Optimal. Leaf size=135 \[ -\frac {a^8 x^{-2 n}}{2 n}-\frac {8 a^7 b x^{-n}}{n}+28 a^6 b^2 \log (x)+\frac {56 a^5 b^3 x^n}{n}+\frac {35 a^4 b^4 x^{2 n}}{n}+\frac {56 a^3 b^5 x^{3 n}}{3 n}+\frac {7 a^2 b^6 x^{4 n}}{n}+\frac {8 a b^7 x^{5 n}}{5 n}+\frac {b^8 x^{6 n}}{6 n} \]
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Rubi [A] time = 0.06, antiderivative size = 135, 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} \[ \frac {56 a^5 b^3 x^n}{n}+\frac {35 a^4 b^4 x^{2 n}}{n}+\frac {56 a^3 b^5 x^{3 n}}{3 n}+\frac {7 a^2 b^6 x^{4 n}}{n}+28 a^6 b^2 \log (x)-\frac {8 a^7 b x^{-n}}{n}-\frac {a^8 x^{-2 n}}{2 n}+\frac {8 a b^7 x^{5 n}}{5 n}+\frac {b^8 x^{6 n}}{6 n} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^{-1-2 n} \left (a+b x^n\right )^8 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^8}{x^3} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (56 a^5 b^3+\frac {a^8}{x^3}+\frac {8 a^7 b}{x^2}+\frac {28 a^6 b^2}{x}+70 a^4 b^4 x+56 a^3 b^5 x^2+28 a^2 b^6 x^3+8 a b^7 x^4+b^8 x^5\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac {a^8 x^{-2 n}}{2 n}-\frac {8 a^7 b x^{-n}}{n}+\frac {56 a^5 b^3 x^n}{n}+\frac {35 a^4 b^4 x^{2 n}}{n}+\frac {56 a^3 b^5 x^{3 n}}{3 n}+\frac {7 a^2 b^6 x^{4 n}}{n}+\frac {8 a b^7 x^{5 n}}{5 n}+\frac {b^8 x^{6 n}}{6 n}+28 a^6 b^2 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.11, size = 116, normalized size = 0.86 \[ \frac {-\frac {1}{2} a^8 x^{-2 n}-8 a^7 b x^{-n}+28 a^6 b^2 n \log (x)+56 a^5 b^3 x^n+35 a^4 b^4 x^{2 n}+\frac {56}{3} a^3 b^5 x^{3 n}+7 a^2 b^6 x^{4 n}+\frac {8}{5} a b^7 x^{5 n}+\frac {1}{6} b^8 x^{6 n}}{n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 116, normalized size = 0.86 \[ \frac {840 \, a^{6} b^{2} n x^{2 \, n} \log \relax (x) + 5 \, b^{8} x^{8 \, n} + 48 \, a b^{7} x^{7 \, n} + 210 \, a^{2} b^{6} x^{6 \, n} + 560 \, a^{3} b^{5} x^{5 \, n} + 1050 \, a^{4} b^{4} x^{4 \, n} + 1680 \, a^{5} b^{3} x^{3 \, n} - 240 \, a^{7} b x^{n} - 15 \, a^{8}}{30 \, n x^{2 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 116, normalized size = 0.86 \[ \frac {840 \, a^{6} b^{2} n x^{2 \, n} \log \relax (x) + 5 \, b^{8} x^{8 \, n} + 48 \, a b^{7} x^{7 \, n} + 210 \, a^{2} b^{6} x^{6 \, n} + 560 \, a^{3} b^{5} x^{5 \, n} + 1050 \, a^{4} b^{4} x^{4 \, n} + 1680 \, a^{5} b^{3} x^{3 \, n} - 240 \, a^{7} b x^{n} - 15 \, a^{8}}{30 \, n x^{2 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 128, normalized size = 0.95 \[ 28 a^{6} b^{2} \ln \relax (x )-\frac {a^{8} x^{-2 n}}{2 n}-\frac {8 a^{7} b \,x^{-n}}{n}+\frac {56 a^{5} b^{3} x^{n}}{n}+\frac {35 a^{4} b^{4} x^{2 n}}{n}+\frac {56 a^{3} b^{5} x^{3 n}}{3 n}+\frac {7 a^{2} b^{6} x^{4 n}}{n}+\frac {8 a \,b^{7} x^{5 n}}{5 n}+\frac {b^{8} x^{6 n}}{6 n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 129, normalized size = 0.96 \[ 28 \, a^{6} b^{2} \log \relax (x) + \frac {b^{8} x^{6 \, n}}{6 \, n} + \frac {8 \, a b^{7} x^{5 \, n}}{5 \, n} + \frac {7 \, a^{2} b^{6} x^{4 \, n}}{n} + \frac {56 \, a^{3} b^{5} x^{3 \, n}}{3 \, n} + \frac {35 \, a^{4} b^{4} x^{2 \, n}}{n} + \frac {56 \, a^{5} b^{3} x^{n}}{n} - \frac {a^{8}}{2 \, n x^{2 \, n}} - \frac {8 \, a^{7} b}{n x^{n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.60, size = 129, normalized size = 0.96 \[ \frac {b^8\,x^{6\,n}}{6\,n}-\frac {a^8}{2\,n\,x^{2\,n}}+28\,a^6\,b^2\,\ln \relax (x)+\frac {35\,a^4\,b^4\,x^{2\,n}}{n}+\frac {56\,a^3\,b^5\,x^{3\,n}}{3\,n}+\frac {7\,a^2\,b^6\,x^{4\,n}}{n}-\frac {8\,a^7\,b}{n\,x^n}+\frac {8\,a\,b^7\,x^{5\,n}}{5\,n}+\frac {56\,a^5\,b^3\,x^n}{n} \]
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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