Integrand size = 16, antiderivative size = 112 \[ \int x^3 (a+b x)^{10} (A+B x) \, dx=-\frac {a^3 (A b-a B) (a+b x)^{11}}{11 b^5}+\frac {a^2 (3 A b-4 a B) (a+b x)^{12}}{12 b^5}-\frac {3 a (A b-2 a B) (a+b x)^{13}}{13 b^5}+\frac {(A b-4 a B) (a+b x)^{14}}{14 b^5}+\frac {B (a+b x)^{15}}{15 b^5} \]
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Time = 0.07 (sec) , antiderivative size = 112, 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} \[ \int x^3 (a+b x)^{10} (A+B x) \, dx=-\frac {a^3 (a+b x)^{11} (A b-a B)}{11 b^5}+\frac {a^2 (a+b x)^{12} (3 A b-4 a B)}{12 b^5}+\frac {(a+b x)^{14} (A b-4 a B)}{14 b^5}-\frac {3 a (a+b x)^{13} (A b-2 a B)}{13 b^5}+\frac {B (a+b x)^{15}}{15 b^5} \]
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Rule 77
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^3 (-A b+a B) (a+b x)^{10}}{b^4}-\frac {a^2 (-3 A b+4 a B) (a+b x)^{11}}{b^4}+\frac {3 a (-A b+2 a B) (a+b x)^{12}}{b^4}+\frac {(A b-4 a B) (a+b x)^{13}}{b^4}+\frac {B (a+b x)^{14}}{b^4}\right ) \, dx \\ & = -\frac {a^3 (A b-a B) (a+b x)^{11}}{11 b^5}+\frac {a^2 (3 A b-4 a B) (a+b x)^{12}}{12 b^5}-\frac {3 a (A b-2 a B) (a+b x)^{13}}{13 b^5}+\frac {(A b-4 a B) (a+b x)^{14}}{14 b^5}+\frac {B (a+b x)^{15}}{15 b^5} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(231\) vs. \(2(112)=224\).
Time = 0.02 (sec) , antiderivative size = 231, normalized size of antiderivative = 2.06 \[ \int x^3 (a+b x)^{10} (A+B x) \, dx=\frac {1}{4} a^{10} A x^4+\frac {1}{5} a^9 (10 A b+a B) x^5+\frac {5}{6} a^8 b (9 A b+2 a B) x^6+\frac {15}{7} a^7 b^2 (8 A b+3 a B) x^7+\frac {15}{4} a^6 b^3 (7 A b+4 a B) x^8+\frac {14}{3} a^5 b^4 (6 A b+5 a B) x^9+\frac {21}{5} a^4 b^5 (5 A b+6 a B) x^{10}+\frac {30}{11} a^3 b^6 (4 A b+7 a B) x^{11}+\frac {5}{4} a^2 b^7 (3 A b+8 a B) x^{12}+\frac {5}{13} a b^8 (2 A b+9 a B) x^{13}+\frac {1}{14} b^9 (A b+10 a B) x^{14}+\frac {1}{15} b^{10} B x^{15} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(235\) vs. \(2(102)=204\).
