Integrand size = 13, antiderivative size = 38 \[ \int (a+b x)^5 (A+B x) \, dx=\frac {(A b-a B) (a+b x)^6}{6 b^2}+\frac {B (a+b x)^7}{7 b^2} \]
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Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {45} \[ \int (a+b x)^5 (A+B x) \, dx=\frac {(a+b x)^6 (A b-a B)}{6 b^2}+\frac {B (a+b x)^7}{7 b^2} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(A b-a B) (a+b x)^5}{b}+\frac {B (a+b x)^6}{b}\right ) \, dx \\ & = \frac {(A b-a B) (a+b x)^6}{6 b^2}+\frac {B (a+b x)^7}{7 b^2} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(109\) vs. \(2(38)=76\).
Time = 0.01 (sec) , antiderivative size = 109, normalized size of antiderivative = 2.87 \[ \int (a+b x)^5 (A+B x) \, dx=a^5 A x+\frac {1}{2} a^4 (5 A b+a B) x^2+\frac {5}{3} a^3 b (2 A b+a B) x^3+\frac {5}{2} a^2 b^2 (A b+a B) x^4+a b^3 (A b+2 a B) x^5+\frac {1}{6} b^4 (A b+5 a B) x^6+\frac {1}{7} b^5 B x^7 \]
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Leaf count of result is larger than twice the leaf count of optimal. \(116\) vs. \(2(34)=68\).
Time = 0.39 (sec) , antiderivative size = 117, normalized size of antiderivative = 3.08
method | result | size |
norman | \(\frac {b^{5} B \,x^{7}}{7}+\left (\frac {1}{6} b^{5} A +\frac {5}{6} a \,b^{4} B \right ) x^{6}+\left (a \,b^{4} A +2 a^{2} b^{3} B \right ) x^{5}+\left (\frac {5}{2} a^{2} b^{3} A +\frac {5}{2} a^{3} b^{2} B \right ) x^{4}+\left (\frac {10}{3} a^{3} b^{2} A +\frac {5}{3} a^{4} b B \right ) x^{3}+\left (\frac {5}{2} a^{4} b A +\frac {1}{2} a^{5} B \right ) x^{2}+a^{5} A x\) | \(117\) |
default | \(\frac {b^{5} B \,x^{7}}{7}+\frac {\left (b^{5} A +5 a \,b^{4} B \right ) x^{6}}{6}+\frac {\left (5 a \,b^{4} A +10 a^{2} b^{3} B \right ) x^{5}}{5}+\frac {\left (10 a^{2} b^{3} A +10 a^{3} b^{2} B \right ) x^{4}}{4}+\frac {\left (10 a^{3} b^{2} A +5 a^{4} b B \right ) x^{3}}{3}+\frac {\left (5 a^{4} b A +a^{5} B \right ) x^{2}}{2}+a^{5} A x\) | \(121\) |
gosper | \(\frac {1}{7} b^{5} B \,x^{7}+\frac {1}{6} x^{6} b^{5} A +\frac {5}{6} x^{6} a \,b^{4} B +A a \,b^{4} x^{5}+2 B \,a^{2} b^{3} x^{5}+\frac {5}{2} x^{4} a^{2} b^{3} A +\frac {5}{2} x^{4} a^{3} b^{2} B +\frac {10}{3} x^{3} a^{3} b^{2} A +\frac {5}{3} x^{3} a^{4} b B +\frac {5}{2} x^{2} a^{4} b A +\frac {1}{2} x^{2} a^{5} B +a^{5} A x\) | \(122\) |
risch | \(\frac {1}{7} b^{5} B \,x^{7}+\frac {1}{6} x^{6} b^{5} A +\frac {5}{6} x^{6} a \,b^{4} B +A a \,b^{4} x^{5}+2 B \,a^{2} b^{3} x^{5}+\frac {5}{2} x^{4} a^{2} b^{3} A +\frac {5}{2} x^{4} a^{3} b^{2} B +\frac {10}{3} x^{3} a^{3} b^{2} A +\frac {5}{3} x^{3} a^{4} b B +\frac {5}{2} x^{2} a^{4} b A +\frac {1}{2} x^{2} a^{5} B +a^{5} A x\) | \(122\) |
parallelrisch | \(\frac {1}{7} b^{5} B \,x^{7}+\frac {1}{6} x^{6} b^{5} A +\frac {5}{6} x^{6} a \,b^{4} B +A a \,b^{4} x^{5}+2 B \,a^{2} b^{3} x^{5}+\frac {5}{2} x^{4} a^{2} b^{3} A +\frac {5}{2} x^{4} a^{3} b^{2} B +\frac {10}{3} x^{3} a^{3} b^{2} A +\frac {5}{3} x^{3} a^{4} b B +\frac {5}{2} x^{2} a^{4} b A +\frac {1}{2} x^{2} a^{5} B +a^{5} A x\) | \(122\) |
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Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (34) = 68\).
