Integrand size = 11, antiderivative size = 64 \[ \int x^3 (a+b x)^{10} \, dx=-\frac {a^3 (a+b x)^{11}}{11 b^4}+\frac {a^2 (a+b x)^{12}}{4 b^4}-\frac {3 a (a+b x)^{13}}{13 b^4}+\frac {(a+b x)^{14}}{14 b^4} \]
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Time = 0.02 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int x^3 (a+b x)^{10} \, dx=-\frac {a^3 (a+b x)^{11}}{11 b^4}+\frac {a^2 (a+b x)^{12}}{4 b^4}+\frac {(a+b x)^{14}}{14 b^4}-\frac {3 a (a+b x)^{13}}{13 b^4} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a^3 (a+b x)^{10}}{b^3}+\frac {3 a^2 (a+b x)^{11}}{b^3}-\frac {3 a (a+b x)^{12}}{b^3}+\frac {(a+b x)^{13}}{b^3}\right ) \, dx \\ & = -\frac {a^3 (a+b x)^{11}}{11 b^4}+\frac {a^2 (a+b x)^{12}}{4 b^4}-\frac {3 a (a+b x)^{13}}{13 b^4}+\frac {(a+b x)^{14}}{14 b^4} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 128, normalized size of antiderivative = 2.00 \[ \int x^3 (a+b x)^{10} \, dx=\frac {a^{10} x^4}{4}+2 a^9 b x^5+\frac {15}{2} a^8 b^2 x^6+\frac {120}{7} a^7 b^3 x^7+\frac {105}{4} a^6 b^4 x^8+28 a^5 b^5 x^9+21 a^4 b^6 x^{10}+\frac {120}{11} a^3 b^7 x^{11}+\frac {15}{4} a^2 b^8 x^{12}+\frac {10}{13} a b^9 x^{13}+\frac {b^{10} x^{14}}{14} \]
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Time = 0.17 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.77
method | result | size |
gosper | \(\frac {1}{4} a^{10} x^{4}+2 a^{9} b \,x^{5}+\frac {15}{2} a^{8} b^{2} x^{6}+\frac {120}{7} a^{7} b^{3} x^{7}+\frac {105}{4} a^{6} b^{4} x^{8}+28 a^{5} b^{5} x^{9}+21 a^{4} b^{6} x^{10}+\frac {120}{11} a^{3} b^{7} x^{11}+\frac {15}{4} a^{2} b^{8} x^{12}+\frac {10}{13} a \,b^{9} x^{13}+\frac {1}{14} b^{10} x^{14}\) | \(113\) |
default | \(\frac {1}{4} a^{10} x^{4}+2 a^{9} b \,x^{5}+\frac {15}{2} a^{8} b^{2} x^{6}+\frac {120}{7} a^{7} b^{3} x^{7}+\frac {105}{4} a^{6} b^{4} x^{8}+28 a^{5} b^{5} x^{9}+21 a^{4} b^{6} x^{10}+\frac {120}{11} a^{3} b^{7} x^{11}+\frac {15}{4} a^{2} b^{8} x^{12}+\frac {10}{13} a \,b^{9} x^{13}+\frac {1}{14} b^{10} x^{14}\) | \(113\) |
norman | \(\frac {1}{4} a^{10} x^{4}+2 a^{9} b \,x^{5}+\frac {15}{2} a^{8} b^{2} x^{6}+\frac {120}{7} a^{7} b^{3} x^{7}+\frac {105}{4} a^{6} b^{4} x^{8}+28 a^{5} b^{5} x^{9}+21 a^{4} b^{6} x^{10}+\frac {120}{11} a^{3} b^{7} x^{11}+\frac {15}{4} a^{2} b^{8} x^{12}+\frac {10}{13} a \,b^{9} x^{13}+\frac {1}{14} b^{10} x^{14}\) | \(113\) |
risch | \(\frac {1}{4} a^{10} x^{4}+2 a^{9} b \,x^{5}+\frac {15}{2} a^{8} b^{2} x^{6}+\frac {120}{7} a^{7} b^{3} x^{7}+\frac {105}{4} a^{6} b^{4} x^{8}+28 a^{5} b^{5} x^{9}+21 a^{4} b^{6} x^{10}+\frac {120}{11} a^{3} b^{7} x^{11}+\frac {15}{4} a^{2} b^{8} x^{12}+\frac {10}{13} a \,b^{9} x^{13}+\frac {1}{14} b^{10} x^{14}\) | \(113\) |
