Optimal. Leaf size=84 \[ \frac{3 a^2 \left (a+b x^n\right )^{10}}{10 b^4 n}-\frac{a^3 \left (a+b x^n\right )^9}{9 b^4 n}+\frac{\left (a+b x^n\right )^{12}}{12 b^4 n}-\frac{3 a \left (a+b x^n\right )^{11}}{11 b^4 n} \]
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Rubi [A] time = 0.0497295, antiderivative size = 84, 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} \[ \frac{3 a^2 \left (a+b x^n\right )^{10}}{10 b^4 n}-\frac{a^3 \left (a+b x^n\right )^9}{9 b^4 n}+\frac{\left (a+b x^n\right )^{12}}{12 b^4 n}-\frac{3 a \left (a+b x^n\right )^{11}}{11 b^4 n} \]
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 x^3 (a+b x)^8 \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a^3 (a+b x)^8}{b^3}+\frac{3 a^2 (a+b x)^9}{b^3}-\frac{3 a (a+b x)^{10}}{b^3}+\frac{(a+b x)^{11}}{b^3}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a^3 \left (a+b x^n\right )^9}{9 b^4 n}+\frac{3 a^2 \left (a+b x^n\right )^{10}}{10 b^4 n}-\frac{3 a \left (a+b x^n\right )^{11}}{11 b^4 n}+\frac{\left (a+b x^n\right )^{12}}{12 b^4 n}\\ \end{align*}
Mathematica [A] time = 0.0385049, size = 53, normalized size = 0.63 \[ -\frac{\left (a+b x^n\right )^9 \left (-9 a^2 b x^n+a^3+45 a b^2 x^{2 n}-165 b^3 x^{3 n}\right )}{1980 b^4 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 136, normalized size = 1.6 \begin{align*}{\frac{{b}^{8} \left ({x}^{n} \right ) ^{12}}{12\,n}}+{\frac{8\,{b}^{7}a \left ({x}^{n} \right ) ^{11}}{11\,n}}+{\frac{14\,{b}^{6}{a}^{2} \left ({x}^{n} \right ) ^{10}}{5\,n}}+{\frac{56\,{a}^{3}{b}^{5} \left ({x}^{n} \right ) ^{9}}{9\,n}}+{\frac{35\,{a}^{4}{b}^{4} \left ({x}^{n} \right ) ^{8}}{4\,n}}+8\,{\frac{{a}^{5}{b}^{3} \left ({x}^{n} \right ) ^{7}}{n}}+{\frac{14\,{a}^{6}{b}^{2} \left ({x}^{n} \right ) ^{6}}{3\,n}}+{\frac{8\,b{a}^{7} \left ({x}^{n} \right ) ^{5}}{5\,n}}+{\frac{{a}^{8} \left ({x}^{n} \right ) ^{4}}{4\,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 = 1.27256, size = 282, normalized size = 3.36 \begin{align*} \frac{165 \, b^{8} x^{12 \, n} + 1440 \, a b^{7} x^{11 \, n} + 5544 \, a^{2} b^{6} x^{10 \, n} + 12320 \, a^{3} b^{5} x^{9 \, n} + 17325 \, a^{4} b^{4} x^{8 \, n} + 15840 \, a^{5} b^{3} x^{7 \, n} + 9240 \, a^{6} b^{2} x^{6 \, n} + 3168 \, a^{7} b x^{5 \, n} + 495 \, a^{8} x^{4 \, n}}{1980 \, 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^{4 \, n - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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