Optimal. Leaf size=50 \[ \frac{3 a^2 b x^n}{n}+a^3 \log (x)+\frac{3 a b^2 x^{2 n}}{2 n}+\frac{b^3 x^{3 n}}{3 n} \]
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Rubi [A] time = 0.0201271, antiderivative size = 50, 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} \[ \frac{3 a^2 b x^n}{n}+a^3 \log (x)+\frac{3 a b^2 x^{2 n}}{2 n}+\frac{b^3 x^{3 n}}{3 n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+b x^n\right )^3}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^3}{x} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (3 a^2 b+\frac{a^3}{x}+3 a b^2 x+b^3 x^2\right ) \, dx,x,x^n\right )}{n}\\ &=\frac{3 a^2 b x^n}{n}+\frac{3 a b^2 x^{2 n}}{2 n}+\frac{b^3 x^{3 n}}{3 n}+a^3 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0242516, size = 41, normalized size = 0.82 \[ \frac{b x^n \left (18 a^2+9 a b x^n+2 b^2 x^{2 n}\right )}{6 n}+a^3 \log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 52, normalized size = 1. \begin{align*}{\frac{{b}^{3} \left ({x}^{n} \right ) ^{3}}{3\,n}}+{\frac{3\, \left ({x}^{n} \right ) ^{2}{b}^{2}a}{2\,n}}+3\,{\frac{b{a}^{2}{x}^{n}}{n}}+{\frac{{a}^{3}\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 = 0.983369, size = 65, normalized size = 1.3 \begin{align*} \frac{a^{3} \log \left (x^{n}\right )}{n} + \frac{2 \, b^{3} x^{3 \, n} + 9 \, a b^{2} x^{2 \, n} + 18 \, a^{2} b x^{n}}{6 \, n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.31316, size = 99, normalized size = 1.98 \begin{align*} \frac{6 \, a^{3} n \log \left (x\right ) + 2 \, b^{3} x^{3 \, n} + 9 \, a b^{2} x^{2 \, n} + 18 \, a^{2} b x^{n}}{6 \, n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.376683, size = 53, normalized size = 1.06 \begin{align*} \begin{cases} a^{3} \log{\left (x \right )} + \frac{3 a^{2} b x^{n}}{n} + \frac{3 a b^{2} x^{2 n}}{2 n} + \frac{b^{3} x^{3 n}}{3 n} & \text{for}\: n \neq 0 \\\left (a + b\right )^{3} \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 )}^{3}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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