Integrand size = 13, antiderivative size = 38 \[ \int (a+b x) (c+d x)^{10} \, dx=-\frac {(b c-a d) (c+d x)^{11}}{11 d^2}+\frac {b (c+d x)^{12}}{12 d^2} \]
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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} \[ \int (a+b x) (c+d x)^{10} \, dx=\frac {b (c+d x)^{12}}{12 d^2}-\frac {(c+d x)^{11} (b c-a d)}{11 d^2} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b c+a d) (c+d x)^{10}}{d}+\frac {b (c+d x)^{11}}{d}\right ) \, dx \\ & = -\frac {(b c-a d) (c+d x)^{11}}{11 d^2}+\frac {b (c+d x)^{12}}{12 d^2} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(220\) vs. \(2(38)=76\).
Time = 0.01 (sec) , antiderivative size = 220, normalized size of antiderivative = 5.79 \[ \int (a+b x) (c+d x)^{10} \, dx=a c^{10} x+\frac {1}{2} c^9 (b c+10 a d) x^2+\frac {5}{3} c^8 d (2 b c+9 a d) x^3+\frac {15}{4} c^7 d^2 (3 b c+8 a d) x^4+6 c^6 d^3 (4 b c+7 a d) x^5+7 c^5 d^4 (5 b c+6 a d) x^6+6 c^4 d^5 (6 b c+5 a d) x^7+\frac {15}{4} c^3 d^6 (7 b c+4 a d) x^8+\frac {5}{3} c^2 d^7 (8 b c+3 a d) x^9+\frac {1}{2} c d^8 (9 b c+2 a d) x^{10}+\frac {1}{11} d^9 (10 b c+a d) x^{11}+\frac {1}{12} b d^{10} x^{12} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(231\) vs. \(2(34)=68\).
Time = 0.19 (sec) , antiderivative size = 232, normalized size of antiderivative = 6.11
method | result | size |
norman | \(\frac {b \,d^{10} x^{12}}{12}+\left (\frac {1}{11} a \,d^{10}+\frac {10}{11} b c \,d^{9}\right ) x^{11}+\left (a c \,d^{9}+\frac {9}{2} b \,c^{2} d^{8}\right ) x^{10}+\left (5 a \,c^{2} d^{8}+\frac {40}{3} b \,c^{3} d^{7}\right ) x^{9}+\left (15 a \,c^{3} d^{7}+\frac {105}{4} b \,c^{4} d^{6}\right ) x^{8}+\left (30 a \,c^{4} d^{6}+36 b \,c^{5} d^{5}\right ) x^{7}+\left (42 a \,c^{5} d^{5}+35 b \,c^{6} d^{4}\right ) x^{6}+\left (42 a \,c^{6} d^{4}+24 b \,c^{7} d^{3}\right ) x^{5}+\left (30 a \,c^{7} d^{3}+\frac {45}{4} b \,c^{8} d^{2}\right ) x^{4}+\left (15 a \,c^{8} d^{2}+\frac {10}{3} b \,c^{9} d \right ) x^{3}+\left (5 a \,c^{9} d +\frac {1}{2} b \,c^{10}\right ) x^{2}+a \,c^{10} x\) | \(232\) |
