Integrand size = 30, antiderivative size = 23 \[ \int \frac {x^{-1+n} \left (b+2 c x^n\right )}{\left (a+b x^n+c x^{2 n}\right )^8} \, dx=-\frac {1}{7 n \left (a+b x^n+c x^{2 n}\right )^7} \]
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Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1482, 643} \[ \int \frac {x^{-1+n} \left (b+2 c x^n\right )}{\left (a+b x^n+c x^{2 n}\right )^8} \, dx=-\frac {1}{7 n \left (a+b x^n+c x^{2 n}\right )^7} \]
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Rule 643
Rule 1482
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {b+2 c x}{\left (a+b x+c x^2\right )^8} \, dx,x,x^n\right )}{n} \\ & = -\frac {1}{7 n \left (a+b x^n+c x^{2 n}\right )^7} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {x^{-1+n} \left (b+2 c x^n\right )}{\left (a+b x^n+c x^{2 n}\right )^8} \, dx=-\frac {1}{7 n \left (a+x^n \left (b+c x^n\right )\right )^7} \]
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Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96
\[-\frac {1}{7 n \left (a +b \,x^{n}+c \,x^{2 n}\right )^{7}}\]
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Leaf count of result is larger than twice the leaf count of optimal. 394 vs. \(2 (21) = 42\).
Time = 0.35 (sec) , antiderivative size = 394, normalized size of antiderivative = 17.13 \[ \int \frac {x^{-1+n} \left (b+2 c x^n\right )}{\left (a+b x^n+c x^{2 n}\right )^8} \, dx=-\frac {1}{7 \, {\left (c^{7} n x^{14 \, n} + 7 \, b c^{6} n x^{13 \, n} + 7 \, a^{6} b n x^{n} + a^{7} n + 7 \, {\left (3 \, b^{2} c^{5} + a c^{6}\right )} n x^{12 \, n} + 7 \, {\left (5 \, b^{3} c^{4} + 6 \, a b c^{5}\right )} n x^{11 \, n} + 7 \, {\left (5 \, b^{4} c^{3} + 15 \, a b^{2} c^{4} + 3 \, a^{2} c^{5}\right )} n x^{10 \, n} + 7 \, {\left (3 \, b^{5} c^{2} + 20 \, a b^{3} c^{3} + 15 \, a^{2} b c^{4}\right )} n x^{9 \, n} + 7 \, {\left (b^{6} c + 15 \, a b^{4} c^{2} + 30 \, a^{2} b^{2} c^{3} + 5 \, a^{3} c^{4}\right )} n x^{8 \, n} + {\left (b^{7} + 42 \, a b^{5} c + 210 \, a^{2} b^{3} c^{2} + 140 \, a^{3} b c^{3}\right )} n x^{7 \, n} + 7 \, {\left (a b^{6} + 15 \, a^{2} b^{4} c + 30 \, a^{3} b^{2} c^{2} + 5 \, a^{4} c^{3}\right )} n x^{6 \, n} + 7 \, {\left (3 \, a^{2} b^{5} + 20 \, a^{3} b^{3} c + 15 \, a^{4} b c^{2}\right )} n x^{5 \, n} + 7 \, {\left (5 \, a^{3} b^{4} + 15 \, a^{4} b^{2} c + 3 \, a^{5} c^{2}\right )} n x^{4 \, n} + 7 \, {\left (5 \, a^{4} b^{3} + 6 \, a^{5} b c\right )} n x^{3 \, n} + 7 \, {\left (3 \, a^{5} b^{2} + a^{6} c\right )} n x^{2 \, n}\right )}} \]
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Timed out. \[ \int \frac {x^{-1+n} \left (b+2 c x^n\right )}{\left (a+b x^n+c x^{2 n}\right )^8} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 416 vs. \(2 (21) = 42\).
