Optimal. Leaf size=133 \[ -\frac {a^8 x^{-5 n}}{5 n}-\frac {2 a^7 b x^{-4 n}}{n}-\frac {28 a^6 b^2 x^{-3 n}}{3 n}-\frac {28 a^5 b^3 x^{-2 n}}{n}-\frac {70 a^4 b^4 x^{-n}}{n}+56 a^3 b^5 \log (x)+\frac {28 a^2 b^6 x^n}{n}+\frac {4 a b^7 x^{2 n}}{n}+\frac {b^8 x^{3 n}}{3 n} \]
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Rubi [A] time = 0.06, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {266, 43} \[ -\frac {28 a^6 b^2 x^{-3 n}}{3 n}-\frac {28 a^5 b^3 x^{-2 n}}{n}-\frac {70 a^4 b^4 x^{-n}}{n}+\frac {28 a^2 b^6 x^n}{n}+56 a^3 b^5 \log (x)-\frac {2 a^7 b x^{-4 n}}{n}-\frac {a^8 x^{-5 n}}{5 n}+\frac {4 a b^7 x^{2 n}}{n}+\frac {b^8 x^{3 n}}{3 n} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^{-1-5 n} \left (a+b x^n\right )^8 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^8}{x^6} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (28 a^2 b^6+\frac {a^8}{x^6}+\frac {8 a^7 b}{x^5}+\frac {28 a^6 b^2}{x^4}+\frac {56 a^5 b^3}{x^3}+\frac {70 a^4 b^4}{x^2}+\frac {56 a^3 b^5}{x}+8 a b^7 x+b^8 x^2\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac {a^8 x^{-5 n}}{5 n}-\frac {2 a^7 b x^{-4 n}}{n}-\frac {28 a^6 b^2 x^{-3 n}}{3 n}-\frac {28 a^5 b^3 x^{-2 n}}{n}-\frac {70 a^4 b^4 x^{-n}}{n}+\frac {28 a^2 b^6 x^n}{n}+\frac {4 a b^7 x^{2 n}}{n}+\frac {b^8 x^{3 n}}{3 n}+56 a^3 b^5 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.10, size = 114, normalized size = 0.86 \[ \frac {-\frac {1}{5} a^8 x^{-5 n}-2 a^7 b x^{-4 n}-\frac {28}{3} a^6 b^2 x^{-3 n}-28 a^5 b^3 x^{-2 n}-70 a^4 b^4 x^{-n}+56 a^3 b^5 n \log (x)+28 a^2 b^6 x^n+4 a b^7 x^{2 n}+\frac {1}{3} b^8 x^{3 n}}{n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 116, normalized size = 0.87 \[ \frac {840 \, a^{3} b^{5} n x^{5 \, n} \log \relax (x) + 5 \, b^{8} x^{8 \, n} + 60 \, a b^{7} x^{7 \, n} + 420 \, a^{2} b^{6} x^{6 \, n} - 1050 \, a^{4} b^{4} x^{4 \, n} - 420 \, a^{5} b^{3} x^{3 \, n} - 140 \, a^{6} b^{2} x^{2 \, n} - 30 \, a^{7} b x^{n} - 3 \, a^{8}}{15 \, n x^{5 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 116, normalized size = 0.87 \[ \frac {840 \, a^{3} b^{5} n x^{5 \, n} \log \relax (x) + 5 \, b^{8} x^{8 \, n} + 60 \, a b^{7} x^{7 \, n} + 420 \, a^{2} b^{6} x^{6 \, n} - 1050 \, a^{4} b^{4} x^{4 \, n} - 420 \, a^{5} b^{3} x^{3 \, n} - 140 \, a^{6} b^{2} x^{2 \, n} - 30 \, a^{7} b x^{n} - 3 \, a^{8}}{15 \, n x^{5 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 128, normalized size = 0.96 \[ 56 a^{3} b^{5} \ln \relax (x )-\frac {a^{8} x^{-5 n}}{5 n}-\frac {2 a^{7} b \,x^{-4 n}}{n}-\frac {28 a^{6} b^{2} x^{-3 n}}{3 n}-\frac {28 a^{5} b^{3} x^{-2 n}}{n}-\frac {70 a^{4} b^{4} x^{-n}}{n}+\frac {28 a^{2} b^{6} x^{n}}{n}+\frac {4 a \,b^{7} x^{2 n}}{n}+\frac {b^{8} x^{3 n}}{3 n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 135, normalized size = 1.02 \[ 56 \, a^{3} b^{5} \log \relax (x) + \frac {b^{8} x^{3 \, n}}{3 \, n} + \frac {4 \, a b^{7} x^{2 \, n}}{n} + \frac {28 \, a^{2} b^{6} x^{n}}{n} - \frac {a^{8}}{5 \, n x^{5 \, n}} - \frac {2 \, a^{7} b}{n x^{4 \, n}} - \frac {28 \, a^{6} b^{2}}{3 \, n x^{3 \, n}} - \frac {28 \, a^{5} b^{3}}{n x^{2 \, n}} - \frac {70 \, a^{4} b^{4}}{n x^{n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.55, size = 135, normalized size = 1.02 \[ \frac {b^8\,x^{3\,n}}{3\,n}-\frac {a^8}{5\,n\,x^{5\,n}}+56\,a^3\,b^5\,\ln \relax (x)-\frac {70\,a^4\,b^4}{n\,x^n}-\frac {28\,a^5\,b^3}{n\,x^{2\,n}}-\frac {28\,a^6\,b^2}{3\,n\,x^{3\,n}}+\frac {4\,a\,b^7\,x^{2\,n}}{n}-\frac {2\,a^7\,b}{n\,x^{4\,n}}+\frac {28\,a^2\,b^6\,x^n}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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