Integrand size = 17, antiderivative size = 77 \[ \int x^{-1-11 n} \left (a+b x^n\right )^8 \, dx=-\frac {x^{-11 n} \left (a+b x^n\right )^9}{11 a n}+\frac {b x^{-10 n} \left (a+b x^n\right )^9}{55 a^2 n}-\frac {b^2 x^{-9 n} \left (a+b x^n\right )^9}{495 a^3 n} \]
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Time = 0.02 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {272, 47, 37} \[ \int x^{-1-11 n} \left (a+b x^n\right )^8 \, dx=-\frac {b^2 x^{-9 n} \left (a+b x^n\right )^9}{495 a^3 n}+\frac {b x^{-10 n} \left (a+b x^n\right )^9}{55 a^2 n}-\frac {x^{-11 n} \left (a+b x^n\right )^9}{11 a n} \]
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Rule 37
Rule 47
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(a+b x)^8}{x^{12}} \, dx,x,x^n\right )}{n} \\ & = -\frac {x^{-11 n} \left (a+b x^n\right )^9}{11 a n}-\frac {(2 b) \text {Subst}\left (\int \frac {(a+b x)^8}{x^{11}} \, dx,x,x^n\right )}{11 a n} \\ & = -\frac {x^{-11 n} \left (a+b x^n\right )^9}{11 a n}+\frac {b x^{-10 n} \left (a+b x^n\right )^9}{55 a^2 n}+\frac {b^2 \text {Subst}\left (\int \frac {(a+b x)^8}{x^{10}} \, dx,x,x^n\right )}{55 a^2 n} \\ & = -\frac {x^{-11 n} \left (a+b x^n\right )^9}{11 a n}+\frac {b x^{-10 n} \left (a+b x^n\right )^9}{55 a^2 n}-\frac {b^2 x^{-9 n} \left (a+b x^n\right )^9}{495 a^3 n} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.47 \[ \int x^{-1-11 n} \left (a+b x^n\right )^8 \, dx=\frac {x^{-11 n} \left (-45 a^8-396 a^7 b x^n-1540 a^6 b^2 x^{2 n}-3465 a^5 b^3 x^{3 n}-4950 a^4 b^4 x^{4 n}-4620 a^3 b^5 x^{5 n}-2772 a^2 b^6 x^{6 n}-990 a b^7 x^{7 n}-165 b^8 x^{8 n}\right )}{495 n} \]
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Time = 8.64 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.77
method | result | size |
risch | \(-\frac {b^{8} x^{-3 n}}{3 n}-\frac {2 a \,b^{7} x^{-4 n}}{n}-\frac {28 a^{2} b^{6} x^{-5 n}}{5 n}-\frac {28 a^{3} b^{5} x^{-6 n}}{3 n}-\frac {10 a^{4} b^{4} x^{-7 n}}{n}-\frac {7 a^{5} b^{3} x^{-8 n}}{n}-\frac {28 a^{6} b^{2} x^{-9 n}}{9 n}-\frac {4 a^{7} b \,x^{-10 n}}{5 n}-\frac {a^{8} x^{-11 n}}{11 n}\) | \(136\) |
parallelrisch | \(\frac {-165 b^{8} x^{-1-11 n} x^{8 n} x -990 a \,b^{7} x^{-1-11 n} x^{7 n} x -2772 a^{2} b^{6} x^{-1-11 n} x^{6 n} x -4620 a^{3} b^{5} x^{-1-11 n} x^{5 n} x -4950 a^{4} b^{4} x^{-1-11 n} x^{4 n} x -3465 a^{5} b^{3} x^{-1-11 n} x^{3 n} x -1540 a^{6} b^{2} x^{-1-11 n} x^{2 n} x -396 a^{7} b \,x^{-1-11 n} x^{n} x -45 a^{8} x^{-1-11 n} x}{495 n}\) | \(179\) |
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Time = 0.28 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.47 \[ \int x^{-1-11 n} \left (a+b x^n\right )^8 \, dx=-\frac {165 \, b^{8} x^{8 \, n} + 990 \, a b^{7} x^{7 \, n} + 2772 \, a^{2} b^{6} x^{6 \, n} + 4620 \, a^{3} b^{5} x^{5 \, n} + 4950 \, a^{4} b^{4} x^{4 \, n} + 3465 \, a^{5} b^{3} x^{3 \, n} + 1540 \, a^{6} b^{2} x^{2 \, n} + 396 \, a^{7} b x^{n} + 45 \, a^{8}}{495 \, n x^{11 \, n}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 228 vs. \(2 (63) = 126\).
