Integrand size = 26, antiderivative size = 143 \[ \int (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {(b d-a e)^5 (a+b x)^7}{7 b^6}+\frac {5 e (b d-a e)^4 (a+b x)^8}{8 b^6}+\frac {10 e^2 (b d-a e)^3 (a+b x)^9}{9 b^6}+\frac {e^3 (b d-a e)^2 (a+b x)^{10}}{b^6}+\frac {5 e^4 (b d-a e) (a+b x)^{11}}{11 b^6}+\frac {e^5 (a+b x)^{12}}{12 b^6} \]
[Out]
Time = 0.21 (sec) , antiderivative size = 143, 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} \[ \int (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {5 e^4 (a+b x)^{11} (b d-a e)}{11 b^6}+\frac {e^3 (a+b x)^{10} (b d-a e)^2}{b^6}+\frac {10 e^2 (a+b x)^9 (b d-a e)^3}{9 b^6}+\frac {5 e (a+b x)^8 (b d-a e)^4}{8 b^6}+\frac {(a+b x)^7 (b d-a e)^5}{7 b^6}+\frac {e^5 (a+b x)^{12}}{12 b^6} \]
[In]
[Out]
Rule 27
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int (a+b x)^6 (d+e x)^5 \, dx \\ & = \int \left (\frac {(b d-a e)^5 (a+b x)^6}{b^5}+\frac {5 e (b d-a e)^4 (a+b x)^7}{b^5}+\frac {10 e^2 (b d-a e)^3 (a+b x)^8}{b^5}+\frac {10 e^3 (b d-a e)^2 (a+b x)^9}{b^5}+\frac {5 e^4 (b d-a e) (a+b x)^{10}}{b^5}+\frac {e^5 (a+b x)^{11}}{b^5}\right ) \, dx \\ & = \frac {(b d-a e)^5 (a+b x)^7}{7 b^6}+\frac {5 e (b d-a e)^4 (a+b x)^8}{8 b^6}+\frac {10 e^2 (b d-a e)^3 (a+b x)^9}{9 b^6}+\frac {e^3 (b d-a e)^2 (a+b x)^{10}}{b^6}+\frac {5 e^4 (b d-a e) (a+b x)^{11}}{11 b^6}+\frac {e^5 (a+b x)^{12}}{12 b^6} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(501\) vs. \(2(143)=286\).
Time = 0.05 (sec) , antiderivative size = 501, normalized size of antiderivative = 3.50 \[ \int (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=a^6 d^5 x+\frac {1}{2} a^5 d^4 (6 b d+5 a e) x^2+\frac {5}{3} a^4 d^3 \left (3 b^2 d^2+6 a b d e+2 a^2 e^2\right ) x^3+\frac {5}{4} a^3 d^2 \left (4 b^3 d^3+15 a b^2 d^2 e+12 a^2 b d e^2+2 a^3 e^3\right ) x^4+a^2 d \left (3 b^4 d^4+20 a b^3 d^3 e+30 a^2 b^2 d^2 e^2+12 a^3 b d e^3+a^4 e^4\right ) x^5+\frac {1}{6} a \left (6 b^5 d^5+75 a b^4 d^4 e+200 a^2 b^3 d^3 e^2+150 a^3 b^2 d^2 e^3+30 a^4 b d e^4+a^5 e^5\right ) x^6+\frac {1}{7} b \left (b^5 d^5+30 a b^4 d^4 e+150 a^2 b^3 d^3 e^2+200 a^3 b^2 d^2 e^3+75 a^4 b d e^4+6 a^5 e^5\right ) x^7+\frac {5}{8} b^2 e \left (b^4 d^4+12 a b^3 d^3 e+30 a^2 b^2 d^2 e^2+20 a^3 b d e^3+3 a^4 e^4\right ) x^8+\frac {5}{9} b^3 e^2 \left (2 b^3 d^3+12 a b^2 d^2 e+15 a^2 b d e^2+4 a^3 e^3\right ) x^9+\frac {1}{2} b^4 e^3 \left (2 b^2 d^2+6 a b d e+3 a^2 e^2\right ) x^{10}+\frac {1}{11} b^5 e^4 (5 b d+6 a e) x^{11}+\frac {1}{12} b^6 e^5 x^{12} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(509\) vs. \(2(133)=266\).
