Optimal. Leaf size=75 \[ \frac {(a+b x)^{12} (-2 a B e+A b e+b B d)}{12 b^3}+\frac {(a+b x)^{11} (A b-a B) (b d-a e)}{11 b^3}+\frac {B e (a+b x)^{13}}{13 b^3} \]
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Rubi [A] time = 0.41, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {77} \[ \frac {(a+b x)^{12} (-2 a B e+A b e+b B d)}{12 b^3}+\frac {(a+b x)^{11} (A b-a B) (b d-a e)}{11 b^3}+\frac {B e (a+b x)^{13}}{13 b^3} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int (a+b x)^{10} (A+B x) (d+e x) \, dx &=\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*}
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Mathematica [B] time = 0.21, size = 383, normalized size = 5.11 \[ \frac {1}{6} a^{10} x (3 A (2 d+e x)+B x (3 d+2 e x))+\frac {5}{6} a^9 b x^2 (A (6 d+4 e x)+B x (4 d+3 e x))+\frac {3}{4} a^8 b^2 x^3 (5 A (4 d+3 e x)+3 B x (5 d+4 e x))+2 a^7 b^3 x^4 (3 A (5 d+4 e x)+2 B x (6 d+5 e x))+a^6 b^4 x^5 (7 A (6 d+5 e x)+5 B x (7 d+6 e x))+\frac {3}{2} a^5 b^5 x^6 (4 A (7 d+6 e x)+3 B x (8 d+7 e x))+\frac {5}{12} a^4 b^6 x^7 (9 A (8 d+7 e x)+7 B x (9 d+8 e x))+\frac {1}{3} a^3 b^7 x^8 (5 A (9 d+8 e x)+4 B x (10 d+9 e x))+\frac {1}{22} a^2 b^8 x^9 \left (110 A d+99 A e x+99 B d x+90 B e x^2\right )+\frac {1}{66} a b^9 x^{10} \left (66 A d+60 A e x+60 B d x+55 B e x^2\right )+\frac {b^{10} x^{11} (13 A (12 d+11 e x)+11 B x (13 d+12 e x))}{1716} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.86, size = 529, normalized size = 7.05 \[ \frac {1}{13} x^{13} e b^{10} B + \frac {1}{12} x^{12} d b^{10} B + \frac {5}{6} x^{12} e b^{9} a B + \frac {1}{12} x^{12} e b^{10} A + \frac {10}{11} x^{11} d b^{9} a B + \frac {45}{11} x^{11} e b^{8} a^{2} B + \frac {1}{11} x^{11} d b^{10} A + \frac {10}{11} x^{11} e b^{9} a A + \frac {9}{2} x^{10} d b^{8} a^{2} B + 12 x^{10} e b^{7} a^{3} B + x^{10} d b^{9} a A + \frac {9}{2} x^{10} e b^{8} a^{2} A + \frac {40}{3} x^{9} d b^{7} a^{3} B + \frac {70}{3} x^{9} e b^{6} a^{4} B + 5 x^{9} d b^{8} a^{2} A + \frac {40}{3} x^{9} e b^{7} a^{3} A + \frac {105}{4} x^{8} d b^{6} a^{4} B + \frac {63}{2} x^{8} e b^{5} a^{5} B + 15 x^{8} d b^{7} a^{3} A + \frac {105}{4} x^{8} e b^{6} a^{4} A + 36 x^{7} d b^{5} a^{5} B + 30 x^{7} e b^{4} a^{6} B + 30 x^{7} d b^{6} a^{4} A + 36 x^{7} e b^{5} a^{5} A + 35 x^{6} d b^{4} a^{6} B + 20 x^{6} e b^{3} a^{7} B + 42 x^{6} d b^{5} a^{5} A + 35 x^{6} e b^{4} a^{6} A + 24 x^{5} d b^{3} a^{7} B + 9 x^{5} e b^{2} a^{8} B + 42 x^{5} d b^{4} a^{6} A + 24 x^{5} e b^{3} a^{7} A + \frac {45}{4} x^{4} d b^{2} a^{8} B + \frac {5}{2} x^{4} e b a^{9} B + 30 x^{4} d b^{3} a^{7} A + \frac {45}{4} x^{4} e b^{2} a^{8} A + \frac {10}{3} x^{3} d b a^{9} B + \frac {1}{3} x^{3} e a^{10} B + 15 x^{3} d b^{2} a^{8} A + \frac {10}{3} x^{3} e b a^{9} A + \frac {1}{2} x^{2} d a^{10} B + 5 x^{2} d b a^{9} A + \frac {1}{2} x^{2} e a^{10} A + x d a^{10} A \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.20, size = 551, normalized size = 7.35 \[ \frac {1}{13} \, B b^{10} x^{13} e + \frac {1}{12} \, B b^{10} d x^{12} + \frac {5}{6} \, B a b^{9} x^{12} e + \frac {1}{12} \, A b^{10} x^{12} e + \frac {10}{11} \, B a b^{9} d x^{11} + \frac {1}{11} \, A b^{10} d x^{11} + \frac {45}{11} \, B a^{2} b^{8} x^{11} e + \frac {10}{11} \, A a b^{9} x^{11} e + \frac {9}{2} \, B a^{2} b^{8} d x^{10} + A a b^{9} d x^{10} + 12 \, B a^{3} b^{7} x^{10} e + \frac {9}{2} \, A a^{2} b^{8} x^{10} e + \frac {40}{3} \, B a^{3} b^{7} d x^{9} + 5 \, A a^{2} b^{8} d x^{9} + \frac {70}{3} \, B a^{4} b^{6} x^{9} e + \frac {40}{3} \, A a^{3} b^{7} x^{9} e + \frac {105}{4} \, B a^{4} b^{6} d x^{8} + 15 \, A a^{3} b^{7} d x^{8} + \frac {63}{2} \, B a^{5} b^{5} x^{8} e + \frac {105}{4} \, A a^{4} b^{6} x^{8} e + 36 \, B a^{5} b^{5} d x^{7} + 30 \, A a^{4} b^{6} d x^{7} + 30 \, B a^{6} b^{4} x^{7} e + 36 \, A a^{5} b^{5} x^{7} e + 35 \, B a^{6} b^{4} d x^{6} + 42 \, A a^{5} b^{5} d x^{6} + 20 \, B a^{7} b^{3} x^{6} e + 35 \, A a^{6} b^{4} x^{6} e + 24 \, B a^{7} b^{3} d x^{5} + 42 \, A a^{6} b^{4} d x^{5} + 9 \, B a^{8} b^{2} x^{5} e + 24 \, A a^{7} b^{3} x^{5} e + \frac {45}{4} \, B a^{8} b^{2} d x^{4} + 30 \, A a^{7} b^{3} d x^{4} + \frac {5}{2} \, B a^{9} b x^{4} e + \frac {45}{4} \, A a^{8} b^{2} x^{4} e + \frac {10}{3} \, B a^{9} b d x^{3} + 15 \, A a^{8} b^{2} d x^{3} + \frac {1}{3} \, B a^{10} x^{3} e + \frac {10}{3} \, A a^{9} b x^{3} e + \frac {1}{2} \, B a^{10} d x^{2} + 5 \, A a^{9} b d x^{2} + \frac {1}{2} \, A a^{10} x^{2} e + A a^{10} d x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 485, normalized size = 6.47 \[ \frac {B \,b^{10} e \,x^{13}}{13}+A \,a^{10} d x +\frac {\left (B \,b^{10} d +\left (b^{10} A +10 a \,b^{9} B \right ) e \right ) x^{12}}{12}+\frac {\left (\left (b^{10} A +10 a \,b^{9} B \right ) d +\left (10 a \,b^{9} A +45 a^{2} b^{8} B \right ) e \right ) x^{11}}{11}+\frac {\left (\left (10 a \,b^{9} A +45 a^{2} b^{8} B \right ) d +\left (45 a^{2} b^{8} A +120 a^{3} b^{7} B \right ) e \right ) x^{10}}{10}+\frac {\left (\left (45 a^{2} b^{8} A +120 a^{3} b^{7} B \right ) d +\left (120 a^{3} b^{7} A +210 a^{4} b^{6} B \right ) e \right ) x^{9}}{9}+\frac {\left (\left (120 a^{3} b^{7} A +210 a^{4} b^{6} B \right ) d +\left (210 a^{4} b^{6} A +252 a^{5} b^{5} B \right ) e \right ) x^{8}}{8}+\frac {\left (\left (210 a^{4} b^{6} A +252 a^{5} b^{5} B \right ) d +\left (252 a^{5} b^{5} A +210 a^{6} b^{4} B \right ) e \right ) x^{7}}{7}+\frac {\left (\left (252 a^{5} b^{5} A +210 a^{6} b^{4} B \right ) d +\left (210 a^{6} b^{4} A +120 a^{7} b^{3} B \right ) e \right ) x^{6}}{6}+\frac {\left (\left (210 a^{6} b^{4} A +120 a^{7} b^{3} B \right ) d +\left (120 a^{7} b^{3} A +45 a^{8} b^{2} B \right ) e \right ) x^{5}}{5}+\frac {\left (\left (120 a^{7} b^{3} A +45 a^{8} b^{2} B \right ) d +\left (45 a^{8} b^{2} A +10 a^{9} b B \right ) e \right ) x^{4}}{4}+\frac {\left (\left (45 a^{8} b^{2} A +10 a^{9} b B \right ) d +\left (10 a^{9} b A +a^{10} B \right ) e \right ) x^{3}}{3}+\frac {\left (A \,a^{10} e +\left (10 a^{9} b A +a^{10} B \right ) d \right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.63, size = 493, normalized size = 6.57 \[ \frac {1}{13} \, B b^{10} e x^{13} + A a^{10} d x + \frac {1}{12} \, {\left (B b^{10} d + {\left (10 \, B a b^{9} + A b^{10}\right )} e\right )} x^{12} + \frac {1}{11} \, {\left ({\left (10 \, B a b^{9} + A b^{10}\right )} d + 5 \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} e\right )} x^{11} + \frac {1}{2} \, {\left ({\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} d + 3 \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} e\right )} x^{10} + \frac {5}{3} \, {\left ({\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} d + 2 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} e\right )} x^{9} + \frac {3}{4} \, {\left (5 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} d + 7 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} e\right )} x^{8} + 6 \, {\left ({\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} d + {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} e\right )} x^{7} + {\left (7 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} d + 5 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} e\right )} x^{6} + 3 \, {\left (2 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} d + {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} e\right )} x^{5} + \frac {5}{4} \, {\left (3 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} d + {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} e\right )} x^{4} + \frac {1}{3} \, {\left (5 \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} d + {\left (B a^{10} + 10 \, A a^{9} b\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (A a^{10} e + {\left (B a^{10} + 10 \, A a^{9} b\right )} d\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.26, size = 409, normalized size = 5.45 \[ x^3\,\left (\frac {B\,a^{10}\,e}{3}+\frac {10\,A\,a^9\,b\,e}{3}+\frac {10\,B\,a^9\,b\,d}{3}+15\,A\,a^8\,b^2\,d\right )+x^{11}\,\left (\frac {A\,b^{10}\,d}{11}+\frac {10\,A\,a\,b^9\,e}{11}+\frac {10\,B\,a\,b^9\,d}{11}+\frac {45\,B\,a^2\,b^8\,e}{11}\right )+x^2\,\left (\frac {A\,a^{10}\,e}{2}+\frac {B\,a^{10}\,d}{2}+5\,A\,a^9\,b\,d\right )+x^{12}\,\left (\frac {A\,b^{10}\,e}{12}+\frac {B\,b^{10}\,d}{12}+\frac {5\,B\,a\,b^9\,e}{6}\right )+6\,a^4\,b^4\,x^7\,\left (5\,A\,b^2\,d+5\,B\,a^2\,e+6\,A\,a\,b\,e+6\,B\,a\,b\,d\right )+3\,a^6\,b^2\,x^5\,\left (14\,A\,b^2\,d+3\,B\,a^2\,e+8\,A\,a\,b\,e+8\,B\,a\,b\,d\right )+\frac {5\,a^2\,b^6\,x^9\,\left (3\,A\,b^2\,d+14\,B\,a^2\,e+8\,A\,a\,b\,e+8\,B\,a\,b\,d\right )}{3}+a^5\,b^3\,x^6\,\left (42\,A\,b^2\,d+20\,B\,a^2\,e+35\,A\,a\,b\,e+35\,B\,a\,b\,d\right )+\frac {3\,a^3\,b^5\,x^8\,\left (20\,A\,b^2\,d+42\,B\,a^2\,e+35\,A\,a\,b\,e+35\,B\,a\,b\,d\right )}{4}+A\,a^{10}\,d\,x+\frac {B\,b^{10}\,e\,x^{13}}{13}+\frac {5\,a^7\,b\,x^4\,\left (24\,A\,b^2\,d+2\,B\,a^2\,e+9\,A\,a\,b\,e+9\,B\,a\,b\,d\right )}{4}+\frac {a\,b^7\,x^{10}\,\left (2\,A\,b^2\,d+24\,B\,a^2\,e+9\,A\,a\,b\,e+9\,B\,a\,b\,d\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.16, size = 549, normalized size = 7.32 \[ A a^{10} d x + \frac {B b^{10} e x^{13}}{13} + x^{12} \left (\frac {A b^{10} e}{12} + \frac {5 B a b^{9} e}{6} + \frac {B b^{10} d}{12}\right ) + x^{11} \left (\frac {10 A a b^{9} e}{11} + \frac {A b^{10} d}{11} + \frac {45 B a^{2} b^{8} e}{11} + \frac {10 B a b^{9} d}{11}\right ) + x^{10} \left (\frac {9 A a^{2} b^{8} e}{2} + A a b^{9} d + 12 B a^{3} b^{7} e + \frac {9 B a^{2} b^{8} d}{2}\right ) + x^{9} \left (\frac {40 A a^{3} b^{7} e}{3} + 5 A a^{2} b^{8} d + \frac {70 B a^{4} b^{6} e}{3} + \frac {40 B a^{3} b^{7} d}{3}\right ) + x^{8} \left (\frac {105 A a^{4} b^{6} e}{4} + 15 A a^{3} b^{7} d + \frac {63 B a^{5} b^{5} e}{2} + \frac {105 B a^{4} b^{6} d}{4}\right ) + x^{7} \left (36 A a^{5} b^{5} e + 30 A a^{4} b^{6} d + 30 B a^{6} b^{4} e + 36 B a^{5} b^{5} d\right ) + x^{6} \left (35 A a^{6} b^{4} e + 42 A a^{5} b^{5} d + 20 B a^{7} b^{3} e + 35 B a^{6} b^{4} d\right ) + x^{5} \left (24 A a^{7} b^{3} e + 42 A a^{6} b^{4} d + 9 B a^{8} b^{2} e + 24 B a^{7} b^{3} d\right ) + x^{4} \left (\frac {45 A a^{8} b^{2} e}{4} + 30 A a^{7} b^{3} d + \frac {5 B a^{9} b e}{2} + \frac {45 B a^{8} b^{2} d}{4}\right ) + x^{3} \left (\frac {10 A a^{9} b e}{3} + 15 A a^{8} b^{2} d + \frac {B a^{10} e}{3} + \frac {10 B a^{9} b d}{3}\right ) + x^{2} \left (\frac {A a^{10} e}{2} + 5 A a^{9} b d + \frac {B a^{10} d}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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