Time = 1.98 (sec) , antiderivative size = 228, normalized size of antiderivative = 2.96 \[ \int x^{-1-11 n} \left (a+b x^n\right )^8 \, dx=\begin {cases} - \frac {a^{8} x x^{- 11 n - 1}}{11 n} - \frac {4 a^{7} b x x^{n} x^{- 11 n - 1}}{5 n} - \frac {28 a^{6} b^{2} x x^{2 n} x^{- 11 n - 1}}{9 n} - \frac {7 a^{5} b^{3} x x^{3 n} x^{- 11 n - 1}}{n} - \frac {10 a^{4} b^{4} x x^{4 n} x^{- 11 n - 1}}{n} - \frac {28 a^{3} b^{5} x x^{5 n} x^{- 11 n - 1}}{3 n} - \frac {28 a^{2} b^{6} x x^{6 n} x^{- 11 n - 1}}{5 n} - \frac {2 a b^{7} x x^{7 n} x^{- 11 n - 1}}{n} - \frac {b^{8} x x^{8 n} x^{- 11 n - 1}}{3 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{8} \log {\left (x \right )} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.99 \[ \int x^{-1-11 n} \left (a+b x^n\right )^8 \, dx=-\frac {a^{8}}{11 \, n x^{11 \, n}} - \frac {4 \, a^{7} b}{5 \, n x^{10 \, n}} - \frac {28 \, a^{6} b^{2}}{9 \, n x^{9 \, n}} - \frac {7 \, a^{5} b^{3}}{n x^{8 \, n}} - \frac {10 \, a^{4} b^{4}}{n x^{7 \, n}} - \frac {28 \, a^{3} b^{5}}{3 \, n x^{6 \, n}} - \frac {28 \, a^{2} b^{6}}{5 \, n x^{5 \, n}} - \frac {2 \, a b^{7}}{n x^{4 \, n}} - \frac {b^{8}}{3 \, n x^{3 \, n}} \]
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Time = 0.32 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.47 \[ \int x^{-1-11 n} \left (a+b x^n\right )^8 \, dx=-\frac {165 \, b^{8} x^{8 \, n} + 990 \, a b^{7} x^{7 \, n} + 2772 \, a^{2} b^{6} x^{6 \, n} + 4620 \, a^{3} b^{5} x^{5 \, n} + 4950 \, a^{4} b^{4} x^{4 \, n} + 3465 \, a^{5} b^{3} x^{3 \, n} + 1540 \, a^{6} b^{2} x^{2 \, n} + 396 \, a^{7} b x^{n} + 45 \, a^{8}}{495 \, n x^{11 \, n}} \]
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Time = 5.90 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.99 \[ \int x^{-1-11 n} \left (a+b x^n\right )^8 \, dx=-\frac {a^8}{11\,n\,x^{11\,n}}-\frac {b^8}{3\,n\,x^{3\,n}}-\frac {28\,a^2\,b^6}{5\,n\,x^{5\,n}}-\frac {28\,a^3\,b^5}{3\,n\,x^{6\,n}}-\frac {10\,a^4\,b^4}{n\,x^{7\,n}}-\frac {7\,a^5\,b^3}{n\,x^{8\,n}}-\frac {28\,a^6\,b^2}{9\,n\,x^{9\,n}}-\frac {2\,a\,b^7}{n\,x^{4\,n}}-\frac {4\,a^7\,b}{5\,n\,x^{10\,n}} \]
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