \, a b^{5} d e^{5} + 5 \, a^{2} b^{4} e^{6}\right )} x^{11} + {\left (2 \, b^{6} d^{3} e^{3} + 9 \, a b^{5} d^{2} e^{4} + 9 \, a^{2} b^{4} d e^{5} + 2 \, a^{3} b^{3} e^{6}\right )} x^{10} + \frac {5}{3} \, {\left (b^{6} d^{4} e^{2} + 8 \, a b^{5} d^{3} e^{3} + 15 \, a^{2} b^{4} d^{2} e^{4} + 8 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{9} + \frac {3}{4} \, {\left (b^{6} d^{5} e + 15 \, a b^{5} d^{4} e^{2} + 50 \, a^{2} b^{4} d^{3} e^{3} + 50 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} d^{6} + 36 \, a b^{5} d^{5} e + 225 \, a^{2} b^{4} d^{4} e^{2} + 400 \, a^{3} b^{3} d^{3} e^{3} + 225 \, a^{4} b^{2} d^{2} e^{4} + 36 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} x^{7} + {\left (a b^{5} d^{6} + 15 \, a^{2} b^{4} d^{5} e + 50 \, a^{3} b^{3} d^{4} e^{2} + 50 \, a^{4} b^{2} d^{3} e^{3} + 15 \, a^{5} b d^{2} e^{4} + a^{6} d e^{5}\right )} x^{6} + 3 \, {\left (a^{2} b^{4} d^{6} + 8 \, a^{3} b^{3} d^{5} e + 15 \, a^{4} b^{2} d^{4} e^{2} + 8 \, a^{5} b d^{3} e^{3} + a^{6} d^{2} e^{4}\right )} x^{5} + \frac {5}{2} \, {\left (2 \, a^{3} b^{3} d^{6} + 9 \, a^{4} b^{2} d^{5} e + 9 \, a^{5} b d^{4} e^{2} + 2 \, a^{6} d^{3} e^{3}\right )} x^{4} + {\left (5 \, a^{4} b^{2} d^{6} + 12 \, a^{5} b d^{5} e + 5 \, a^{6} d^{4} e^{2}\right )} x^{3} + 3 \, {\left (a^{5} b d^{6} + a^{6} d^{5} e\right )} x^{2} \]
[In]
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Leaf count of result is larger than twice the leaf count of optimal. 677 vs. \(2 (155) = 310\).
Time = 0.06 (sec) , antiderivative size = 677, normalized size of antiderivative = 3.96 \[ \int (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=a^{6} d^{6} x + \frac {b^{6} e^{6} x^{13}}{13} + x^{12} \left (\frac {a b^{5} e^{6}}{2} + \frac {b^{6} d e^{5}}{2}\right ) + x^{11} \cdot \left (\frac {15 a^{2} b^{4} e^{6}}{11} + \frac {36 a b^{5} d e^{5}}{11} + \frac {15 b^{6} d^{2} e^{4}}{11}\right ) + x^{10} \cdot \left (2 a^{3} b^{3} e^{6} + 9 a^{2} b^{4} d e^{5} + 9 a b^{5} d^{2} e^{4} + 2 b^{6} d^{3} e^{3}\right ) + x^{9} \cdot \left (\frac {5 a^{4} b^{2} e^{6}}{3} + \frac {40 a^{3} b^{3} d e^{5}}{3} + 25 a^{2} b^{4} d^{2} e^{4} + \frac {40 a b^{5} d^{3} e^{3}}{3} + \frac {5 b^{6} d^{4} e^{2}}{3}\right ) + x^{8} \cdot \left (\frac {3 a^{5} b e^{6}}{4} + \frac {45 a^{4} b^{2} d e^{5}}{4} + \frac {75 a^{3} b^{3} d^{2} e^{4}}{2} + \frac {75 a^{2} b^{4} d^{3} e^{3}}{2} + \frac {45 a b^{5} d^{4} e^{2}}{4} + \frac {3 b^{6} d^{5} e}{4}\right ) + x^{7} \left (\frac {a^{6} e^{6}}{7} + \frac {36 a^{5} b d e^{5}}{7} + \frac {225 a^{4} b^{2} d^{2} e^{4}}{7} + \frac {400 a^{3} b^{3} d^{3} e^{3}}{7} + \frac {225 a^{2} b^{4} d^{4} e^{2}}{7} + \frac {36 a b^{5} d^{5} e}{7} + \frac {b^{6} d^{6}}{7}\right ) + x^{6} \left (a^{6} d e^{5} + 15 a^{5} b d^{2} e^{4} + 50 a^{4} b^{2} d^{3} e^{3} + 50 a^{3} b^{3} d^{4} e^{2} + 15 a^{2} b^{4} d^{5} e + a b^{5} d^{6}\right ) + x^{5} \cdot \left (3 a^{6} d^{2} e^{4} + 24 a^{5} b d^{3} e^{3} + 45 a^{4} b^{2} d^{4} e^{2} + 24 a^{3} b^{3} d^{5} e + 3 a^{2} b^{4} d^{6}\right ) + x^{4} \cdot \left (5 a^{6} d^{3} e^{3} + \frac {45 a^{5} b d^{4} e^{2}}{2} + \frac {45 a^{4} b^{2} d^{5} e}{2} + 5 a^{3} b^{3} d^{6}\right ) + x^{3} \cdot \left (5 a^{6} d^{4} e^{2} + 12 a^{5} b d^{5} e + 5 a^{4} b^{2} d^{6}\right ) + x^{2} \cdot \left (3 a^{6} d^{5} e + 3 a^{5} b d^{6}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 599 vs. \(2 (159) = 318\).
