Integrand size = 13, antiderivative size = 56 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^4} \, dx=-\frac {(b+a x)^9}{11 b x^{11}}+\frac {a (b+a x)^9}{55 b^2 x^{10}}-\frac {a^2 (b+a x)^9}{495 b^3 x^9} \]
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Time = 0.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {269, 47, 37} \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^4} \, dx=-\frac {a^2 (a x+b)^9}{495 b^3 x^9}+\frac {a (a x+b)^9}{55 b^2 x^{10}}-\frac {(a x+b)^9}{11 b x^{11}} \]
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Rule 37
Rule 47
Rule 269
Rubi steps \begin{align*} \text {integral}& = \int \frac {(b+a x)^8}{x^{12}} \, dx \\ & = -\frac {(b+a x)^9}{11 b x^{11}}-\frac {(2 a) \int \frac {(b+a x)^8}{x^{11}} \, dx}{11 b} \\ & = -\frac {(b+a x)^9}{11 b x^{11}}+\frac {a (b+a x)^9}{55 b^2 x^{10}}+\frac {a^2 \int \frac {(b+a x)^8}{x^{10}} \, dx}{55 b^2} \\ & = -\frac {(b+a x)^9}{11 b x^{11}}+\frac {a (b+a x)^9}{55 b^2 x^{10}}-\frac {a^2 (b+a x)^9}{495 b^3 x^9} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.82 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^4} \, dx=-\frac {b^8}{11 x^{11}}-\frac {4 a b^7}{5 x^{10}}-\frac {28 a^2 b^6}{9 x^9}-\frac {7 a^3 b^5}{x^8}-\frac {10 a^4 b^4}{x^7}-\frac {28 a^5 b^3}{3 x^6}-\frac {28 a^6 b^2}{5 x^5}-\frac {2 a^7 b}{x^4}-\frac {a^8}{3 x^3} \]
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Time = 0.02 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.61
method | result | size |
norman | \(\frac {-\frac {1}{3} a^{8} x^{8}-2 x^{7} b \,a^{7}-\frac {28}{5} a^{6} b^{2} x^{6}-\frac {28}{3} a^{5} b^{3} x^{5}-10 a^{4} x^{4} b^{4}-7 a^{3} b^{5} x^{3}-\frac {28}{9} a^{2} b^{6} x^{2}-\frac {4}{5} a \,b^{7} x -\frac {1}{11} b^{8}}{x^{11}}\) | \(90\) |
risch | \(\frac {-\frac {1}{3} a^{8} x^{8}-2 x^{7} b \,a^{7}-\frac {28}{5} a^{6} b^{2} x^{6}-\frac {28}{3} a^{5} b^{3} x^{5}-10 a^{4} x^{4} b^{4}-7 a^{3} b^{5} x^{3}-\frac {28}{9} a^{2} b^{6} x^{2}-\frac {4}{5} a \,b^{7} x -\frac {1}{11} b^{8}}{x^{11}}\) | \(90\) |
gosper | \(-\frac {165 a^{8} x^{8}+990 x^{7} b \,a^{7}+2772 a^{6} b^{2} x^{6}+4620 a^{5} b^{3} x^{5}+4950 a^{4} x^{4} b^{4}+3465 a^{3} b^{5} x^{3}+1540 a^{2} b^{6} x^{2}+396 a \,b^{7} x +45 b^{8}}{495 x^{11}}\) | \(91\) |
default | \(-\frac {28 a^{5} b^{3}}{3 x^{6}}-\frac {10 a^{4} b^{4}}{x^{7}}-\frac {7 a^{3} b^{5}}{x^{8}}-\frac {4 a \,b^{7}}{5 x^{10}}-\frac {a^{8}}{3 x^{3}}-\frac {b^{8}}{11 x^{11}}-\frac {2 a^{7} b}{x^{4}}-\frac {28 a^{6} b^{2}}{5 x^{5}}-\frac {28 a^{2} b^{6}}{9 x^{9}}\) | \(91\) |
parallelrisch | \(\frac {-165 a^{8} x^{8}-990 x^{7} b \,a^{7}-2772 a^{6} b^{2} x^{6}-4620 a^{5} b^{3} x^{5}-4950 a^{4} x^{4} b^{4}-3465 a^{3} b^{5} x^{3}-1540 a^{2} b^{6} x^{2}-396 a \,b^{7} x -45 b^{8}}{495 x^{11}}\) | \(91\) |
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Time = 0.25 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.61 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^4} \, dx=-\frac {165 \, a^{8} x^{8} + 990 \, a^{7} b x^{7} + 2772 \, a^{6} b^{2} x^{6} + 4620 \, a^{5} b^{3} x^{5} + 4950 \, a^{4} b^{4} x^{4} + 3465 \, a^{3} b^{5} x^{3} + 1540 \, a^{2} b^{6} x^{2} + 396 \, a b^{7} x + 45 \, b^{8}}{495 \, x^{11}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (48) = 96\).
Time = 0.36 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.73 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^4} \, dx=\frac {- 165 a^{8} x^{8} - 990 a^{7} b x^{7} - 2772 a^{6} b^{2} x^{6} - 4620 a^{5} b^{3} x^{5} - 4950 a^{4} b^{4} x^{4} - 3465 a^{3} b^{5} x^{3} - 1540 a^{2} b^{6} x^{2} - 396 a b^{7} x - 45 b^{8}}{495 x^{11}} \]
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Time = 0.19 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.61 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^4} \, dx=-\frac {165 \, a^{8} x^{8} + 990 \, a^{7} b x^{7} + 2772 \, a^{6} b^{2} x^{6} + 4620 \, a^{5} b^{3} x^{5} + 4950 \, a^{4} b^{4} x^{4} + 3465 \, a^{3} b^{5} x^{3} + 1540 \, a^{2} b^{6} x^{2} + 396 \, a b^{7} x + 45 \, b^{8}}{495 \, x^{11}} \]
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Time = 0.27 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.61 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^4} \, dx=-\frac {165 \, a^{8} x^{8} + 990 \, a^{7} b x^{7} + 2772 \, a^{6} b^{2} x^{6} + 4620 \, a^{5} b^{3} x^{5} + 4950 \, a^{4} b^{4} x^{4} + 3465 \, a^{3} b^{5} x^{3} + 1540 \, a^{2} b^{6} x^{2} + 396 \, a b^{7} x + 45 \, b^{8}}{495 \, x^{11}} \]
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Time = 5.54 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.61 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^4} \, dx=-\frac {\frac {a^8\,x^8}{3}+2\,a^7\,b\,x^7+\frac {28\,a^6\,b^2\,x^6}{5}+\frac {28\,a^5\,b^3\,x^5}{3}+10\,a^4\,b^4\,x^4+7\,a^3\,b^5\,x^3+\frac {28\,a^2\,b^6\,x^2}{9}+\frac {4\,a\,b^7\,x}{5}+\frac {b^8}{11}}{x^{11}} \]
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