default | \(\frac {b \,d^{10} x^{12}}{12}+\frac {\left (a \,d^{10}+10 b c \,d^{9}\right ) x^{11}}{11}+\frac {\left (10 a c \,d^{9}+45 b \,c^{2} d^{8}\right ) x^{10}}{10}+\frac {\left (45 a \,c^{2} d^{8}+120 b \,c^{3} d^{7}\right ) x^{9}}{9}+\frac {\left (120 a \,c^{3} d^{7}+210 b \,c^{4} d^{6}\right ) x^{8}}{8}+\frac {\left (210 a \,c^{4} d^{6}+252 b \,c^{5} d^{5}\right ) x^{7}}{7}+\frac {\left (252 a \,c^{5} d^{5}+210 b \,c^{6} d^{4}\right ) x^{6}}{6}+\frac {\left (210 a \,c^{6} d^{4}+120 b \,c^{7} d^{3}\right ) x^{5}}{5}+\frac {\left (120 a \,c^{7} d^{3}+45 b \,c^{8} d^{2}\right ) x^{4}}{4}+\frac {\left (45 a \,c^{8} d^{2}+10 b \,c^{9} d \right ) x^{3}}{3}+\frac {\left (10 a \,c^{9} d +b \,c^{10}\right ) x^{2}}{2}+a \,c^{10} x\) | \(241\) |
gosper | \(\frac {1}{12} b \,d^{10} x^{12}+\frac {1}{11} x^{11} a \,d^{10}+\frac {10}{11} x^{11} b c \,d^{9}+x^{10} a c \,d^{9}+\frac {9}{2} x^{10} b \,c^{2} d^{8}+5 x^{9} a \,c^{2} d^{8}+\frac {40}{3} x^{9} b \,c^{3} d^{7}+15 x^{8} a \,c^{3} d^{7}+\frac {105}{4} x^{8} b \,c^{4} d^{6}+30 a \,c^{4} d^{6} x^{7}+36 b \,c^{5} d^{5} x^{7}+42 a \,c^{5} d^{5} x^{6}+35 b \,c^{6} d^{4} x^{6}+42 a \,c^{6} d^{4} x^{5}+24 b \,c^{7} d^{3} x^{5}+30 x^{4} a \,c^{7} d^{3}+\frac {45}{4} x^{4} b \,c^{8} d^{2}+15 x^{3} a \,c^{8} d^{2}+\frac {10}{3} x^{3} b \,c^{9} d +5 x^{2} a \,c^{9} d +\frac {1}{2} x^{2} b \,c^{10}+a \,c^{10} x\) | \(242\) |
risch | \(\frac {1}{12} b \,d^{10} x^{12}+\frac {1}{11} x^{11} a \,d^{10}+\frac {10}{11} x^{11} b c \,d^{9}+x^{10} a c \,d^{9}+\frac {9}{2} x^{10} b \,c^{2} d^{8}+5 x^{9} a \,c^{2} d^{8}+\frac {40}{3} x^{9} b \,c^{3} d^{7}+15 x^{8} a \,c^{3} d^{7}+\frac {105}{4} x^{8} b \,c^{4} d^{6}+30 a \,c^{4} d^{6} x^{7}+36 b \,c^{5} d^{5} x^{7}+42 a \,c^{5} d^{5} x^{6}+35 b \,c^{6} d^{4} x^{6}+42 a \,c^{6} d^{4} x^{5}+24 b \,c^{7} d^{3} x^{5}+30 x^{4} a \,c^{7} d^{3}+\frac {45}{4} x^{4} b \,c^{8} d^{2}+15 x^{3} a \,c^{8} d^{2}+\frac {10}{3} x^{3} b \,c^{9} d +5 x^{2} a \,c^{9} d +\frac {1}{2} x^{2} b \,c^{10}+a \,c^{10} x\) | \(242\) |
parallelrisch | \(\frac {1}{12} b \,d^{10} x^{12}+\frac {1}{11} x^{11} a \,d^{10}+\frac {10}{11} x^{11} b c \,d^{9}+x^{10} a c \,d^{9}+\frac {9}{2} x^{10} b \,c^{2} d^{8}+5 x^{9} a \,c^{2} d^{8}+\frac {40}{3} x^{9} b \,c^{3} d^{7}+15 x^{8} a \,c^{3} d^{7}+\frac {105}{4} x^{8} b \,c^{4} d^{6}+30 a \,c^{4} d^{6} x^{7}+36 b \,c^{5} d^{5} x^{7}+42 a \,c^{5} d^{5} x^{6}+35 b \,c^{6} d^{4} x^{6}+42 a \,c^{6} d^{4} x^{5}+24 b \,c^{7} d^{3} x^{5}+30 x^{4} a \,c^{7} d^{3}+\frac {45}{4} x^{4} b \,c^{8} d^{2}+15 x^{3} a \,c^{8} d^{2}+\frac {10}{3} x^{3} b \,c^{9} d +5 x^{2} a \,c^{9} d +\frac {1}{2} x^{2} b \,c^{10}+a \,c^{10} x\) | \(242\) |
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Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (34) = 68\).