Time = 0.20 (sec) , antiderivative size = 599, normalized size of antiderivative = 3.50 \[ \int (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{13} \, b^{6} e^{6} x^{13} + a^{6} d^{6} x + \frac {1}{2} \, {\left (b^{6} d e^{5} + a b^{5} e^{6}\right )} x^{12} + \frac {3}{11} \, {\left (5 \, b^{6} d^{2} e^{4} + 12 \, a b^{5} d e^{5} + 5 \, a^{2} b^{4} e^{6}\right )} x^{11} + {\left (2 \, b^{6} d^{3} e^{3} + 9 \, a b^{5} d^{2} e^{4} + 9 \, a^{2} b^{4} d e^{5} + 2 \, a^{3} b^{3} e^{6}\right )} x^{10} + \frac {5}{3} \, {\left (b^{6} d^{4} e^{2} + 8 \, a b^{5} d^{3} e^{3} + 15 \, a^{2} b^{4} d^{2} e^{4} + 8 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{9} + \frac {3}{4} \, {\left (b^{6} d^{5} e + 15 \, a b^{5} d^{4} e^{2} + 50 \, a^{2} b^{4} d^{3} e^{3} + 50 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} d^{6} + 36 \, a b^{5} d^{5} e + 225 \, a^{2} b^{4} d^{4} e^{2} + 400 \, a^{3} b^{3} d^{3} e^{3} + 225 \, a^{4} b^{2} d^{2} e^{4} + 36 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} x^{7} + {\left (a b^{5} d^{6} + 15 \, a^{2} b^{4} d^{5} e + 50 \, a^{3} b^{3} d^{4} e^{2} + 50 \, a^{4} b^{2} d^{3} e^{3} + 15 \, a^{5} b d^{2} e^{4} + a^{6} d e^{5}\right )} x^{6} + 3 \, {\left (a^{2} b^{4} d^{6} + 8 \, a^{3} b^{3} d^{5} e + 15 \, a^{4} b^{2} d^{4} e^{2} + 8 \, a^{5} b d^{3} e^{3} + a^{6} d^{2} e^{4}\right )} x^{5} + \frac {5}{2} \, {\left (2 \, a^{3} b^{3} d^{6} + 9 \, a^{4} b^{2} d^{5} e + 9 \, a^{5} b d^{4} e^{2} + 2 \, a^{6} d^{3} e^{3}\right )} x^{4} + {\left (5 \, a^{4} b^{2} d^{6} + 12 \, a^{5} b d^{5} e + 5 \, a^{6} d^{4} e^{2}\right )} x^{3} + 3 \, {\left (a^{5} b d^{6} + a^{6} d^{5} e\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 689 vs. \(2 (159) = 318\).
Time = 0.27 (sec) , antiderivative size = 689, normalized size of antiderivative = 4.03 \[ \int (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{13} \, b^{6} e^{6} x^{13} + \frac {1}{2} \, b^{6} d e^{5} x^{12} + \frac {1}{2} \, a b^{5} e^{6} x^{12} + \frac {15}{11} \, b^{6} d^{2} e^{4} x^{11} + \frac {36}{11} \, a b^{5} d e^{5} x^{11} + \frac {15}{11} \, a^{2} b^{4} e^{6} x^{11} + 2 \, b^{6} d^{3} e^{3} x^{10} + 9 \, a b^{5} d^{2} e^{4} x^{10} + 9 \, a^{2} b^{4} d e^{5} x^{10} + 2 \, a^{3} b^{3} e^{6} x^{10} + \frac {5}{3} \, b^{6} d^{4} e^{2} x^{9} + \frac {40}{3} \, a b^{5} d^{3} e^{3} x^{9} + 25 \, a^{2} b^{4} d^{2} e^{4} x^{9} + \frac {40}{3} \, a^{3} b^{3} d e^{5} x^{9} + \frac {5}{3} \, a^{4} b^{2} e^{6} x^{9} + \frac {3}{4} \, b^{6} d^{5} e x^{8} + \frac {45}{4} \, a b^{5} d^{4} e^{2} x^{8} + \frac {75}{2} \, a^{2} b^{4} d^{3} e^{3} x^{8} + \frac {75}{2} \, a^{3} b^{3} d^{2} e^{4} x^{8} + \frac {45}{4} \, a^{4} b^{2} d e^{5} x^{8} + \frac {3}{4} \, a^{5} b e^{6} x^{8} + \frac {1}{7} \, b^{6} d^{6} x^{7} + \frac {36}{7} \, a b^{5} d^{5} e x^{7} + \frac {225}{7} \, a^{2} b^{4} d^{4} e^{2} x^{7} + \frac {400}{7} \, a^{3} b^{3} d^{3} e^{3} x^{7} + \frac {225}{7} \, a^{4} b^{2} d^{2} e^{4} x^{7} + \frac {36}{7} \, a^{5} b d e^{5} x^{7} + \frac {1}{7} \, a^{6} e^{6} x^{7} + a b^{5} d^{6} x^{6} + 15 \, a^{2} b^{4} d^{5} e x^{6} + 50 \, a^{3} b^{3} d^{4} e^{2} x^{6} + 50 \, a^{4} b^{2} d^{3} e^{3} x^{6} + 15 \, a^{5} b d^{2} e^{4} x^{6} + a^{6} d e^{5} x^{6} + 3 \, a^{2} b^{4} d^{6} x^{5} + 24 \, a^{3} b^{3} d^{5} e x^{5} + 45 \, a^{4} b^{2} d^{4} e^{2} x^{5} + 24 \, a^{5} b d^{3} e^{3} x^{5} + 3 \, a^{6} d^{2} e^{4} x^{5} + 5 \, a^{3} b^{3} d^{6} x^{4} + \frac {45}{2} \, a^{4} b^{2} d^{5} e x^{4} + \frac {45}{2} \, a^{5} b d^{4} e^{2} x^{4} + 5 \, a^{6} d^{3} e^{3} x^{4} + 5 \, a^{4} b^{2} d^{6} x^{3} + 12 \, a^{5} b d^{5} e x^{3} + 5 \, a^{6} d^{4} e^{2} x^{3} + 3 \, a^{5} b d^{6} x^{2} + 3 \, a^{6} d^{5} e x^{2} + a^{6} d^{6} x \]
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Time = 10.29 (sec) , antiderivative size = 579, normalized size of antiderivative = 3.39 \[ \int (d+e x)^6 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=x^6\,\left (a^6\,d\,e^5+15\,a^5\,b\,d^2\,e^4+50\,a^4\,b^2\,d^3\,e^3+50\,a^3\,b^3\,d^4\,e^2+15\,a^2\,b^4\,d^5\,e+a\,b^5\,d^6\right )+x^8\,\left (\frac {3\,a^5\,b\,e^6}{4}+\frac {45\,a^4\,b^2\,d\,e^5}{4}+\frac {75\,a^3\,b^3\,d^2\,e^4}{2}+\frac {75\,a^2\,b^4\,d^3\,e^3}{2}+\frac {45\,a\,b^5\,d^4\,e^2}{4}+\frac {3\,b^6\,d^5\,e}{4}\right )+x^5\,\left (3\,a^6\,d^2\,e^4+24\,a^5\,b\,d^3\,e^3+45\,a^4\,b^2\,d^4\,e^2+24\,a^3\,b^3\,d^5\,e+3\,a^2\,b^4\,d^6\right )+x^9\,\left (\frac {5\,a^4\,b^2\,e^6}{3}+\frac {40\,a^3\,b^3\,d\,e^5}{3}+25\,a^2\,b^4\,d^2\,e^4+\frac {40\,a\,b^5\,d^3\,e^3}{3}+\frac {5\,b^6\,d^4\,e^2}{3}\right )+x^7\,\left (\frac {a^6\,e^6}{7}+\frac {36\,a^5\,b\,d\,e^5}{7}+\frac {225\,a^4\,b^2\,d^2\,e^4}{7}+\frac {400\,a^3\,b^3\,d^3\,e^3}{7}+\frac {225\,a^2\,b^4\,d^4\,e^2}{7}+\frac {36\,a\,b^5\,d^5\,e}{7}+\frac {b^6\,d^6}{7}\right )+a^6\,d^6\,x+\frac {b^6\,e^6\,x^{13}}{13}+\frac {5\,a^3\,d^3\,x^4\,\left (2\,a^3\,e^3+9\,a^2\,b\,d\,e^2+9\,a\,b^2\,d^2\,e+2\,b^3\,d^3\right )}{2}+b^3\,e^3\,x^{10}\,\left (2\,a^3\,e^3+9\,a^2\,b\,d\,e^2+9\,a\,b^2\,d^2\,e+2\,b^3\,d^3\right )+3\,a^5\,d^5\,x^2\,\left (a\,e+b\,d\right )+\frac {b^5\,e^5\,x^{12}\,\left (a\,e+b\,d\right )}{2}+a^4\,d^4\,x^3\,\left (5\,a^2\,e^2+12\,a\,b\,d\,e+5\,b^2\,d^2\right )+\frac {3\,b^4\,e^4\,x^{11}\,\left (5\,a^2\,e^2+12\,a\,b\,d\,e+5\,b^2\,d^2\right )}{11} \]
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