Time = 0.54 (sec) , antiderivative size = 416, normalized size of antiderivative = 18.09 \[ \int \frac {x^{-1+n} \left (b+2 c x^n\right )}{\left (a+b x^n+c x^{2 n}\right )^8} \, dx=-\frac {1}{7 \, {\left (c^{7} n x^{14 \, n} + 7 \, b c^{6} n x^{13 \, n} + 7 \, a^{6} b n x^{n} + a^{7} n + 7 \, {\left (3 \, b^{2} c^{5} n + a c^{6} n\right )} x^{12 \, n} + 7 \, {\left (5 \, b^{3} c^{4} n + 6 \, a b c^{5} n\right )} x^{11 \, n} + 7 \, {\left (5 \, b^{4} c^{3} n + 15 \, a b^{2} c^{4} n + 3 \, a^{2} c^{5} n\right )} x^{10 \, n} + 7 \, {\left (3 \, b^{5} c^{2} n + 20 \, a b^{3} c^{3} n + 15 \, a^{2} b c^{4} n\right )} x^{9 \, n} + 7 \, {\left (b^{6} c n + 15 \, a b^{4} c^{2} n + 30 \, a^{2} b^{2} c^{3} n + 5 \, a^{3} c^{4} n\right )} x^{8 \, n} + {\left (b^{7} n + 42 \, a b^{5} c n + 210 \, a^{2} b^{3} c^{2} n + 140 \, a^{3} b c^{3} n\right )} x^{7 \, n} + 7 \, {\left (a b^{6} n + 15 \, a^{2} b^{4} c n + 30 \, a^{3} b^{2} c^{2} n + 5 \, a^{4} c^{3} n\right )} x^{6 \, n} + 7 \, {\left (3 \, a^{2} b^{5} n + 20 \, a^{3} b^{3} c n + 15 \, a^{4} b c^{2} n\right )} x^{5 \, n} + 7 \, {\left (5 \, a^{3} b^{4} n + 15 \, a^{4} b^{2} c n + 3 \, a^{5} c^{2} n\right )} x^{4 \, n} + 7 \, {\left (5 \, a^{4} b^{3} n + 6 \, a^{5} b c n\right )} x^{3 \, n} + 7 \, {\left (3 \, a^{5} b^{2} n + a^{6} c n\right )} x^{2 \, n}\right )}} \]
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Time = 0.31 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {x^{-1+n} \left (b+2 c x^n\right )}{\left (a+b x^n+c x^{2 n}\right )^8} \, dx=-\frac {1}{7 \, {\left (c x^{2 \, n} + b x^{n} + a\right )}^{7} n} \]
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Time = 22.40 (sec) , antiderivative size = 496, normalized size of antiderivative = 21.57 \[ \int \frac {x^{-1+n} \left (b+2 c x^n\right )}{\left (a+b x^n+c x^{2 n}\right )^8} \, dx=-\frac {1}{7\,a^7\,n+7\,b^7\,n\,x^{7\,n}+7\,c^7\,n\,x^{14\,n}+49\,a^6\,b\,n\,x^n+49\,a\,b^6\,n\,x^{6\,n}+49\,a^6\,c\,n\,x^{2\,n}+49\,a\,c^6\,n\,x^{12\,n}+49\,b^6\,c\,n\,x^{8\,n}+49\,b\,c^6\,n\,x^{13\,n}+147\,a^5\,b^2\,n\,x^{2\,n}+245\,a^4\,b^3\,n\,x^{3\,n}+245\,a^3\,b^4\,n\,x^{4\,n}+147\,a^2\,b^5\,n\,x^{5\,n}+147\,a^5\,c^2\,n\,x^{4\,n}+245\,a^4\,c^3\,n\,x^{6\,n}+245\,a^3\,c^4\,n\,x^{8\,n}+147\,a^2\,c^5\,n\,x^{10\,n}+147\,b^5\,c^2\,n\,x^{9\,n}+245\,b^4\,c^3\,n\,x^{10\,n}+245\,b^3\,c^4\,n\,x^{11\,n}+147\,b^2\,c^5\,n\,x^{12\,n}+735\,a^4\,b^2\,c\,n\,x^{4\,n}+980\,a^3\,b^3\,c\,n\,x^{5\,n}+735\,a^4\,b\,c^2\,n\,x^{5\,n}+735\,a^2\,b^4\,c\,n\,x^{6\,n}+980\,a^3\,b\,c^3\,n\,x^{7\,n}+735\,a\,b^4\,c^2\,n\,x^{8\,n}+980\,a\,b^3\,c^3\,n\,x^{9\,n}+735\,a^2\,b\,c^4\,n\,x^{9\,n}+735\,a\,b^2\,c^4\,n\,x^{10\,n}+1470\,a^3\,b^2\,c^2\,n\,x^{6\,n}+1470\,a^2\,b^3\,c^2\,n\,x^{7\,n}+1470\,a^2\,b^2\,c^3\,n\,x^{8\,n}+294\,a^5\,b\,c\,n\,x^{3\,n}+294\,a\,b^5\,c\,n\,x^{7\,n}+294\,a\,b\,c^5\,n\,x^{11\,n}} \]
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