Time = 0.40 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.11
method | result | size |
norman | \(\frac {a^{10} A \,x^{4}}{4}+\left (2 a^{9} b A +\frac {1}{5} a^{10} B \right ) x^{5}+\left (\frac {15}{2} a^{8} b^{2} A +\frac {5}{3} a^{9} b B \right ) x^{6}+\left (\frac {120}{7} a^{7} b^{3} A +\frac {45}{7} a^{8} b^{2} B \right ) x^{7}+\left (\frac {105}{4} a^{6} b^{4} A +15 a^{7} b^{3} B \right ) x^{8}+\left (28 a^{5} b^{5} A +\frac {70}{3} a^{6} b^{4} B \right ) x^{9}+\left (21 a^{4} b^{6} A +\frac {126}{5} a^{5} b^{5} B \right ) x^{10}+\left (\frac {120}{11} a^{3} b^{7} A +\frac {210}{11} a^{4} b^{6} B \right ) x^{11}+\left (\frac {15}{4} a^{2} b^{8} A +10 a^{3} b^{7} B \right ) x^{12}+\left (\frac {10}{13} a \,b^{9} A +\frac {45}{13} a^{2} b^{8} B \right ) x^{13}+\left (\frac {1}{14} b^{10} A +\frac {5}{7} a \,b^{9} B \right ) x^{14}+\frac {b^{10} B \,x^{15}}{15}\) | \(236\) |
default | \(\frac {b^{10} B \,x^{15}}{15}+\frac {\left (b^{10} A +10 a \,b^{9} B \right ) x^{14}}{14}+\frac {\left (10 a \,b^{9} A +45 a^{2} b^{8} B \right ) x^{13}}{13}+\frac {\left (45 a^{2} b^{8} A +120 a^{3} b^{7} B \right ) x^{12}}{12}+\frac {\left (120 a^{3} b^{7} A +210 a^{4} b^{6} B \right ) x^{11}}{11}+\frac {\left (210 a^{4} b^{6} A +252 a^{5} b^{5} B \right ) x^{10}}{10}+\frac {\left (252 a^{5} b^{5} A +210 a^{6} b^{4} B \right ) x^{9}}{9}+\frac {\left (210 a^{6} b^{4} A +120 a^{7} b^{3} B \right ) x^{8}}{8}+\frac {\left (120 a^{7} b^{3} A +45 a^{8} b^{2} B \right ) x^{7}}{7}+\frac {\left (45 a^{8} b^{2} A +10 a^{9} b B \right ) x^{6}}{6}+\frac {\left (10 a^{9} b A +a^{10} B \right ) x^{5}}{5}+\frac {a^{10} A \,x^{4}}{4}\) | \(244\) |
gosper | \(\frac {1}{4} a^{10} A \,x^{4}+2 x^{5} a^{9} b A +\frac {1}{5} x^{5} a^{10} B +\frac {15}{2} x^{6} a^{8} b^{2} A +\frac {5}{3} x^{6} a^{9} b B +\frac {120}{7} x^{7} a^{7} b^{3} A +\frac {45}{7} x^{7} a^{8} b^{2} B +\frac {105}{4} x^{8} a^{6} b^{4} A +15 x^{8} a^{7} b^{3} B +28 x^{9} a^{5} b^{5} A +\frac {70}{3} x^{9} a^{6} b^{4} B +21 x^{10} a^{4} b^{6} A +\frac {126}{5} x^{10} a^{5} b^{5} B +\frac {120}{11} x^{11} a^{3} b^{7} A +\frac {210}{11} x^{11} a^{4} b^{6} B +\frac {15}{4} x^{12} a^{2} b^{8} A +10 x^{12} a^{3} b^{7} B +\frac {10}{13} x^{13} a \,b^{9} A +\frac {45}{13} x^{13} a^{2} b^{8} B +\frac {1}{14} x^{14} b^{10} A +\frac {5}{7} x^{14} a \,b^{9} B +\frac {1}{15} b^{10} B \,x^{15}\) | \(246\) |
risch | \(\frac {1}{4} a^{10} A \,x^{4}+2 x^{5} a^{9} b A +\frac {1}{5} x^{5} a^{10} B +\frac {15}{2} x^{6} a^{8} b^{2} A +\frac {5}{3} x^{6} a^{9} b B +\frac {120}{7} x^{7} a^{7} b^{3} A +\frac {45}{7} x^{7} a^{8} b^{2} B +\frac {105}{4} x^{8} a^{6} b^{4} A +15 x^{8} a^{7} b^{3} B +28 x^{9} a^{5} b^{5} A +\frac {70}{3} x^{9} a^{6} b^{4} B +21 x^{10} a^{4} b^{6} A +\frac {126}{5} x^{10} a^{5} b^{5} B +\frac {120}{11} x^{11} a^{3} b^{7} A +\frac {210}{11} x^{11} a^{4} b^{6} B +\frac {15}{4} x^{12} a^{2} b^{8} A +10 x^{12} a^{3} b^{7} B +\frac {10}{13} x^{13} a \,b^{9} A +\frac {45}{13} x^{13} a^{2} b^{8} B +\frac {1}{14} x^{14} b^{10} A +\frac {5}{7} x^{14} a \,b^{9} B +\frac {1}{15} b^{10} B \,x^{15}\) | \(246\) |
