Integrand size = 18, antiderivative size = 75 \[ \int (a+b x)^{10} (A+B x) (d+e x) \, dx=\frac {(A b-a B) (b d-a e) (a+b x)^{11}}{11 b^3}+\frac {(b B d+A b e-2 a B e) (a+b x)^{12}}{12 b^3}+\frac {B e (a+b x)^{13}}{13 b^3} \]
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Time = 0.27 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {78} \[ \int (a+b x)^{10} (A+B x) (d+e x) \, dx=\frac {(a+b x)^{12} (-2 a B e+A b e+b B d)}{12 b^3}+\frac {(a+b x)^{11} (A b-a B) (b d-a e)}{11 b^3}+\frac {B e (a+b x)^{13}}{13 b^3} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(A b-a B) (b d-a e) (a+b x)^{10}}{b^2}+\frac {(b B d+A b e-2 a B e) (a+b x)^{11}}{b^2}+\frac {B e (a+b x)^{12}}{b^2}\right ) \, dx \\ & = \frac {(A b-a B) (b d-a e) (a+b x)^{11}}{11 b^3}+\frac {(b B d+A b e-2 a B e) (a+b x)^{12}}{12 b^3}+\frac {B e (a+b x)^{13}}{13 b^3} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(383\) vs. \(2(75)=150\).
Time = 0.11 (sec) , antiderivative size = 383, normalized size of antiderivative = 5.11 \[ \int (a+b x)^{10} (A+B x) (d+e x) \, dx=\frac {1}{66} a b^9 x^{10} \left (66 A d+60 B d x+60 A e x+55 B e x^2\right )+\frac {1}{22} a^2 b^8 x^9 \left (110 A d+99 B d x+99 A e x+90 B e x^2\right )+\frac {1}{6} a^{10} x (3 A (2 d+e x)+B x (3 d+2 e x))+\frac {3}{4} a^8 b^2 x^3 (5 A (4 d+3 e x)+3 B x (5 d+4 e x))+\frac {5}{6} a^9 b x^2 (B x (4 d+3 e x)+A (6 d+4 e x))+2 a^7 b^3 x^4 (3 A (5 d+4 e x)+2 B x (6 d+5 e x))+a^6 b^4 x^5 (7 A (6 d+5 e x)+5 B x (7 d+6 e x))+\frac {3}{2} a^5 b^5 x^6 (4 A (7 d+6 e x)+3 B x (8 d+7 e x))+\frac {5}{12} a^4 b^6 x^7 (9 A (8 d+7 e x)+7 B x (9 d+8 e x))+\frac {1}{3} a^3 b^7 x^8 (5 A (9 d+8 e x)+4 B x (10 d+9 e x))+\frac {b^{10} x^{11} (13 A (12 d+11 e x)+11 B x (13 d+12 e x))}{1716} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(458\) vs. \(2(69)=138\).
