Optimal. Leaf size=62 \[ \frac{a^2 \left (a+b x^n\right )^9}{9 b^3 n}+\frac{\left (a+b x^n\right )^{11}}{11 b^3 n}-\frac{a \left (a+b x^n\right )^{10}}{5 b^3 n} \]
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Rubi [A] time = 0.0388772, antiderivative size = 62, 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} \[ \frac{a^2 \left (a+b x^n\right )^9}{9 b^3 n}+\frac{\left (a+b x^n\right )^{11}}{11 b^3 n}-\frac{a \left (a+b x^n\right )^{10}}{5 b^3 n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^{-1+3 n} \left (a+b x^n\right )^8 \, dx &=\frac{\operatorname{Subst}\left (\int x^2 (a+b x)^8 \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^2 (a+b x)^8}{b^2}-\frac{2 a (a+b x)^9}{b^2}+\frac{(a+b x)^{10}}{b^2}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac{a^2 \left (a+b x^n\right )^9}{9 b^3 n}-\frac{a \left (a+b x^n\right )^{10}}{5 b^3 n}+\frac{\left (a+b x^n\right )^{11}}{11 b^3 n}\\ \end{align*}
Mathematica [A] time = 0.0322741, size = 40, normalized size = 0.65 \[ \frac{\left (a+b x^n\right )^9 \left (a^2-9 a b x^n+45 b^2 x^{2 n}\right )}{495 b^3 n} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.023, size = 136, normalized size = 2.2 \begin{align*}{\frac{{b}^{8} \left ({x}^{n} \right ) ^{11}}{11\,n}}+{\frac{4\,{b}^{7}a \left ({x}^{n} \right ) ^{10}}{5\,n}}+{\frac{28\,{b}^{6}{a}^{2} \left ({x}^{n} \right ) ^{9}}{9\,n}}+7\,{\frac{{a}^{3}{b}^{5} \left ({x}^{n} \right ) ^{8}}{n}}+10\,{\frac{{a}^{4}{b}^{4} \left ({x}^{n} \right ) ^{7}}{n}}+{\frac{28\,{a}^{5}{b}^{3} \left ({x}^{n} \right ) ^{6}}{3\,n}}+{\frac{28\,{a}^{6}{b}^{2} \left ({x}^{n} \right ) ^{5}}{5\,n}}+2\,{\frac{b{a}^{7} \left ({x}^{n} \right ) ^{4}}{n}}+{\frac{{a}^{8} \left ({x}^{n} \right ) ^{3}}{3\,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 = 1.39633, size = 271, normalized size = 4.37 \begin{align*} \frac{45 \, b^{8} x^{11 \, n} + 396 \, a b^{7} x^{10 \, n} + 1540 \, a^{2} b^{6} x^{9 \, n} + 3465 \, a^{3} b^{5} x^{8 \, n} + 4950 \, a^{4} b^{4} x^{7 \, n} + 4620 \, a^{5} b^{3} x^{6 \, n} + 2772 \, a^{6} b^{2} x^{5 \, n} + 990 \, a^{7} b x^{4 \, n} + 165 \, a^{8} x^{3 \, n}}{495 \, 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^{3 \, n - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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