parallelrisch | \(\frac {1}{4} a^{10} x^{4}+2 a^{9} b \,x^{5}+\frac {15}{2} a^{8} b^{2} x^{6}+\frac {120}{7} a^{7} b^{3} x^{7}+\frac {105}{4} a^{6} b^{4} x^{8}+28 a^{5} b^{5} x^{9}+21 a^{4} b^{6} x^{10}+\frac {120}{11} a^{3} b^{7} x^{11}+\frac {15}{4} a^{2} b^{8} x^{12}+\frac {10}{13} a \,b^{9} x^{13}+\frac {1}{14} b^{10} x^{14}\) | \(113\) |
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Time = 0.22 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.75 \[ \int x^3 (a+b x)^{10} \, dx=\frac {1}{14} \, b^{10} x^{14} + \frac {10}{13} \, a b^{9} x^{13} + \frac {15}{4} \, a^{2} b^{8} x^{12} + \frac {120}{11} \, a^{3} b^{7} x^{11} + 21 \, a^{4} b^{6} x^{10} + 28 \, a^{5} b^{5} x^{9} + \frac {105}{4} \, a^{6} b^{4} x^{8} + \frac {120}{7} \, a^{7} b^{3} x^{7} + \frac {15}{2} \, a^{8} b^{2} x^{6} + 2 \, a^{9} b x^{5} + \frac {1}{4} \, a^{10} x^{4} \]
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Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (56) = 112\).
Time = 0.03 (sec) , antiderivative size = 129, normalized size of antiderivative = 2.02 \[ \int x^3 (a+b x)^{10} \, dx=\frac {a^{10} x^{4}}{4} + 2 a^{9} b x^{5} + \frac {15 a^{8} b^{2} x^{6}}{2} + \frac {120 a^{7} b^{3} x^{7}}{7} + \frac {105 a^{6} b^{4} x^{8}}{4} + 28 a^{5} b^{5} x^{9} + 21 a^{4} b^{6} x^{10} + \frac {120 a^{3} b^{7} x^{11}}{11} + \frac {15 a^{2} b^{8} x^{12}}{4} + \frac {10 a b^{9} x^{13}}{13} + \frac {b^{10} x^{14}}{14} \]
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Time = 0.21 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.75 \[ \int x^3 (a+b x)^{10} \, dx=\frac {1}{14} \, b^{10} x^{14} + \frac {10}{13} \, a b^{9} x^{13} + \frac {15}{4} \, a^{2} b^{8} x^{12} + \frac {120}{11} \, a^{3} b^{7} x^{11} + 21 \, a^{4} b^{6} x^{10} + 28 \, a^{5} b^{5} x^{9} + \frac {105}{4} \, a^{6} b^{4} x^{8} + \frac {120}{7} \, a^{7} b^{3} x^{7} + \frac {15}{2} \, a^{8} b^{2} x^{6} + 2 \, a^{9} b x^{5} + \frac {1}{4} \, a^{10} x^{4} \]
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Time = 0.31 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.75 \[ \int x^3 (a+b x)^{10} \, dx=\frac {1}{14} \, b^{10} x^{14} + \frac {10}{13} \, a b^{9} x^{13} + \frac {15}{4} \, a^{2} b^{8} x^{12} + \frac {120}{11} \, a^{3} b^{7} x^{11} + 21 \, a^{4} b^{6} x^{10} + 28 \, a^{5} b^{5} x^{9} + \frac {105}{4} \, a^{6} b^{4} x^{8} + \frac {120}{7} \, a^{7} b^{3} x^{7} + \frac {15}{2} \, a^{8} b^{2} x^{6} + 2 \, a^{9} b x^{5} + \frac {1}{4} \, a^{10} x^{4} \]
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Time = 0.08 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.75 \[ \int x^3 (a+b x)^{10} \, dx=\frac {a^{10}\,x^4}{4}+2\,a^9\,b\,x^5+\frac {15\,a^8\,b^2\,x^6}{2}+\frac {120\,a^7\,b^3\,x^7}{7}+\frac {105\,a^6\,b^4\,x^8}{4}+28\,a^5\,b^5\,x^9+21\,a^4\,b^6\,x^{10}+\frac {120\,a^3\,b^7\,x^{11}}{11}+\frac {15\,a^2\,b^8\,x^{12}}{4}+\frac {10\,a\,b^9\,x^{13}}{13}+\frac {b^{10}\,x^{14}}{14} \]
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