Time = 0.68 (sec) , antiderivative size = 459, normalized size of antiderivative = 6.12
method | result | size |
norman | \(\frac {b^{10} B e \,x^{13}}{13}+\left (\frac {1}{12} A \,b^{10} e +\frac {5}{6} B a \,b^{9} e +\frac {1}{12} b^{10} B d \right ) x^{12}+\left (\frac {10}{11} A a \,b^{9} e +\frac {1}{11} A \,b^{10} d +\frac {45}{11} B \,a^{2} b^{8} e +\frac {10}{11} B a \,b^{9} d \right ) x^{11}+\left (\frac {9}{2} A \,a^{2} b^{8} e +A a \,b^{9} d +12 B \,a^{3} b^{7} e +\frac {9}{2} B \,a^{2} b^{8} d \right ) x^{10}+\left (\frac {40}{3} A \,a^{3} b^{7} e +5 A \,a^{2} b^{8} d +\frac {70}{3} B \,a^{4} b^{6} e +\frac {40}{3} B \,a^{3} b^{7} d \right ) x^{9}+\left (\frac {105}{4} A \,a^{4} b^{6} e +15 A \,a^{3} b^{7} d +\frac {63}{2} B \,a^{5} b^{5} e +\frac {105}{4} B \,a^{4} b^{6} d \right ) x^{8}+\left (36 A \,a^{5} b^{5} e +30 A \,a^{4} b^{6} d +30 B \,a^{6} b^{4} e +36 B \,a^{5} b^{5} d \right ) x^{7}+\left (35 A \,a^{6} b^{4} e +42 A \,a^{5} b^{5} d +20 B \,a^{7} b^{3} e +35 B \,a^{6} b^{4} d \right ) x^{6}+\left (24 A \,a^{7} b^{3} e +42 A \,a^{6} b^{4} d +9 B \,a^{8} b^{2} e +24 B \,a^{7} b^{3} d \right ) x^{5}+\left (\frac {45}{4} A \,a^{8} b^{2} e +30 A \,a^{7} b^{3} d +\frac {5}{2} B \,a^{9} b e +\frac {45}{4} B \,a^{8} b^{2} d \right ) x^{4}+\left (\frac {10}{3} A \,a^{9} b e +15 A \,a^{8} b^{2} d +\frac {1}{3} B \,a^{10} e +\frac {10}{3} B \,a^{9} b d \right ) x^{3}+\left (\frac {1}{2} a^{10} A e +5 A \,a^{9} b d +\frac {1}{2} B \,a^{10} d \right ) x^{2}+a^{10} A d x\) | \(459\) |
default | \(\frac {b^{10} B e \,x^{13}}{13}+\frac {\left (\left (b^{10} A +10 a \,b^{9} B \right ) e +b^{10} B d \right ) x^{12}}{12}+\frac {\left (\left (10 a \,b^{9} A +45 a^{2} b^{8} B \right ) e +\left (b^{10} A +10 a \,b^{9} B \right ) d \right ) x^{11}}{11}+\frac {\left (\left (45 a^{2} b^{8} A +120 a^{3} b^{7} B \right ) e +\left (10 a \,b^{9} A +45 a^{2} b^{8} B \right ) d \right ) x^{10}}{10}+\frac {\left (\left (120 a^{3} b^{7} A +210 a^{4} b^{6} B \right ) e +\left (45 a^{2} b^{8} A +120 a^{3} b^{7} B \right ) d \right ) x^{9}}{9}+\frac {\left (\left (210 a^{4} b^{6} A +252 a^{5} b^{5} B \right ) e +\left (120 a^{3} b^{7} A +210 a^{4} b^{6} B \right ) d \right ) x^{8}}{8}+\frac {\left (\left (252 a^{5} b^{5} A +210 a^{6} b^{4} B \right ) e +\left (210 a^{4} b^{6} A +252 a^{5} b^{5} B \right ) d \right ) x^{7}}{7}+\frac {\left (\left (210 a^{6} b^{4} A +120 a^{7} b^{3} B \right ) e +\left (252 a^{5} b^{5} A +210 a^{6} b^{4} B \right ) d \right ) x^{6}}{6}+\frac {\left (\left (120 a^{7} b^{3} A +45 a^{8} b^{2} B \right ) e +\left (210 a^{6} b^{4} A +120 a^{7} b^{3} B \right ) d \right ) x^{5}}{5}+\frac {\left (\left (45 a^{8} b^{2} A +10 a^{9} b B \right ) e +\left (120 a^{7} b^{3} A +45 a^{8} b^{2} B \right ) d \right ) x^{4}}{4}+\frac {\left (\left (10 a^{9} b A +a^{10} B \right ) e +\left (45 a^{8} b^{2} A +10 a^{9} b B \right ) d \right ) x^{3}}{3}+\frac {\left (a^{10} A e +\left (10 a^{9} b A +a^{10} B \right ) d \right ) x^{2}}{2}+a^{10} A d x\) | \(485\) |
gosper | \(\frac {45}{11} x^{11} B \,a^{2} b^{8} e +\frac {1}{13} b^{10} B e \,x^{13}+\frac {45}{4} x^{4} A \,a^{8} b^{2} e +5 x^{9} A \,a^{2} b^{8} d +\frac {70}{3} x^{9} B \,a^{4} b^{6} e +35 A \,a^{6} b^{4} e \,x^{6}+35 B \,a^{6} b^{4} d \,x^{6}+24 A \,a^{7} b^{3} e \,x^{5}+\frac {10}{3} x^{3} A \,a^{9} b e +\frac {10}{11} x^{11} B a \,b^{9} d +\frac {1}{2} x^{2} B \,a^{10} d +a^{10} A d x +\frac {10}{11} x^{11} A a \,b^{9} e +\frac {45}{4} x^{4} B \,a^{8} b^{2} d +\frac {5}{6} x^{12} B a \,b^{9} e +\frac {1}{3} x^{3} B \,a^{10} e +42 A \,a^{5} b^{5} d \,x^{6}+20 B \,a^{7} b^{3} e \,x^{6}+\frac {40}{3} x^{9} A \,a^{3} b^{7} e +15 x^{3} A \,a^{8} b^{2} d +\frac {10}{3} x^{3} B \,a^{9} b d +5 x^{2} A \,a^{9} b d +\frac {1}{2} x^{2} a^{10} A e +42 A \,a^{6} b^{4} d \,x^{5}+9 B \,a^{8} b^{2} e \,x^{5}+\frac {105}{4} x^{8} A \,a^{4} b^{6} e +15 x^{8} A \,a^{3} b^{7} d +\frac {63}{2} x^{8} B \,a^{5} b^{5} e +\frac {105}{4} x^{8} B \,a^{4} b^{6} d +\frac {9}{2} x^{10} B \,a^{2} b^{8} d +36 B \,a^{5} b^{5} d \,x^{7}+x^{10} A a \,b^{9} d +12 x^{10} B \,a^{3} b^{7} e +\frac {5}{2} x^{4} B \,a^{9} b e +30 x^{4} A \,a^{7} b^{3} d +36 A \,a^{5} b^{5} e \,x^{7}+30 A \,a^{4} b^{6} d \,x^{7}+30 B \,a^{6} b^{4} e \,x^{7}+\frac {1}{11} x^{11} A \,b^{10} d +\frac {40}{3} x^{9} B \,a^{3} b^{7} d +\frac {1}{12} x^{12} A \,b^{10} e +\frac {1}{12} x^{12} b^{10} B d +\frac {9}{2} x^{10} A \,a^{2} b^{8} e +24 B \,a^{7} b^{3} d \,x^{5}\) | \(530\) |
risch | \(\frac {45}{11} x^{11} B \,a^{2} b^{8} e +\frac {1}{13} b^{10} B e \,x^{13}+\frac {45}{4} x^{4} A \,a^{8} b^{2} e +5 x^{9} A \,a^{2} b^{8} d +\frac {70}{3} x^{9} B \,a^{4} b^{6} e +35 A \,a^{6} b^{4} e \,x^{6}+35 B \,a^{6} b^{4} d \,x^{6}+24 A \,a^{7} b^{3} e \,x^{5}+\frac {10}{3} x^{3} A \,a^{9} b e +\frac {10}{11} x^{11} B a \,b^{9} d +\frac {1}{2} x^{2} B \,a^{10} d +a^{10} A d x +\frac {10}{11} x^{11} A a \,b^{9} e +\frac {45}{4} x^{4} B \,a^{8} b^{2} d +\frac {5}{6} x^{12} B a \,b^{9} e +\frac {1}{3} x^{3} B \,a^{10} e +42 A \,a^{5} b^{5} d \,x^{6}+20 B \,a^{7} b^{3} e \,x^{6}+\frac {40}{3} x^{9} A \,a^{3} b^{7} e +15 x^{3} A \,a^{8} b^{2} d +\frac {10}{3} x^{3} B \,a^{9} b d +5 x^{2} A \,a^{9} b d +\frac {1}{2} x^{2} a^{10} A e +42 A \,a^{6} b^{4} d \,x^{5}+9 B \,a^{8} b^{2} e \,x^{5}+\frac {105}{4} x^{8} A \,a^{4} b^{6} e +15 x^{8} A \,a^{3} b^{7} d +\frac {63}{2} x^{8} B \,a^{5} b^{5} e +\frac {105}{4} x^{8} B \,a^{4} b^{6} d +\frac {9}{2} x^{10} B \,a^{2} b^{8} d +36 B \,a^{5} b^{5} d \,x^{7}+x^{10} A a \,b^{9} d +12 x^{10} B \,a^{3} b^{7} e +\frac {5}{2} x^{4} B \,a^{9} b e +30 x^{4} A \,a^{7} b^{3} d +36 A \,a^{5} b^{5} e \,x^{7}+30 A \,a^{4} b^{6} d \,x^{7}+30 B \,a^{6} b^{4} e \,x^{7}+\frac {1}{11} x^{11} A \,b^{10} d +\frac {40}{3} x^{9} B \,a^{3} b^{7} d +\frac {1}{12} x^{12} A \,b^{10} e +\frac {1}{12} x^{12} b^{10} B d +\frac {9}{2} x^{10} A \,a^{2} b^{8} e +24 B \,a^{7} b^{3} d \,x^{5}\) | \(530\) |
parallelrisch | \(\frac {45}{11} x^{11} B \,a^{2} b^{8} e +\frac {1}{13} b^{10} B e \,x^{13}+\frac {45}{4} x^{4} A \,a^{8} b^{2} e +5 x^{9} A \,a^{2} b^{8} d +\frac {70}{3} x^{9} B \,a^{4} b^{6} e +35 A \,a^{6} b^{4} e \,x^{6}+35 B \,a^{6} b^{4} d \,x^{6}+24 A \,a^{7} b^{3} e \,x^{5}+\frac {10}{3} x^{3} A \,a^{9} b e +\frac {10}{11} x^{11} B a \,b^{9} d +\frac {1}{2} x^{2} B \,a^{10} d +a^{10} A d x +\frac {10}{11} x^{11} A a \,b^{9} e +\frac {45}{4} x^{4} B \,a^{8} b^{2} d +\frac {5}{6} x^{12} B a \,b^{9} e +\frac {1}{3} x^{3} B \,a^{10} e +42 A \,a^{5} b^{5} d \,x^{6}+20 B \,a^{7} b^{3} e \,x^{6}+\frac {40}{3} x^{9} A \,a^{3} b^{7} e +15 x^{3} A \,a^{8} b^{2} d +\frac {10}{3} x^{3} B \,a^{9} b d +5 x^{2} A \,a^{9} b d +\frac {1}{2} x^{2} a^{10} A e +42 A \,a^{6} b^{4} d \,x^{5}+9 B \,a^{8} b^{2} e \,x^{5}+\frac {105}{4} x^{8} A \,a^{4} b^{6} e +15 x^{8} A \,a^{3} b^{7} d +\frac {63}{2} x^{8} B \,a^{5} b^{5} e +\frac {105}{4} x^{8} B \,a^{4} b^{6} d +\frac {9}{2} x^{10} B \,a^{2} b^{8} d +36 B \,a^{5} b^{5} d \,x^{7}+x^{10} A a \,b^{9} d +12 x^{10} B \,a^{3} b^{7} e +\frac {5}{2} x^{4} B \,a^{9} b e +30 x^{4} A \,a^{7} b^{3} d +36 A \,a^{5} b^{5} e \,x^{7}+30 A \,a^{4} b^{6} d \,x^{7}+30 B \,a^{6} b^{4} e \,x^{7}+\frac {1}{11} x^{11} A \,b^{10} d +\frac {40}{3} x^{9} B \,a^{3} b^{7} d +\frac {1}{12} x^{12} A \,b^{10} e +\frac {1}{12} x^{12} b^{10} B d +\frac {9}{2} x^{10} A \,a^{2} b^{8} e +24 B \,a^{7} b^{3} d \,x^{5}\) | \(530\) |
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Leaf count of result is larger than twice the leaf count of optimal. 493 vs. \(2 (69) = 138\).
