Integrand size = 16, antiderivative size = 117 \[ \int x^4 (a+b x)^5 (A+B x) \, dx=\frac {1}{5} a^5 A x^5+\frac {1}{6} a^4 (5 A b+a B) x^6+\frac {5}{7} a^3 b (2 A b+a B) x^7+\frac {5}{4} a^2 b^2 (A b+a B) x^8+\frac {5}{9} a b^3 (A b+2 a B) x^9+\frac {1}{10} b^4 (A b+5 a B) x^{10}+\frac {1}{11} b^5 B x^{11} \]
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Time = 0.05 (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} \[ \int x^4 (a+b x)^5 (A+B x) \, dx=\frac {1}{5} a^5 A x^5+\frac {1}{6} a^4 x^6 (a B+5 A b)+\frac {5}{7} a^3 b x^7 (a B+2 A b)+\frac {5}{4} a^2 b^2 x^8 (a B+A b)+\frac {1}{10} b^4 x^{10} (5 a B+A b)+\frac {5}{9} a b^3 x^9 (2 a B+A b)+\frac {1}{11} b^5 B x^{11} \]
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Rule 77
Rubi steps \begin{align*} \text {integral}& = \int \left (a^5 A x^4+a^4 (5 A b+a B) x^5+5 a^3 b (2 A b+a B) x^6+10 a^2 b^2 (A b+a B) x^7+5 a b^3 (A b+2 a B) x^8+b^4 (A b+5 a B) x^9+b^5 B x^{10}\right ) \, dx \\ & = \frac {1}{5} a^5 A x^5+\frac {1}{6} a^4 (5 A b+a B) x^6+\frac {5}{7} a^3 b (2 A b+a B) x^7+\frac {5}{4} a^2 b^2 (A b+a B) x^8+\frac {5}{9} a b^3 (A b+2 a B) x^9+\frac {1}{10} b^4 (A b+5 a B) x^{10}+\frac {1}{11} b^5 B x^{11} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00 \[ \int x^4 (a+b x)^5 (A+B x) \, dx=\frac {1}{5} a^5 A x^5+\frac {1}{6} a^4 (5 A b+a B) x^6+\frac {5}{7} a^3 b (2 A b+a B) x^7+\frac {5}{4} a^2 b^2 (A b+a B) x^8+\frac {5}{9} a b^3 (A b+2 a B) x^9+\frac {1}{10} b^4 (A b+5 a B) x^{10}+\frac {1}{11} b^5 B x^{11} \]
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Time = 0.39 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.03
method | result | size |
norman | \(\frac {b^{5} B \,x^{11}}{11}+\left (\frac {1}{10} b^{5} A +\frac {1}{2} a \,b^{4} B \right ) x^{10}+\left (\frac {5}{9} a \,b^{4} A +\frac {10}{9} a^{2} b^{3} B \right ) x^{9}+\left (\frac {5}{4} a^{2} b^{3} A +\frac {5}{4} a^{3} b^{2} B \right ) x^{8}+\left (\frac {10}{7} a^{3} b^{2} A +\frac {5}{7} a^{4} b B \right ) x^{7}+\left (\frac {5}{6} a^{4} b A +\frac {1}{6} a^{5} B \right ) x^{6}+\frac {a^{5} A \,x^{5}}{5}\) | \(121\) |
default | \(\frac {b^{5} B \,x^{11}}{11}+\frac {\left (b^{5} A +5 a \,b^{4} B \right ) x^{10}}{10}+\frac {\left (5 a \,b^{4} A +10 a^{2} b^{3} B \right ) x^{9}}{9}+\frac {\left (10 a^{2} b^{3} A +10 a^{3} b^{2} B \right ) x^{8}}{8}+\frac {\left (10 a^{3} b^{2} A +5 a^{4} b B \right ) x^{7}}{7}+\frac {\left (5 a^{4} b A +a^{5} B \right ) x^{6}}{6}+\frac {a^{5} A \,x^{5}}{5}\) | \(124\) |
gosper | \(\frac {1}{11} b^{5} B \,x^{11}+\frac {1}{10} x^{10} b^{5} A +\frac {1}{2} x^{10} a \,b^{4} B +\frac {5}{9} x^{9} a \,b^{4} A +\frac {10}{9} x^{9} a^{2} b^{3} B +\frac {5}{4} x^{8} a^{2} b^{3} A +\frac {5}{4} x^{8} a^{3} b^{2} B +\frac {10}{7} x^{7} a^{3} b^{2} A +\frac {5}{7} x^{7} a^{4} b B +\frac {5}{6} x^{6} a^{4} b A +\frac {1}{6} x^{6} a^{5} B +\frac {1}{5} a^{5} A \,x^{5}\) | \(126\) |
risch | \(\frac {1}{11} b^{5} B \,x^{11}+\frac {1}{10} x^{10} b^{5} A +\frac {1}{2} x^{10} a \,b^{4} B +\frac {5}{9} x^{9} a \,b^{4} A +\frac {10}{9} x^{9} a^{2} b^{3} B +\frac {5}{4} x^{8} a^{2} b^{3} A +\frac {5}{4} x^{8} a^{3} b^{2} B +\frac {10}{7} x^{7} a^{3} b^{2} A +\frac {5}{7} x^{7} a^{4} b B +\frac {5}{6} x^{6} a^{4} b A +\frac {1}{6} x^{6} a^{5} B +\frac {1}{5} a^{5} A \,x^{5}\) | \(126\) |