Time = 0.22 (sec) , antiderivative size = 240, normalized size of antiderivative = 6.32 \[ \int (a+b x) (c+d x)^{10} \, dx=\frac {1}{12} \, b d^{10} x^{12} + a c^{10} x + \frac {1}{11} \, {\left (10 \, b c d^{9} + a d^{10}\right )} x^{11} + \frac {1}{2} \, {\left (9 \, b c^{2} d^{8} + 2 \, a c d^{9}\right )} x^{10} + \frac {5}{3} \, {\left (8 \, b c^{3} d^{7} + 3 \, a c^{2} d^{8}\right )} x^{9} + \frac {15}{4} \, {\left (7 \, b c^{4} d^{6} + 4 \, a c^{3} d^{7}\right )} x^{8} + 6 \, {\left (6 \, b c^{5} d^{5} + 5 \, a c^{4} d^{6}\right )} x^{7} + 7 \, {\left (5 \, b c^{6} d^{4} + 6 \, a c^{5} d^{5}\right )} x^{6} + 6 \, {\left (4 \, b c^{7} d^{3} + 7 \, a c^{6} d^{4}\right )} x^{5} + \frac {15}{4} \, {\left (3 \, b c^{8} d^{2} + 8 \, a c^{7} d^{3}\right )} x^{4} + \frac {5}{3} \, {\left (2 \, b c^{9} d + 9 \, a c^{8} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (b c^{10} + 10 \, a c^{9} d\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (32) = 64\).
Time = 0.05 (sec) , antiderivative size = 248, normalized size of antiderivative = 6.53 \[ \int (a+b x) (c+d x)^{10} \, dx=a c^{10} x + \frac {b d^{10} x^{12}}{12} + x^{11} \left (\frac {a d^{10}}{11} + \frac {10 b c d^{9}}{11}\right ) + x^{10} \left (a c d^{9} + \frac {9 b c^{2} d^{8}}{2}\right ) + x^{9} \cdot \left (5 a c^{2} d^{8} + \frac {40 b c^{3} d^{7}}{3}\right ) + x^{8} \cdot \left (15 a c^{3} d^{7} + \frac {105 b c^{4} d^{6}}{4}\right ) + x^{7} \cdot \left (30 a c^{4} d^{6} + 36 b c^{5} d^{5}\right ) + x^{6} \cdot \left (42 a c^{5} d^{5} + 35 b c^{6} d^{4}\right ) + x^{5} \cdot \left (42 a c^{6} d^{4} + 24 b c^{7} d^{3}\right ) + x^{4} \cdot \left (30 a c^{7} d^{3} + \frac {45 b c^{8} d^{2}}{4}\right ) + x^{3} \cdot \left (15 a c^{8} d^{2} + \frac {10 b c^{9} d}{3}\right ) + x^{2} \cdot \left (5 a c^{9} d + \frac {b c^{10}}{2}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (34) = 68\).
