Integrand size = 13, antiderivative size = 38 \[ \int (a+b x)^{10} (A+B x) \, dx=\frac {(A b-a B) (a+b x)^{11}}{11 b^2}+\frac {B (a+b x)^{12}}{12 b^2} \]
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Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {45} \[ \int (a+b x)^{10} (A+B x) \, dx=\frac {(a+b x)^{11} (A b-a B)}{11 b^2}+\frac {B (a+b x)^{12}}{12 b^2} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(A b-a B) (a+b x)^{10}}{b}+\frac {B (a+b x)^{11}}{b}\right ) \, dx \\ & = \frac {(A b-a B) (a+b x)^{11}}{11 b^2}+\frac {B (a+b x)^{12}}{12 b^2} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(198\) vs. \(2(38)=76\).
Time = 0.04 (sec) , antiderivative size = 198, normalized size of antiderivative = 5.21 \[ \int (a+b x)^{10} (A+B x) \, dx=\frac {1}{132} x \left (66 a^{10} (2 A+B x)+220 a^9 b x (3 A+2 B x)+495 a^8 b^2 x^2 (4 A+3 B x)+792 a^7 b^3 x^3 (5 A+4 B x)+924 a^6 b^4 x^4 (6 A+5 B x)+792 a^5 b^5 x^5 (7 A+6 B x)+495 a^4 b^6 x^6 (8 A+7 B x)+220 a^3 b^7 x^7 (9 A+8 B x)+66 a^2 b^8 x^8 (10 A+9 B x)+12 a b^9 x^9 (11 A+10 B x)+b^{10} x^{10} (12 A+11 B x)\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(231\) vs. \(2(34)=68\).
Time = 0.67 (sec) , antiderivative size = 232, normalized size of antiderivative = 6.11
method | result | size |
norman | \(\frac {b^{10} B \,x^{12}}{12}+\left (\frac {1}{11} b^{10} A +\frac {10}{11} a \,b^{9} B \right ) x^{11}+\left (a \,b^{9} A +\frac {9}{2} a^{2} b^{8} B \right ) x^{10}+\left (5 a^{2} b^{8} A +\frac {40}{3} a^{3} b^{7} B \right ) x^{9}+\left (15 a^{3} b^{7} A +\frac {105}{4} a^{4} b^{6} B \right ) x^{8}+\left (30 a^{4} b^{6} A +36 a^{5} b^{5} B \right ) x^{7}+\left (42 a^{5} b^{5} A +35 a^{6} b^{4} B \right ) x^{6}+\left (42 a^{6} b^{4} A +24 a^{7} b^{3} B \right ) x^{5}+\left (30 a^{7} b^{3} A +\frac {45}{4} a^{8} b^{2} B \right ) x^{4}+\left (15 a^{8} b^{2} A +\frac {10}{3} a^{9} b B \right ) x^{3}+\left (5 a^{9} b A +\frac {1}{2} a^{10} B \right ) x^{2}+a^{10} A x\) | \(232\) |
default | \(\frac {b^{10} B \,x^{12}}{12}+\frac {\left (b^{10} A +10 a \,b^{9} B \right ) x^{11}}{11}+\frac {\left (10 a \,b^{9} A +45 a^{2} b^{8} B \right ) x^{10}}{10}+\frac {\left (45 a^{2} b^{8} A +120 a^{3} b^{7} B \right ) x^{9}}{9}+\frac {\left (120 a^{3} b^{7} A +210 a^{4} b^{6} B \right ) x^{8}}{8}+\frac {\left (210 a^{4} b^{6} A +252 a^{5} b^{5} B \right ) x^{7}}{7}+\frac {\left (252 a^{5} b^{5} A +210 a^{6} b^{4} B \right ) x^{6}}{6}+\frac {\left (210 a^{6} b^{4} A +120 a^{7} b^{3} B \right ) x^{5}}{5}+\frac {\left (120 a^{7} b^{3} A +45 a^{8} b^{2} B \right ) x^{4}}{4}+\frac {\left (45 a^{8} b^{2} A +10 a^{9} b B \right ) x^{3}}{3}+\frac {\left (10 a^{9} b A +a^{10} B \right ) x^{2}}{2}+a^{10} A x\) | \(241\) |
