Optimal. Leaf size=82 \[ -\frac{5 a^3 b^2 x^{-2 n}}{n}-\frac{10 a^2 b^3 x^{-n}}{n}-\frac{5 a^4 b x^{-3 n}}{3 n}-\frac{a^5 x^{-4 n}}{4 n}+5 a b^4 \log (x)+\frac{b^5 x^n}{n} \]
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Rubi [A] time = 0.0350119, antiderivative size = 82, 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} \[ -\frac{5 a^3 b^2 x^{-2 n}}{n}-\frac{10 a^2 b^3 x^{-n}}{n}-\frac{5 a^4 b x^{-3 n}}{3 n}-\frac{a^5 x^{-4 n}}{4 n}+5 a b^4 \log (x)+\frac{b^5 x^n}{n} \]
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 )^5 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^5} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (b^5+\frac{a^5}{x^5}+\frac{5 a^4 b}{x^4}+\frac{10 a^3 b^2}{x^3}+\frac{10 a^2 b^3}{x^2}+\frac{5 a b^4}{x}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a^5 x^{-4 n}}{4 n}-\frac{5 a^4 b x^{-3 n}}{3 n}-\frac{5 a^3 b^2 x^{-2 n}}{n}-\frac{10 a^2 b^3 x^{-n}}{n}+\frac{b^5 x^n}{n}+5 a b^4 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0407851, size = 72, normalized size = 0.88 \[ \frac{-5 a^3 b^2 x^{-2 n}-10 a^2 b^3 x^{-n}-\frac{5}{3} a^4 b x^{-3 n}-\frac{1}{4} a^5 x^{-4 n}+5 a b^4 n \log (x)+b^5 x^n}{n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 97, normalized size = 1.2 \begin{align*}{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}} \left ({\frac{{b}^{5} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{5}}{n}}+5\,a{b}^{4}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}-{\frac{{a}^{5}}{4\,n}}-10\,{\frac{{a}^{2}{b}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{n}}-5\,{\frac{{a}^{3}{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{n}}-{\frac{5\,{a}^{4}b{{\rm e}^{n\ln \left ( x \right ) }}}{3\,n}} \right ) } \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 = 1.29766, size = 174, normalized size = 2.12 \begin{align*} \frac{60 \, a b^{4} n x^{4 \, n} \log \left (x\right ) + 12 \, b^{5} x^{5 \, n} - 120 \, a^{2} b^{3} x^{3 \, n} - 60 \, a^{3} b^{2} x^{2 \, n} - 20 \, a^{4} b x^{n} - 3 \, a^{5}}{12 \, n x^{4 \, 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 = 1.2369, size = 104, normalized size = 1.27 \begin{align*} \frac{60 \, a b^{4} n x^{4 \, n} \log \left (x\right ) + 12 \, b^{5} x^{5 \, n} - 120 \, a^{2} b^{3} x^{3 \, n} - 60 \, a^{3} b^{2} x^{2 \, n} - 20 \, a^{4} b x^{n} - 3 \, a^{5}}{12 \, n x^{4 \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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