Optimal. Leaf size=133 \[ -\frac{28 a^6 b^2 x^{-3 n}}{3 n}-\frac{28 a^5 b^3 x^{-2 n}}{n}-\frac{70 a^4 b^4 x^{-n}}{n}+\frac{28 a^2 b^6 x^n}{n}+56 a^3 b^5 \log (x)-\frac{2 a^7 b x^{-4 n}}{n}-\frac{a^8 x^{-5 n}}{5 n}+\frac{4 a b^7 x^{2 n}}{n}+\frac{b^8 x^{3 n}}{3 n} \]
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Rubi [A] time = 0.0583786, antiderivative size = 133, 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} \[ -\frac{28 a^6 b^2 x^{-3 n}}{3 n}-\frac{28 a^5 b^3 x^{-2 n}}{n}-\frac{70 a^4 b^4 x^{-n}}{n}+\frac{28 a^2 b^6 x^n}{n}+56 a^3 b^5 \log (x)-\frac{2 a^7 b x^{-4 n}}{n}-\frac{a^8 x^{-5 n}}{5 n}+\frac{4 a b^7 x^{2 n}}{n}+\frac{b^8 x^{3 n}}{3 n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
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*}
Mathematica [A] time = 0.0619468, size = 114, normalized size = 0.86 \[ \frac{-\frac{28}{3} a^6 b^2 x^{-3 n}-28 a^5 b^3 x^{-2 n}-70 a^4 b^4 x^{-n}+28 a^2 b^6 x^n+56 a^3 b^5 n \log (x)-2 a^7 b x^{-4 n}-\frac{1}{5} a^8 x^{-5 n}+4 a b^7 x^{2 n}+\frac{1}{3} b^8 x^{3 n}}{n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 128, normalized size = 1. \begin{align*} 56\,{a}^{3}{b}^{5}\ln \left ( x \right ) +{\frac{{b}^{8} \left ({x}^{n} \right ) ^{3}}{3\,n}}+4\,{\frac{{b}^{7}a \left ({x}^{n} \right ) ^{2}}{n}}+28\,{\frac{{x}^{n}{a}^{2}{b}^{6}}{n}}-70\,{\frac{{a}^{4}{b}^{4}}{n{x}^{n}}}-28\,{\frac{{a}^{5}{b}^{3}}{n \left ({x}^{n} \right ) ^{2}}}-{\frac{28\,{a}^{6}{b}^{2}}{3\,n \left ({x}^{n} \right ) ^{3}}}-2\,{\frac{b{a}^{7}}{n \left ({x}^{n} \right ) ^{4}}}-{\frac{{a}^{8}}{5\,n \left ({x}^{n} \right ) ^{5}}} \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 = 1.3377, size = 265, normalized size = 1.99 \begin{align*} \frac{840 \, a^{3} b^{5} n x^{5 \, n} \log \left (x\right ) + 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}} \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 = 1.20692, size = 157, normalized size = 1.18 \begin{align*} \frac{840 \, a^{3} b^{5} n x^{5 \, n} \log \left (x\right ) + 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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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