Integrand size = 14, antiderivative size = 61 \[ \int x (a+b x)^{10} (A+B x) \, dx=-\frac {a (A b-a B) (a+b x)^{11}}{11 b^3}+\frac {(A b-2 a B) (a+b x)^{12}}{12 b^3}+\frac {B (a+b x)^{13}}{13 b^3} \]
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Time = 0.05 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {77} \[ \int x (a+b x)^{10} (A+B x) \, dx=\frac {(a+b x)^{12} (A b-2 a B)}{12 b^3}-\frac {a (a+b x)^{11} (A b-a B)}{11 b^3}+\frac {B (a+b x)^{13}}{13 b^3} \]
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Rule 77
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a (-A b+a B) (a+b x)^{10}}{b^2}+\frac {(A b-2 a B) (a+b x)^{11}}{b^2}+\frac {B (a+b x)^{12}}{b^2}\right ) \, dx \\ & = -\frac {a (A b-a B) (a+b x)^{11}}{11 b^3}+\frac {(A b-2 a B) (a+b x)^{12}}{12 b^3}+\frac {B (a+b x)^{13}}{13 b^3} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(218\) vs. \(2(61)=122\).
Time = 0.02 (sec) , antiderivative size = 218, normalized size of antiderivative = 3.57 \[ \int x (a+b x)^{10} (A+B x) \, dx=\frac {1}{6} a^{10} x^2 (3 A+2 B x)+\frac {5}{6} a^9 b x^3 (4 A+3 B x)+\frac {9}{4} a^8 b^2 x^4 (5 A+4 B x)+4 a^7 b^3 x^5 (6 A+5 B x)+5 a^6 b^4 x^6 (7 A+6 B x)+\frac {9}{2} a^5 b^5 x^7 (8 A+7 B x)+\frac {35}{12} a^4 b^6 x^8 (9 A+8 B x)+\frac {4}{3} a^3 b^7 x^9 (10 A+9 B x)+\frac {9}{22} a^2 b^8 x^{10} (11 A+10 B x)+\frac {5}{66} a b^9 x^{11} (12 A+11 B x)+\frac {1}{156} b^{10} x^{12} (13 A+12 B x) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(235\) vs. \(2(55)=110\).
Time = 0.40 (sec) , antiderivative size = 236, normalized size of antiderivative = 3.87
method | result | size |
norman | \(\frac {b^{10} B \,x^{13}}{13}+\left (\frac {1}{12} b^{10} A +\frac {5}{6} a \,b^{9} B \right ) x^{12}+\left (\frac {10}{11} a \,b^{9} A +\frac {45}{11} a^{2} b^{8} B \right ) x^{11}+\left (\frac {9}{2} a^{2} b^{8} A +12 a^{3} b^{7} B \right ) x^{10}+\left (\frac {40}{3} a^{3} b^{7} A +\frac {70}{3} a^{4} b^{6} B \right ) x^{9}+\left (\frac {105}{4} a^{4} b^{6} A +\frac {63}{2} a^{5} b^{5} B \right ) x^{8}+\left (36 a^{5} b^{5} A +30 a^{6} b^{4} B \right ) x^{7}+\left (35 a^{6} b^{4} A +20 a^{7} b^{3} B \right ) x^{6}+\left (24 a^{7} b^{3} A +9 a^{8} b^{2} B \right ) x^{5}+\left (\frac {45}{4} a^{8} b^{2} A +\frac {5}{2} a^{9} b B \right ) x^{4}+\left (\frac {10}{3} a^{9} b A +\frac {1}{3} a^{10} B \right ) x^{3}+\frac {a^{10} A \,x^{2}}{2}\) | \(236\) |
default | \(\frac {b^{10} B \,x^{13}}{13}+\frac {\left (b^{10} A +10 a \,b^{9} B \right ) x^{12}}{12}+\frac {\left (10 a \,b^{9} A +45 a^{2} b^{8} B \right ) x^{11}}{11}+\frac {\left (45 a^{2} b^{8} A +120 a^{3} b^{7} B \right ) x^{10}}{10}+\frac {\left (120 a^{3} b^{7} A +210 a^{4} b^{6} B \right ) x^{9}}{9}+\frac {\left (210 a^{4} b^{6} A +252 a^{5} b^{5} B \right ) x^{8}}{8}+\frac {\left (252 a^{5} b^{5} A +210 a^{6} b^{4} B \right ) x^{7}}{7}+\frac {\left (210 a^{6} b^{4} A +120 a^{7} b^{3} B \right ) x^{6}}{6}+\frac {\left (120 a^{7} b^{3} A +45 a^{8} b^{2} B \right ) x^{5}}{5}+\frac {\left (45 a^{8} b^{2} A +10 a^{9} b B \right ) x^{4}}{4}+\frac {\left (10 a^{9} b A +a^{10} B \right ) x^{3}}{3}+\frac {a^{10} A \,x^{2}}{2}\) | \(244\) |
