Optimal. Leaf size=128 \[ \frac{5 a^2 \left (a+b x^n\right )^{12}}{6 b^6 n}-\frac{10 a^3 \left (a+b x^n\right )^{11}}{11 b^6 n}+\frac{a^4 \left (a+b x^n\right )^{10}}{2 b^6 n}-\frac{a^5 \left (a+b x^n\right )^9}{9 b^6 n}+\frac{\left (a+b x^n\right )^{14}}{14 b^6 n}-\frac{5 a \left (a+b x^n\right )^{13}}{13 b^6 n} \]
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Rubi [A] time = 0.0676856, antiderivative size = 128, 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} \[ \frac{5 a^2 \left (a+b x^n\right )^{12}}{6 b^6 n}-\frac{10 a^3 \left (a+b x^n\right )^{11}}{11 b^6 n}+\frac{a^4 \left (a+b x^n\right )^{10}}{2 b^6 n}-\frac{a^5 \left (a+b x^n\right )^9}{9 b^6 n}+\frac{\left (a+b x^n\right )^{14}}{14 b^6 n}-\frac{5 a \left (a+b x^n\right )^{13}}{13 b^6 n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^{-1+6 n} \left (a+b x^n\right )^8 \, dx &=\frac{\operatorname{Subst}\left (\int x^5 (a+b x)^8 \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a^5 (a+b x)^8}{b^5}+\frac{5 a^4 (a+b x)^9}{b^5}-\frac{10 a^3 (a+b x)^{10}}{b^5}+\frac{10 a^2 (a+b x)^{11}}{b^5}-\frac{5 a (a+b x)^{12}}{b^5}+\frac{(a+b x)^{13}}{b^5}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a^5 \left (a+b x^n\right )^9}{9 b^6 n}+\frac{a^4 \left (a+b x^n\right )^{10}}{2 b^6 n}-\frac{10 a^3 \left (a+b x^n\right )^{11}}{11 b^6 n}+\frac{5 a^2 \left (a+b x^n\right )^{12}}{6 b^6 n}-\frac{5 a \left (a+b x^n\right )^{13}}{13 b^6 n}+\frac{\left (a+b x^n\right )^{14}}{14 b^6 n}\\ \end{align*}
Mathematica [A] time = 0.049978, size = 79, normalized size = 0.62 \[ -\frac{\left (a+b x^n\right )^9 \left (45 a^3 b^2 x^{2 n}-165 a^2 b^3 x^{3 n}-9 a^4 b x^n+a^5+495 a b^4 x^{4 n}-1287 b^5 x^{5 n}\right )}{18018 b^6 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 136, normalized size = 1.1 \begin{align*}{\frac{{b}^{8} \left ({x}^{n} \right ) ^{14}}{14\,n}}+{\frac{8\,{b}^{7}a \left ({x}^{n} \right ) ^{13}}{13\,n}}+{\frac{7\,{b}^{6}{a}^{2} \left ({x}^{n} \right ) ^{12}}{3\,n}}+{\frac{56\,{a}^{3}{b}^{5} \left ({x}^{n} \right ) ^{11}}{11\,n}}+7\,{\frac{{a}^{4}{b}^{4} \left ({x}^{n} \right ) ^{10}}{n}}+{\frac{56\,{a}^{5}{b}^{3} \left ({x}^{n} \right ) ^{9}}{9\,n}}+{\frac{7\,{a}^{6}{b}^{2} \left ({x}^{n} \right ) ^{8}}{2\,n}}+{\frac{8\,b{a}^{7} \left ({x}^{n} \right ) ^{7}}{7\,n}}+{\frac{{a}^{8} \left ({x}^{n} \right ) ^{6}}{6\,n}} \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 = 1.30295, size = 297, normalized size = 2.32 \begin{align*} \frac{1287 \, b^{8} x^{14 \, n} + 11088 \, a b^{7} x^{13 \, n} + 42042 \, a^{2} b^{6} x^{12 \, n} + 91728 \, a^{3} b^{5} x^{11 \, n} + 126126 \, a^{4} b^{4} x^{10 \, n} + 112112 \, a^{5} b^{3} x^{9 \, n} + 63063 \, a^{6} b^{2} x^{8 \, n} + 20592 \, a^{7} b x^{7 \, n} + 3003 \, a^{8} x^{6 \, n}}{18018 \, 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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{n} + a\right )}^{8} x^{6 \, n - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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