Time = 2.25 (sec) , antiderivative size = 510, normalized size of antiderivative = 3.57
method | result | size |
norman | \(\frac {b^{6} e^{5} x^{12}}{12}+\left (\frac {6}{11} e^{5} a \,b^{5}+\frac {5}{11} d \,e^{4} b^{6}\right ) x^{11}+\left (\frac {3}{2} e^{5} a^{2} b^{4}+3 d \,e^{4} a \,b^{5}+d^{2} e^{3} b^{6}\right ) x^{10}+\left (\frac {20}{9} e^{5} a^{3} b^{3}+\frac {25}{3} a^{2} b^{4} d \,e^{4}+\frac {20}{3} d^{2} e^{3} a \,b^{5}+\frac {10}{9} d^{3} e^{2} b^{6}\right ) x^{9}+\left (\frac {15}{8} e^{5} a^{4} b^{2}+\frac {25}{2} d \,e^{4} a^{3} b^{3}+\frac {75}{4} d^{2} e^{3} a^{2} b^{4}+\frac {15}{2} d^{3} e^{2} a \,b^{5}+\frac {5}{8} d^{4} e \,b^{6}\right ) x^{8}+\left (\frac {6}{7} a^{5} b \,e^{5}+\frac {75}{7} d \,e^{4} a^{4} b^{2}+\frac {200}{7} d^{2} e^{3} a^{3} b^{3}+\frac {150}{7} d^{3} e^{2} a^{2} b^{4}+\frac {30}{7} d^{4} e a \,b^{5}+\frac {1}{7} d^{5} b^{6}\right ) x^{7}+\left (\frac {1}{6} e^{5} a^{6}+5 d \,e^{4} a^{5} b +25 d^{2} e^{3} a^{4} b^{2}+\frac {100}{3} d^{3} e^{2} a^{3} b^{3}+\frac {25}{2} d^{4} e \,a^{2} b^{4}+d^{5} a \,b^{5}\right ) x^{6}+\left (d \,e^{4} a^{6}+12 d^{2} e^{3} a^{5} b +30 d^{3} e^{2} a^{4} b^{2}+20 d^{4} e \,a^{3} b^{3}+3 d^{5} a^{2} b^{4}\right ) x^{5}+\left (\frac {5}{2} d^{2} e^{3} a^{6}+15 d^{3} e^{2} a^{5} b +\frac {75}{4} d^{4} e \,a^{4} b^{2}+5 a^{3} b^{3} d^{5}\right ) x^{4}+\left (\frac {10}{3} d^{3} e^{2} a^{6}+10 d^{4} e \,a^{5} b +5 d^{5} a^{4} b^{2}\right ) x^{3}+\left (\frac {5}{2} d^{4} e \,a^{6}+3 d^{5} a^{5} b \right ) x^{2}+d^{5} a^{6} x\) | \(510\) |
default | \(\frac {b^{6} e^{5} x^{12}}{12}+\frac {\left (6 e^{5} a \,b^{5}+5 d \,e^{4} b^{6}\right ) x^{11}}{11}+\frac {\left (15 e^{5} a^{2} b^{4}+30 d \,e^{4} a \,b^{5}+10 d^{2} e^{3} b^{6}\right ) x^{10}}{10}+\frac {\left (20 e^{5} a^{3} b^{3}+75 a^{2} b^{4} d \,e^{4}+60 d^{2} e^{3} a \,b^{5}+10 d^{3} e^{2} b^{6}\right ) x^{9}}{9}+\frac {\left (15 e^{5} a^{4} b^{2}+100 d \,e^{4} a^{3} b^{3}+150 d^{2} e^{3} a^{2} b^{4}+60 d^{3} e^{2} a \,b^{5}+5 d^{4} e \,b^{6}\right ) x^{8}}{8}+\frac {\left (6 a^{5} b \,e^{5}+75 d \,e^{4} a^{4} b^{2}+200 d^{2} e^{3} a^{3} b^{3}+150 d^{3} e^{2} a^{2} b^{4}+30 d^{4} e a \,b^{5}+d^{5} b^{6}\right ) x^{7}}{7}+\frac {\left (e^{5} a^{6}+30 