Integrand size = 9, antiderivative size = 30 \[ \int x (a+b x)^{10} \, dx=-\frac {a (a+b x)^{11}}{11 b^2}+\frac {(a+b x)^{12}}{12 b^2} \]
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Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {45} \[ \int x (a+b x)^{10} \, dx=\frac {(a+b x)^{12}}{12 b^2}-\frac {a (a+b x)^{11}}{11 b^2} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a (a+b x)^{10}}{b}+\frac {(a+b x)^{11}}{b}\right ) \, dx \\ & = -\frac {a (a+b x)^{11}}{11 b^2}+\frac {(a+b x)^{12}}{12 b^2} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(128\) vs. \(2(30)=60\).
Time = 0.00 (sec) , antiderivative size = 128, normalized size of antiderivative = 4.27 \[ \int x (a+b x)^{10} \, dx=\frac {a^{10} x^2}{2}+\frac {10}{3} a^9 b x^3+\frac {45}{4} a^8 b^2 x^4+24 a^7 b^3 x^5+35 a^6 b^4 x^6+36 a^5 b^5 x^7+\frac {105}{4} a^4 b^6 x^8+\frac {40}{3} a^3 b^7 x^9+\frac {9}{2} a^2 b^8 x^{10}+\frac {10}{11} a b^9 x^{11}+\frac {b^{10} x^{12}}{12} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(112\) vs. \(2(26)=52\).
Time = 0.18 (sec) , antiderivative size = 113, normalized size of antiderivative = 3.77
method | result | size |
gosper | \(\frac {1}{12} b^{10} x^{12}+\frac {10}{11} a \,b^{9} x^{11}+\frac {9}{2} a^{2} b^{8} x^{10}+\frac {40}{3} a^{3} b^{7} x^{9}+\frac {105}{4} a^{4} b^{6} x^{8}+36 a^{5} b^{5} x^{7}+35 a^{6} b^{4} x^{6}+24 a^{7} b^{3} x^{5}+\frac {45}{4} a^{8} b^{2} x^{4}+\frac {10}{3} a^{9} b \,x^{3}+\frac {1}{2} a^{10} x^{2}\) | \(113\) |
default | \(\frac {1}{12} b^{10} x^{12}+\frac {10}{11} a \,b^{9} x^{11}+\frac {9}{2} a^{2} b^{8} x^{10}+\frac {40}{3} a^{3} b^{7} x^{9}+\frac {105}{4} a^{4} b^{6} x^{8}+36 a^{5} b^{5} x^{7}+35 a^{6} b^{4} x^{6}+24 a^{7} b^{3} x^{5}+\frac {45}{4} a^{8} b^{2} x^{4}+\frac {10}{3} a^{9} b \,x^{3}+\frac {1}{2} a^{10} x^{2}\) | \(113\) |
norman | \(\frac {1}{12} b^{10} x^{12}+\frac {10}{11} a \,b^{9} x^{11}+\frac {9}{2} a^{2} b^{8} x^{10}+\frac {40}{3} a^{3} b^{7} x^{9}+\frac {105}{4} a^{4} b^{6} x^{8}+36 a^{5} b^{5} x^{7}+35 a^{6} b^{4} x^{6}+24 a^{7} b^{3} x^{5}+\frac {45}{4} a^{8} b^{2} x^{4}+\frac {10}{3} a^{9} b \,x^{3}+\frac {1}{2} a^{10} x^{2}\) | \(113\) |
risch | \(\frac {1}{12} b^{10} x^{12}+\frac {10}{11} a \,b^{9} x^{11}+\frac {9}{2} a^{2} b^{8} x^{10}+\frac {40}{3} a^{3} b^{7} x^{9}+\frac {105}{4} a^{4} b^{6} x^{8}+36 a^{5} b^{5} x^{7}+35 a^{6} b^{4} x^{6}+24 a^{7} b^{3} x^{5}+\frac {45}{4} a^{8} b^{2} x^{4}+\frac {10}{3} a^{9} b \,x^{3}+\frac {1}{2} a^{10} x^{2}\) | \(113\) |
parallelrisch | \(\frac {1}{12} b^{10} x^{12}+\frac {10}{11} a \,b^{9} x^{11}+\frac {9}{2} a^{2} b^{8} x^{10}+\frac {40}{3} a^{3} b^{7} x^{9}+\frac {105}{4} a^{4} b^{6} x^{8}+36 a^{5} b^{5} x^{7}+35 a^{6} b^{4} x^{6}+24 a^{7} b^{3} x^{5}+\frac {45}{4} a^{8} b^{2} x^{4}+\frac {10}{3} a^{9} b \,x^{3}+\frac {1}{2} a^{10} x^{2}\) | \(113\) |
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Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (26) = 52\).
