Integrand size = 16, antiderivative size = 117 \[ \int x^5 (a+b x)^5 (A+B x) \, dx=\frac {1}{6} a^5 A x^6+\frac {1}{7} a^4 (5 A b+a B) x^7+\frac {5}{8} a^3 b (2 A b+a B) x^8+\frac {10}{9} a^2 b^2 (A b+a B) x^9+\frac {1}{2} a b^3 (A b+2 a B) x^{10}+\frac {1}{11} b^4 (A b+5 a B) x^{11}+\frac {1}{12} b^5 B x^{12} \]
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Time = 0.08 (sec) , antiderivative size = 117, 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} \[ \int x^5 (a+b x)^5 (A+B x) \, dx=\frac {1}{6} a^5 A x^6+\frac {1}{7} a^4 x^7 (a B+5 A b)+\frac {5}{8} a^3 b x^8 (a B+2 A b)+\frac {10}{9} a^2 b^2 x^9 (a B+A b)+\frac {1}{11} b^4 x^{11} (5 a B+A b)+\frac {1}{2} a b^3 x^{10} (2 a B+A b)+\frac {1}{12} b^5 B x^{12} \]
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Rule 77
Rubi steps \begin{align*} \text {integral}& = \int \left (a^5 A x^5+a^4 (5 A b+a B) x^6+5 a^3 b (2 A b+a B) x^7+10 a^2 b^2 (A b+a B) x^8+5 a b^3 (A b+2 a B) x^9+b^4 (A b+5 a B) x^{10}+b^5 B x^{11}\right ) \, dx \\ & = \frac {1}{6} a^5 A x^6+\frac {1}{7} a^4 (5 A b+a B) x^7+\frac {5}{8} a^3 b (2 A b+a B) x^8+\frac {10}{9} a^2 b^2 (A b+a B) x^9+\frac {1}{2} a b^3 (A b+2 a B) x^{10}+\frac {1}{11} b^4 (A b+5 a B) x^{11}+\frac {1}{12} b^5 B x^{12} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00 \[ \int x^5 (a+b x)^5 (A+B x) \, dx=\frac {1}{6} a^5 A x^6+\frac {1}{7} a^4 (5 A b+a B) x^7+\frac {5}{8} a^3 b (2 A b+a B) x^8+\frac {10}{9} a^2 b^2 (A b+a B) x^9+\frac {1}{2} a b^3 (A b+2 a B) x^{10}+\frac {1}{11} b^4 (A b+5 a B) x^{11}+\frac {1}{12} b^5 B x^{12} \]
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Time = 0.39 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.03
method | result | size |
norman | \(\frac {b^{5} B \,x^{12}}{12}+\left (\frac {1}{11} b^{5} A +\frac {5}{11} a \,b^{4} B \right ) x^{11}+\left (\frac {1}{2} a \,b^{4} A +a^{2} b^{3} B \right ) x^{10}+\left (\frac {10}{9} a^{2} b^{3} A +\frac {10}{9} a^{3} b^{2} B \right ) x^{9}+\left (\frac {5}{4} a^{3} b^{2} A +\frac {5}{8} a^{4} b B \right ) x^{8}+\left (\frac {5}{7} a^{4} b A +\frac {1}{7} a^{5} B \right ) x^{7}+\frac {a^{5} A \,x^{6}}{6}\) | \(120\) |
default | \(\frac {b^{5} B \,x^{12}}{12}+\frac {\left (b^{5} A +5 a \,b^{4} B \right ) x^{11}}{11}+\frac {\left (5 a \,b^{4} A +10 a^{2} b^{3} B \right ) x^{10}}{10}+\frac {\left (10 a^{2} b^{3} A +10 a^{3} b^{2} B \right ) x^{9}}{9}+\frac {\left (10 a^{3} b^{2} A +5 a^{4} b B \right ) x^{8}}{8}+\frac {\left (5 a^{4} b A +a^{5} B \right ) x^{7}}{7}+\frac {a^{5} A \,x^{6}}{6}\) | \(124\) |
gosper | \(\frac {1}{12} b^{5} B \,x^{12}+\frac {1}{11} x^{11} b^{5} A +\frac {5}{11} x^{11} a \,b^{4} B +\frac {1}{2} x^{10} a \,b^{4} A +x^{10} a^{2} b^{3} B +\frac {10}{9} x^{9} a^{2} b^{3} A +\frac {10}{9} x^{9} a^{3} b^{2} B +\frac {5}{4} x^{8} a^{3} b^{2} A +\frac {5}{8} x^{8} a^{4} b B +\frac {5}{7} x^{7} a^{4} b A +\frac {1}{7} x^{7} a^{5} B +\frac {1}{6} a^{5} A \,x^{6}\) | \(125\) |
risch | \(\frac {1}{12} b^{5} B \,x^{12}+\frac {1}{11} x^{11} b^{5} A +\frac {5}{11} x^{11} a \,b^{4} B +\frac {1}{2} x^{10} a \,b^{4} A +x^{10} a^{2} b^{3} B +\frac {10}{9} x^{9} a^{2} b^{3} A +\frac {10}{9} x^{9} a^{3} b^{2} B +\frac {5}{4} x^{8} a^{3} b^{2} A +\frac {5}{8} x^{8} a^{4} b B +\frac {5}{7} x^{7} a^{4} b A +\frac {1}{7} x^{7} a^{5} B +\frac {1}{6} a^{5} A \,x^{6}\) | \(125\) |