Time = 0.22 (sec) , antiderivative size = 493, normalized size of antiderivative = 6.57 \[ \int (a+b x)^{10} (A+B x) (d+e x) \, dx=\frac {1}{13} \, B b^{10} e x^{13} + A a^{10} d x + \frac {1}{12} \, {\left (B b^{10} d + {\left (10 \, B a b^{9} + A b^{10}\right )} e\right )} x^{12} + \frac {1}{11} \, {\left ({\left (10 \, B a b^{9} + A b^{10}\right )} d + 5 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} e\right )} x^{11} + \frac {1}{2} \, {\left ({\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} d + 3 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} e\right )} x^{10} + \frac {5}{3} \, {\left ({\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} d + 2 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} e\right )} x^{9} + \frac {3}{4} \, {\left (5 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} d + 7 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} e\right )} x^{8} + 6 \, {\left ({\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} d + {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} e\right )} x^{7} + {\left (7 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} d + 5 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} e\right )} x^{6} + 3 \, {\left (2 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} d + {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} e\right )} x^{5} + \frac {5}{4} \, {\left (3 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} d + {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} e\right )} x^{4} + \frac {1}{3} \, {\left (5 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} d + {\left (B a^{10} + 10 \, A a^{9} b\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (A a^{10} e + {\left (B a^{10} + 10 \, A a^{9} b\right )} d\right )} x^{2} \]
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[Out]
Leaf count of result is larger than twice the leaf count of optimal. 549 vs. \(2 (71) = 142\).
Time = 0.06 (sec) , antiderivative size = 549, normalized size of antiderivative = 7.32 \[ \int (a+b x)^{10} (A+B x) (d+e x) \, dx=A a^{10} d x + \frac {B b^{10} e x^{13}}{13} + x^{12} \left (\frac {A b^{10} e}{12} + \frac {5 B a b^{9} e}{6} + \frac {B b^{10} d}{12}\right ) + x^{11} \cdot \left (\frac {10 A a b^{9} e}{11} + \frac {A b^{10} d}{11} + \frac {45 B a^{2} b^{8} e}{11} + \frac {10 B a b^{9} d}{11}\right ) + x^{10} \cdot \left (\frac {9 A a^{2} b^{8} e}{2} + A a b^{9} d + 12 B a^{3} b^{7} e + \frac {9 B a^{2} b^{8} d}{2}\right ) + x^{9} \cdot \left (\frac {40 A a^{3} b^{7} e}{3} + 5 A a^{2} b^{8} d + \frac {70 B a^{4} b^{6} e}{3} + \frac {40 B a^{3} b^{7} d}{3}\right ) + x^{8} \cdot \left (\frac {105 A a^{4} b^{6} e}{4} + 15 A a^{3} b^{7} d + \frac {63 B a^{5} b^{5} e}{2} + \frac {105 B a^{4} b^{6} d}{4}\right ) + x^{7} \cdot \left (36 A a^{5} b^{5} e + 30 A a^{4} b^{6} d + 30 B a^{6} b^{4} e + 36 B a^{5} b^{5} d\right ) + x^{6} \cdot \left (35 A a^{6} b^{4} e + 42 A a^{5} b^{5} d + 20 B a^{7} b^{3} e + 35 B a^{6} b^{4} d\right ) + x^{5} \cdot \left (24 A a^{7} b^{3} e + 42 A a^{6} b^{4} d + 9 B a^{8} b^{2} e + 24 B a^{7} b^{3} d\right ) + x^{4} \cdot \left (\frac {45 A a^{8} b^{2} e}{4} + 30 A a^{7} b^{3} d + \frac {5 B a^{9} b e}{2} + \frac {45 B a^{8} b^{2} d}{4}\right ) + x^{3} \cdot \left (\frac {10 A a^{9} b e}{3} + 15 A a^{8} b^{2} d + \frac {B a^{10} e}{3} + \frac {10 B a^{9} b d}{3}\right ) + x^{2} \left (\frac {A a^{10} e}{2} + 5 A a^{9} b d + \frac {B a^{10} d}{2}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 493 vs. \(2 (69) = 138\).
