Integrand size = 25, antiderivative size = 304 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx=-\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^5}{5 e^6}-\frac {\left (c d^2-b d e+a e^2\right ) \left (2 A e (2 c d-b e)-B \left (5 c d^2-e (3 b d-a e)\right )\right ) (d+e x)^6}{6 e^6}-\frac {\left (B \left (10 c^2 d^3+b e^2 (3 b d-2 a e)-6 c d e (2 b d-a e)\right )-A e \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )\right ) (d+e x)^7}{7 e^6}-\frac {\left (2 A c e (2 c d-b e)-B \left (10 c^2 d^2+b^2 e^2-2 c e (4 b d-a e)\right )\right ) (d+e x)^8}{8 e^6}-\frac {c (5 B c d-2 b B e-A c e) (d+e x)^9}{9 e^6}+\frac {B c^2 (d+e x)^{10}}{10 e^6} \]
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Time = 0.39 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {785} \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx=-\frac {(d+e x)^8 \left (2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )\right )}{8 e^6}-\frac {(d+e x)^7 \left (B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )\right )}{7 e^6}+\frac {(d+e x)^6 \left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{6 e^6}-\frac {(d+e x)^5 (B d-A e) \left (a e^2-b d e+c d^2\right )^2}{5 e^6}-\frac {c (d+e x)^9 (-A c e-2 b B e+5 B c d)}{9 e^6}+\frac {B c^2 (d+e x)^{10}}{10 e^6} \]
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Rule 785
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-B d+A e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^4}{e^5}+\frac {\left (c d^2-b d e+a e^2\right ) \left (5 B c d^2-B e (3 b d-a e)-2 A e (2 c d-b e)\right ) (d+e x)^5}{e^5}+\frac {\left (-B \left (10 c^2 d^3+b e^2 (3 b d-2 a e)-6 c d e (2 b d-a e)\right )+A e \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )\right ) (d+e x)^6}{e^5}+\frac {\left (-2 A c e (2 c d-b e)+B \left (10 c^2 d^2+b^2 e^2-2 c e (4 b d-a e)\right )\right ) (d+e x)^7}{e^5}+\frac {c (-5 B c d+2 b B e+A c e) (d+e x)^8}{e^5}+\frac {B c^2 (d+e x)^9}{e^5}\right ) \, dx \\ & = -\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^5}{5 e^6}+\frac {\left (c d^2-b d e+a e^2\right ) \left (5 B c d^2-B e (3 b d-a e)-2 A e (2 c d-b e)\right ) (d+e x)^6}{6 e^6}-\frac {\left (B \left (10 c^2 d^3+b e^2 (3 b d-2 a e)-6 c d e (2 b d-a e)\right )-A e \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )\right ) (d+e x)^7}{7 e^6}-\frac {\left (2 A c e (2 c d-b e)-B \left (10 c^2 d^2+b^2 e^2-2 c e (4 b d-a e)\right )\right ) (d+e x)^8}{8 e^6}-\frac {c (5 B c d-2 b B e-A c e) (d+e x)^9}{9 e^6}+\frac {B c^2 (d+e x)^{10}}{10 e^6} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 550, normalized size of antiderivative = 1.81 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx=a^2 A d^4 x+\frac {1}{2} a d^3 (2 A b d+a B d+4 a A e) x^2+\frac {1}{3} d^2 \left (2 a B d (b d+2 a e)+A \left (b^2 d^2+8 a b d e+2 a \left (c d^2+3 a e^2\right )\right )\right ) x^3+\frac {1}{4} d \left (b^2 d^2 (B d+4 A e)+2 b d \left (A c d^2+4 a