Integrand size = 16, antiderivative size = 112 \[ \int x^3 (a+b x)^5 (A+B x) \, dx=-\frac {a^3 (A b-a B) (a+b x)^6}{6 b^5}+\frac {a^2 (3 A b-4 a B) (a+b x)^7}{7 b^5}-\frac {3 a (A b-2 a B) (a+b x)^8}{8 b^5}+\frac {(A b-4 a B) (a+b x)^9}{9 b^5}+\frac {B (a+b x)^{10}}{10 b^5} \]
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Time = 0.05 (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} \[ \int x^3 (a+b x)^5 (A+B x) \, dx=-\frac {a^3 (a+b x)^6 (A b-a B)}{6 b^5}+\frac {a^2 (a+b x)^7 (3 A b-4 a B)}{7 b^5}+\frac {(a+b x)^9 (A b-4 a B)}{9 b^5}-\frac {3 a (a+b x)^8 (A b-2 a B)}{8 b^5}+\frac {B (a+b x)^{10}}{10 b^5} \]
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Rule 77
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^3 (-A b+a B) (a+b x)^5}{b^4}-\frac {a^2 (-3 A b+4 a B) (a+b x)^6}{b^4}+\frac {3 a (-A b+2 a B) (a+b x)^7}{b^4}+\frac {(A b-4 a B) (a+b x)^8}{b^4}+\frac {B (a+b x)^9}{b^4}\right ) \, dx \\ & = -\frac {a^3 (A b-a B) (a+b x)^6}{6 b^5}+\frac {a^2 (3 A b-4 a B) (a+b x)^7}{7 b^5}-\frac {3 a (A b-2 a B) (a+b x)^8}{8 b^5}+\frac {(A b-4 a B) (a+b x)^9}{9 b^5}+\frac {B (a+b x)^{10}}{10 b^5} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.04 \[ \int x^3 (a+b x)^5 (A+B x) \, dx=\frac {1}{4} a^5 A x^4+\frac {1}{5} a^4 (5 A b+a B) x^5+\frac {5}{6} a^3 b (2 A b+a B) x^6+\frac {10}{7} a^2 b^2 (A b+a B) x^7+\frac {5}{8} a b^3 (A b+2 a B) x^8+\frac {1}{9} b^4 (A b+5 a B) x^9+\frac {1}{10} b^5 B x^{10} \]
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Time = 0.39 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.07
method | result | size |
norman | \(\frac {b^{5} B \,x^{10}}{10}+\left (\frac {1}{9} b^{5} A +\frac {5}{9} a \,b^{4} B \right ) x^{9}+\left (\frac {5}{8} a \,b^{4} A +\frac {5}{4} a^{2} b^{3} B \right ) x^{8}+\left (\frac {10}{7} a^{2} b^{3} A +\frac {10}{7} a^{3} b^{2} B \right ) x^{7}+\left (\frac {5}{3} a^{3} b^{2} A +\frac {5}{6} a^{4} b B \right ) x^{6}+\left (a^{4} b A +\frac {1}{5} a^{5} B \right ) x^{5}+\frac {a^{5} A \,x^{4}}{4}\) | \(120\) |
default | \(\frac {b^{5} B \,x^{10}}{10}+\frac {\left (b^{5} A +5 a \,b^{4} B \right ) x^{9}}{9}+\frac {\left (5 a \,b^{4} A +10 a^{2} b^{3} B \right ) x^{8}}{8}+\frac {\left (10 a^{2} b^{3} A +10 a^{3} b^{2} B \right ) x^{7}}{7}+\frac {\left (10 a^{3} b^{2} A +5 a^{4} b B \right ) x^{6}}{6}+\frac {\left (5 a^{4} b A +a^{5} B \right ) x^{5}}{5}+\frac {a^{5} A \,x^{4}}{4}\) | \(124\) |
gosper | \(\frac {1}{10} b^{5} B \,x^{10}+\frac {1}{9} x^{9} b^{5} A +\frac {5}{9} x^{9} a \,b^{4} B +\frac {5}{8} x^{8} a \,b^{4} A +\frac {5}{4} x^{8} a^{2} b^{3} B +\frac {10}{7} x^{7} a^{2} b^{3} A +\frac {10}{7} x^{7} a^{3} b^{2} B +\frac {5}{3} x^{6} a^{3} b^{2} A +\frac {5}{6} x^{6} a^{4} b B +x^{5} a^{4} b A +\frac {1}{5} x^{5} a^{5} B +\frac {1}{4} a^{5} A \,x^{4}\) | \(125\) |
risch | \(\frac {1}{10} b^{5} B \,x^{10}+\frac {1}{9} x^{9} b^{5} A +\frac {5}{9} x^{9} a \,b^{4} B +\frac {5}{8} x^{8} a \,b^{4} A +\frac {5}{4} x^{8} a^{2} b^{3} B +\frac {10}{7} x^{7} a^{2} b^{3} A +\frac {10}{7} x^{7} a^{3} b^{2} B +\frac {5}{3} x^{6} a^{3} b^{2} A +\frac {5}{6} x^{6} a^{4} b B +x^{5} a^{4} b A +\frac {1}{5} x^{5} a^{5} B +\frac {1}{4} a^{5} A \,x^{4}\) | \(125\) |
