Integrand size = 16, antiderivative size = 87 \[ \int x^2 (a+b x)^{10} (A+B x) \, dx=\frac {a^2 (A b-a B) (a+b x)^{11}}{11 b^4}-\frac {a (2 A b-3 a B) (a+b x)^{12}}{12 b^4}+\frac {(A b-3 a B) (a+b x)^{13}}{13 b^4}+\frac {B (a+b x)^{14}}{14 b^4} \]
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Time = 0.06 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {77} \[ \int x^2 (a+b x)^{10} (A+B x) \, dx=\frac {a^2 (a+b x)^{11} (A b-a B)}{11 b^4}+\frac {(a+b x)^{13} (A b-3 a B)}{13 b^4}-\frac {a (a+b x)^{12} (2 A b-3 a B)}{12 b^4}+\frac {B (a+b x)^{14}}{14 b^4} \]
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Rule 77
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a^2 (-A b+a B) (a+b x)^{10}}{b^3}+\frac {a (-2 A b+3 a B) (a+b x)^{11}}{b^3}+\frac {(A b-3 a B) (a+b x)^{12}}{b^3}+\frac {B (a+b x)^{13}}{b^3}\right ) \, dx \\ & = \frac {a^2 (A b-a B) (a+b x)^{11}}{11 b^4}-\frac {a (2 A b-3 a B) (a+b x)^{12}}{12 b^4}+\frac {(A b-3 a B) (a+b x)^{13}}{13 b^4}+\frac {B (a+b x)^{14}}{14 b^4} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(226\) vs. \(2(87)=174\).
Time = 0.02 (sec) , antiderivative size = 226, normalized size of antiderivative = 2.60 \[ \int x^2 (a+b x)^{10} (A+B x) \, dx=\frac {1}{3} a^{10} A x^3+\frac {1}{4} a^9 (10 A b+a B) x^4+a^8 b (9 A b+2 a B) x^5+\frac {5}{2} a^7 b^2 (8 A b+3 a B) x^6+\frac {30}{7} a^6 b^3 (7 A b+4 a B) x^7+\frac {21}{4} a^5 b^4 (6 A b+5 a B) x^8+\frac {14}{3} a^4 b^5 (5 A b+6 a B) x^9+3 a^3 b^6 (4 A b+7 a B) x^{10}+\frac {15}{11} a^2 b^7 (3 A b+8 a B) x^{11}+\frac {5}{12} a b^8 (2 A b+9 a B) x^{12}+\frac {1}{13} b^9 (A b+10 a B) x^{13}+\frac {1}{14} b^{10} B x^{14} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(235\) vs. \(2(79)=158\).
Time = 0.40 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.71
method | result | size |
norman | \(\frac {a^{10} A \,x^{3}}{3}+\left (\frac {5}{2} a^{9} b A +\frac {1}{4} a^{10} B \right ) x^{4}+\left (9 a^{8} b^{2} A +2 a^{9} b B \right ) x^{5}+\left (20 a^{7} b^{3} A +\frac {15}{2} a^{8} b^{2} B \right ) x^{6}+\left (30 a^{6} b^{4} A +\frac {120}{7} a^{7} b^{3} B \right ) x^{7}+\left (\frac {63}{2} a^{5} b^{5} A +\frac {105}{4} a^{6} b^{4} B \right ) x^{8}+\left (\frac {70}{3} a^{4} b^{6} A +28 a^{5} b^{5} B \right ) x^{9}+\left (12 a^{3} b^{7} A +21 a^{4} b^{6} B \right ) x^{10}+\left (\frac {45}{11} a^{2} b^{8} A +\frac {120}{11} a^{3} b^{7} B \right ) x^{11}+\left (\frac {5}{6} a \,b^{9} A +\frac {15}{4} a^{2} b^{8} B \right ) x^{12}+\left (\frac {1}{13} b^{10} A +\frac {10}{13} a \,b^{9} B \right ) x^{13}+\frac {b^{10} B \,x^{14}}{14}\) | \(236\) |
default | \(\frac {b^{10} B \,x^{14}}{14}+\frac {\left (b^{10} A +10 a \,b^{9} B \right ) x^{13}}{13}+\frac {\left (10 a \,b^{9} A +45 a^{2} b^{8} B \right ) x^{12}}{12}+\frac {\left (45 a^{2} b^{8} A +120 a^{3} b^{7} B \right ) x^{11}}{11}+\frac {\left (120 a^{3} b^{7} A +210 a^{4} b^{6} B \right ) x^{10}}{10}+\frac {\left (210 a^{4} b^{6} A +252 a^{5} b^{5} B \right ) x^{9}}{9}+\frac {\left (252 a^{5} b^{5} A +210 a^{6} b^{4} B \right ) x^{8}}{8}+\frac {\left (210 a^{6} b^{4} A +120 a^{7} b^{3} B \right ) x^{7}}{7}+\frac {\left (120 a^{7} b^{3} A +45 a^{8} b^{2} B \right ) x^{6}}{6}+\frac {\left (45 a^{8} b^{2} A +10 a^{9} b B \right ) x^{5}}{5}+\frac {\left (10 a^{9} b A +a^{10} B \right ) x^{4}}{4}+\frac {a^{10} A \,x^{3}}{3}\) | \(244\) |