gosper | \(\frac {1}{12} b^{10} B \,x^{12}+\frac {1}{11} x^{11} b^{10} A +\frac {10}{11} x^{11} a \,b^{9} B +x^{10} a \,b^{9} A +\frac {9}{2} x^{10} a^{2} b^{8} B +5 x^{9} a^{2} b^{8} A +\frac {40}{3} x^{9} a^{3} b^{7} B +15 x^{8} a^{3} b^{7} A +\frac {105}{4} x^{8} a^{4} b^{6} B +30 A \,a^{4} b^{6} x^{7}+36 B \,a^{5} b^{5} x^{7}+42 A \,a^{5} b^{5} x^{6}+35 B \,a^{6} b^{4} x^{6}+42 A \,a^{6} b^{4} x^{5}+24 B \,a^{7} b^{3} x^{5}+30 x^{4} a^{7} b^{3} A +\frac {45}{4} x^{4} a^{8} b^{2} B +15 x^{3} a^{8} b^{2} A +\frac {10}{3} x^{3} a^{9} b B +5 x^{2} a^{9} b A +\frac {1}{2} x^{2} a^{10} B +a^{10} A x\) | \(242\) |
risch | \(\frac {1}{12} b^{10} B \,x^{12}+\frac {1}{11} x^{11} b^{10} A +\frac {10}{11} x^{11} a \,b^{9} B +x^{10} a \,b^{9} A +\frac {9}{2} x^{10} a^{2} b^{8} B +5 x^{9} a^{2} b^{8} A +\frac {40}{3} x^{9} a^{3} b^{7} B +15 x^{8} a^{3} b^{7} A +\frac {105}{4} x^{8} a^{4} b^{6} B +30 A \,a^{4} b^{6} x^{7}+36 B \,a^{5} b^{5} x^{7}+42 A \,a^{5} b^{5} x^{6}+35 B \,a^{6} b^{4} x^{6}+42 A \,a^{6} b^{4} x^{5}+24 B \,a^{7} b^{3} x^{5}+30 x^{4} a^{7} b^{3} A +\frac {45}{4} x^{4} a^{8} b^{2} B +15 x^{3} a^{8} b^{2} A +\frac {10}{3} x^{3} a^{9} b B +5 x^{2} a^{9} b A +\frac {1}{2} x^{2} a^{10} B +a^{10} A x\) | \(242\) |
parallelrisch | \(\frac {1}{12} b^{10} B \,x^{12}+\frac {1}{11} x^{11} b^{10} A +\frac {10}{11} x^{11} a \,b^{9} B +x^{10} a \,b^{9} A +\frac {9}{2} x^{10} a^{2} b^{8} B +5 x^{9} a^{2} b^{8} A +\frac {40}{3} x^{9} a^{3} b^{7} B +15 x^{8} a^{3} b^{7} A +\frac {105}{4} x^{8} a^{4} b^{6} B +30 A \,a^{4} b^{6} x^{7}+36 B \,a^{5} b^{5} x^{7}+42 A \,a^{5} b^{5} x^{6}+35 B \,a^{6} b^{4} x^{6}+42 A \,a^{6} b^{4} x^{5}+24 B \,a^{7} b^{3} x^{5}+30 x^{4} a^{7} b^{3} A +\frac {45}{4} x^{4} a^{8} b^{2} B +15 x^{3} a^{8} b^{2} A +\frac {10}{3} x^{3} a^{9} b B +5 x^{2} a^{9} b A +\frac {1}{2} x^{2} a^{10} B +a^{10} A x\) | \(242\) |
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Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (34) = 68\).
Time = 0.21 (sec) , antiderivative size = 240, normalized size of antiderivative = 6.32 \[ \int (a+b x)^{10} (A+B x) \, dx=\frac {1}{12} \, B b^{10} x^{12} + A a^{10} x + \frac {1}{11} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{11} + \frac {1}{2} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{10} + \frac {5}{3} \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{9} + \frac {15}{4} \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{8} + 6 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{7} + 7 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{6} + 6 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{5} + \frac {15}{4} \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{4} + \frac {5}{3} \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (32) = 64\).