Time = 0.21 (sec) , antiderivative size = 493, normalized size of antiderivative = 6.57 \[ \int (a+b x)^{10} (A+B x) (d+e x) \, dx=\frac {1}{13} \, B b^{10} e x^{13} + A a^{10} d x + \frac {1}{12} \, {\left (B b^{10} d + {\left (10 \, B a b^{9} + A b^{10}\right )} e\right )} x^{12} + \frac {1}{11} \, {\left ({\left (10 \, B a b^{9} + A b^{10}\right )} d + 5 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} e\right )} x^{11} + \frac {1}{2} \, {\left ({\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} d + 3 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} e\right )} x^{10} + \frac {5}{3} \, {\left ({\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} d + 2 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} e\right )} x^{9} + \frac {3}{4} \, {\left (5 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} d + 7 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} e\right )} x^{8} + 6 \, {\left ({\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} d + {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} e\right )} x^{7} + {\left (7 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} d + 5 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} e\right )} x^{6} + 3 \, {\left (2 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} d + {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} e\right )} x^{5} + \frac {5}{4} \, {\left (3 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} d + {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} e\right )} x^{4} + \frac {1}{3} \, {\left (5 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} d + {\left (B a^{10} + 10 \, A a^{9} b\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (A a^{10} e + {\left (B a^{10} + 10 \, A a^{9} b\right )} d\right )} x^{2} \]
[In]
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Leaf count of result is larger than twice the leaf count of optimal. 529 vs. \(2 (69) = 138\).
Time = 0.30 (sec) , antiderivative size = 529, normalized size of antiderivative = 7.05 \[ \int (a+b x)^{10} (A+B x) (d+e x) \, dx=\frac {1}{13} \, B b^{10} e x^{13} + \frac {1}{12} \, B b^{10} d x^{12} + \frac {5}{6} \, B a b^{9} e x^{12} + \frac {1}{12} \, A b^{10} e x^{12} + \frac {10}{11} \, B a b^{9} d x^{11} + \frac {1}{11} \, A b^{10} d x^{11} + \frac {45}{11} \, B a^{2} b^{8} e x^{11} + \frac {10}{11} \, A a b^{9} e x^{11} + \frac {9}{2} \, B a^{2} b^{8} d x^{10} + A a b^{9} d x^{10} + 12 \, B a^{3} b^{7} e x^{10} + \frac {9}{2} \, A