Time = 0.21 (sec) , antiderivative size = 112, normalized size of antiderivative = 3.73 \[ \int x (a+b x)^{10} \, dx=\frac {1}{12} \, b^{10} x^{12} + \frac {10}{11} \, a b^{9} x^{11} + \frac {9}{2} \, a^{2} b^{8} x^{10} + \frac {40}{3} \, a^{3} b^{7} x^{9} + \frac {105}{4} \, a^{4} b^{6} x^{8} + 36 \, a^{5} b^{5} x^{7} + 35 \, a^{6} b^{4} x^{6} + 24 \, a^{7} b^{3} x^{5} + \frac {45}{4} \, a^{8} b^{2} x^{4} + \frac {10}{3} \, a^{9} b x^{3} + \frac {1}{2} \, a^{10} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (24) = 48\).
Time = 0.04 (sec) , antiderivative size = 129, normalized size of antiderivative = 4.30 \[ \int x (a+b x)^{10} \, dx=\frac {a^{10} x^{2}}{2} + \frac {10 a^{9} b x^{3}}{3} + \frac {45 a^{8} b^{2} x^{4}}{4} + 24 a^{7} b^{3} x^{5} + 35 a^{6} b^{4} x^{6} + 36 a^{5} b^{5} x^{7} + \frac {105 a^{4} b^{6} x^{8}}{4} + \frac {40 a^{3} b^{7} x^{9}}{3} + \frac {9 a^{2} b^{8} x^{10}}{2} + \frac {10 a b^{9} x^{11}}{11} + \frac {b^{10} x^{12}}{12} \]
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Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (26) = 52\).
Time = 0.21 (sec) , antiderivative size = 112, normalized size of antiderivative = 3.73 \[ \int x (a+b x)^{10} \, dx=\frac {1}{12} \, b^{10} x^{12} + \frac {10}{11} \, a b^{9} x^{11} + \frac {9}{2} \, a^{2} b^{8} x^{10} + \frac {40}{3} \, a^{3} b^{7} x^{9} + \frac {105}{4} \, a^{4} b^{6} x^{8} + 36 \, a^{5} b^{5} x^{7} + 35 \, a^{6} b^{4} x^{6} + 24 \, a^{7} b^{3} x^{5} + \frac {45}{4} \, a^{8} b^{2} x^{4} + \frac {10}{3} \, a^{9} b x^{3} + \frac {1}{2} \, a^{10} x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (26) = 52\).
Time = 0.30 (sec) , antiderivative size = 112, normalized size of antiderivative = 3.73 \[ \int x (a+b x)^{10} \, dx=\frac {1}{12} \, b^{10} x^{12} + \frac {10}{11} \, a b^{9} x^{11} + \frac {9}{2} \, a^{2} b^{8} x^{10} + \frac {40}{3} \, a^{3} b^{7} x^{9} + \frac {105}{4} \, a^{4} b^{6} x^{8} + 36 \, a^{5} b^{5} x^{7} + 35 \, a^{6} b^{4} x^{6} + 24 \, a^{7} b^{3} x^{5} + \frac {45}{4} \, a^{8} b^{2} x^{4} + \frac {10}{3} \, a^{9} b x^{3} + \frac {1}{2} \, a^{10} x^{2} \]
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Time = 0.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83 \[ \int x (a+b x)^{10} \, dx=-\frac {2\,\left (\frac {a\,{\left (a+b\,x\right )}^{11}}{22}-\frac {{\left (a+b\,x\right )}^{12}}{24}\right )}{b^2} \]
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