parallelrisch | \(\frac {1}{12} b^{5} B \,x^{12}+\frac {1}{11} x^{11} b^{5} A +\frac {5}{11} x^{11} a \,b^{4} B +\frac {1}{2} x^{10} a \,b^{4} A +x^{10} a^{2} b^{3} B +\frac {10}{9} x^{9} a^{2} b^{3} A +\frac {10}{9} x^{9} a^{3} b^{2} B +\frac {5}{4} x^{8} a^{3} b^{2} A +\frac {5}{8} x^{8} a^{4} b B +\frac {5}{7} x^{7} a^{4} b A +\frac {1}{7} x^{7} a^{5} B +\frac {1}{6} a^{5} A \,x^{6}\) | \(125\) |
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Time = 0.22 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.02 \[ \int x^5 (a+b x)^5 (A+B x) \, dx=\frac {1}{12} \, B b^{5} x^{12} + \frac {1}{6} \, A a^{5} x^{6} + \frac {1}{11} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{11} + \frac {1}{2} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{10} + \frac {10}{9} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + \frac {5}{8} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{8} + \frac {1}{7} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{7} \]
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Time = 0.03 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.14 \[ \int x^5 (a+b x)^5 (A+B x) \, dx=\frac {A a^{5} x^{6}}{6} + \frac {B b^{5} x^{12}}{12} + x^{11} \left (\frac {A b^{5}}{11} + \frac {5 B a b^{4}}{11}\right ) + x^{10} \left (\frac {A a b^{4}}{2} + B a^{2} b^{3}\right ) + x^{9} \cdot \left (\frac {10 A a^{2} b^{3}}{9} + \frac {10 B a^{3} b^{2}}{9}\right ) + x^{8} \cdot \left (\frac {5 A a^{3} b^{2}}{4} + \frac {5 B a^{4} b}{8}\right ) + x^{7} \cdot \left (\frac {5 A a^{4} b}{7} + \frac {B a^{5}}{7}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.02 \[ \int x^5 (a+b x)^5 (A+B x) \, dx=\frac {1}{12} \, B b^{5} x^{12} + \frac {1}{6} \, A a^{5} x^{6} + \frac {1}{11} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{11} + \frac {1}{2} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{10} + \frac {10}{9} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + \frac {5}{8} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{8} + \frac {1}{7} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{7} \]
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Time = 0.28 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.06 \[ \int x^5 (a+b x)^5 (A+B x) \, dx=\frac {1}{12} \, B b^{5} x^{12} + \frac {5}{11} \, B a b^{4} x^{11} + \frac {1}{11} \, A b^{5} x^{11} + B a^{2} b^{3} x^{10} + \frac {1}{2} \, A a b^{4} x^{10} + \frac {10}{9} \, B a^{3} b^{2} x^{9} + \frac {10}{9} \, A a^{2} b^{3} x^{9} + \frac {5}{8} \, B a^{4} b x^{8} + \frac {5}{4} \, A a^{3} b^{2} x^{8} + \frac {1}{7} \, B a^{5} x^{7} + \frac {5}{7} \, A a^{4} b x^{7} + \frac {1}{6} \, A a^{5} x^{6} \]
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Time = 0.37 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.91 \[ \int x^5 (a+b x)^5 (A+B x) \, dx=x^7\,\left (\frac {B\,a^5}{7}+\frac {5\,A\,b\,a^4}{7}\right )+x^{11}\,\left (\frac {A\,b^5}{11}+\frac {5\,B\,a\,b^4}{11}\right )+\frac {A\,a^5\,x^6}{6}+\frac {B\,b^5\,x^{12}}{12}+\frac {10\,a^2\,b^2\,x^9\,\left (A\,b+B\,a\right )}{9}+\frac {5\,a^3\,b\,x^8\,\left (2\,A\,b+B\,a\right )}{8}+\frac {a\,b^3\,x^{10}\,\left (A\,b+2\,B\,a\right )}{2} \]
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