d \,e^{4} a^{5} b +150 d^{2} e^{3} a^{4} b^{2}+200 d^{3} e^{2} a^{3} b^{3}+75 d^{4} e \,a^{2} b^{4}+6 d^{5} a \,b^{5}\right ) x^{6}}{6}+\frac {\left (5 d \,e^{4} a^{6}+60 d^{2} e^{3} a^{5} b +150 d^{3} e^{2} a^{4} b^{2}+100 d^{4} e \,a^{3} b^{3}+15 d^{5} a^{2} b^{4}\right ) x^{5}}{5}+\frac {\left (10 d^{2} e^{3} a^{6}+60 d^{3} e^{2} a^{5} b +75 d^{4} e \,a^{4} b^{2}+20 a^{3} b^{3} d^{5}\right ) x^{4}}{4}+\frac {\left (10 d^{3} e^{2} a^{6}+30 d^{4} e \,a^{5} b +15 d^{5} a^{4} b^{2}\right ) x^{3}}{3}+\frac {\left (5 d^{4} e \,a^{6}+6 d^{5} a^{5} b \right ) x^{2}}{2}+d^{5} a^{6} x\) | \(521\) |
risch | \(\frac {1}{6} x^{6} e^{5} a^{6}+\frac {1}{12} b^{6} e^{5} x^{12}+d^{5} a^{6} x +3 x^{2} d^{5} a^{5} b +a^{6} d \,e^{4} x^{5}+3 a^{2} b^{4} d^{5} x^{5}+\frac {1}{7} x^{7} d^{5} b^{6}+\frac {6}{11} x^{11} e^{5} a \,b^{5}+\frac {5}{11} x^{11} d \,e^{4} b^{6}+\frac {3}{2} x^{10} e^{5} a^{2} b^{4}+x^{10} d^{2} e^{3} b^{6}+\frac {20}{9} x^{9} e^{5} a^{3} b^{3}+\frac {10}{9} x^{9} d^{3} e^{2} b^{6}+\frac {15}{8} x^{8} e^{5} a^{4} b^{2}+3 x^{10} d \,e^{4} a \,b^{5}+\frac {25}{3} x^{9} a^{2} b^{4} d \,e^{4}+\frac {20}{3} x^{9} d^{2} e^{3} a \,b^{5}+\frac {25}{2} x^{8} d \,e^{4} a^{3} b^{3}+\frac {75}{4} x^{8} d^{2} e^{3} a^{2} b^{4}+\frac {15}{2} x^{8} d^{3} e^{2} a \,b^{5}+\frac {75}{7} x^{7} d \,e^{4} a^{4} b^{2}+\frac {200}{7} x^{7} d^{2} e^{3} a^{3} b^{3}+\frac {150}{7} x^{7} d^{3} e^{2} a^{2} b^{4}+\frac {30}{7} x^{7} d^{4} e a \,b^{5}+5 x^{6} d \,e^{4} a^{5} b +25 x^{6} d^{2} e^{3} a^{4} b^{2}+\frac {5}{8} x^{8} d^{4} e \,b^{6}+\frac {6}{7} x^{7} a^{5} b \,e^{5}+x^{6} d^{5} a \,b^{5}+\frac {5}{2} x^{4} d^{2} e^{3} a^{6}+5 x^{4} a^{3} b^{3} d^{5}+\frac {100}{3} x^{6} d^{3} e^{2} a^{3} b^{3}+\frac {25}{2} x^{6} d^{4} e \,a^{2} b^{4}+15 x^{4} d^{3} e^{2} a^{5} b +\frac {75}{4} x^{4} d^{4} e \,a^{4} b^{2}+10 x^{3} d^{4} e \,a^{5} b +12 a^{5} b \,d^{2} e^{3} x^{5}+30 a^{4} b^{2} d^{3} e^{2} x^{5}+20 a^{3} b^{3} d^{4} e \,x^{5}+\frac {10}{3} x^{3} d^{3} e^{2} a^{6}+5 x^{3} d^{5} a^{4} b^{2}+\frac {5}{2} x^{2} d^{4} e \,a^{6}\) | \(580\) |
