Integrand size = 13, antiderivative size = 38 \[ \int (a+b x)^6 (A+B x) \, dx=\frac {(A b-a B) (a+b x)^7}{7 b^2}+\frac {B (a+b x)^8}{8 b^2} \]
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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} \[ \int (a+b x)^6 (A+B x) \, dx=\frac {(a+b x)^7 (A b-a B)}{7 b^2}+\frac {B (a+b x)^8}{8 b^2} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(A b-a B) (a+b x)^6}{b}+\frac {B (a+b x)^7}{b}\right ) \, dx \\ & = \frac {(A b-a B) (a+b x)^7}{7 b^2}+\frac {B (a+b x)^8}{8 b^2} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(122\) vs. \(2(38)=76\).
Time = 0.03 (sec) , antiderivative size = 122, normalized size of antiderivative = 3.21 \[ \int (a+b x)^6 (A+B x) \, dx=\frac {1}{56} x \left (28 a^6 (2 A+B x)+56 a^5 b x (3 A+2 B x)+70 a^4 b^2 x^2 (4 A+3 B x)+56 a^3 b^3 x^3 (5 A+4 B x)+28 a^2 b^4 x^4 (6 A+5 B x)+8 a b^5 x^5 (7 A+6 B x)+b^6 x^6 (8 A+7 B x)\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(139\) vs. \(2(34)=68\).
Time = 0.70 (sec) , antiderivative size = 140, normalized size of antiderivative = 3.68
method | result | size |
norman | \(\frac {b^{6} B \,x^{8}}{8}+\left (\frac {1}{7} b^{6} A +\frac {6}{7} a \,b^{5} B \right ) x^{7}+\left (a \,b^{5} A +\frac {5}{2} a^{2} b^{4} B \right ) x^{6}+\left (3 a^{2} b^{4} A +4 a^{3} b^{3} B \right ) x^{5}+\left (5 a^{3} b^{3} A +\frac {15}{4} a^{4} b^{2} B \right ) x^{4}+\left (5 a^{4} b^{2} A +2 a^{5} b B \right ) x^{3}+\left (3 A \,a^{5} b +\frac {1}{2} B \,a^{6}\right ) x^{2}+A \,a^{6} x\) | \(140\) |
default | \(\frac {b^{6} B \,x^{8}}{8}+\frac {\left (b^{6} A +6 a \,b^{5} B \right ) x^{7}}{7}+\frac {\left (6 a \,b^{5} A +15 a^{2} b^{4} B \right ) x^{6}}{6}+\frac {\left (15 a^{2} b^{4} A +20 a^{3} b^{3} B \right ) x^{5}}{5}+\frac {\left (20 a^{3} b^{3} A +15 a^{4} b^{2} B \right ) x^{4}}{4}+\frac {\left (15 a^{4} b^{2} A +6 a^{5} b B \right ) x^{3}}{3}+\frac {\left (6 A \,a^{5} b +B \,a^{6}\right ) x^{2}}{2}+A \,a^{6} x\) | \(145\) |
gosper | \(\frac {1}{8} b^{6} B \,x^{8}+\frac {1}{7} x^{7} b^{6} A +\frac {6}{7} x^{7} a \,b^{5} B +x^{6} a \,b^{5} A +\frac {5}{2} x^{6} a^{2} b^{4} B +3 A \,a^{2} b^{4} x^{5}+4 B \,a^{3} b^{3} x^{5}+5 x^{4} a^{3} b^{3} A +\frac {15}{4} x^{4} a^{4} b^{2} B +5 A \,a^{4} b^{2} x^{3}+2 B \,a^{5} b \,x^{3}+3 x^{2} A \,a^{5} b +\frac {1}{2} x^{2} B \,a^{6}+A \,a^{6} x\) | \(146\) |
risch | \(\frac {1}{8} b^{6} B \,x^{8}+\frac {1}{7} x^{7} b^{6} A +\frac {6}{7} x^{7} a \,b^{5} B +x^{6} a \,b^{5} A +\frac {5}{2} x^{6} a^{2} b^{4} B +3 A \,a^{2} b^{4} x^{5}+4 B \,a^{3} b^{3} x^{5}+5 x^{4} a^{3} b^{3} A +\frac {15}{4} x^{4} a^{4} b^{2} B +5 A \,a^{4} b^{2} x^{3}+2 B \,a^{5} b \,x^{3}+3 x^{2} A \,a^{5} b +\frac {1}{2} x^{2} B \,a^{6}+A \,a^{6} x\) | \(146\) |
parallelrisch | \(\frac {1}{8} b^{6} B \,x^{8}+\frac {1}{7} x^{7} b^{6} A +\frac {6}{7} x^{7} a \,b^{5} B +x^{6} a \,b^{5} A +\frac {5}{2} x^{6} a^{2} b^{4} B +3 A \,a^{2} b^{4} x^{5}+4 B \,a^{3} b^{3} x^{5}+5 x^{4} a^{3} b^{3} A +\frac {15}{4} x^{4} a^{4} b^{2} B +5 A \,a^{4} b^{2} x^{3}+2 B \,a^{5} b \,x^{3}+3 x^{2} A \,a^{5} b +\frac {1}{2} x^{2} B \,a^{6}+A \,a^{6} x\) | \(146\) |
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Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (34) = 68\).