gosper | \(\frac {1}{3} a^{10} A \,x^{3}+\frac {5}{2} x^{4} a^{9} b A +\frac {1}{4} x^{4} a^{10} B +9 A \,a^{8} b^{2} x^{5}+2 B \,a^{9} b \,x^{5}+20 x^{6} a^{7} b^{3} A +\frac {15}{2} x^{6} a^{8} b^{2} B +30 x^{7} a^{6} b^{4} A +\frac {120}{7} x^{7} a^{7} b^{3} B +\frac {63}{2} x^{8} a^{5} b^{5} A +\frac {105}{4} x^{8} a^{6} b^{4} B +\frac {70}{3} x^{9} a^{4} b^{6} A +28 x^{9} a^{5} b^{5} B +12 A \,a^{3} b^{7} x^{10}+21 B \,a^{4} b^{6} x^{10}+\frac {45}{11} x^{11} a^{2} b^{8} A +\frac {120}{11} x^{11} a^{3} b^{7} B +\frac {5}{6} x^{12} a \,b^{9} A +\frac {15}{4} x^{12} a^{2} b^{8} B +\frac {1}{13} x^{13} b^{10} A +\frac {10}{13} x^{13} a \,b^{9} B +\frac {1}{14} b^{10} B \,x^{14}\) | \(246\) |
risch | \(\frac {1}{3} a^{10} A \,x^{3}+\frac {5}{2} x^{4} a^{9} b A +\frac {1}{4} x^{4} a^{10} B +9 A \,a^{8} b^{2} x^{5}+2 B \,a^{9} b \,x^{5}+20 x^{6} a^{7} b^{3} A +\frac {15}{2} x^{6} a^{8} b^{2} B +30 x^{7} a^{6} b^{4} A +\frac {120}{7} x^{7} a^{7} b^{3} B +\frac {63}{2} x^{8} a^{5} b^{5} A +\frac {105}{4} x^{8} a^{6} b^{4} B +\frac {70}{3} x^{9} a^{4} b^{6} A +28 x^{9} a^{5} b^{5} B +12 A \,a^{3} b^{7} x^{10}+21 B \,a^{4} b^{6} x^{10}+\frac {45}{11} x^{11} a^{2} b^{8} A +\frac {120}{11} x^{11} a^{3} b^{7} B +\frac {5}{6} x^{12} a \,b^{9} A +\frac {15}{4} x^{12} a^{2} b^{8} B +\frac {1}{13} x^{13} b^{10} A +\frac {10}{13} x^{13} a \,b^{9} B +\frac {1}{14} b^{10} B \,x^{14}\) | \(246\) |
parallelrisch | \(\frac {1}{3} a^{10} A \,x^{3}+\frac {5}{2} x^{4} a^{9} b A +\frac {1}{4} x^{4} a^{10} B +9 A \,a^{8} b^{2} x^{5}+2 B \,a^{9} b \,x^{5}+20 x^{6} a^{7} b^{3} A +\frac {15}{2} x^{6} a^{8} b^{2} B +30 x^{7} a^{6} b^{4} A +\frac {120}{7} x^{7} a^{7} b^{3} B +\frac {63}{2} x^{8} a^{5} b^{5} A +\frac {105}{4} x^{8} a^{6} b^{4} B +\frac {70}{3} x^{9} a^{4} b^{6} A +28 x^{9} a^{5} b^{5} B +12 A \,a^{3} b^{7} x^{10}+21 B \,a^{4} b^{6} x^{10}+\frac {45}{11} x^{11} a^{2} b^{8} A +\frac {120}{11} x^{11} a^{3} b^{7} B +\frac {5}{6} x^{12} a \,b^{9} A +\frac {15}{4} x^{12} a^{2} b^{8} B +\frac {1}{13} x^{13} b^{10} A +\frac {10}{13} x^{13} a \,b^{9} B +\frac {1}{14} b^{10} B \,x^{14}\) | \(246\) |
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Leaf count of result is larger than twice the leaf count of optimal. 242 vs. \(2 (80) = 160\).
Time = 0.22 (sec) , antiderivative size = 242, normalized size of antiderivative = 2.78 \[ \int x^2 (a+b x)^{10} (A+B x) \, dx=\frac {1}{14} \, B b^{10} x^{14} + \frac {1}{3} \, A a^{10} x^{3} + \frac {1}{13} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{13} + \frac {5}{12} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{12} + \frac {15}{11} \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{11} + 3 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{10} + \frac {14}{3} \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{9} + \frac {21}{4} \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{8} + \frac {30}{7} \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{7} + \frac {5}{2} \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{6} + {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x^{4} \]
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Leaf count of result is larger than twice the leaf count of optimal. 262 vs. \(2 (80) = 160\).
