Integrand size = 13, antiderivative size = 98 \[ \int \frac {\left (a+b x^3\right )^8}{x^8} \, dx=-\frac {a^8}{7 x^7}-\frac {2 a^7 b}{x^4}-\frac {28 a^6 b^2}{x}+28 a^5 b^3 x^2+14 a^4 b^4 x^5+7 a^3 b^5 x^8+\frac {28}{11} a^2 b^6 x^{11}+\frac {4}{7} a b^7 x^{14}+\frac {b^8 x^{17}}{17} \]
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Time = 0.03 (sec) , antiderivative size = 98, 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 = {276} \[ \int \frac {\left (a+b x^3\right )^8}{x^8} \, dx=-\frac {a^8}{7 x^7}-\frac {2 a^7 b}{x^4}-\frac {28 a^6 b^2}{x}+28 a^5 b^3 x^2+14 a^4 b^4 x^5+7 a^3 b^5 x^8+\frac {28}{11} a^2 b^6 x^{11}+\frac {4}{7} a b^7 x^{14}+\frac {b^8 x^{17}}{17} \]
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Rule 276
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^8}{x^8}+\frac {8 a^7 b}{x^5}+\frac {28 a^6 b^2}{x^2}+56 a^5 b^3 x+70 a^4 b^4 x^4+56 a^3 b^5 x^7+28 a^2 b^6 x^{10}+8 a b^7 x^{13}+b^8 x^{16}\right ) \, dx \\ & = -\frac {a^8}{7 x^7}-\frac {2 a^7 b}{x^4}-\frac {28 a^6 b^2}{x}+28 a^5 b^3 x^2+14 a^4 b^4 x^5+7 a^3 b^5 x^8+\frac {28}{11} a^2 b^6 x^{11}+\frac {4}{7} a b^7 x^{14}+\frac {b^8 x^{17}}{17} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^3\right )^8}{x^8} \, dx=-\frac {a^8}{7 x^7}-\frac {2 a^7 b}{x^4}-\frac {28 a^6 b^2}{x}+28 a^5 b^3 x^2+14 a^4 b^4 x^5+7 a^3 b^5 x^8+\frac {28}{11} a^2 b^6 x^{11}+\frac {4}{7} a b^7 x^{14}+\frac {b^8 x^{17}}{17} \]
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Time = 3.68 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.93
method | result | size |
default | \(-\frac {a^{8}}{7 x^{7}}-\frac {2 a^{7} b}{x^{4}}-\frac {28 a^{6} b^{2}}{x}+28 a^{5} b^{3} x^{2}+14 a^{4} b^{4} x^{5}+7 a^{3} b^{5} x^{8}+\frac {28 a^{2} b^{6} x^{11}}{11}+\frac {4 x^{14} b^{7} a}{7}+\frac {b^{8} x^{17}}{17}\) | \(91\) |
norman | \(\frac {-\frac {1}{7} a^{8}+\frac {28}{11} a^{2} b^{6} x^{18}+\frac {4}{7} a \,b^{7} x^{21}+\frac {1}{17} b^{8} x^{24}-2 x^{3} b \,a^{7}+14 a^{4} b^{4} x^{12}+7 a^{3} b^{5} x^{15}-28 a^{6} b^{2} x^{6}+28 x^{9} b^{3} a^{5}}{x^{7}}\) | \(92\) |
gosper | \(-\frac {-77 b^{8} x^{24}-748 a \,b^{7} x^{21}-3332 a^{2} b^{6} x^{18}-9163 a^{3} b^{5} x^{15}-18326 a^{4} b^{4} x^{12}-36652 x^{9} b^{3} a^{5}+36652 a^{6} b^{2} x^{6}+2618 x^{3} b \,a^{7}+187 a^{8}}{1309 x^{7}}\) | \(93\) |
risch | \(\frac {b^{8} x^{17}}{17}+\frac {4 x^{14} b^{7} a}{7}+\frac {28 a^{2} b^{6} x^{11}}{11}+7 a^{3} b^{5} x^{8}+14 a^{4} b^{4} x^{5}+28 a^{5} b^{3} x^{2}+\frac {-28 a^{6} b^{2} x^{6}-2 x^{3} b \,a^{7}-\frac {1}{7} a^{8}}{x^{7}}\) | \(93\) |
parallelrisch | \(\frac {77 b^{8} x^{24}+748 a \,b^{7} x^{21}+3332 a^{2} b^{6} x^{18}+9163 a^{3} b^{5} x^{15}+18326 a^{4} b^{4} x^{12}+36652 x^{9} b^{3} a^{5}-36652 a^{6} b^{2} x^{6}-2618 x^{3} b \,a^{7}-187 a^{8}}{1309 x^{7}}\) | \(93\) |
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Time = 0.28 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b x^3\right )^8}{x^8} \, dx=\frac {77 \, b^{8} x^{24} + 748 \, a b^{7} x^{21} + 3332 \, a^{2} b^{6} x^{18} + 9163 \, a^{3} b^{5} x^{15} + 18326 \, a^{4} b^{4} x^{12} + 36652 \, a^{5} b^{3} x^{9} - 36652 \, a^{6} b^{2} x^{6} - 2618 \, a^{7} b x^{3} - 187 \, a^{8}}{1309 \, x^{7}} \]
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Time = 0.14 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b x^3\right )^8}{x^8} \, dx=28 a^{5} b^{3} x^{2} + 14 a^{4} b^{4} x^{5} + 7 a^{3} b^{5} x^{8} + \frac {28 a^{2} b^{6} x^{11}}{11} + \frac {4 a b^{7} x^{14}}{7} + \frac {b^{8} x^{17}}{17} + \frac {- a^{8} - 14 a^{7} b x^{3} - 196 a^{6} b^{2} x^{6}}{7 x^{7}} \]
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Time = 0.19 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b x^3\right )^8}{x^8} \, dx=\frac {1}{17} \, b^{8} x^{17} + \frac {4}{7} \, a b^{7} x^{14} + \frac {28}{11} \, a^{2} b^{6} x^{11} + 7 \, a^{3} b^{5} x^{8} + 14 \, a^{4} b^{4} x^{5} + 28 \, a^{5} b^{3} x^{2} - \frac {196 \, a^{6} b^{2} x^{6} + 14 \, a^{7} b x^{3} + a^{8}}{7 \, x^{7}} \]
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Time = 0.28 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b x^3\right )^8}{x^8} \, dx=\frac {1}{17} \, b^{8} x^{17} + \frac {4}{7} \, a b^{7} x^{14} + \frac {28}{11} \, a^{2} b^{6} x^{11} + 7 \, a^{3} b^{5} x^{8} + 14 \, a^{4} b^{4} x^{5} + 28 \, a^{5} b^{3} x^{2} - \frac {196 \, a^{6} b^{2} x^{6} + 14 \, a^{7} b x^{3} + a^{8}}{7 \, x^{7}} \]
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Time = 0.05 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b x^3\right )^8}{x^8} \, dx=\frac {b^8\,x^{17}}{17}-\frac {\frac {a^8}{7}+2\,a^7\,b\,x^3+28\,a^6\,b^2\,x^6}{x^7}+\frac {4\,a\,b^7\,x^{14}}{7}+28\,a^5\,b^3\,x^2+14\,a^4\,b^4\,x^5+7\,a^3\,b^5\,x^8+\frac {28\,a^2\,b^6\,x^{11}}{11} \]
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