Time = 0.22 (sec) , antiderivative size = 142, normalized size of antiderivative = 3.74 \[ \int (a+b x)^6 (A+B x) \, dx=\frac {1}{8} \, B b^{6} x^{8} + A a^{6} x + \frac {1}{7} \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{7} + \frac {1}{2} \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{6} + {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{5} + \frac {5}{4} \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{4} + {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (32) = 64\).
Time = 0.03 (sec) , antiderivative size = 148, normalized size of antiderivative = 3.89 \[ \int (a+b x)^6 (A+B x) \, dx=A a^{6} x + \frac {B b^{6} x^{8}}{8} + x^{7} \left (\frac {A b^{6}}{7} + \frac {6 B a b^{5}}{7}\right ) + x^{6} \left (A a b^{5} + \frac {5 B a^{2} b^{4}}{2}\right ) + x^{5} \cdot \left (3 A a^{2} b^{4} + 4 B a^{3} b^{3}\right ) + x^{4} \cdot \left (5 A a^{3} b^{3} + \frac {15 B a^{4} b^{2}}{4}\right ) + x^{3} \cdot \left (5 A a^{4} b^{2} + 2 B a^{5} b\right ) + x^{2} \cdot \left (3 A a^{5} b + \frac {B a^{6}}{2}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (34) = 68\).
Time = 0.20 (sec) , antiderivative size = 142, normalized size of antiderivative = 3.74 \[ \int (a+b x)^6 (A+B x) \, dx=\frac {1}{8} \, B b^{6} x^{8} + A a^{6} x + \frac {1}{7} \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{7} + \frac {1}{2} \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{6} + {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{5} + \frac {5}{4} \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{4} + {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 145 vs. \(2 (34) = 68\).
Time = 0.27 (sec) , antiderivative size = 145, normalized size of antiderivative = 3.82 \[ \int (a+b x)^6 (A+B x) \, dx=\frac {1}{8} \, B b^{6} x^{8} + \frac {6}{7} \, B a b^{5} x^{7} + \frac {1}{7} \, A b^{6} x^{7} + \frac {5}{2} \, B a^{2} b^{4} x^{6} + A a b^{5} x^{6} + 4 \, B a^{3} b^{3} x^{5} + 3 \, A a^{2} b^{4} x^{5} + \frac {15}{4} \, B a^{4} b^{2} x^{4} + 5 \, A a^{3} b^{3} x^{4} + 2 \, B a^{5} b x^{3} + 5 \, A a^{4} b^{2} x^{3} + \frac {1}{2} \, B a^{6} x^{2} + 3 \, A a^{5} b x^{2} + A a^{6} x \]
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Time = 0.08 (sec) , antiderivative size = 126, normalized size of antiderivative = 3.32 \[ \int (a+b x)^6 (A+B x) \, dx=x^2\,\left (\frac {B\,a^6}{2}+3\,A\,b\,a^5\right )+x^7\,\left (\frac {A\,b^6}{7}+\frac {6\,B\,a\,b^5}{7}\right )+\frac {B\,b^6\,x^8}{8}+A\,a^6\,x+\frac {5\,a^3\,b^2\,x^4\,\left (4\,A\,b+3\,B\,a\right )}{4}+a^2\,b^3\,x^5\,\left (3\,A\,b+4\,B\,a\right )+a^4\,b\,x^3\,\left (5\,A\,b+2\,B\,a\right )+\frac {a\,b^4\,x^6\,\left (2\,A\,b+5\,B\,a\right )}{2} \]
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