parallelrisch | \(\frac {1}{6} x^{6} e^{5} a^{6}+\frac {1}{12} b^{6} e^{5} x^{12}+d^{5} a^{6} x +3 x^{2} d^{5} a^{5} b +a^{6} d \,e^{4} x^{5}+3 a^{2} b^{4} d^{5} x^{5}+\frac {1}{7} x^{7} d^{5} b^{6}+\frac {6}{11} x^{11} e^{5} a \,b^{5}+\frac {5}{11} x^{11} d \,e^{4} b^{6}+\frac {3}{2} x^{10} e^{5} a^{2} b^{4}+x^{10} d^{2} e^{3} b^{6}+\frac {20}{9} x^{9} e^{5} a^{3} b^{3}+\frac {10}{9} x^{9} d^{3} e^{2} b^{6}+\frac {15}{8} x^{8} e^{5} a^{4} b^{2}+3 x^{10} d \,e^{4} a \,b^{5}+\frac {25}{3} x^{9} a^{2} b^{4} d \,e^{4}+\frac {20}{3} x^{9} d^{2} e^{3} a \,b^{5}+\frac {25}{2} x^{8} d \,e^{4} a^{3} b^{3}+\frac {75}{4} x^{8} d^{2} e^{3} a^{2} b^{4}+\frac {15}{2} x^{8} d^{3} e^{2} a \,b^{5}+\frac {75}{7} x^{7} d \,e^{4} a^{4} b^{2}+\frac {200}{7} x^{7} d^{2} e^{3} a^{3} b^{3}+\frac {150}{7} x^{7} d^{3} e^{2} a^{2} b^{4}+\frac {30}{7} x^{7} d^{4} e a \,b^{5}+5 x^{6} d \,e^{4} a^{5} b +25 x^{6} d^{2} e^{3} a^{4} b^{2}+\frac {5}{8} x^{8} d^{4} e \,b^{6}+\frac {6}{7} x^{7} a^{5} b \,e^{5}+x^{6} d^{5} a \,b^{5}+\frac {5}{2} x^{4} d^{2} e^{3} a^{6}+5 x^{4} a^{3} b^{3} d^{5}+\frac {100}{3} x^{6} d^{3} e^{2} a^{3} b^{3}+\frac {25}{2} x^{6} d^{4} e \,a^{2} b^{4}+15 x^{4} d^{3} e^{2} a^{5} b +\frac {75}{4} x^{4} d^{4} e \,a^{4} b^{2}+10 x^{3} d^{4} e \,a^{5} b +12 a^{5} b \,d^{2} e^{3} x^{5}+30 a^{4} b^{2} d^{3} e^{2} x^{5}+20 a^{3} b^{3} d^{4} e \,x^{5}+\frac {10}{3} x^{3} d^{3} e^{2} a^{6}+5 x^{3} d^{5} a^{4} b^{2}+\frac {5}{2} x^{2} d^{4} e \,a^{6}\) | \(580\) |
gosper | \(\frac {x \left (462 b^{6} e^{5} x^{11}+3024 x^{10} e^{5} a \,b^{5}+2520 x^{10} d \,e^{4} b^{6}+8316 x^{9} e^{5} a^{2} b^{4}+16632 x^{9} d \,e^{4} a \,b^{5}+5544 x^{9} d^{2} e^{3} b^{6}+12320 x^{8} e^{5} a^{3} b^{3}+46200 x^{8} a^{2} b^{4} d \,e^{4}+36960 x^{8} d^{2} e^{3} a \,b^{5}+6160 x^{8} d^{3} e^{2} b^{6}+10395 x^{7} e^{5} a^{4} b^{2}+69300 x^{7} d \,e^{4} a^{3} b^{3}+103950 x^{7} d^{2} e^{3} a^{2} b^{4}+41580 x^{7} d^{3} e^{2} a \,b^{5}+3465 x^{7} d^{4} e \,b^{6}+4752 x^{6} a^{5} b \,e^{5}+59400 x^{6} d \,e^{4} a^{4} b^{2}+158400 x^{6} d^{2} e^{3} a^{3} b^{3}+118800 x^{6} d^{3} e^{2} a^{2} b^{4}+23760 x^{6} d^{4} e a \,b^{5}+792 x^{6} d^{5} b^{6}+924 x^{5} e^{5} a^{6}+27720 x^{5} d \,e^{4} a^{5} b +138600 x^{5} d^{2} e^{3} a^{4} b^{2}+184800 x^{5} d^{3} e^{2} a^{3} b^{3}+69300 x^{5} d^{4} e \,a^{2} b^{4}+5544 x^{5} d^{5} a \,b^{5}+5544 a^{6} d \,e^{4} x^{4}+66528 a^{5} b \,d^{2} e^{3} x^{4}+166320 a^{4} b^{2} d^{3} e^{2} x^{4}+110880 a^{3} b^{3} d^{4} e \,x^{4}+16632 a^{2} b^{4} d^{5} x^{4}+13860 x^{3} d^{2} e^{3} a^{6}+83160 x^{3} d^{3} e^{2} a^{5} b +103950 x^{3} d^{4} e \,a^{4} b^{2}+27720 x^{3} a^{3} b^{3} d^{5}+18480 x^{2} d^{3} e^{2} a^{6}+55440 x^{2} d^{4} e \,a^{5} b +27720 x^{2} d^{5} a^{4} b^{2}+13860 x \,d^{4} e \,a^{6}+16632 x \,d^{5} a^{5} b +5544 d^{5} a^{6}\right )}{5544}\) | \(582\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 517 vs. \(2 (133) = 266\).
Time = 0.30 (sec) , antiderivative size = 517, normalized size of antiderivative = 3.62 \[ \int (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{12} \, b^{6} e^{5} x^{12} + a^{6} d^{5} x + \frac {1}{11} \, {\left (5 \, b^{6} d e^{4} + 6 \, a b^{5} e^{5}\right )} x^{11} + \frac {1}{2} \, {\left (2 \, b^{6} d^{2} e^{3} + 6 \, a b^{5} d e^{4} + 3 \, a^{2} b^{4} e^{5}\right )} x^{10} + \frac {5}{9} \, {\left (2 \, b^{6} d^{3} e^{2} + 12 \, a b^{5} d^{2} e^{3} + 15 \, a^{2} b^{4} d e^{4} + 4 \, a^{3} b^{3} e^{5}\right )} x^{9} + \frac {5}{8} \, {\left (b^{6} d^{4} e + 12 \, a b^{5} d^{3} e^{2} + 30 \, a^{2} b^{4} d^{2} e^{3} + 20 \, a^{3} b^{3} d e^{4} + 3 \, a^{4} b^{2} e^{5}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} d^{5} + 30 \, a b^{5} d^{4} e + 150 \, a^{2} b^{4} d^{3} e^{2} + 200 \, a^{3} b^{3} d^{2} e^{3} + 75 \, a^{4} b^{2} d e^{4} + 6 \, a^{5} b e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (6 \, a b^{5} d^{5} + 75 \, a^{2} b^{4} d^{4} e + 200 \, a^{3} b^{3} d^{3} e^{2} + 150 \, a^{4} b^{2} d^{2} e^{3} + 30 \, a^{5} b d e^{4} + a^{6} e^{5}\right )} x^{6} + {\left (3 \, a^{2} b^{4} d^{5} + 20 \, a^{3} b^{3} d^{4} e + 30 \, a^{4} b^{2} d^{3} e^{2} + 12 \, a^{5} b d^{2} e^{3} + a^{6} d e^{4}\right )} x^{5} + \frac {5}{4} \, {\left (4 \, a^{3} b^{3} d^{5} + 15 \, a^{4} b^{2} d^{4} e + 12 \, a^{5} b d^{3} e^{2} + 2 \, a^{6} d^{2} e^{3}\right )} x^{4} + \frac {5}{3} \, {\left (3 \, a^{4} b^{2} d^{5} + 6 \, a^{5} b d^{4} e + 2 \, a^{6} d^{3} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (6 \, a^{5} b d^{5} + 5 \, a^{6} d^{4} e\right )} x^{2} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 580 vs. \(2 (129) = 258\).