gosper | \(\frac {1}{13} b^{10} B \,x^{13}+\frac {1}{12} x^{12} b^{10} A +\frac {5}{6} x^{12} a \,b^{9} B +\frac {10}{11} x^{11} a \,b^{9} A +\frac {45}{11} x^{11} a^{2} b^{8} B +\frac {9}{2} x^{10} a^{2} b^{8} A +12 x^{10} a^{3} b^{7} B +\frac {40}{3} x^{9} a^{3} b^{7} A +\frac {70}{3} x^{9} a^{4} b^{6} B +\frac {105}{4} x^{8} a^{4} b^{6} A +\frac {63}{2} x^{8} a^{5} b^{5} B +36 A \,a^{5} b^{5} x^{7}+30 B \,a^{6} b^{4} x^{7}+35 A \,a^{6} b^{4} x^{6}+20 B \,a^{7} b^{3} x^{6}+24 A \,a^{7} b^{3} x^{5}+9 B \,a^{8} b^{2} x^{5}+\frac {45}{4} x^{4} a^{8} b^{2} A +\frac {5}{2} x^{4} a^{9} b B +\frac {10}{3} x^{3} a^{9} b A +\frac {1}{3} x^{3} a^{10} B +\frac {1}{2} a^{10} A \,x^{2}\) | \(246\) |
risch | \(\frac {1}{13} b^{10} B \,x^{13}+\frac {1}{12} x^{12} b^{10} A +\frac {5}{6} x^{12} a \,b^{9} B +\frac {10}{11} x^{11} a \,b^{9} A +\frac {45}{11} x^{11} a^{2} b^{8} B +\frac {9}{2} x^{10} a^{2} b^{8} A +12 x^{10} a^{3} b^{7} B +\frac {40}{3} x^{9} a^{3} b^{7} A +\frac {70}{3} x^{9} a^{4} b^{6} B +\frac {105}{4} x^{8} a^{4} b^{6} A +\frac {63}{2} x^{8} a^{5} b^{5} B +36 A \,a^{5} b^{5} x^{7}+30 B \,a^{6} b^{4} x^{7}+35 A \,a^{6} b^{4} x^{6}+20 B \,a^{7} b^{3} x^{6}+24 A \,a^{7} b^{3} x^{5}+9 B \,a^{8} b^{2} x^{5}+\frac {45}{4} x^{4} a^{8} b^{2} A +\frac {5}{2} x^{4} a^{9} b B +\frac {10}{3} x^{3} a^{9} b A +\frac {1}{3} x^{3} a^{10} B +\frac {1}{2} a^{10} A \,x^{2}\) | \(246\) |
parallelrisch | \(\frac {1}{13} b^{10} B \,x^{13}+\frac {1}{12} x^{12} b^{10} A +\frac {5}{6} x^{12} a \,b^{9} B +\frac {10}{11} x^{11} a \,b^{9} A +\frac {45}{11} x^{11} a^{2} b^{8} B +\frac {9}{2} x^{10} a^{2} b^{8} A +12 x^{10} a^{3} b^{7} B +\frac {40}{3} x^{9} a^{3} b^{7} A +\frac {70}{3} x^{9} a^{4} b^{6} B +\frac {105}{4} x^{8} a^{4} b^{6} A +\frac {63}{2} x^{8} a^{5} b^{5} B +36 A \,a^{5} b^{5} x^{7}+30 B \,a^{6} b^{4} x^{7}+35 A \,a^{6} b^{4} x^{6}+20 B \,a^{7} b^{3} x^{6}+24 A \,a^{7} b^{3} x^{5}+9 B \,a^{8} b^{2} x^{5}+\frac {45}{4} x^{4} a^{8} b^{2} A +\frac {5}{2} x^{4} a^{9} b B +\frac {10}{3} x^{3} a^{9} b A +\frac {1}{3} x^{3} a^{10} B +\frac {1}{2} a^{10} A \,x^{2}\) | \(246\) |
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Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (56) = 112\).
Time = 0.22 (sec) , antiderivative size = 243, normalized size of antiderivative = 3.98 \[ \int x (a+b x)^{10} (A+B x) \, dx=\frac {1}{13} \, B b^{10} x^{13} + \frac {1}{2} \, A a^{10} x^{2} + \frac {1}{12} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{12} + \frac {5}{11} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{11} + \frac {3}{2} \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{10} + \frac {10}{3} \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{9} + \frac {21}{4} \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{8} + 6 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{7} + 5 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{6} + 3 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{5} + \frac {5}{4} \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{4} + \frac {1}{3} \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x^{3} \]
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Leaf count of result is larger than twice the leaf count of optimal. 262 vs. \(2 (53) = 106\).
