Optimal. Leaf size=99 \[ -\frac{5 a^3 b^2 x^{-8 n}}{4 n}-\frac{10 a^2 b^3 x^{-7 n}}{7 n}-\frac{5 a^4 b x^{-9 n}}{9 n}-\frac{a^5 x^{-10 n}}{10 n}-\frac{5 a b^4 x^{-6 n}}{6 n}-\frac{b^5 x^{-5 n}}{5 n} \]
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Rubi [A] time = 0.040428, antiderivative size = 99, 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} \[ -\frac{5 a^3 b^2 x^{-8 n}}{4 n}-\frac{10 a^2 b^3 x^{-7 n}}{7 n}-\frac{5 a^4 b x^{-9 n}}{9 n}-\frac{a^5 x^{-10 n}}{10 n}-\frac{5 a b^4 x^{-6 n}}{6 n}-\frac{b^5 x^{-5 n}}{5 n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^{-1-10 n} \left (a+b x^n\right )^5 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^{11}} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^5}{x^{11}}+\frac{5 a^4 b}{x^{10}}+\frac{10 a^3 b^2}{x^9}+\frac{10 a^2 b^3}{x^8}+\frac{5 a b^4}{x^7}+\frac{b^5}{x^6}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a^5 x^{-10 n}}{10 n}-\frac{5 a^4 b x^{-9 n}}{9 n}-\frac{5 a^3 b^2 x^{-8 n}}{4 n}-\frac{10 a^2 b^3 x^{-7 n}}{7 n}-\frac{5 a b^4 x^{-6 n}}{6 n}-\frac{b^5 x^{-5 n}}{5 n}\\ \end{align*}
Mathematica [A] time = 0.032089, size = 74, normalized size = 0.75 \[ -\frac{x^{-10 n} \left (1575 a^3 b^2 x^{2 n}+1800 a^2 b^3 x^{3 n}+700 a^4 b x^n+126 a^5+1050 a b^4 x^{4 n}+252 b^5 x^{5 n}\right )}{1260 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 88, normalized size = 0.9 \begin{align*} -{\frac{{b}^{5}}{5\,n \left ({x}^{n} \right ) ^{5}}}-{\frac{5\,a{b}^{4}}{6\,n \left ({x}^{n} \right ) ^{6}}}-{\frac{10\,{a}^{2}{b}^{3}}{7\,n \left ({x}^{n} \right ) ^{7}}}-{\frac{5\,{a}^{3}{b}^{2}}{4\,n \left ({x}^{n} \right ) ^{8}}}-{\frac{5\,{a}^{4}b}{9\,n \left ({x}^{n} \right ) ^{9}}}-{\frac{{a}^{5}}{10\,n \left ({x}^{n} \right ) ^{10}}} \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 = 1.25522, size = 180, normalized size = 1.82 \begin{align*} -\frac{252 \, b^{5} x^{5 \, n} + 1050 \, a b^{4} x^{4 \, n} + 1800 \, a^{2} b^{3} x^{3 \, n} + 1575 \, a^{3} b^{2} x^{2 \, n} + 700 \, a^{4} b x^{n} + 126 \, a^{5}}{1260 \, n x^{10 \, 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 [A] time = 1.23201, size = 100, normalized size = 1.01 \begin{align*} -\frac{252 \, b^{5} x^{5 \, n} + 1050 \, a b^{4} x^{4 \, n} + 1800 \, a^{2} b^{3} x^{3 \, n} + 1575 \, a^{3} b^{2} x^{2 \, n} + 700 \, a^{4} b x^{n} + 126 \, a^{5}}{1260 \, n x^{10 \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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