parallelrisch | \(\frac {1}{4} a^{10} A \,x^{4}+2 x^{5} a^{9} b A +\frac {1}{5} x^{5} a^{10} B +\frac {15}{2} x^{6} a^{8} b^{2} A +\frac {5}{3} x^{6} a^{9} b B +\frac {120}{7} x^{7} a^{7} b^{3} A +\frac {45}{7} x^{7} a^{8} b^{2} B +\frac {105}{4} x^{8} a^{6} b^{4} A +15 x^{8} a^{7} b^{3} B +28 x^{9} a^{5} b^{5} A +\frac {70}{3} x^{9} a^{6} b^{4} B +21 x^{10} a^{4} b^{6} A +\frac {126}{5} x^{10} a^{5} b^{5} B +\frac {120}{11} x^{11} a^{3} b^{7} A +\frac {210}{11} x^{11} a^{4} b^{6} B +\frac {15}{4} x^{12} a^{2} b^{8} A +10 x^{12} a^{3} b^{7} B +\frac {10}{13} x^{13} a \,b^{9} A +\frac {45}{13} x^{13} a^{2} b^{8} B +\frac {1}{14} x^{14} b^{10} A +\frac {5}{7} x^{14} a \,b^{9} B +\frac {1}{15} b^{10} B \,x^{15}\) | \(246\) |
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Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (104) = 208\).
Time = 0.23 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.17 \[ \int x^3 (a+b x)^{10} (A+B x) \, dx=\frac {1}{15} \, B b^{10} x^{15} + \frac {1}{4} \, A a^{10} x^{4} + \frac {1}{14} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{14} + \frac {5}{13} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{13} + \frac {5}{4} \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{12} + \frac {30}{11} \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{11} + \frac {21}{5} \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{10} + \frac {14}{3} \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{9} + \frac {15}{4} \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{8} + \frac {15}{7} \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{7} + \frac {5}{6} \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{6} + \frac {1}{5} \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x^{5} \]
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Leaf count of result is larger than twice the leaf count of optimal. 265 vs. \(2 (105) = 210\).
Time = 0.04 (sec) , antiderivative size = 265, normalized size of antiderivative = 2.37 \[ \int x^3 (a+b x)^{10} (A+B x) \, dx=\frac {A a^{10} x^{4}}{4} + \frac {B b^{10} x^{15}}{15} + x^{14} \left (\frac {A b^{10}}{14} + \frac {5 B a b^{9}}{7}\right ) + x^{13} \cdot \left (\frac {10 A a b^{9}}{13} + \frac {45 B a^{2} b^{8}}{13}\right ) + x^{12} \cdot \left (\frac {15 A a^{2} b^{8}}{4} + 10 B a^{3} b^{7}\right ) + x^{11} \cdot \left (\frac {120 A a^{3} b^{7}}{11} + \frac {210 B a^{4} b^{6}}{11}\right ) + x^{10} \cdot \left (21 A a^{4} b^{6} + \frac {126 B a^{5} b^{5}}{5}\right ) + x^{9} \cdot \left (28 A a^{5} b^{5} + \frac {70 B a^{6} b^{4}}{3}\right ) + x^{8} \cdot \left (\frac {105 A a^{6} b^{4}}{4} + 15 B a^{7} b^{3}\right ) + x^{7} \cdot \left (\frac {120 A a^{7} b^{3}}{7} + \frac {45 B a^{8} b^{2}}{7}\right ) + x^{6} \cdot \left (\frac {15 A a^{8} b^{2}}{2} + \frac {5 B a^{9} b}{3}\right ) + x^{5} \cdot \left (2 A a^{9} b + \frac {B a^{10}}{5}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (104) = 208\).