Time = 0.22 (sec) , antiderivative size = 115, normalized size of antiderivative = 3.03 \[ \int (a+b x)^5 (A+B x) \, dx=\frac {1}{7} \, B b^{5} x^{7} + A a^{5} x + \frac {1}{6} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{6} + {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{5} + \frac {5}{2} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{4} + \frac {5}{3} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (32) = 64\).
Time = 0.03 (sec) , antiderivative size = 129, normalized size of antiderivative = 3.39 \[ \int (a+b x)^5 (A+B x) \, dx=A a^{5} x + \frac {B b^{5} x^{7}}{7} + x^{6} \left (\frac {A b^{5}}{6} + \frac {5 B a b^{4}}{6}\right ) + x^{5} \left (A a b^{4} + 2 B a^{2} b^{3}\right ) + x^{4} \cdot \left (\frac {5 A a^{2} b^{3}}{2} + \frac {5 B a^{3} b^{2}}{2}\right ) + x^{3} \cdot \left (\frac {10 A a^{3} b^{2}}{3} + \frac {5 B a^{4} b}{3}\right ) + x^{2} \cdot \left (\frac {5 A a^{4} b}{2} + \frac {B a^{5}}{2}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (34) = 68\).
Time = 0.20 (sec) , antiderivative size = 115, normalized size of antiderivative = 3.03 \[ \int (a+b x)^5 (A+B x) \, dx=\frac {1}{7} \, B b^{5} x^{7} + A a^{5} x + \frac {1}{6} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{6} + {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{5} + \frac {5}{2} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{4} + \frac {5}{3} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (34) = 68\).
Time = 0.31 (sec) , antiderivative size = 121, normalized size of antiderivative = 3.18 \[ \int (a+b x)^5 (A+B x) \, dx=\frac {1}{7} \, B b^{5} x^{7} + \frac {5}{6} \, B a b^{4} x^{6} + \frac {1}{6} \, A b^{5} x^{6} + 2 \, B a^{2} b^{3} x^{5} + A a b^{4} x^{5} + \frac {5}{2} \, B a^{3} b^{2} x^{4} + \frac {5}{2} \, A a^{2} b^{3} x^{4} + \frac {5}{3} \, B a^{4} b x^{3} + \frac {10}{3} \, A a^{3} b^{2} x^{3} + \frac {1}{2} \, B a^{5} x^{2} + \frac {5}{2} \, A a^{4} b x^{2} + A a^{5} x \]
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Time = 0.10 (sec) , antiderivative size = 103, normalized size of antiderivative = 2.71 \[ \int (a+b x)^5 (A+B x) \, dx=x^2\,\left (\frac {B\,a^5}{2}+\frac {5\,A\,b\,a^4}{2}\right )+x^6\,\left (\frac {A\,b^5}{6}+\frac {5\,B\,a\,b^4}{6}\right )+\frac {B\,b^5\,x^7}{7}+A\,a^5\,x+\frac {5\,a^2\,b^2\,x^4\,\left (A\,b+B\,a\right )}{2}+\frac {5\,a^3\,b\,x^3\,\left (2\,A\,b+B\,a\right )}{3}+a\,b^3\,x^5\,\left (A\,b+2\,B\,a\right ) \]
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