Optimal. Leaf size=135 \[ -\frac{14 a^6 b^2 x^{-2 n}}{n}-\frac{56 a^5 b^3 x^{-n}}{n}+\frac{56 a^3 b^5 x^n}{n}+\frac{14 a^2 b^6 x^{2 n}}{n}+70 a^4 b^4 \log (x)-\frac{8 a^7 b x^{-3 n}}{3 n}-\frac{a^8 x^{-4 n}}{4 n}+\frac{8 a b^7 x^{3 n}}{3 n}+\frac{b^8 x^{4 n}}{4 n} \]
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Rubi [A] time = 0.059085, antiderivative size = 135, 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} \[ -\frac{14 a^6 b^2 x^{-2 n}}{n}-\frac{56 a^5 b^3 x^{-n}}{n}+\frac{56 a^3 b^5 x^n}{n}+\frac{14 a^2 b^6 x^{2 n}}{n}+70 a^4 b^4 \log (x)-\frac{8 a^7 b x^{-3 n}}{3 n}-\frac{a^8 x^{-4 n}}{4 n}+\frac{8 a b^7 x^{3 n}}{3 n}+\frac{b^8 x^{4 n}}{4 n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^{-1-4 n} \left (a+b x^n\right )^8 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^5} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (56 a^3 b^5+\frac{a^8}{x^5}+\frac{8 a^7 b}{x^4}+\frac{28 a^6 b^2}{x^3}+\frac{56 a^5 b^3}{x^2}+\frac{70 a^4 b^4}{x}+28 a^2 b^6 x+8 a b^7 x^2+b^8 x^3\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a^8 x^{-4 n}}{4 n}-\frac{8 a^7 b x^{-3 n}}{3 n}-\frac{14 a^6 b^2 x^{-2 n}}{n}-\frac{56 a^5 b^3 x^{-n}}{n}+\frac{56 a^3 b^5 x^n}{n}+\frac{14 a^2 b^6 x^{2 n}}{n}+\frac{8 a b^7 x^{3 n}}{3 n}+\frac{b^8 x^{4 n}}{4 n}+70 a^4 b^4 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0584527, size = 116, normalized size = 0.86 \[ \frac{-14 a^6 b^2 x^{-2 n}-56 a^5 b^3 x^{-n}+56 a^3 b^5 x^n+14 a^2 b^6 x^{2 n}+70 a^4 b^4 n \log (x)-\frac{8}{3} a^7 b x^{-3 n}-\frac{1}{4} a^8 x^{-4 n}+\frac{8}{3} a b^7 x^{3 n}+\frac{1}{4} b^8 x^{4 n}}{n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 128, normalized size = 1. \begin{align*} 70\,{a}^{4}{b}^{4}\ln \left ( x \right ) +{\frac{{b}^{8} \left ({x}^{n} \right ) ^{4}}{4\,n}}+{\frac{8\,{b}^{7}a \left ({x}^{n} \right ) ^{3}}{3\,n}}+14\,{\frac{{a}^{2}{b}^{6} \left ({x}^{n} \right ) ^{2}}{n}}+56\,{\frac{{x}^{n}{a}^{3}{b}^{5}}{n}}-56\,{\frac{{a}^{5}{b}^{3}}{n{x}^{n}}}-14\,{\frac{{a}^{6}{b}^{2}}{n \left ({x}^{n} \right ) ^{2}}}-{\frac{8\,b{a}^{7}}{3\,n \left ({x}^{n} \right ) ^{3}}}-{\frac{{a}^{8}}{4\,n \left ({x}^{n} \right ) ^{4}}} \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 = 1.38866, size = 263, normalized size = 1.95 \begin{align*} \frac{840 \, a^{4} b^{4} n x^{4 \, n} \log \left (x\right ) + 3 \, b^{8} x^{8 \, n} + 32 \, a b^{7} x^{7 \, n} + 168 \, a^{2} b^{6} x^{6 \, n} + 672 \, a^{3} b^{5} x^{5 \, n} - 672 \, a^{5} b^{3} x^{3 \, n} - 168 \, a^{6} b^{2} x^{2 \, n} - 32 \, a^{7} b x^{n} - 3 \, a^{8}}{12 \, n x^{4 \, 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 = 1.22448, size = 157, normalized size = 1.16 \begin{align*} \frac{840 \, a^{4} b^{4} n x^{4 \, n} \log \left (x\right ) + 3 \, b^{8} x^{8 \, n} + 32 \, a b^{7} x^{7 \, n} + 168 \, a^{2} b^{6} x^{6 \, n} + 672 \, a^{3} b^{5} x^{5 \, n} - 672 \, a^{5} b^{3} x^{3 \, n} - 168 \, a^{6} b^{2} x^{2 \, n} - 32 \, a^{7} b x^{n} - 3 \, a^{8}}{12 \, n x^{4 \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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