Optimal. Leaf size=84 \[ \frac{5 a^3 b^2 x^{2 n}}{n}+\frac{10 a^2 b^3 x^{3 n}}{3 n}+\frac{5 a^4 b x^n}{n}+a^5 \log (x)+\frac{5 a b^4 x^{4 n}}{4 n}+\frac{b^5 x^{5 n}}{5 n} \]
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Rubi [A] time = 0.0352064, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{5 a^3 b^2 x^{2 n}}{n}+\frac{10 a^2 b^3 x^{3 n}}{3 n}+\frac{5 a^4 b x^n}{n}+a^5 \log (x)+\frac{5 a b^4 x^{4 n}}{4 n}+\frac{b^5 x^{5 n}}{5 n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+b x^n\right )^5}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^5}{x} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (5 a^4 b+\frac{a^5}{x}+10 a^3 b^2 x+10 a^2 b^3 x^2+5 a b^4 x^3+b^5 x^4\right ) \, dx,x,x^n\right )}{n}\\ &=\frac{5 a^4 b x^n}{n}+\frac{5 a^3 b^2 x^{2 n}}{n}+\frac{10 a^2 b^3 x^{3 n}}{3 n}+\frac{5 a b^4 x^{4 n}}{4 n}+\frac{b^5 x^{5 n}}{5 n}+a^5 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0391113, size = 74, normalized size = 0.88 \[ \frac{5 a^3 b^2 x^{2 n}+\frac{10}{3} a^2 b^3 x^{3 n}+5 a^4 b x^n+a^5 n \log (x)+\frac{5}{4} a b^4 x^{4 n}+\frac{1}{5} b^5 x^{5 n}}{n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 84, normalized size = 1. \begin{align*}{\frac{{b}^{5} \left ({x}^{n} \right ) ^{5}}{5\,n}}+{\frac{5\,a{b}^{4} \left ({x}^{n} \right ) ^{4}}{4\,n}}+{\frac{10\,{a}^{2}{b}^{3} \left ({x}^{n} \right ) ^{3}}{3\,n}}+5\,{\frac{{a}^{3}{b}^{2} \left ({x}^{n} \right ) ^{2}}{n}}+5\,{\frac{{a}^{4}b{x}^{n}}{n}}+{\frac{{a}^{5}\ln \left ({x}^{n} \right ) }{n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.98316, size = 100, normalized size = 1.19 \begin{align*} \frac{a^{5} \log \left (x^{n}\right )}{n} + \frac{12 \, b^{5} x^{5 \, n} + 75 \, a b^{4} x^{4 \, n} + 200 \, a^{2} b^{3} x^{3 \, n} + 300 \, a^{3} b^{2} x^{2 \, n} + 300 \, a^{4} b x^{n}}{60 \, n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28355, size = 165, normalized size = 1.96 \begin{align*} \frac{60 \, a^{5} n \log \left (x\right ) + 12 \, b^{5} x^{5 \, n} + 75 \, a b^{4} x^{4 \, n} + 200 \, a^{2} b^{3} x^{3 \, n} + 300 \, a^{3} b^{2} x^{2 \, n} + 300 \, a^{4} b x^{n}}{60 \, n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.778728, size = 85, normalized size = 1.01 \begin{align*} \begin{cases} a^{5} \log{\left (x \right )} + \frac{5 a^{4} b x^{n}}{n} + \frac{5 a^{3} b^{2} x^{2 n}}{n} + \frac{10 a^{2} b^{3} x^{3 n}}{3 n} + \frac{5 a b^{4} x^{4 n}}{4 n} + \frac{b^{5} x^{5 n}}{5 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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{n} + a\right )}^{5}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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