Optimal. Leaf size=140 \[ -\frac{14 a^6 b^2 x^{-6 n}}{3 n}-\frac{56 a^5 b^3 x^{-5 n}}{5 n}-\frac{35 a^4 b^4 x^{-4 n}}{2 n}-\frac{56 a^3 b^5 x^{-3 n}}{3 n}-\frac{14 a^2 b^6 x^{-2 n}}{n}-\frac{8 a^7 b x^{-7 n}}{7 n}-\frac{a^8 x^{-8 n}}{8 n}-\frac{8 a b^7 x^{-n}}{n}+b^8 \log (x) \]
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Rubi [A] time = 0.0588352, antiderivative size = 140, normalized size of antiderivative = 1., 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} \[ -\frac{14 a^6 b^2 x^{-6 n}}{3 n}-\frac{56 a^5 b^3 x^{-5 n}}{5 n}-\frac{35 a^4 b^4 x^{-4 n}}{2 n}-\frac{56 a^3 b^5 x^{-3 n}}{3 n}-\frac{14 a^2 b^6 x^{-2 n}}{n}-\frac{8 a^7 b x^{-7 n}}{7 n}-\frac{a^8 x^{-8 n}}{8 n}-\frac{8 a b^7 x^{-n}}{n}+b^8 \log (x) \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^{-1-8 n} \left (a+b x^n\right )^8 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^9} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^8}{x^9}+\frac{8 a^7 b}{x^8}+\frac{28 a^6 b^2}{x^7}+\frac{56 a^5 b^3}{x^6}+\frac{70 a^4 b^4}{x^5}+\frac{56 a^3 b^5}{x^4}+\frac{28 a^2 b^6}{x^3}+\frac{8 a b^7}{x^2}+\frac{b^8}{x}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a^8 x^{-8 n}}{8 n}-\frac{8 a^7 b x^{-7 n}}{7 n}-\frac{14 a^6 b^2 x^{-6 n}}{3 n}-\frac{56 a^5 b^3 x^{-5 n}}{5 n}-\frac{35 a^4 b^4 x^{-4 n}}{2 n}-\frac{56 a^3 b^5 x^{-3 n}}{3 n}-\frac{14 a^2 b^6 x^{-2 n}}{n}-\frac{8 a b^7 x^{-n}}{n}+b^8 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0902527, size = 108, normalized size = 0.77 \[ b^8 \log (x)-\frac{a x^{-8 n} \left (3920 a^5 b^2 x^{2 n}+9408 a^4 b^3 x^{3 n}+14700 a^3 b^4 x^{4 n}+15680 a^2 b^5 x^{5 n}+960 a^6 b x^n+105 a^7+11760 a b^6 x^{6 n}+6720 b^7 x^{7 n}\right )}{840 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 129, normalized size = 0.9 \begin{align*}{b}^{8}\ln \left ( x \right ) -8\,{\frac{{b}^{7}a}{n{x}^{n}}}-14\,{\frac{{a}^{2}{b}^{6}}{n \left ({x}^{n} \right ) ^{2}}}-{\frac{56\,{a}^{3}{b}^{5}}{3\,n \left ({x}^{n} \right ) ^{3}}}-{\frac{35\,{a}^{4}{b}^{4}}{2\,n \left ({x}^{n} \right ) ^{4}}}-{\frac{56\,{a}^{5}{b}^{3}}{5\,n \left ({x}^{n} \right ) ^{5}}}-{\frac{14\,{a}^{6}{b}^{2}}{3\,n \left ({x}^{n} \right ) ^{6}}}-{\frac{8\,b{a}^{7}}{7\,n \left ({x}^{n} \right ) ^{7}}}-{\frac{{a}^{8}}{8\,n \left ({x}^{n} \right ) ^{8}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.04961, size = 285, normalized size = 2.04 \begin{align*} \frac{840 \, b^{8} n x^{8 \, n} \log \left (x\right ) - 6720 \, a b^{7} x^{7 \, n} - 11760 \, a^{2} b^{6} x^{6 \, n} - 15680 \, a^{3} b^{5} x^{5 \, n} - 14700 \, a^{4} b^{4} x^{4 \, n} - 9408 \, a^{5} b^{3} x^{3 \, n} - 3920 \, a^{6} b^{2} x^{2 \, n} - 960 \, a^{7} b x^{n} - 105 \, a^{8}}{840 \, n x^{8 \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21854, size = 157, normalized size = 1.12 \begin{align*} \frac{840 \, b^{8} n x^{8 \, n} \log \left (x\right ) - 6720 \, a b^{7} x^{7 \, n} - 11760 \, a^{2} b^{6} x^{6 \, n} - 15680 \, a^{3} b^{5} x^{5 \, n} - 14700 \, a^{4} b^{4} x^{4 \, n} - 9408 \, a^{5} b^{3} x^{3 \, n} - 3920 \, a^{6} b^{2} x^{2 \, n} - 960 \, a^{7} b x^{n} - 105 \, a^{8}}{840 \, n x^{8 \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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