Time = 0.21 (sec) , antiderivative size = 240, normalized size of antiderivative = 6.32 \[ \int (a+b x) (c+d x)^{10} \, dx=\frac {1}{12} \, b d^{10} x^{12} + a c^{10} x + \frac {1}{11} \, {\left (10 \, b c d^{9} + a d^{10}\right )} x^{11} + \frac {1}{2} \, {\left (9 \, b c^{2} d^{8} + 2 \, a c d^{9}\right )} x^{10} + \frac {5}{3} \, {\left (8 \, b c^{3} d^{7} + 3 \, a c^{2} d^{8}\right )} x^{9} + \frac {15}{4} \, {\left (7 \, b c^{4} d^{6} + 4 \, a c^{3} d^{7}\right )} x^{8} + 6 \, {\left (6 \, b c^{5} d^{5} + 5 \, a c^{4} d^{6}\right )} x^{7} + 7 \, {\left (5 \, b c^{6} d^{4} + 6 \, a c^{5} d^{5}\right )} x^{6} + 6 \, {\left (4 \, b c^{7} d^{3} + 7 \, a c^{6} d^{4}\right )} x^{5} + \frac {15}{4} \, {\left (3 \, b c^{8} d^{2} + 8 \, a c^{7} d^{3}\right )} x^{4} + \frac {5}{3} \, {\left (2 \, b c^{9} d + 9 \, a c^{8} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (b c^{10} + 10 \, a c^{9} d\right )} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 241 vs. \(2 (34) = 68\).
Time = 0.30 (sec) , antiderivative size = 241, normalized size of antiderivative = 6.34 \[ \int (a+b x) (c+d x)^{10} \, dx=\frac {1}{12} \, b d^{10} x^{12} + \frac {10}{11} \, b c d^{9} x^{11} + \frac {1}{11} \, a d^{10} x^{11} + \frac {9}{2} \, b c^{2} d^{8} x^{10} + a c d^{9} x^{10} + \frac {40}{3} \, b c^{3} d^{7} x^{9} + 5 \, a c^{2} d^{8} x^{9} + \frac {105}{4} \, b c^{4} d^{6} x^{8} + 15 \, a c^{3} d^{7} x^{8} + 36 \, b c^{5} d^{5} x^{7} + 30 \, a c^{4} d^{6} x^{7} + 35 \, b c^{6} d^{4} x^{6} + 42 \, a c^{5} d^{5} x^{6} + 24 \, b c^{7} d^{3} x^{5} + 42 \, a c^{6} d^{4} x^{5} + \frac {45}{4} \, b c^{8} d^{2} x^{4} + 30 \, a c^{7} d^{3} x^{4} + \frac {10}{3} \, b c^{9} d x^{3} + 15 \, a c^{8} d^{2} x^{3} + \frac {1}{2} \, b c^{10} x^{2} + 5 \, a c^{9} d x^{2} + a c^{10} x \]
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Time = 0.13 (sec) , antiderivative size = 208, normalized size of antiderivative = 5.47 \[ \int (a+b x) (c+d x)^{10} \, dx=x^2\,\left (\frac {b\,c^{10}}{2}+5\,a\,d\,c^9\right )+x^{11}\,\left (\frac {a\,d^{10}}{11}+\frac {10\,b\,c\,d^9}{11}\right )+\frac {b\,d^{10}\,x^{12}}{12}+a\,c^{10}\,x+\frac {5\,c^8\,d\,x^3\,\left (9\,a\,d+2\,b\,c\right )}{3}+\frac {c\,d^8\,x^{10}\,\left (2\,a\,d+9\,b\,c\right )}{2}+\frac {15\,c^7\,d^2\,x^4\,\left (8\,a\,d+3\,b\,c\right )}{4}+6\,c^6\,d^3\,x^5\,\left (7\,a\,d+4\,b\,c\right )+7\,c^5\,d^4\,x^6\,\left (6\,a\,d+5\,b\,c\right )+6\,c^4\,d^5\,x^7\,\left (5\,a\,d+6\,b\,c\right )+\frac {15\,c^3\,d^6\,x^8\,\left (4\,a\,d+7\,b\,c\right )}{4}+\frac {5\,c^2\,d^7\,x^9\,\left (3\,a\,d+8\,b\,c\right )}{3} \]
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