parallelrisch | \(\frac {1}{11} b^{5} B \,x^{11}+\frac {1}{10} x^{10} b^{5} A +\frac {1}{2} x^{10} a \,b^{4} B +\frac {5}{9} x^{9} a \,b^{4} A +\frac {10}{9} x^{9} a^{2} b^{3} B +\frac {5}{4} x^{8} a^{2} b^{3} A +\frac {5}{4} x^{8} a^{3} b^{2} B +\frac {10}{7} x^{7} a^{3} b^{2} A +\frac {5}{7} x^{7} a^{4} b B +\frac {5}{6} x^{6} a^{4} b A +\frac {1}{6} x^{6} a^{5} B +\frac {1}{5} a^{5} A \,x^{5}\) | \(126\) |
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Time = 0.21 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.02 \[ \int x^4 (a+b x)^5 (A+B x) \, dx=\frac {1}{11} \, B b^{5} x^{11} + \frac {1}{5} \, A a^{5} x^{5} + \frac {1}{10} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + \frac {5}{9} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{9} + \frac {5}{4} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{8} + \frac {5}{7} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{7} + \frac {1}{6} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{6} \]
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Time = 0.03 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.16 \[ \int x^4 (a+b x)^5 (A+B x) \, dx=\frac {A a^{5} x^{5}}{5} + \frac {B b^{5} x^{11}}{11} + x^{10} \left (\frac {A b^{5}}{10} + \frac {B a b^{4}}{2}\right ) + x^{9} \cdot \left (\frac {5 A a b^{4}}{9} + \frac {10 B a^{2} b^{3}}{9}\right ) + x^{8} \cdot \left (\frac {5 A a^{2} b^{3}}{4} + \frac {5 B a^{3} b^{2}}{4}\right ) + x^{7} \cdot \left (\frac {10 A a^{3} b^{2}}{7} + \frac {5 B a^{4} b}{7}\right ) + x^{6} \cdot \left (\frac {5 A a^{4} b}{6} + \frac {B a^{5}}{6}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.02 \[ \int x^4 (a+b x)^5 (A+B x) \, dx=\frac {1}{11} \, B b^{5} x^{11} + \frac {1}{5} \, A a^{5} x^{5} + \frac {1}{10} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + \frac {5}{9} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{9} + \frac {5}{4} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{8} + \frac {5}{7} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{7} + \frac {1}{6} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{6} \]
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Time = 0.27 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.07 \[ \int x^4 (a+b x)^5 (A+B x) \, dx=\frac {1}{11} \, B b^{5} x^{11} + \frac {1}{2} \, B a b^{4} x^{10} + \frac {1}{10} \, A b^{5} x^{10} + \frac {10}{9} \, B a^{2} b^{3} x^{9} + \frac {5}{9} \, A a b^{4} x^{9} + \frac {5}{4} \, B a^{3} b^{2} x^{8} + \frac {5}{4} \, A a^{2} b^{3} x^{8} + \frac {5}{7} \, B a^{4} b x^{7} + \frac {10}{7} \, A a^{3} b^{2} x^{7} + \frac {1}{6} \, B a^{5} x^{6} + \frac {5}{6} \, A a^{4} b x^{6} + \frac {1}{5} \, A a^{5} x^{5} \]
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Time = 0.04 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.91 \[ \int x^4 (a+b x)^5 (A+B x) \, dx=x^6\,\left (\frac {B\,a^5}{6}+\frac {5\,A\,b\,a^4}{6}\right )+x^{10}\,\left (\frac {A\,b^5}{10}+\frac {B\,a\,b^4}{2}\right )+\frac {A\,a^5\,x^5}{5}+\frac {B\,b^5\,x^{11}}{11}+\frac {5\,a^2\,b^2\,x^8\,\left (A\,b+B\,a\right )}{4}+\frac {5\,a^3\,b\,x^7\,\left (2\,A\,b+B\,a\right )}{7}+\frac {5\,a\,b^3\,x^9\,\left (A\,b+2\,B\,a\right )}{9} \]
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