B d e+6 a A e^2\right )+2 a \left (B c d^3+4 A c d^2 e+3 a B d e^2+2 a A e^3\right )\right ) x^4+\frac {1}{5} \left (2 b^2 d^2 e (2 B d+3 A e)+4 a B d e \left (2 c d^2+a e^2\right )+2 b d \left (B c d^3+4 A c d^2 e+6 a B d e^2+4 a A e^3\right )+A \left (c^2 d^4+12 a c d^2 e^2+a^2 e^4\right )\right ) x^5+\frac {1}{6} \left (2 A e \left (2 c^2 d^3+b e^2 (2 b d+a e)+2 c d e (3 b d+2 a e)\right )+B \left (c^2 d^4+4 c d^2 e (2 b d+3 a e)+e^2 \left (6 b^2 d^2+8 a b d e+a^2 e^2\right )\right )\right ) x^6+\frac {1}{7} e \left (A e \left (6 c^2 d^2+b^2 e^2+2 c e (4 b d+a e)\right )+2 B \left (2 c^2 d^3+b e^2 (2 b d+a e)+2 c d e (3 b d+2 a e)\right )\right ) x^7+\frac {1}{8} e^2 \left (2 A c e (2 c d+b e)+B \left (6 c^2 d^2+b^2 e^2+2 c e (4 b d+a e)\right )\right ) x^8+\frac {1}{9} c e^3 (4 B c d+2 b B e+A c e) x^9+\frac {1}{10} B c^2 e^4 x^{10} \]
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Time = 0.36 (sec) , antiderivative size = 545, normalized size of antiderivative = 1.79
method | result | size |
default | \(\frac {B \,e^{4} c^{2} x^{10}}{10}+\frac {\left (\left (A \,e^{4}+4 B \,e^{3} d \right ) c^{2}+2 B \,e^{4} b c \right ) x^{9}}{9}+\frac {\left (\left (4 A \,e^{3} d +6 B \,d^{2} e^{2}\right ) c^{2}+2 \left (A \,e^{4}+4 B \,e^{3} d \right ) b c +B \,e^{4} \left (2 a c +b^{2}\right )\right ) x^{8}}{8}+\frac {\left (\left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) c^{2}+2 \left (4 A \,e^{3} d +6 B \,d^{2} e^{2}\right ) b c +\left (A \,e^{4}+4 B \,e^{3} d \right ) \left (2 a c +b^{2}\right )+2 B a b \,e^{4}\right ) x^{7}}{7}+\frac {\left (\left (4 A \,d^{3} e +B \,d^{4}\right ) c^{2}+2 \left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) b c +\left (4 A \,e^{3} d +6 B \,d^{2} e^{2}\right ) \left (2 a c +b^{2}\right )+2 \left (A \,e^{4}+4 B \,e^{3} d \right ) b a +B \,e^{4} a^{2}\right ) x^{6}}{6}+\frac {\left (d^{4} A \,c^{2}+2 \left (4 A \,d^{3} e +B \,d^{4}\right ) b c +\left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) \left (2 a c +b^{2}\right )+2 \left (4 A \,e^{3} d +6 B \,d^{2} e^{2}\right ) b a +\left (A \,e^{4}+4 B \,e^{3} d \right ) a^{2}\right ) x^{5}}{5}+\frac {\left (2 A \,d^{4} b c +\left (4 A \,d^{3} e +B \,d^{4}\right ) \left (2 a c +b^{2}\right )+2 \left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) b a +\left (4 A \,e^{3} d +6 B \,d^{2} e^{2}\right ) a^{2}\right ) x^{4}}{4}+\frac {\left (A \,d^{4} \left (2 a c +b^{2}\right )+2 \left (4 A \,d^{3} e +B \,d^{4}\right ) b a +\left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) a^{2}\right ) x^{3}}{3}+\frac {\left (2 A a b \,d^{4}+\left (4 A \,d^{3} e +B \,d^{4}\right ) a^{2}\right ) x^{2}}{2}+d^{4} A \,a^{2} x\) | \(545\) |