parallelrisch | \(\frac {1}{10} b^{5} B \,x^{10}+\frac {1}{9} x^{9} b^{5} A +\frac {5}{9} x^{9} a \,b^{4} B +\frac {5}{8} x^{8} a \,b^{4} A +\frac {5}{4} x^{8} a^{2} b^{3} B +\frac {10}{7} x^{7} a^{2} b^{3} A +\frac {10}{7} x^{7} a^{3} b^{2} B +\frac {5}{3} x^{6} a^{3} b^{2} A +\frac {5}{6} x^{6} a^{4} b B +x^{5} a^{4} b A +\frac {1}{5} x^{5} a^{5} B +\frac {1}{4} a^{5} A \,x^{4}\) | \(125\) |
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Time = 0.22 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.06 \[ \int x^3 (a+b x)^5 (A+B x) \, dx=\frac {1}{10} \, B b^{5} x^{10} + \frac {1}{4} \, A a^{5} x^{4} + \frac {1}{9} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{9} + \frac {5}{8} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + \frac {10}{7} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{7} + \frac {5}{6} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + \frac {1}{5} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{5} \]
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Time = 0.03 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.20 \[ \int x^3 (a+b x)^5 (A+B x) \, dx=\frac {A a^{5} x^{4}}{4} + \frac {B b^{5} x^{10}}{10} + x^{9} \left (\frac {A b^{5}}{9} + \frac {5 B a b^{4}}{9}\right ) + x^{8} \cdot \left (\frac {5 A a b^{4}}{8} + \frac {5 B a^{2} b^{3}}{4}\right ) + x^{7} \cdot \left (\frac {10 A a^{2} b^{3}}{7} + \frac {10 B a^{3} b^{2}}{7}\right ) + x^{6} \cdot \left (\frac {5 A a^{3} b^{2}}{3} + \frac {5 B a^{4} b}{6}\right ) + x^{5} \left (A a^{4} b + \frac {B a^{5}}{5}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.06 \[ \int x^3 (a+b x)^5 (A+B x) \, dx=\frac {1}{10} \, B b^{5} x^{10} + \frac {1}{4} \, A a^{5} x^{4} + \frac {1}{9} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{9} + \frac {5}{8} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + \frac {10}{7} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{7} + \frac {5}{6} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + \frac {1}{5} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{5} \]
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Time = 0.27 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.11 \[ \int x^3 (a+b x)^5 (A+B x) \, dx=\frac {1}{10} \, B b^{5} x^{10} + \frac {5}{9} \, B a b^{4} x^{9} + \frac {1}{9} \, A b^{5} x^{9} + \frac {5}{4} \, B a^{2} b^{3} x^{8} + \frac {5}{8} \, A a b^{4} x^{8} + \frac {10}{7} \, B a^{3} b^{2} x^{7} + \frac {10}{7} \, A a^{2} b^{3} x^{7} + \frac {5}{6} \, B a^{4} b x^{6} + \frac {5}{3} \, A a^{3} b^{2} x^{6} + \frac {1}{5} \, B a^{5} x^{5} + A a^{4} b x^{5} + \frac {1}{4} \, A a^{5} x^{4} \]
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Time = 0.04 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.95 \[ \int x^3 (a+b x)^5 (A+B x) \, dx=x^5\,\left (\frac {B\,a^5}{5}+A\,b\,a^4\right )+x^9\,\left (\frac {A\,b^5}{9}+\frac {5\,B\,a\,b^4}{9}\right )+\frac {A\,a^5\,x^4}{4}+\frac {B\,b^5\,x^{10}}{10}+\frac {10\,a^2\,b^2\,x^7\,\left (A\,b+B\,a\right )}{7}+\frac {5\,a^3\,b\,x^6\,\left (2\,A\,b+B\,a\right )}{6}+\frac {5\,a\,b^3\,x^8\,\left (A\,b+2\,B\,a\right )}{8} \]
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