Optimal. Leaf size=97 \[ -\frac{10 a^3 b^2 x^{-7 n}}{7 n}-\frac{5 a^2 b^3 x^{-6 n}}{3 n}-\frac{5 a^4 b x^{-8 n}}{8 n}-\frac{a^5 x^{-9 n}}{9 n}-\frac{a b^4 x^{-5 n}}{n}-\frac{b^5 x^{-4 n}}{4 n} \]
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Rubi [A] time = 0.0385956, antiderivative size = 97, 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} \[ -\frac{10 a^3 b^2 x^{-7 n}}{7 n}-\frac{5 a^2 b^3 x^{-6 n}}{3 n}-\frac{5 a^4 b x^{-8 n}}{8 n}-\frac{a^5 x^{-9 n}}{9 n}-\frac{a b^4 x^{-5 n}}{n}-\frac{b^5 x^{-4 n}}{4 n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^{-1-9 n} \left (a+b x^n\right )^5 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^{10}} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^5}{x^{10}}+\frac{5 a^4 b}{x^9}+\frac{10 a^3 b^2}{x^8}+\frac{10 a^2 b^3}{x^7}+\frac{5 a b^4}{x^6}+\frac{b^5}{x^5}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a^5 x^{-9 n}}{9 n}-\frac{5 a^4 b x^{-8 n}}{8 n}-\frac{10 a^3 b^2 x^{-7 n}}{7 n}-\frac{5 a^2 b^3 x^{-6 n}}{3 n}-\frac{a b^4 x^{-5 n}}{n}-\frac{b^5 x^{-4 n}}{4 n}\\ \end{align*}
Mathematica [A] time = 0.0310675, size = 74, normalized size = 0.76 \[ -\frac{x^{-9 n} \left (720 a^3 b^2 x^{2 n}+840 a^2 b^3 x^{3 n}+315 a^4 b x^n+56 a^5+504 a b^4 x^{4 n}+126 b^5 x^{5 n}\right )}{504 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 88, normalized size = 0.9 \begin{align*} -{\frac{{b}^{5}}{4\,n \left ({x}^{n} \right ) ^{4}}}-{\frac{a{b}^{4}}{n \left ({x}^{n} \right ) ^{5}}}-{\frac{5\,{a}^{2}{b}^{3}}{3\,n \left ({x}^{n} \right ) ^{6}}}-{\frac{10\,{a}^{3}{b}^{2}}{7\,n \left ({x}^{n} \right ) ^{7}}}-{\frac{5\,{a}^{4}b}{8\,n \left ({x}^{n} \right ) ^{8}}}-{\frac{{a}^{5}}{9\,n \left ({x}^{n} \right ) ^{9}}} \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 = 1.35517, size = 171, normalized size = 1.76 \begin{align*} -\frac{126 \, b^{5} x^{5 \, n} + 504 \, a b^{4} x^{4 \, n} + 840 \, a^{2} b^{3} x^{3 \, n} + 720 \, a^{3} b^{2} x^{2 \, n} + 315 \, a^{4} b x^{n} + 56 \, a^{5}}{504 \, n x^{9 \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 174.117, size = 94, normalized size = 0.97 \begin{align*} \begin{cases} - \frac{a^{5} x^{- 9 n}}{9 n} - \frac{5 a^{4} b x^{- 8 n}}{8 n} - \frac{10 a^{3} b^{2} x^{- 7 n}}{7 n} - \frac{5 a^{2} b^{3} x^{- 6 n}}{3 n} - \frac{a b^{4} x^{- 5 n}}{n} - \frac{b^{5} x^{- 4 n}}{4 n} & \text{for}\: n \neq 0 \\\left (a + b\right )^{5} \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 [A] time = 1.22793, size = 100, normalized size = 1.03 \begin{align*} -\frac{126 \, b^{5} x^{5 \, n} + 504 \, a b^{4} x^{4 \, n} + 840 \, a^{2} b^{3} x^{3 \, n} + 720 \, a^{3} b^{2} x^{2 \, n} + 315 \, a^{4} b x^{n} + 56 \, a^{5}}{504 \, n x^{9 \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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