Optimal. Leaf size=19 \[ \frac{\left (a+b x^n\right )^9}{9 b n} \]
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Rubi [A] time = 0.0045803, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {261} \[ \frac{\left (a+b x^n\right )^9}{9 b n} \]
Antiderivative was successfully verified.
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Rule 261
Rubi steps
\begin{align*} \int x^{-1+n} \left (a+b x^n\right )^8 \, dx &=\frac{\left (a+b x^n\right )^9}{9 b n}\\ \end{align*}
Mathematica [A] time = 0.004583, size = 19, normalized size = 1. \[ \frac{\left (a+b x^n\right )^9}{9 b n} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.023, size = 132, normalized size = 7. \begin{align*}{\frac{{b}^{8} \left ({x}^{n} \right ) ^{9}}{9\,n}}+{\frac{{b}^{7}a \left ({x}^{n} \right ) ^{8}}{n}}+4\,{\frac{{b}^{6}{a}^{2} \left ({x}^{n} \right ) ^{7}}{n}}+{\frac{28\,{a}^{3}{b}^{5} \left ({x}^{n} \right ) ^{6}}{3\,n}}+14\,{\frac{{a}^{4}{b}^{4} \left ({x}^{n} \right ) ^{5}}{n}}+14\,{\frac{{a}^{5}{b}^{3} \left ({x}^{n} \right ) ^{4}}{n}}+{\frac{28\,{a}^{6}{b}^{2} \left ({x}^{n} \right ) ^{3}}{3\,n}}+4\,{\frac{b{a}^{7} \left ({x}^{n} \right ) ^{2}}{n}}+{\frac{{a}^{8}{x}^{n}}{n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.964729, size = 23, normalized size = 1.21 \begin{align*} \frac{{\left (b x^{n} + a\right )}^{9}}{9 \, b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.3705, size = 239, normalized size = 12.58 \begin{align*} \frac{b^{8} x^{9 \, n} + 9 \, a b^{7} x^{8 \, n} + 36 \, a^{2} b^{6} x^{7 \, n} + 84 \, a^{3} b^{5} x^{6 \, n} + 126 \, a^{4} b^{4} x^{5 \, n} + 126 \, a^{5} b^{3} x^{4 \, n} + 84 \, a^{6} b^{2} x^{3 \, n} + 36 \, a^{7} b x^{2 \, n} + 9 \, a^{8} x^{n}}{9 \, n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 86.3691, size = 133, normalized size = 7. \begin{align*} \begin{cases} \frac{a^{8} x^{n}}{n} + \frac{4 a^{7} b x^{2 n}}{n} + \frac{28 a^{6} b^{2} x^{3 n}}{3 n} + \frac{14 a^{5} b^{3} x^{4 n}}{n} + \frac{14 a^{4} b^{4} x^{5 n}}{n} + \frac{28 a^{3} b^{5} x^{6 n}}{3 n} + \frac{4 a^{2} b^{6} x^{7 n}}{n} + \frac{a b^{7} x^{8 n}}{n} + \frac{b^{8} x^{9 n}}{9 n} & \text{for}\: n \neq 0 \\\left (a + b\right )^{8} \log{\left (x \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.21871, size = 149, normalized size = 7.84 \begin{align*} \frac{b^{8} x^{9 \, n} + 9 \, a b^{7} x^{8 \, n} + 36 \, a^{2} b^{6} x^{7 \, n} + 84 \, a^{3} b^{5} x^{6 \, n} + 126 \, a^{4} b^{4} x^{5 \, n} + 126 \, a^{5} b^{3} x^{4 \, n} + 84 \, a^{6} b^{2} x^{3 \, n} + 36 \, a^{7} b x^{2 \, n} + 9 \, a^{8} x^{n}}{9 \, n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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