Time = 0.06 (sec) , antiderivative size = 580, normalized size of antiderivative = 4.06 \[ \int (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=a^{6} d^{5} x + \frac {b^{6} e^{5} x^{12}}{12} + x^{11} \cdot \left (\frac {6 a b^{5} e^{5}}{11} + \frac {5 b^{6} d e^{4}}{11}\right ) + x^{10} \cdot \left (\frac {3 a^{2} b^{4} e^{5}}{2} + 3 a b^{5} d e^{4} + b^{6} d^{2} e^{3}\right ) + x^{9} \cdot \left (\frac {20 a^{3} b^{3} e^{5}}{9} + \frac {25 a^{2} b^{4} d e^{4}}{3} + \frac {20 a b^{5} d^{2} e^{3}}{3} + \frac {10 b^{6} d^{3} e^{2}}{9}\right ) + x^{8} \cdot \left (\frac {15 a^{4} b^{2} e^{5}}{8} + \frac {25 a^{3} b^{3} d e^{4}}{2} + \frac {75 a^{2} b^{4} d^{2} e^{3}}{4} + \frac {15 a b^{5} d^{3} e^{2}}{2} + \frac {5 b^{6} d^{4} e}{8}\right ) + x^{7} \cdot \left (\frac {6 a^{5} b e^{5}}{7} + \frac {75 a^{4} b^{2} d e^{4}}{7} + \frac {200 a^{3} b^{3} d^{2} e^{3}}{7} + \frac {150 a^{2} b^{4} d^{3} e^{2}}{7} + \frac {30 a b^{5} d^{4} e}{7} + \frac {b^{6} d^{5}}{7}\right ) + x^{6} \left (\frac {a^{6} e^{5}}{6} + 5 a^{5} b d e^{4} + 25 a^{4} b^{2} d^{2} e^{3} + \frac {100 a^{3} b^{3} d^{3} e^{2}}{3} + \frac {25 a^{2} b^{4} d^{4} e}{2} + a b^{5} d^{5}\right ) + x^{5} \left (a^{6} d e^{4} + 12 a^{5} b d^{2} e^{3} + 30 a^{4} b^{2} d^{3} e^{2} + 20 a^{3} b^{3} d^{4} e + 3 a^{2} b^{4} d^{5}\right ) + x^{4} \cdot \left (\frac {5 a^{6} d^{2} e^{3}}{2} + 15 a^{5} b d^{3} e^{2} + \frac {75 a^{4} b^{2} d^{4} e}{4} + 5 a^{3} b^{3} d^{5}\right ) + x^{3} \cdot \left (\frac {10 a^{6} d^{3} e^{2}}{3} + 10 a^{5} b d^{4} e + 5 a^{4} b^{2} d^{5}\right ) + x^{2} \cdot \left (\frac {5 a^{6} d^{4} e}{2} + 3 a^{5} b d^{5}\right ) \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 517 vs. \(2 (133) = 266\).