Time = 0.05 (sec) , antiderivative size = 248, normalized size of antiderivative = 6.53 \[ \int (a+b x)^{10} (A+B x) \, dx=A a^{10} x + \frac {B b^{10} x^{12}}{12} + x^{11} \left (\frac {A b^{10}}{11} + \frac {10 B a b^{9}}{11}\right ) + x^{10} \left (A a b^{9} + \frac {9 B a^{2} b^{8}}{2}\right ) + x^{9} \cdot \left (5 A a^{2} b^{8} + \frac {40 B a^{3} b^{7}}{3}\right ) + x^{8} \cdot \left (15 A a^{3} b^{7} + \frac {105 B a^{4} b^{6}}{4}\right ) + x^{7} \cdot \left (30 A a^{4} b^{6} + 36 B a^{5} b^{5}\right ) + x^{6} \cdot \left (42 A a^{5} b^{5} + 35 B a^{6} b^{4}\right ) + x^{5} \cdot \left (42 A a^{6} b^{4} + 24 B a^{7} b^{3}\right ) + x^{4} \cdot \left (30 A a^{7} b^{3} + \frac {45 B a^{8} b^{2}}{4}\right ) + x^{3} \cdot \left (15 A a^{8} b^{2} + \frac {10 B a^{9} b}{3}\right ) + x^{2} \cdot \left (5 A a^{9} b + \frac {B a^{10}}{2}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (34) = 68\).
Time = 0.19 (sec) , antiderivative size = 240, normalized size of antiderivative = 6.32 \[ \int (a+b x)^{10} (A+B x) \, dx=\frac {1}{12} \, B b^{10} x^{12} + A a^{10} x + \frac {1}{11} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{11} + \frac {1}{2} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{10} + \frac {5}{3} \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{9} + \frac {15}{4} \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{8} + 6 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{7} + 7 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{6} + 6 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{5} + \frac {15}{4} \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{4} + \frac {5}{3} \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 241 vs. \(2 (34) = 68\).
Time = 0.28 (sec) , antiderivative size = 241, normalized size of antiderivative = 6.34 \[ \int (a+b x)^{10} (A+B x) \, dx=\frac {1}{12} \, B b^{10} x^{12} + \frac {10}{11} \, B a b^{9} x^{11} + \frac {1}{11} \, A b^{10} x^{11} + \frac {9}{2} \, B a^{2} b^{8} x^{10} + A a b^{9} x^{10} + \frac {40}{3} \, B a^{3} b^{7} x^{9} + 5 \, A a^{2} b^{8} x^{9} + \frac {105}{4} \, B a^{4} b^{6} x^{8} + 15 \, A a^{3} b^{7} x^{8} + 36 \, B a^{5} b^{5} x^{7} + 30 \, A a^{4} b^{6} x^{7} + 35 \, B a^{6} b^{4} x^{6} + 42 \, A a^{5} b^{5} x^{6} + 24 \, B a^{7} b^{3} x^{5} + 42 \, A a^{6} b^{4} x^{5} + \frac {45}{4} \, B a^{8} b^{2} x^{4} + 30 \, A a^{7} b^{3} x^{4} + \frac {10}{3} \, B a^{9} b x^{3} + 15 \, A a^{8} b^{2} x^{3} + \frac {1}{2} \, B a^{10} x^{2} + 5 \, A a^{9} b x^{2} + A a^{10} x \]
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Time = 0.14 (sec) , antiderivative size = 208, normalized size of antiderivative = 5.47 \[ \int (a+b x)^{10} (A+B x) \, dx=x^2\,\left (\frac {B\,a^{10}}{2}+5\,A\,b\,a^9\right )+x^{11}\,\left (\frac {A\,b^{10}}{11}+\frac {10\,B\,a\,b^9}{11}\right )+\frac {B\,b^{10}\,x^{12}}{12}+A\,a^{10}\,x+\frac {15\,a^7\,b^2\,x^4\,\left (8\,A\,b+3\,B\,a\right )}{4}+6\,a^6\,b^3\,x^5\,\left (7\,A\,b+4\,B\,a\right )+7\,a^5\,b^4\,x^6\,\left (6\,A\,b+5\,B\,a\right )+6\,a^4\,b^5\,x^7\,\left (5\,A\,b+6\,B\,a\right )+\frac {15\,a^3\,b^6\,x^8\,\left (4\,A\,b+7\,B\,a\right )}{4}+\frac {5\,a^2\,b^7\,x^9\,\left (3\,A\,b+8\,B\,a\right )}{3}+\frac {5\,a^8\,b\,x^3\,\left (9\,A\,b+2\,B\,a\right )}{3}+\frac {a\,b^8\,x^{10}\,\left (2\,A\,b+9\,B\,a\right )}{2} \]
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