Optimal. Leaf size=151 \[ -\frac{28 a^6 b^2 x^{-13 n}}{13 n}-\frac{14 a^5 b^3 x^{-12 n}}{3 n}-\frac{70 a^4 b^4 x^{-11 n}}{11 n}-\frac{28 a^3 b^5 x^{-10 n}}{5 n}-\frac{28 a^2 b^6 x^{-9 n}}{9 n}-\frac{4 a^7 b x^{-14 n}}{7 n}-\frac{a^8 x^{-15 n}}{15 n}-\frac{a b^7 x^{-8 n}}{n}-\frac{b^8 x^{-7 n}}{7 n} \]
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Rubi [A] time = 0.0608719, antiderivative size = 151, 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} \[ -\frac{28 a^6 b^2 x^{-13 n}}{13 n}-\frac{14 a^5 b^3 x^{-12 n}}{3 n}-\frac{70 a^4 b^4 x^{-11 n}}{11 n}-\frac{28 a^3 b^5 x^{-10 n}}{5 n}-\frac{28 a^2 b^6 x^{-9 n}}{9 n}-\frac{4 a^7 b x^{-14 n}}{7 n}-\frac{a^8 x^{-15 n}}{15 n}-\frac{a b^7 x^{-8 n}}{n}-\frac{b^8 x^{-7 n}}{7 n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^{-1-15 n} \left (a+b x^n\right )^8 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^{16}} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^8}{x^{16}}+\frac{8 a^7 b}{x^{15}}+\frac{28 a^6 b^2}{x^{14}}+\frac{56 a^5 b^3}{x^{13}}+\frac{70 a^4 b^4}{x^{12}}+\frac{56 a^3 b^5}{x^{11}}+\frac{28 a^2 b^6}{x^{10}}+\frac{8 a b^7}{x^9}+\frac{b^8}{x^8}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a^8 x^{-15 n}}{15 n}-\frac{4 a^7 b x^{-14 n}}{7 n}-\frac{28 a^6 b^2 x^{-13 n}}{13 n}-\frac{14 a^5 b^3 x^{-12 n}}{3 n}-\frac{70 a^4 b^4 x^{-11 n}}{11 n}-\frac{28 a^3 b^5 x^{-10 n}}{5 n}-\frac{28 a^2 b^6 x^{-9 n}}{9 n}-\frac{a b^7 x^{-8 n}}{n}-\frac{b^8 x^{-7 n}}{7 n}\\ \end{align*}
Mathematica [A] time = 0.0472497, size = 113, normalized size = 0.75 \[ -\frac{x^{-15 n} \left (97020 a^6 b^2 x^{2 n}+210210 a^5 b^3 x^{3 n}+286650 a^4 b^4 x^{4 n}+252252 a^3 b^5 x^{5 n}+140140 a^2 b^6 x^{6 n}+25740 a^7 b x^n+3003 a^8+45045 a b^7 x^{7 n}+6435 b^8 x^{8 n}\right )}{45045 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 136, normalized size = 0.9 \begin{align*} -{\frac{{b}^{8}}{7\,n \left ({x}^{n} \right ) ^{7}}}-{\frac{{b}^{7}a}{n \left ({x}^{n} \right ) ^{8}}}-{\frac{28\,{a}^{2}{b}^{6}}{9\,n \left ({x}^{n} \right ) ^{9}}}-{\frac{28\,{a}^{3}{b}^{5}}{5\,n \left ({x}^{n} \right ) ^{10}}}-{\frac{70\,{a}^{4}{b}^{4}}{11\,n \left ({x}^{n} \right ) ^{11}}}-{\frac{14\,{a}^{5}{b}^{3}}{3\,n \left ({x}^{n} \right ) ^{12}}}-{\frac{28\,{a}^{6}{b}^{2}}{13\,n \left ({x}^{n} \right ) ^{13}}}-{\frac{4\,b{a}^{7}}{7\,n \left ({x}^{n} \right ) ^{14}}}-{\frac{{a}^{8}}{15\,n \left ({x}^{n} \right ) ^{15}}} \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 = 1.08145, size = 293, normalized size = 1.94 \begin{align*} -\frac{6435 \, b^{8} x^{8 \, n} + 45045 \, a b^{7} x^{7 \, n} + 140140 \, a^{2} b^{6} x^{6 \, n} + 252252 \, a^{3} b^{5} x^{5 \, n} + 286650 \, a^{4} b^{4} x^{4 \, n} + 210210 \, a^{5} b^{3} x^{3 \, n} + 97020 \, a^{6} b^{2} x^{2 \, n} + 25740 \, a^{7} b x^{n} + 3003 \, a^{8}}{45045 \, n x^{15 \, 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 = 1.24264, size = 153, normalized size = 1.01 \begin{align*} -\frac{6435 \, b^{8} x^{8 \, n} + 45045 \, a b^{7} x^{7 \, n} + 140140 \, a^{2} b^{6} x^{6 \, n} + 252252 \, a^{3} b^{5} x^{5 \, n} + 286650 \, a^{4} b^{4} x^{4 \, n} + 210210 \, a^{5} b^{3} x^{3 \, n} + 97020 \, a^{6} b^{2} x^{2 \, n} + 25740 \, a^{7} b x^{n} + 3003 \, a^{8}}{45045 \, n x^{15 \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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