Optimal. Leaf size=40 \[ \frac{\left (a+b x^n\right )^{10}}{10 b^2 n}-\frac{a \left (a+b x^n\right )^9}{9 b^2 n} \]
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Rubi [A] time = 0.0185763, antiderivative size = 40, 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} \[ \frac{\left (a+b x^n\right )^{10}}{10 b^2 n}-\frac{a \left (a+b x^n\right )^9}{9 b^2 n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^{-1+2 n} \left (a+b x^n\right )^8 \, dx &=\frac{\operatorname{Subst}\left (\int x (a+b x)^8 \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a (a+b x)^8}{b}+\frac{(a+b x)^9}{b}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a \left (a+b x^n\right )^9}{9 b^2 n}+\frac{\left (a+b x^n\right )^{10}}{10 b^2 n}\\ \end{align*}
Mathematica [A] time = 0.0266571, size = 27, normalized size = 0.68 \[ -\frac{\left (a-9 b x^n\right ) \left (a+b x^n\right )^9}{90 b^2 n} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.022, size = 136, normalized size = 3.4 \begin{align*}{\frac{{b}^{8} \left ({x}^{n} \right ) ^{10}}{10\,n}}+{\frac{8\,{b}^{7}a \left ({x}^{n} \right ) ^{9}}{9\,n}}+{\frac{7\,{b}^{6}{a}^{2} \left ({x}^{n} \right ) ^{8}}{2\,n}}+8\,{\frac{{a}^{3}{b}^{5} \left ({x}^{n} \right ) ^{7}}{n}}+{\frac{35\,{a}^{4}{b}^{4} \left ({x}^{n} \right ) ^{6}}{3\,n}}+{\frac{56\,{a}^{5}{b}^{3} \left ({x}^{n} \right ) ^{5}}{5\,n}}+7\,{\frac{{a}^{6}{b}^{2} \left ({x}^{n} \right ) ^{4}}{n}}+{\frac{8\,b{a}^{7} \left ({x}^{n} \right ) ^{3}}{3\,n}}+{\frac{{a}^{8} \left ({x}^{n} \right ) ^{2}}{2\,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 [B] time = 1.29129, size = 261, normalized size = 6.52 \begin{align*} \frac{9 \, b^{8} x^{10 \, n} + 80 \, a b^{7} x^{9 \, n} + 315 \, a^{2} b^{6} x^{8 \, n} + 720 \, a^{3} b^{5} x^{7 \, n} + 1050 \, a^{4} b^{4} x^{6 \, n} + 1008 \, a^{5} b^{3} x^{5 \, n} + 630 \, a^{6} b^{2} x^{4 \, n} + 240 \, a^{7} b x^{3 \, n} + 45 \, a^{8} x^{2 \, n}}{90 \, 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^{2 \, n - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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