Optimal. Leaf size=49 \[ 3 a^2 b \log (x)-\frac{a^3 x^{-n}}{n}+\frac{3 a b^2 x^n}{n}+\frac{b^3 x^{2 n}}{2 n} \]
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Rubi [A] time = 0.0222159, antiderivative size = 49, 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} \[ 3 a^2 b \log (x)-\frac{a^3 x^{-n}}{n}+\frac{3 a b^2 x^n}{n}+\frac{b^3 x^{2 n}}{2 n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^{-1-n} \left (a+b x^n\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^3}{x^2} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (3 a b^2+\frac{a^3}{x^2}+\frac{3 a^2 b}{x}+b^3 x\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a^3 x^{-n}}{n}+\frac{3 a b^2 x^n}{n}+\frac{b^3 x^{2 n}}{2 n}+3 a^2 b \log (x)\\ \end{align*}
Mathematica [A] time = 0.0250209, size = 45, normalized size = 0.92 \[ \frac{3 a^2 b n \log (x)-a^3 x^{-n}+3 a b^2 x^n+\frac{1}{2} b^3 x^{2 n}}{n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 62, normalized size = 1.3 \begin{align*}{\frac{1}{{{\rm e}^{n\ln \left ( x \right ) }}} \left ( 3\,b{a}^{2}\ln \left ( x \right ){{\rm e}^{n\ln \left ( x \right ) }}-{\frac{{a}^{3}}{n}}+{\frac{{b}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{2\,n}}+3\,{\frac{{b}^{2}a \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{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 = 1.27646, size = 103, normalized size = 2.1 \begin{align*} \frac{6 \, a^{2} b n x^{n} \log \left (x\right ) + b^{3} x^{3 \, n} + 6 \, a b^{2} x^{2 \, n} - 2 \, a^{3}}{2 \, n x^{n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 79.7773, size = 277, normalized size = 5.65 \begin{align*} \begin{cases} a^{3} x + 3 a^{2} b \log{\left (x \right )} - \frac{3 a b^{2}}{x} - \frac{b^{3}}{2 x^{2}} & \text{for}\: n = -1 \\\left (a + b\right )^{3} \log{\left (x \right )} & \text{for}\: n = 0 \\- \frac{2 a^{3} n}{2 n^{2} x^{n} + 2 n x^{n}} - \frac{2 a^{3}}{2 n^{2} x^{n} + 2 n x^{n}} + \frac{6 a^{2} b n^{2} x^{n} \log{\left (x \right )}}{2 n^{2} x^{n} + 2 n x^{n}} + \frac{6 a^{2} b n x^{n} \log{\left (x \right )}}{2 n^{2} x^{n} + 2 n x^{n}} + \frac{6 a^{2} b n x^{n}}{2 n^{2} x^{n} + 2 n x^{n}} + \frac{6 a b^{2} n x^{2 n}}{2 n^{2} x^{n} + 2 n x^{n}} + \frac{6 a b^{2} x^{2 n}}{2 n^{2} x^{n} + 2 n x^{n}} + \frac{b^{3} n x^{3 n}}{2 n^{2} x^{n} + 2 n x^{n}} + \frac{b^{3} x^{3 n}}{2 n^{2} x^{n} + 2 n x^{n}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20743, size = 65, normalized size = 1.33 \begin{align*} \frac{6 \, a^{2} b n x^{n} \log \left (x\right ) + b^{3} x^{3 \, n} + 6 \, a b^{2} x^{2 \, n} - 2 \, a^{3}}{2 \, n x^{n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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