Time = 0.04 (sec) , antiderivative size = 262, normalized size of antiderivative = 3.01 \[ \int x^2 (a+b x)^{10} (A+B x) \, dx=\frac {A a^{10} x^{3}}{3} + \frac {B b^{10} x^{14}}{14} + x^{13} \left (\frac {A b^{10}}{13} + \frac {10 B a b^{9}}{13}\right ) + x^{12} \cdot \left (\frac {5 A a b^{9}}{6} + \frac {15 B a^{2} b^{8}}{4}\right ) + x^{11} \cdot \left (\frac {45 A a^{2} b^{8}}{11} + \frac {120 B a^{3} b^{7}}{11}\right ) + x^{10} \cdot \left (12 A a^{3} b^{7} + 21 B a^{4} b^{6}\right ) + x^{9} \cdot \left (\frac {70 A a^{4} b^{6}}{3} + 28 B a^{5} b^{5}\right ) + x^{8} \cdot \left (\frac {63 A a^{5} b^{5}}{2} + \frac {105 B a^{6} b^{4}}{4}\right ) + x^{7} \cdot \left (30 A a^{6} b^{4} + \frac {120 B a^{7} b^{3}}{7}\right ) + x^{6} \cdot \left (20 A a^{7} b^{3} + \frac {15 B a^{8} b^{2}}{2}\right ) + x^{5} \cdot \left (9 A a^{8} b^{2} + 2 B a^{9} b\right ) + x^{4} \cdot \left (\frac {5 A a^{9} b}{2} + \frac {B a^{10}}{4}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 242 vs. \(2 (80) = 160\).
Time = 0.20 (sec) , antiderivative size = 242, normalized size of antiderivative = 2.78 \[ \int x^2 (a+b x)^{10} (A+B x) \, dx=\frac {1}{14} \, B b^{10} x^{14} + \frac {1}{3} \, A a^{10} x^{3} + \frac {1}{13} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{13} + \frac {5}{12} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{12} + \frac {15}{11} \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{11} + 3 \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{10} + \frac {14}{3} \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{9} + \frac {21}{4} \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{8} + \frac {30}{7} \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{7} + \frac {5}{2} \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{6} + {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x^{4} \]
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Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (80) = 160\).
Time = 0.29 (sec) , antiderivative size = 245, normalized size of antiderivative = 2.82 \[ \int x^2 (a+b x)^{10} (A+B x) \, dx=\frac {1}{14} \, B b^{10} x^{14} + \frac {10}{13} \, B a b^{9} x^{13} + \frac {1}{13} \, A b^{10} x^{13} + \frac {15}{4} \, B a^{2} b^{8} x^{12} + \frac {5}{6} \, A a b^{9} x^{12} + \frac {120}{11} \, B a^{3} b^{7} x^{11} + \frac {45}{11} \, A a^{2} b^{8} x^{11} + 21 \, B a^{4} b^{6} x^{10} + 12 \, A a^{3} b^{7} x^{10} + 28 \, B a^{5} b^{5} x^{9} + \frac {70}{3} \, A a^{4} b^{6} x^{9} + \frac {105}{4} \, B a^{6} b^{4} x^{8} + \frac {63}{2} \, A a^{5} b^{5} x^{8} + \frac {120}{7} \, B a^{7} b^{3} x^{7} + 30 \, A a^{6} b^{4} x^{7} + \frac {15}{2} \, B a^{8} b^{2} x^{6} + 20 \, A a^{7} b^{3} x^{6} + 2 \, B a^{9} b x^{5} + 9 \, A a^{8} b^{2} x^{5} + \frac {1}{4} \, B a^{10} x^{4} + \frac {5}{2} \, A a^{9} b x^{4} + \frac {1}{3} \, A a^{10} x^{3} \]
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Time = 0.52 (sec) , antiderivative size = 210, normalized size of antiderivative = 2.41 \[ \int x^2 (a+b x)^{10} (A+B x) \, dx=x^4\,\left (\frac {B\,a^{10}}{4}+\frac {5\,A\,b\,a^9}{2}\right )+x^{13}\,\left (\frac {A\,b^{10}}{13}+\frac {10\,B\,a\,b^9}{13}\right )+\frac {A\,a^{10}\,x^3}{3}+\frac {B\,b^{10}\,x^{14}}{14}+\frac {5\,a^7\,b^2\,x^6\,\left (8\,A\,b+3\,B\,a\right )}{2}+\frac {30\,a^6\,b^3\,x^7\,\left (7\,A\,b+4\,B\,a\right )}{7}+\frac {21\,a^5\,b^4\,x^8\,\left (6\,A\,b+5\,B\,a\right )}{4}+\frac {14\,a^4\,b^5\,x^9\,\left (5\,A\,b+6\,B\,a\right )}{3}+3\,a^3\,b^6\,x^{10}\,\left (4\,A\,b+7\,B\,a\right )+\frac {15\,a^2\,b^7\,x^{11}\,\left (3\,A\,b+8\,B\,a\right )}{11}+a^8\,b\,x^5\,\left (9\,A\,b+2\,B\,a\right )+\frac {5\,a\,b^8\,x^{12}\,\left (2\,A\,b+9\,B\,a\right )}{12} \]
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