norman | \(\frac {B \,e^{4} c^{2} x^{10}}{10}+\left (\frac {1}{9} A \,c^{2} e^{4}+\frac {2}{9} B \,e^{4} b c +\frac {4}{9} B \,c^{2} d \,e^{3}\right ) x^{9}+\left (\frac {1}{4} A b c \,e^{4}+\frac {1}{2} A \,c^{2} d \,e^{3}+\frac {1}{4} B \,e^{4} a c +\frac {1}{8} B \,b^{2} e^{4}+B b c d \,e^{3}+\frac {3}{4} B \,c^{2} d^{2} e^{2}\right ) x^{8}+\left (\frac {2}{7} A a c \,e^{4}+\frac {1}{7} A \,b^{2} e^{4}+\frac {8}{7} A b c d \,e^{3}+\frac {6}{7} A \,c^{2} d^{2} e^{2}+\frac {2}{7} B a b \,e^{4}+\frac {8}{7} B a c d \,e^{3}+\frac {4}{7} B \,b^{2} e^{3} d +\frac {12}{7} B b c \,d^{2} e^{2}+\frac {4}{7} B \,c^{2} d^{3} e \right ) x^{7}+\left (\frac {1}{3} A a b \,e^{4}+\frac {4}{3} A a c d \,e^{3}+\frac {2}{3} A \,b^{2} d \,e^{3}+2 A b c \,d^{2} e^{2}+\frac {2}{3} A \,c^{2} d^{3} e +\frac {1}{6} B \,e^{4} a^{2}+\frac {4}{3} B a b d \,e^{3}+2 B a c \,d^{2} e^{2}+B \,b^{2} d^{2} e^{2}+\frac {4}{3} B b c \,d^{3} e +\frac {1}{6} B \,c^{2} d^{4}\right ) x^{6}+\left (\frac {1}{5} A \,a^{2} e^{4}+\frac {8}{5} A a b d \,e^{3}+\frac {12}{5} A a c \,d^{2} e^{2}+\frac {6}{5} A \,b^{2} d^{2} e^{2}+\frac {8}{5} A b c \,d^{3} e +\frac {1}{5} d^{4} A \,c^{2}+\frac {4}{5} B \,a^{2} d \,e^{3}+\frac {12}{5} B a b \,d^{2} e^{2}+\frac {8}{5} B a c \,d^{3} e +\frac {4}{5} B \,b^{2} d^{3} e +\frac {2}{5} B b c \,d^{4}\right ) x^{5}+\left (A \,a^{2} d \,e^{3}+3 A a b \,d^{2} e^{2}+2 A a c \,d^{3} e +A \,b^{2} d^{3} e +\frac {1}{2} A \,d^{4} b c +\frac {3}{2} B \,a^{2} d^{2} e^{2}+2 B a b \,d^{3} e +\frac {1}{2} B a c \,d^{4}+\frac {1}{4} B \,b^{2} d^{4}\right ) x^{4}+\left (2 A \,a^{2} d^{2} e^{2}+\frac {8}{3} A a b \,d^{3} e +\frac {2}{3} d^{4} A a c +\frac {1}{3} A \,b^{2} d^{4}+\frac {4}{3} B \,a^{2} d^{3} e +\frac {2}{3} B a b \,d^{4}\right ) x^{3}+\left (2 A \,a^{2} d^{3} e +A a b \,d^{4}+\frac {1}{2} B \,a^{2} d^{4}\right ) x^{2}+d^{4} A \,a^{2} x\) | \(610\) |
gosper | \(2 x^{4} A a c \,d^{3} e +\frac {8}{5} x^{5} B a c \,d^{3} e +\frac {12}{5} x^{5} A a c \,d^{2} e^{2}+\frac {8}{5} x^{5} A b c \,d^{3} e +2 x^{6} A b c \,d^{2} e^{2}+2 x^{6} B a c \,d^{2} e^{2}+\frac {4}{3} x^{6} B b c \,d^{3} e +\frac {4}{3} x^{6} A a c d \,e^{3}+\frac {12}{7} x^{7} B b c \,d^{2} e^{2}+\frac {8}{7} x^{7} B a c d \,e^{3}+\frac {8}{5} x^{5} A a b d \,e^{3}+\frac {12}{5} x^{5} B a b \,d^{2} e^{2}+3 x^{4} A a b \,d^{2} e^{2}+2 x^{4} B a b \,d^{3} e +\frac {8}{3} x^{3} A a b \,d^{3} e +\frac {4}{3} x^{6} B a b d \,e^{3}+\frac {8}{7} x^{7} A b c d \,e^{3}+x^{8} B b c d \,e^{3}+\frac {1}{3} x^{3} A \,b^{2} d^{4}+\frac {1}{7} x^{7} A \,b^{2} e^{4}+\frac {1}{4} x^{4} B \,b^{2} d^{4}+\frac {1}{6} x^{6} B \,c^{2} d^{4}+\frac {1}{5} x^{5} d^{4} A \,c^{2}+\frac {1}{9} x^{9} A \,c^{2} e^{4}+\frac {1}{8} B \,b^{2} e^{4} x^{8}+\frac {1}{5} x^{5} A \,a^{2} e^{4}+\frac {1}{2} x^{2} B \,a^{2} d^{4}+\frac {1}{6} x^{6} B \,e^{4} a^{2}+\frac {1}{10} B \,e^{4} c^{2} x^{10}+d^{4} A \,a^{2} x +\frac {4}{9} x^{9} B \,c^{2} d \,e^{3}+\frac {1}{4} x^{8} A b c \,e^{4}+\frac {1}{2} x^{8} A \,c^{2} d \,e^{3}+\frac {1}{4} x^{8} B \,e^{4} a c +\frac {3}{4} x^{8} B \,c^{2} d^{2} e^{2}+\frac {2}{7} x^{7} A a c \,e^{4}+\frac {6}{7} x^{7} A \,c^{2} d^{2} e^{2}+\frac {2}{5} x^{5} B b c \,d^{4}+\frac {1}{2} x^{4} A \,d^{4} b c +\frac {1}{2} x^{4} B a c \,d^{4}+\frac {2}{3} x^{3} d^{4} A a c +\frac {2}{9} x^{9} B \,e^{4} b c +\frac {4}{7} x^{7} B \,c^{2} d^{3} e +\frac {2}{3} x^{6} A \,c^{2} d^{3} e +\frac {4}{5} x^{5} B \,a^{2} d \,e^{3}+x^{4} A \,a^{2} d \,e^{3}+\frac {3}{2} x^{4} B \,a^{2} d^{2} e^{2}+2 x^{3} A \,a^{2} d^{2} e^{2}+\frac {4}{3} x^{3} B \,a^{2} d^{3} e +2 x^{2} A \,a^{2} d^{3} e +x^{4} A \,b^{2} d^{3} e +\frac {2}{3} x^{3} B a b \,d^{4}+x^{2} A a b \,d^{4}+\frac {2}{7} x^{7} B a b \,e^{4}+\frac {4}{7} x^{7} B \,b^{2} e^{3} d +\frac {1}{3} x^{6} A a b \,e^{4}+\frac {2}{3} x^{6} A \,b^{2} d \,e^{3}+x^{6} B \,b^{2} d^{2} e^{2}+\frac {6}{5} x^{5} A \,b^{2} d^{2} e^{2}+\frac {4}{5} x^{5} B \,b^{2} d^{3} e\) | \(744\) |
risch | \(2 x^{4} A a c \,d^{3} e +\frac {8}{5} x^{5} B a c \,d^{3} e +\frac {12}{5} x^{5} A a c \,d^{2} e^{2}+\frac {8}{5} x^{5} A b c \,d^{3} e +2 x^{6} A b c \,d^{2} e^{2}+2 x^{6} B a c \,d^{2} e^{2}+\frac {4}{3} x^{6} B b c \,d^{3} e +\frac {4}{3} x^{6} A a c d \,e^{3}+\frac {12}{7} x^{7} B b c \,d^{2} e^{2}+\frac {8}{7} x^{7} B a c d \,e^{3}+\frac {8}{5} x^{5} A a b d \,e^{3}+\frac {12}{5} x^{5} B a b \,d^{2} e^{2}+3 x^{4} A a b \,d^{2} e^{2}+2 x^{4} B a b \,d^{3} e +\frac {8}{3} x^{3} A a b \,d^{3} e +\frac {4}{3} x^{6} B a b d \,e^{3}+\frac {8}{7} x^{7} A b c d \,e^{3}+x^{8} B b c d \,e^{3}+\frac {1}{3} x^{3} A \,b^{2} d^{4}+\frac {1}{7} x^{7} A \,b^{2} e^{4}+\frac {1}{4} x^{4} B \,b^{2} d^{4}+\frac {1}{6} x^{6} B \,c^{2} d^{4}+\frac {1}{5} x^{5} d^{4} A \,c^{2}+\frac {1}{9} x^{9} A \,c^{2} e^{4}+\frac {1}{8} B \,b^{2} e^{4} x^{8}+\frac {1}{5} x^{5} A \,a^{2} e^{4}+\frac {1}{2} x^{2} B \,a^{2} d^{4}+\frac {1}{6} x^{6} B \,e^{4} a^{2}+\frac {1}{10} B \,e^{4} c^{2} x^{10}+d^{4} A \,a^{2} x +\frac {4}{9} x^{9} B \,c^{2} d \,e^{3}+\frac {1}{4} x^{8} A b c \,e^{4}+\frac {1}{2} x^{8} A \,c^{2} d \,e^{3}+\frac {1}{4} x^{8} B \,e^{4} a c +\frac {3}{4} x^{8} B \,c^{2} d^{2} e^{2}+\frac {2}{7} x^{7} A a c \,e^{4}+\frac {6}{7} x^{7} A \,c^{2} d^{2} e^{2}+\frac {2}{5} x^{5} B b c \,d^{4}+\frac {1}{2} x^{4} A \,d^{4} b c +\frac {1}{2} x^{4} B a c \,d^{4}+\frac {2}{3} x^{3} d^{4} A a c +\frac {2}{9} x^{9} B \,e^{4} b c +\frac {4}{7} x^{7} B \,c^{2} d^{3} e +\frac {2}{3} x^{6} A \,c^{2} d^{3} e +\frac {4}{5} x^{5} B \,a^{2} d \,e^{3}+x^{4} A \,a^{2} d \,e^{3}+\frac {3}{2} x^{4} B \,a^{2} d^{2} e^{2}+2 x^{3} A \,a^{2} d^{2} e^{2}+\frac {4}{3} x^{3} B \,a^{2} d^{3} e +2 x^{2} A \,a^{2} d^{3} e +x^{4} A \,b^{2} d^{3} e +\frac {2}{3} x^{3} B a b \,d^{4}+x^{2} A a b \,d^{4}+\frac {2}{7} x^{7} B a b \,e^{4}+\frac {4}{7} x^{7} B \,b^{2} e^{3} d +\frac {1}{3} x^{6} A a b \,e^{4}+\frac {2}{3} x^{6} A \,b^{2} d \,e^{3}+x^{6} B \,b^{2} d^{2} e^{2}+\frac {6}{5} x^{5} A \,b^{2} d^{2} e^{2}+\frac {4}{5} x^{5} B \,b^{2} d^{3} e\) | \(744\) |