Time = 0.04 (sec) , antiderivative size = 262, normalized size of antiderivative = 4.30 \[ \int x (a+b x)^{10} (A+B x) \, dx=\frac {A a^{10} x^{2}}{2} + \frac {B b^{10} x^{13}}{13} + x^{12} \left (\frac {A b^{10}}{12} + \frac {5 B a b^{9}}{6}\right ) + x^{11} \cdot \left (\frac {10 A a b^{9}}{11} + \frac {45 B a^{2} b^{8}}{11}\right ) + x^{10} \cdot \left (\frac {9 A a^{2} b^{8}}{2} + 12 B a^{3} b^{7}\right ) + x^{9} \cdot \left (\frac {40 A a^{3} b^{7}}{3} + \frac {70 B a^{4} b^{6}}{3}\right ) + x^{8} \cdot \left (\frac {105 A a^{4} b^{6}}{4} + \frac {63 B a^{5} b^{5}}{2}\right ) + x^{7} \cdot \left (36 A a^{5} b^{5} + 30 B a^{6} b^{4}\right ) + x^{6} \cdot \left (35 A a^{6} b^{4} + 20 B a^{7} b^{3}\right ) + x^{5} \cdot \left (24 A a^{7} b^{3} + 9 B a^{8} b^{2}\right ) + x^{4} \cdot \left (\frac {45 A a^{8} b^{2}}{4} + \frac {5 B a^{9} b}{2}\right ) + x^{3} \cdot \left (\frac {10 A a^{9} b}{3} + \frac {B a^{10}}{3}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (56) = 112\).
Time = 0.20 (sec) , antiderivative size = 243, normalized size of antiderivative = 3.98 \[ \int x (a+b x)^{10} (A+B x) \, dx=\frac {1}{13} \, B b^{10} x^{13} + \frac {1}{2} \, A a^{10} x^{2} + \frac {1}{12} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{12} + \frac {5}{11} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{11} + \frac {3}{2} \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{10} + \frac {10}{3} \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{9} + \frac {21}{4} \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{8} + 6 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{7} + 5 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{6} + 3 \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{5} + \frac {5}{4} \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{4} + \frac {1}{3} \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x^{3} \]
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Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (56) = 112\).
Time = 0.27 (sec) , antiderivative size = 245, normalized size of antiderivative = 4.02 \[ \int x (a+b x)^{10} (A+B x) \, dx=\frac {1}{13} \, B b^{10} x^{13} + \frac {5}{6} \, B a b^{9} x^{12} + \frac {1}{12} \, A b^{10} x^{12} + \frac {45}{11} \, B a^{2} b^{8} x^{11} + \frac {10}{11} \, A a b^{9} x^{11} + 12 \, B a^{3} b^{7} x^{10} + \frac {9}{2} \, A a^{2} b^{8} x^{10} + \frac {70}{3} \, B a^{4} b^{6} x^{9} + \frac {40}{3} \, A a^{3} b^{7} x^{9} + \frac {63}{2} \, B a^{5} b^{5} x^{8} + \frac {105}{4} \, A a^{4} b^{6} x^{8} + 30 \, B a^{6} b^{4} x^{7} + 36 \, A a^{5} b^{5} x^{7} + 20 \, B a^{7} b^{3} x^{6} + 35 \, A a^{6} b^{4} x^{6} + 9 \, B a^{8} b^{2} x^{5} + 24 \, A a^{7} b^{3} x^{5} + \frac {5}{2} \, B a^{9} b x^{4} + \frac {45}{4} \, A a^{8} b^{2} x^{4} + \frac {1}{3} \, B a^{10} x^{3} + \frac {10}{3} \, A a^{9} b x^{3} + \frac {1}{2} \, A a^{10} x^{2} \]
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Time = 0.10 (sec) , antiderivative size = 211, normalized size of antiderivative = 3.46 \[ \int x (a+b x)^{10} (A+B x) \, dx=x^3\,\left (\frac {B\,a^{10}}{3}+\frac {10\,A\,b\,a^9}{3}\right )+x^{12}\,\left (\frac {A\,b^{10}}{12}+\frac {5\,B\,a\,b^9}{6}\right )+\frac {A\,a^{10}\,x^2}{2}+\frac {B\,b^{10}\,x^{13}}{13}+3\,a^7\,b^2\,x^5\,\left (8\,A\,b+3\,B\,a\right )+5\,a^6\,b^3\,x^6\,\left (7\,A\,b+4\,B\,a\right )+6\,a^5\,b^4\,x^7\,\left (6\,A\,b+5\,B\,a\right )+\frac {21\,a^4\,b^5\,x^8\,\left (5\,A\,b+6\,B\,a\right )}{4}+\frac {10\,a^3\,b^6\,x^9\,\left (4\,A\,b+7\,B\,a\right )}{3}+\frac {3\,a^2\,b^7\,x^{10}\,\left (3\,A\,b+8\,B\,a\right )}{2}+\frac {5\,a^8\,b\,x^4\,\left (9\,A\,b+2\,B\,a\right )}{4}+\frac {5\,a\,b^8\,x^{11}\,\left (2\,A\,b+9\,B\,a\right )}{11} \]
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