Time = 0.20 (sec) , antiderivative size = 517, normalized size of antiderivative = 3.62 \[ \int (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{12} \, b^{6} e^{5} x^{12} + a^{6} d^{5} x + \frac {1}{11} \, {\left (5 \, b^{6} d e^{4} + 6 \, a b^{5} e^{5}\right )} x^{11} + \frac {1}{2} \, {\left (2 \, b^{6} d^{2} e^{3} + 6 \, a b^{5} d e^{4} + 3 \, a^{2} b^{4} e^{5}\right )} x^{10} + \frac {5}{9} \, {\left (2 \, b^{6} d^{3} e^{2} + 12 \, a b^{5} d^{2} e^{3} + 15 \, a^{2} b^{4} d e^{4} + 4 \, a^{3} b^{3} e^{5}\right )} x^{9} + \frac {5}{8} \, {\left (b^{6} d^{4} e + 12 \, a b^{5} d^{3} e^{2} + 30 \, a^{2} b^{4} d^{2} e^{3} + 20 \, a^{3} b^{3} d e^{4} + 3 \, a^{4} b^{2} e^{5}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} d^{5} + 30 \, a b^{5} d^{4} e + 150 \, a^{2} b^{4} d^{3} e^{2} + 200 \, a^{3} b^{3} d^{2} e^{3} + 75 \, a^{4} b^{2} d e^{4} + 6 \, a^{5} b e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (6 \, a b^{5} d^{5} + 75 \, a^{2} b^{4} d^{4} e + 200 \, a^{3} b^{3} d^{3} e^{2} + 150 \, a^{4} b^{2} d^{2} e^{3} + 30 \, a^{5} b d e^{4} + a^{6} e^{5}\right )} x^{6} + {\left (3 \, a^{2} b^{4} d^{5} + 20 \, a^{3} b^{3} d^{4} e + 30 \, a^{4} b^{2} d^{3} e^{2} + 12 \, a^{5} b d^{2} e^{3} + a^{6} d e^{4}\right )} x^{5} + \frac {5}{4} \, {\left (4 \, a^{3} b^{3} d^{5} + 15 \, a^{4} b^{2} d^{4} e + 12 \, a^{5} b d^{3} e^{2} + 2 \, a^{6} d^{2} e^{3}\right )} x^{4} + \frac {5}{3} \, {\left (3 \, a^{4} b^{2} d^{5} + 6 \, a^{5} b d^{4} e + 2 \, a^{6} d^{3} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (6 \, a^{5} b d^{5} + 5 \, a^{6} d^{4} e\right )} x^{2} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 579 vs. \(2 (133) = 266\).
Time = 0.26 (sec) , antiderivative size = 579, normalized size of antiderivative = 4.05 \[ \int (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{12} \, b^{6} e^{5} x^{12} + \frac {5}{11} \, b^{6} d e^{4} x^{11} + \frac {6}{11} \, a b^{5} e^{5} x^{11} + b^{6} d^{2} e^{3} x^{10} + 3 \, a b^{5} d e^{4} x^{10} + \frac {3}{2} \, a^{2} b^{4} e^{5} x^{10} + \frac {10}{9} \, b^{6} d^{3} e^{2} x^{9} + \frac {20}{3} \, a b^{5} d^{2} e^{3} x^{9} + \frac {25}{3} \, a^{2} b^{4} d e^{4} x^{9} + \frac {20}{9} \, a^{3} b^{3} e^{5} x^{9} + \frac {5}{8} \, b^{6} d^{4} e x^{8} + \frac {15}{2} \, a b^{5} d^{3} e^{2} x^{8} + \frac {75}{4} \, a^{2} b^{4} d^{2} e^{3} x^{8} + \frac {25}{2} \, a^{3} b^{3} d