Optimal. Leaf size=134 \[ -\frac{28 a^6 b^2 x^{-5 n}}{5 n}-\frac{14 a^5 b^3 x^{-4 n}}{n}-\frac{70 a^4 b^4 x^{-3 n}}{3 n}-\frac{28 a^3 b^5 x^{-2 n}}{n}-\frac{28 a^2 b^6 x^{-n}}{n}-\frac{4 a^7 b x^{-6 n}}{3 n}-\frac{a^8 x^{-7 n}}{7 n}+8 a b^7 \log (x)+\frac{b^8 x^n}{n} \]
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Rubi [A] time = 0.0614759, antiderivative size = 134, 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} \[ -\frac{28 a^6 b^2 x^{-5 n}}{5 n}-\frac{14 a^5 b^3 x^{-4 n}}{n}-\frac{70 a^4 b^4 x^{-3 n}}{3 n}-\frac{28 a^3 b^5 x^{-2 n}}{n}-\frac{28 a^2 b^6 x^{-n}}{n}-\frac{4 a^7 b x^{-6 n}}{3 n}-\frac{a^8 x^{-7 n}}{7 n}+8 a b^7 \log (x)+\frac{b^8 x^n}{n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^{-1-7 n} \left (a+b x^n\right )^8 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^8} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (b^8+\frac{a^8}{x^8}+\frac{8 a^7 b}{x^7}+\frac{28 a^6 b^2}{x^6}+\frac{56 a^5 b^3}{x^5}+\frac{70 a^4 b^4}{x^4}+\frac{56 a^3 b^5}{x^3}+\frac{28 a^2 b^6}{x^2}+\frac{8 a b^7}{x}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a^8 x^{-7 n}}{7 n}-\frac{4 a^7 b x^{-6 n}}{3 n}-\frac{28 a^6 b^2 x^{-5 n}}{5 n}-\frac{14 a^5 b^3 x^{-4 n}}{n}-\frac{70 a^4 b^4 x^{-3 n}}{3 n}-\frac{28 a^3 b^5 x^{-2 n}}{n}-\frac{28 a^2 b^6 x^{-n}}{n}+\frac{b^8 x^n}{n}+8 a b^7 \log (x)\\ \end{align*}
Mathematica [A] time = 0.101153, size = 115, normalized size = 0.86 \[ \frac{-\frac{28}{5} a^6 b^2 x^{-5 n}-14 a^5 b^3 x^{-4 n}-\frac{70}{3} a^4 b^4 x^{-3 n}-28 a^3 b^5 x^{-2 n}-28 a^2 b^6 x^{-n}-\frac{4}{3} a^7 b x^{-6 n}-\frac{1}{7} a^8 x^{-7 n}+8 a b^7 n \log (x)+b^8 x^n}{n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 127, normalized size = 1. \begin{align*} 8\,a{b}^{7}\ln \left ( x \right ) +{\frac{{x}^{n}{b}^{8}}{n}}-28\,{\frac{{a}^{2}{b}^{6}}{n{x}^{n}}}-28\,{\frac{{a}^{3}{b}^{5}}{n \left ({x}^{n} \right ) ^{2}}}-{\frac{70\,{a}^{4}{b}^{4}}{3\,n \left ({x}^{n} \right ) ^{3}}}-14\,{\frac{{a}^{5}{b}^{3}}{n \left ({x}^{n} \right ) ^{4}}}-{\frac{28\,{a}^{6}{b}^{2}}{5\,n \left ({x}^{n} \right ) ^{5}}}-{\frac{4\,b{a}^{7}}{3\,n \left ({x}^{n} \right ) ^{6}}}-{\frac{{a}^{8}}{7\,n \left ({x}^{n} \right ) ^{7}}} \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 = 1.25366, size = 277, normalized size = 2.07 \begin{align*} \frac{840 \, a b^{7} n x^{7 \, n} \log \left (x\right ) + 105 \, b^{8} x^{8 \, n} - 2940 \, a^{2} b^{6} x^{6 \, n} - 2940 \, a^{3} b^{5} x^{5 \, n} - 2450 \, a^{4} b^{4} x^{4 \, n} - 1470 \, a^{5} b^{3} x^{3 \, n} - 588 \, a^{6} b^{2} x^{2 \, n} - 140 \, a^{7} b x^{n} - 15 \, a^{8}}{105 \, n x^{7 \, 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 = 1.28006, size = 157, normalized size = 1.17 \begin{align*} \frac{840 \, a b^{7} n x^{7 \, n} \log \left (x\right ) + 105 \, b^{8} x^{8 \, n} - 2940 \, a^{2} b^{6} x^{6 \, n} - 2940 \, a^{3} b^{5} x^{5 \, n} - 2450 \, a^{4} b^{4} x^{4 \, n} - 1470 \, a^{5} b^{3} x^{3 \, n} - 588 \, a^{6} b^{2} x^{2 \, n} - 140 \, a^{7} b x^{n} - 15 \, a^{8}}{105 \, n x^{7 \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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