Optimal. Leaf size=150 \[ \frac{15 a^2 \left (a+b x^n\right )^{13}}{13 b^7 n}-\frac{5 a^3 \left (a+b x^n\right )^{12}}{3 b^7 n}+\frac{15 a^4 \left (a+b x^n\right )^{11}}{11 b^7 n}-\frac{3 a^5 \left (a+b x^n\right )^{10}}{5 b^7 n}+\frac{a^6 \left (a+b x^n\right )^9}{9 b^7 n}+\frac{\left (a+b x^n\right )^{15}}{15 b^7 n}-\frac{3 a \left (a+b x^n\right )^{14}}{7 b^7 n} \]
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Rubi [A] time = 0.0780009, antiderivative size = 150, 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} \[ \frac{15 a^2 \left (a+b x^n\right )^{13}}{13 b^7 n}-\frac{5 a^3 \left (a+b x^n\right )^{12}}{3 b^7 n}+\frac{15 a^4 \left (a+b x^n\right )^{11}}{11 b^7 n}-\frac{3 a^5 \left (a+b x^n\right )^{10}}{5 b^7 n}+\frac{a^6 \left (a+b x^n\right )^9}{9 b^7 n}+\frac{\left (a+b x^n\right )^{15}}{15 b^7 n}-\frac{3 a \left (a+b x^n\right )^{14}}{7 b^7 n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^{-1+7 n} \left (a+b x^n\right )^8 \, dx &=\frac{\operatorname{Subst}\left (\int x^6 (a+b x)^8 \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^6 (a+b x)^8}{b^6}-\frac{6 a^5 (a+b x)^9}{b^6}+\frac{15 a^4 (a+b x)^{10}}{b^6}-\frac{20 a^3 (a+b x)^{11}}{b^6}+\frac{15 a^2 (a+b x)^{12}}{b^6}-\frac{6 a (a+b x)^{13}}{b^6}+\frac{(a+b x)^{14}}{b^6}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac{a^6 \left (a+b x^n\right )^9}{9 b^7 n}-\frac{3 a^5 \left (a+b x^n\right )^{10}}{5 b^7 n}+\frac{15 a^4 \left (a+b x^n\right )^{11}}{11 b^7 n}-\frac{5 a^3 \left (a+b x^n\right )^{12}}{3 b^7 n}+\frac{15 a^2 \left (a+b x^n\right )^{13}}{13 b^7 n}-\frac{3 a \left (a+b x^n\right )^{14}}{7 b^7 n}+\frac{\left (a+b x^n\right )^{15}}{15 b^7 n}\\ \end{align*}
Mathematica [A] time = 0.054637, size = 127, normalized size = 0.85 \[ \frac{\frac{28}{9} a^6 b^2 x^{9 n}+\frac{28}{5} a^5 b^3 x^{10 n}+\frac{70}{11} a^4 b^4 x^{11 n}+\frac{14}{3} a^3 b^5 x^{12 n}+\frac{28}{13} a^2 b^6 x^{13 n}+a^7 b x^{8 n}+\frac{1}{7} a^8 x^{7 n}+\frac{4}{7} a b^7 x^{14 n}+\frac{1}{15} b^8 x^{15 n}}{n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 135, normalized size = 0.9 \begin{align*}{\frac{{b}^{8} \left ({x}^{n} \right ) ^{15}}{15\,n}}+{\frac{4\,{b}^{7}a \left ({x}^{n} \right ) ^{14}}{7\,n}}+{\frac{28\,{b}^{6}{a}^{2} \left ({x}^{n} \right ) ^{13}}{13\,n}}+{\frac{14\,{a}^{3}{b}^{5} \left ({x}^{n} \right ) ^{12}}{3\,n}}+{\frac{70\,{a}^{4}{b}^{4} \left ({x}^{n} \right ) ^{11}}{11\,n}}+{\frac{28\,{a}^{5}{b}^{3} \left ({x}^{n} \right ) ^{10}}{5\,n}}+{\frac{28\,{a}^{6}{b}^{2} \left ({x}^{n} \right ) ^{9}}{9\,n}}+{\frac{b{a}^{7} \left ({x}^{n} \right ) ^{8}}{n}}+{\frac{{a}^{8} \left ({x}^{n} \right ) ^{7}}{7\,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 = 1.34162, size = 301, normalized size = 2.01 \begin{align*} \frac{3003 \, b^{8} x^{15 \, n} + 25740 \, a b^{7} x^{14 \, n} + 97020 \, a^{2} b^{6} x^{13 \, n} + 210210 \, a^{3} b^{5} x^{12 \, n} + 286650 \, a^{4} b^{4} x^{11 \, n} + 252252 \, a^{5} b^{3} x^{10 \, n} + 140140 \, a^{6} b^{2} x^{9 \, n} + 45045 \, a^{7} b x^{8 \, n} + 6435 \, a^{8} x^{7 \, n}}{45045 \, 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^{7 \, n - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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