Time = 0.22 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.17 \[ \int x^3 (a+b x)^{10} (A+B x) \, dx=\frac {1}{15} \, B b^{10} x^{15} + \frac {1}{4} \, A a^{10} x^{4} + \frac {1}{14} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{14} + \frac {5}{13} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{13} + \frac {5}{4} \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{12} + \frac {30}{11} \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{11} + \frac {21}{5} \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{10} + \frac {14}{3} \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{9} + \frac {15}{4} \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{8} + \frac {15}{7} \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{7} + \frac {5}{6} \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{6} + \frac {1}{5} \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x^{5} \]
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Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (104) = 208\).
Time = 0.28 (sec) , antiderivative size = 245, normalized size of antiderivative = 2.19 \[ \int x^3 (a+b x)^{10} (A+B x) \, dx=\frac {1}{15} \, B b^{10} x^{15} + \frac {5}{7} \, B a b^{9} x^{14} + \frac {1}{14} \, A b^{10} x^{14} + \frac {45}{13} \, B a^{2} b^{8} x^{13} + \frac {10}{13} \, A a b^{9} x^{13} + 10 \, B a^{3} b^{7} x^{12} + \frac {15}{4} \, A a^{2} b^{8} x^{12} + \frac {210}{11} \, B a^{4} b^{6} x^{11} + \frac {120}{11} \, A a^{3} b^{7} x^{11} + \frac {126}{5} \, B a^{5} b^{5} x^{10} + 21 \, A a^{4} b^{6} x^{10} + \frac {70}{3} \, B a^{6} b^{4} x^{9} + 28 \, A a^{5} b^{5} x^{9} + 15 \, B a^{7} b^{3} x^{8} + \frac {105}{4} \, A a^{6} b^{4} x^{8} + \frac {45}{7} \, B a^{8} b^{2} x^{7} + \frac {120}{7} \, A a^{7} b^{3} x^{7} + \frac {5}{3} \, B a^{9} b x^{6} + \frac {15}{2} \, A a^{8} b^{2} x^{6} + \frac {1}{5} \, B a^{10} x^{5} + 2 \, A a^{9} b x^{5} + \frac {1}{4} \, A a^{10} x^{4} \]
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Time = 0.10 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.88 \[ \int x^3 (a+b x)^{10} (A+B x) \, dx=x^5\,\left (\frac {B\,a^{10}}{5}+2\,A\,b\,a^9\right )+x^{14}\,\left (\frac {A\,b^{10}}{14}+\frac {5\,B\,a\,b^9}{7}\right )+\frac {A\,a^{10}\,x^4}{4}+\frac {B\,b^{10}\,x^{15}}{15}+\frac {15\,a^7\,b^2\,x^7\,\left (8\,A\,b+3\,B\,a\right )}{7}+\frac {15\,a^6\,b^3\,x^8\,\left (7\,A\,b+4\,B\,a\right )}{4}+\frac {14\,a^5\,b^4\,x^9\,\left (6\,A\,b+5\,B\,a\right )}{3}+\frac {21\,a^4\,b^5\,x^{10}\,\left (5\,A\,b+6\,B\,a\right )}{5}+\frac {30\,a^3\,b^6\,x^{11}\,\left (4\,A\,b+7\,B\,a\right )}{11}+\frac {5\,a^2\,b^7\,x^{12}\,\left (3\,A\,b+8\,B\,a\right )}{4}+\frac {5\,a^8\,b\,x^6\,\left (9\,A\,b+2\,B\,a\right )}{6}+\frac {5\,a\,b^8\,x^{13}\,\left (2\,A\,b+9\,B\,a\right )}{13} \]
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