a^{2} b^{8} e x^{10} + \frac {40}{3} \, B a^{3} b^{7} d x^{9} + 5 \, A a^{2} b^{8} d x^{9} + \frac {70}{3} \, B a^{4} b^{6} e x^{9} + \frac {40}{3} \, A a^{3} b^{7} e x^{9} + \frac {105}{4} \, B a^{4} b^{6} d x^{8} + 15 \, A a^{3} b^{7} d x^{8} + \frac {63}{2} \, B a^{5} b^{5} e x^{8} + \frac {105}{4} \, A a^{4} b^{6} e x^{8} + 36 \, B a^{5} b^{5} d x^{7} + 30 \, A a^{4} b^{6} d x^{7} + 30 \, B a^{6} b^{4} e x^{7} + 36 \, A a^{5} b^{5} e x^{7} + 35 \, B a^{6} b^{4} d x^{6} + 42 \, A a^{5} b^{5} d x^{6} + 20 \, B a^{7} b^{3} e x^{6} + 35 \, A a^{6} b^{4} e x^{6} + 24 \, B a^{7} b^{3} d x^{5} + 42 \, A a^{6} b^{4} d x^{5} + 9 \, B a^{8} b^{2} e x^{5} + 24 \, A a^{7} b^{3} e x^{5} + \frac {45}{4} \, B a^{8} b^{2} d x^{4} + 30 \, A a^{7} b^{3} d x^{4} + \frac {5}{2} \, B a^{9} b e x^{4} + \frac {45}{4} \, A a^{8} b^{2} e x^{4} + \frac {10}{3} \, B a^{9} b d x^{3} + 15 \, A a^{8} b^{2} d x^{3} + \frac {1}{3} \, B a^{10} e x^{3} + \frac {10}{3} \, A a^{9} b e x^{3} + \frac {1}{2} \, B a^{10} d x^{2} + 5 \, A a^{9} b d x^{2} + \frac {1}{2} \, A a^{10} e x^{2} + A a^{10} d x \]
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Time = 1.42 (sec) , antiderivative size = 409, normalized size of antiderivative = 5.45 \[ \int (a+b x)^{10} (A+B x) (d+e x) \, dx=x^3\,\left (\frac {B\,a^{10}\,e}{3}+\frac {10\,A\,a^9\,b\,e}{3}+\frac {10\,B\,a^9\,b\,d}{3}+15\,A\,a^8\,b^2\,d\right )+x^{11}\,\left (\frac {A\,b^{10}\,d}{11}+\frac {10\,A\,a\,b^9\,e}{11}+\frac {10\,B\,a\,b^9\,d}{11}+\frac {45\,B\,a^2\,b^8\,e}{11}\right )+x^2\,\left (\frac {A\,a^{10}\,e}{2}+\frac {B\,a^{10}\,d}{2}+5\,A\,a^9\,b\,d\right )+x^{12}\,\left (\frac {A\,b^{10}\,e}{12}+\frac {B\,b^{10}\,d}{12}+\frac {5\,B\,a\,b^9\,e}{6}\right )+6\,a^4\,b^4\,x^7\,\left (5\,A\,b^2\,d+5\,B\,a^2\,e+6\,A\,a\,b\,e+6\,B\,a\,b\,d\right )+3\,a^6\,b^2\,x^5\,\left (14\,A\,b^2\,d+3\,B\,a^2\,e+8\,A\,a\,b\,e+8\,B\,a\,b\,d\right )+\frac {5\,a^2\,b^6\,x^9\,\left (3\,A\,b^2\,d+14\,B\,a^2\,e+8\,A\,a\,b\,e+8\,B\,a\,b\,d\right )}{3}+a^5\,b^3\,x^6\,\left (42\,A\,b^2\,d+20\,B\,a^2\,e+35\,A\,a\,b\,e+35\,B\,a\,b\,d\right )+\frac {3\,a^3\,b^5\,x^8\,\left (20\,A\,b^2\,d+42\,B\,a^2\,e+35\,A\,a\,b\,e+35\,B\,a\,b\,d\right )}{4}+A\,a^{10}\,d\,x+\frac {B\,b^{10}\,e\,x^{13}}{13}+\frac {5\,a^7\,b\,x^4\,\left (24\,A\,b^2\,d+2\,B\,a^2\,e+9\,A\,a\,b\,e+9\,B\,a\,b\,d\right )}{4}+\frac {a\,b^7\,x^{10}\,\left (2\,A\,b^2\,d+24\,B\,a^2\,e+9\,A\,a\,b\,e+9\,B\,a\,b\,d\right )}{2} \]
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