parallelrisch | \(2 x^{4} A a c \,d^{3} e +\frac {8}{5} x^{5} B a c \,d^{3} e +\frac {12}{5} x^{5} A a c \,d^{2} e^{2}+\frac {8}{5} x^{5} A b c \,d^{3} e +2 x^{6} A b c \,d^{2} e^{2}+2 x^{6} B a c \,d^{2} e^{2}+\frac {4}{3} x^{6} B b c \,d^{3} e +\frac {4}{3} x^{6} A a c d \,e^{3}+\frac {12}{7} x^{7} B b c \,d^{2} e^{2}+\frac {8}{7} x^{7} B a c d \,e^{3}+\frac {8}{5} x^{5} A a b d \,e^{3}+\frac {12}{5} x^{5} B a b \,d^{2} e^{2}+3 x^{4} A a b \,d^{2} e^{2}+2 x^{4} B a b \,d^{3} e +\frac {8}{3} x^{3} A a b \,d^{3} e +\frac {4}{3} x^{6} B a b d \,e^{3}+\frac {8}{7} x^{7} A b c d \,e^{3}+x^{8} B b c d \,e^{3}+\frac {1}{3} x^{3} A \,b^{2} d^{4}+\frac {1}{7} x^{7} A \,b^{2} e^{4}+\frac {1}{4} x^{4} B \,b^{2} d^{4}+\frac {1}{6} x^{6} B \,c^{2} d^{4}+\frac {1}{5} x^{5} d^{4} A \,c^{2}+\frac {1}{9} x^{9} A \,c^{2} e^{4}+\frac {1}{8} B \,b^{2} e^{4} x^{8}+\frac {1}{5} x^{5} A \,a^{2} e^{4}+\frac {1}{2} x^{2} B \,a^{2} d^{4}+\frac {1}{6} x^{6} B \,e^{4} a^{2}+\frac {1}{10} B \,e^{4} c^{2} x^{10}+d^{4} A \,a^{2} x +\frac {4}{9} x^{9} B \,c^{2} d \,e^{3}+\frac {1}{4} x^{8} A b c \,e^{4}+\frac {1}{2} x^{8} A \,c^{2} d \,e^{3}+\frac {1}{4} x^{8} B \,e^{4} a c +\frac {3}{4} x^{8} B \,c^{2} d^{2} e^{2}+\frac {2}{7} x^{7} A a c \,e^{4}+\frac {6}{7} x^{7} A \,c^{2} d^{2} e^{2}+\frac {2}{5} x^{5} B b c \,d^{4}+\frac {1}{2} x^{4} A \,d^{4} b c +\frac {1}{2} x^{4} B a c \,d^{4}+\frac {2}{3} x^{3} d^{4} A a c +\frac {2}{9} x^{9} B \,e^{4} b c +\frac {4}{7} x^{7} B \,c^{2} d^{3} e +\frac {2}{3} x^{6} A \,c^{2} d^{3} e +\frac {4}{5} x^{5} B \,a^{2} d \,e^{3}+x^{4} A \,a^{2} d \,e^{3}+\frac {3}{2} x^{4} B \,a^{2} d^{2} e^{2}+2 x^{3} A \,a^{2} d^{2} e^{2}+\frac {4}{3} x^{3} B \,a^{2} d^{3} e +2 x^{2} A \,a^{2} d^{3} e +x^{4} A \,b^{2} d^{3} e +\frac {2}{3} x^{3} B a b \,d^{4}+x^{2} A a b \,d^{4}+\frac {2}{7} x^{7} B a b \,e^{4}+\frac {4}{7} x^{7} B \,b^{2} e^{3} d +\frac {1}{3} x^{6} A a b \,e^{4}+\frac {2}{3} x^{6} A \,b^{2} d \,e^{3}+x^{6} B \,b^{2} d^{2} e^{2}+\frac {6}{5} x^{5} A \,b^{2} d^{2} e^{2}+\frac {4}{5} x^{5} B \,b^{2} d^{3} e\) | \(744\) |
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Time = 0.32 (sec) , antiderivative size = 532, normalized size of antiderivative = 1.75 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{10} \, B c^{2} e^{4} x^{10} + \frac {1}{9} \, {\left (4 \, B c^{2} d e^{3} + {\left (2 \, B b c + A c^{2}\right )} e^{4}\right )} x^{9} + \frac {1}{8} \, {\left (6 \, B c^{2} d^{2} e^{2} + 4 \, {\left (2 \, B b c + A c^{2}\right )} d e^{3} + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e^{4}\right )} x^{8} + A a^{2} d^{4} x + \frac {1}{7} \, {\left (4 \, B c^{2} d^{3} e + 6 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{2} + 4 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e^{3} + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (B c^{2} d^{4} + 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e + 6 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} e^{2} + 4 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{3} + {\left (B a^{2} + 2 \, A a b\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (A a^{2} e^{4} + {\left (2 \, B b c + A c^{2}\right )} d^{4} + 4 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3} e + 6 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{2} + 4 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (4 \, A a^{2} d e^{3} + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{4} + 4 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{3} e + 6 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, A a^{2} d^{2} e^{2} + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{4} + 4 \, {\left (B a^{2} + 2 \, A a b\right )} d^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, A a^{2} d^{3} e + {\left (B a^{2} + 2 \, A a b\right )} d^{4}\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 765 vs. \(2 (292) = 584\).
Time = 0.06 (sec) , antiderivative size = 765, normalized size of antiderivative = 2.52 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx=A a^{2} d^{4} x + \frac {B c^{2} e^{4} x^{10}}{10} + x^{9} \left (\frac {A c^{2} e^{4}}{9} + \frac {2 B b c e^{4}}{9} + \frac {4 B c^{2} d e^{3}}{9}\right ) + x^{8} \left (\frac {A b c e^{4}}{4} + \frac {A c^{2} d e^{3}}{2} + \frac {B a c e^{4}}{4} + \frac {B b^{2} e^{4}}{8} + B b c d e^{3} + \frac {3 B c^{2} d^{2} e^{2}}{4}\right ) + x^{7} \cdot \left (\frac {2 A a c e^{4}}{7} + \frac {A b^{2} e^{4}}{7} + \frac {8 A b c d e^{3}}{7} + \frac {6 A c^{2} d^{2} e^{2}}{7} + \frac {2 B a b e^{4}}{7} + \frac {8 B a c d e^{3}}{7} + \frac {4 B b^{2} d e^{3}}{7} + \frac {12 B b c d^{2} e^{2}}{7} + \frac {4 B c^{2} d^{3} e}{7}\right ) + x^{6} \left (\frac {A a b e^{4}}{3} + \frac {4 A a c d e^{3}}{3} + \frac {2 A b^{2} d e^{3}}{3} + 2 A b c d^{2} e^{2} + \frac {2 A c^{2} d^{3} e}{3} + \frac {B a^{2} e^{4}}{6} + \frac {4 B a b d e^{3}}{3} + 2 B a c d^{2} e^{2} + B b^{2} d^{2} e^{2} + \frac {4 B b c d^{3} e}{3} + \frac {B c^{2} d^{4}}{6}\right ) + x^{5} \left (\frac {A a^{2} e^{4}}{5} + \frac {8 A a b d e^{3}}{5} + \frac {12 A a c d^{2} e^{2}}{5} + \frac {6 A b^{2} d^{2} e^{2}}{5} + \frac {8 A b c d^{3} e}{5} + \frac {A c^{2} d^{4}}{5} + \frac {4 B a^{2} d e^{3}}{5} + \frac {12 B a b d^{2} e^{2}}{5} + \frac {8 B a c d^{3} e}{5} + \frac {4 B b^{2} d^{3} e}{5} + \frac {2 B b c d^{4}}{5}\right ) + x^{4} \left (A a^{2} d e^{3} + 3 A a b d^{2} e^{2} + 2 A a c d^{3} e + A b^{2} d^{3} e + \frac {A b c d^{4}}{2} + \frac {3 B a^{2} d^{2} e^{2}}{2} + 2 B a b d^{3} e + \frac {B a c d^{4}}{2} + \frac {B b^{2} d^{4}}{4}\right ) + x^{3} \cdot \left (2 A a^{2} d^{2} e^{2} + \frac {8 A a b d^{3} e}{3} + \frac {2 A a c d^{4}}{3} + \frac {A b^{2} d^{4}}{3} + \frac {4 B a^{2} d^{3} e}{3} + \frac {2 B a b d^{4}}{3}\right ) + x^{2} \cdot \left (2 A a^{2} d^{3} e + A a b d^{4} + \frac {B a^{2} d^{4}}{2}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 532, normalized size of antiderivative = 1.75 