e^{4} x^{8} + \frac {15}{8} \, a^{4} b^{2} e^{5} x^{8} + \frac {1}{7} \, b^{6} d^{5} x^{7} + \frac {30}{7} \, a b^{5} d^{4} e x^{7} + \frac {150}{7} \, a^{2} b^{4} d^{3} e^{2} x^{7} + \frac {200}{7} \, a^{3} b^{3} d^{2} e^{3} x^{7} + \frac {75}{7} \, a^{4} b^{2} d e^{4} x^{7} + \frac {6}{7} \, a^{5} b e^{5} x^{7} + a b^{5} d^{5} x^{6} + \frac {25}{2} \, a^{2} b^{4} d^{4} e x^{6} + \frac {100}{3} \, a^{3} b^{3} d^{3} e^{2} x^{6} + 25 \, a^{4} b^{2} d^{2} e^{3} x^{6} + 5 \, a^{5} b d e^{4} x^{6} + \frac {1}{6} \, a^{6} e^{5} x^{6} + 3 \, a^{2} b^{4} d^{5} x^{5} + 20 \, a^{3} b^{3} d^{4} e x^{5} + 30 \, a^{4} b^{2} d^{3} e^{2} x^{5} + 12 \, a^{5} b d^{2} e^{3} x^{5} + a^{6} d e^{4} x^{5} + 5 \, a^{3} b^{3} d^{5} x^{4} + \frac {75}{4} \, a^{4} b^{2} d^{4} e x^{4} + 15 \, a^{5} b d^{3} e^{2} x^{4} + \frac {5}{2} \, a^{6} d^{2} e^{3} x^{4} + 5 \, a^{4} b^{2} d^{5} x^{3} + 10 \, a^{5} b d^{4} e x^{3} + \frac {10}{3} \, a^{6} d^{3} e^{2} x^{3} + 3 \, a^{5} b d^{5} x^{2} + \frac {5}{2} \, a^{6} d^{4} e x^{2} + a^{6} d^{5} x \]
[In]
[Out]
Time = 0.21 (sec) , antiderivative size = 492, normalized size of antiderivative = 3.44 \[ \int (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=x^5\,\left (a^6\,d\,e^4+12\,a^5\,b\,d^2\,e^3+30\,a^4\,b^2\,d^3\,e^2+20\,a^3\,b^3\,d^4\,e+3\,a^2\,b^4\,d^5\right )+x^8\,\left (\frac {15\,a^4\,b^2\,e^5}{8}+\frac {25\,a^3\,b^3\,d\,e^4}{2}+\frac {75\,a^2\,b^4\,d^2\,e^3}{4}+\frac {15\,a\,b^5\,d^3\,e^2}{2}+\frac {5\,b^6\,d^4\,e}{8}\right )+x^6\,\left (\frac {a^6\,e^5}{6}+5\,a^5\,b\,d\,e^4+25\,a^4\,b^2\,d^2\,e^3+\frac {100\,a^3\,b^3\,d^3\,e^2}{3}+\frac {25\,a^2\,b^4\,d^4\,e}{2}+a\,b^5\,d^5\right )+x^7\,\left (\frac {6\,a^5\,b\,e^5}{7}+\frac {75\,a^4\,b^2\,d\,e^4}{7}+\frac {200\,a^3\,b^3\,d^2\,e^3}{7}+\frac {150\,a^2\,b^4\,d^3\,e^2}{7}+\frac {30\,a\,b^5\,d^4\,e}{7}+\frac {b^6\,d^5}{7}\right )+a^6\,d^5\,x+\frac {b^6\,e^5\,x^{12}}{12}+\frac {5\,a^3\,d^2\,x^4\,\left (2\,a^3\,e^3+12\,a^2\,b\,d\,e^2+15\,a\,b^2\,d^2\,e+4\,b^3\,d^3\right )}{4}+\frac {5\,b^3\,e^2\,x^9\,\left (4\,a^3\,e^3+15\,a^2\,b\,d\,e^2+12\,a\,b^2\,d^2\,e+2\,b^3\,d^3\right )}{9}+\frac {a^5\,d^4\,x^2\,\left (5\,a\,e+6\,b\,d\right )}{2}+\frac {b^5\,e^4\,x^{11}\,\left (6\,a\,e+5\,b\,d\right )}{11}+\frac {5\,a^4\,d^3\,x^3\,\left (2\,a^2\,e^2+6\,a\,b\,d\,e+3\,b^2\,d^2\right )}{3}+\frac {b^4\,e^3\,x^{10}\,\left (3\,a^2\,e^2+6\,a\,b\,d\,e+2\,b^2\,d^2\right )}{2} \]
[In]
[Out]