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{10} \, B c^{2} e^{4} x^{10} + \frac {1}{9} \, {\left (4 \, B c^{2} d e^{3} + {\left (2 \, B b c + A c^{2}\right )} e^{4}\right )} x^{9} + \frac {1}{8} \, {\left (6 \, B c^{2} d^{2} e^{2} + 4 \, {\left (2 \, B b c + A c^{2}\right )} d e^{3} + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e^{4}\right )} x^{8} + A a^{2} d^{4} x + \frac {1}{7} \, {\left (4 \, B c^{2} d^{3} e + 6 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{2} + 4 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e^{3} + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (B c^{2} d^{4} + 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e + 6 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} e^{2} + 4 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{3} + {\left (B a^{2} + 2 \, A a b\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (A a^{2} e^{4} + {\left (2 \, B b c + A c^{2}\right )} d^{4} + 4 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3} e + 6 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{2} + 4 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (4 \, A a^{2} d e^{3} + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{4} + 4 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{3} e + 6 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, A a^{2} d^{2} e^{2} + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{4} + 4 \, {\left (B a^{2} + 2 \, A a b\right )} d^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, A a^{2} d^{3} e + {\left (B a^{2} + 2 \, A a b\right )} d^{4}\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 743 vs. \(2 (292) = 584\).
Time = 0.28 (sec) , antiderivative size = 743, normalized size of antiderivative = 2.44 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{10} \, B c^{2} e^{4} x^{10} + \frac {4}{9} \, B c^{2} d e^{3} x^{9} + \frac {2}{9} \, B b c e^{4} x^{9} + \frac {1}{9} \, A c^{2} e^{4} x^{9} + \frac {3}{4} \, B c^{2} d^{2} e^{2} x^{8} + B b c d e^{3} x^{8} + \frac {1}{2} \, A c^{2} d e^{3} x^{8} + \frac {1}{8} \, B b^{2} e^{4} x^{8} + \frac {1}{4} \, B a c e^{4} x^{8} + \frac {1}{4} \, A b c e^{4} x^{8} + \frac {4}{7} \, B c^{2} d^{3} e x^{7} + \frac {12}{7} \, B b c d^{2} e^{2} x^{7} + \frac {6}{7} \, A c^{2} d^{2} e^{2} x^{7} + \frac {4}{7} \, B b^{2} d e^{3} x^{7} + \frac {8}{7} \, B a c d e^{3} x^{7} + \frac {8}{7} \, A b c d e^{3} x^{7} + \frac {2}{7} \, B a b e^{4} x^{7} + \frac {1}{7} \, A b^{2} e^{4} x^{7} + \frac {2}{7} \, A a c e^{4} x^{7} + \frac {1}{6} \, B c^{2} d^{4} x^{6} + \frac {4}{3} \, B b c d^{3} e x^{6} + \frac {2}{3} \, A c^{2} d^{3} e x^{6} + B b^{2} d^{2} e^{2} x^{6} + 2 \, B a c d^{2} e^{2} x^{6} + 2 \, A b c d^{2} e^{2} x^{6} + \frac {4}{3} \, B a b d e^{3} x^{6} + \frac {2}{3} \, A b^{2} d e^{3} x^{6} + \frac {4}{3} \, A a c d e^{3} x^{6} + \frac {1}{6} \, B a^{2} e^{4} x^{6} + \frac {1}{3} \, A a b e^{4} x^{6} + \frac {2}{5} \, B b c d^{4} x^{5} + \frac {1}{5} \, A c^{2} d^{4} x^{5} + \frac {4}{5} \, B b^{2} d^{3} e x^{5} + \frac {8}{5} \, B a c d^{3} e x^{5} + \frac {8}{5} \, A b c d^{3} e x^{5} + \frac {12}{5} \, B a b d^{2} e^{2} x^{5} + \frac {6}{5} \, A b^{2} d^{2} e^{2} x^{5} + \frac {12}{5} \, A a c d^{2} e^{2} x^{5} + \frac {4}{5} \, B a^{2} d e^{3} x^{5} + \frac {8}{5} \, A a b d e^{3} x^{5} + \frac {1}{5} \, A a^{2} e^{4} x^{5} + \frac {1}{4} \, B b^{2} d^{4} x^{4} + \frac {1}{2} \, B a c d^{4} x^{4} + \frac {1}{2} \, A b c d^{4} x^{4} + 2 \, B a b d^{3} e x^{4} + A b^{2} d^{3} e x^{4} + 2 \, A a c d^{3} e x^{4} + \frac {3}{2} \, B a^{2} d^{2} e^{2} x^{4} + 3 \, A a b d^{2} e^{2} x^{4} + A a^{2} d e^{3} x^{4} + \frac {2}{3} \, B a b d^{4} x^{3} + \frac {1}{3} \, A b^{2} d^{4} x^{3} + \frac {2}{3} \, A a c d^{4} x^{3} + \frac {4}{3} \, B a^{2} d^{3} e x^{3} + \frac {8}{3} \, A a b d^{3} e x^{3} + 2 \, A a^{2} d^{2} e^{2} x^{3} + \frac {1}{2} \, B a^{2} d^{4} x^{2} + A a b d^{4} x^{2} + 2 \, A a^{2} d^{3} e x^{2} + A a^{2} d^{4} x \]
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Time = 11.28 (sec) , antiderivative size = 594, normalized size of antiderivative = 1.95 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx=x^3\,\left (\frac {4\,B\,a^2\,d^3\,e}{3}+2\,A\,a^2\,d^2\,e^2+\frac {2\,B\,a\,b\,d^4}{3}+\frac {8\,A\,a\,b\,d^3\,e}{3}+\frac {2\,A\,c\,a\,d^4}{3}+\frac {A\,b^2\,d^4}{3}\right )+x^4\,\left (\frac {3\,B\,a^2\,d^2\,e^2}{2}+A\,a^2\,d\,e^3+2\,B\,a\,b\,d^3\,e+3\,A\,a\,b\,d^2\,e^2+\frac {B\,c\,a\,d^4}{2}+2\,A\,c\,a\,d^3\,e+\frac {B\,b^2\,d^4}{4}+A\,b^2\,d^3\,e+\frac {A\,c\,b\,d^4}{2}\right )+x^8\,\left (\frac {B\,b^2\,e^4}{8}+B\,b\,c\,d\,e^3+\frac {A\,b\,c\,e^4}{4}+\frac {3\,B\,c^2\,d^2\,e^2}{4}+\frac {A\,c^2\,d\,e^3}{2}+\frac {B\,a\,c\,e^4}{4}\right )+x^7\,\left (\frac {4\,B\,b^2\,d\,e^3}{7}+\frac {A\,b^2\,e^4}{7}+\frac {12\,B\,b\,c\,d^2\,e^2}{7}+\frac {8\,A\,b\,c\,d\,e^3}{7}+\frac {2\,B\,a\,b\,e^4}{7}+\frac {4\,B\,c^2\,d^3\,e}{7}+\frac {6\,A\,c^2\,d^2\,e^2}{7}+\frac {8\,B\,a\,c\,d\,e^3}{7}+\frac {2\,A\,a\,c\,e^4}{7}\right )+x^5\,\left (\frac {4\,B\,a^2\,d\,e^3}{5}+\frac {A\,a^2\,e^4}{5}+\frac {12\,B\,a\,b\,d^2\,e^2}{5}+\frac {8\,A\,a\,b\,d\,e^3}{5}+\frac {8\,B\,a\,c\,d^3\,e}{5}+\frac {12\,A\,a\,c\,d^2\,e^2}{5}+\frac {4\,B\,b^2\,d^3\,e}{5}+\frac {6\,A\,b^2\,d^2\,e^2}{5}+\frac {2\,B\,b\,c\,d^4}{5}+\frac {8\,A\,b\,c\,d^3\,e}{5}+\frac {A\,c^2\,d^4}{5}\right )+x^6\,\left (\frac {B\,a^2\,e^4}{6}+\frac {4\,B\,a\,b\,d\,e^3}{3}+\frac {A\,a\,b\,e^4}{3}+2\,B\,a\,c\,d^2\,e^2+\frac {4\,A\,a\,c\,d\,e^3}{3}+B\,b^2\,d^2\,e^2+\frac {2\,A\,b^2\,d\,e^3}{3}+\frac {4\,B\,b\,c\,d^3\,e}{3}+2\,A\,b\,c\,d^2\,e^2+\frac {B\,c^2\,d^4}{6}+\frac {2\,A\,c^2\,d^3\,e}{3}\right )+A\,a^2\,d^4\,x+\frac {a\,d^3\,x^2\,\left (4\,A\,a\,e+2\,A\,b\,d+B\,a\,d\right )}{2}+\frac {c\,e^3\,x^9\,\left (A\,c\,e+2\,B\,b\,e+4\,B\,c\,d\right )}{9}+\frac {B\,c^2\,e^4\,x^{10}}{10} \]
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