Optimal. Leaf size=46 \[ \frac{a^2 \log \left (a+b x^n\right )}{b^3 n}-\frac{a x^n}{b^2 n}+\frac{x^{2 n}}{2 b n} \]
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Rubi [A] time = 0.0242214, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {266, 43} \[ \frac{a^2 \log \left (a+b x^n\right )}{b^3 n}-\frac{a x^n}{b^2 n}+\frac{x^{2 n}}{2 b n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{2+3 (-1+n)}}{a+b x^n} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{a+b x} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a}{b^2}+\frac{x}{b}+\frac{a^2}{b^2 (a+b x)}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a x^n}{b^2 n}+\frac{x^{2 n}}{2 b n}+\frac{a^2 \log \left (a+b x^n\right )}{b^3 n}\\ \end{align*}
Mathematica [A] time = 0.0123571, size = 38, normalized size = 0.83 \[ \frac{2 a^2 \log \left (a+b x^n\right )+b x^n \left (b x^n-2 a\right )}{2 b^3 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 51, normalized size = 1.1 \begin{align*}{\frac{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{2\,bn}}-{\frac{a{{\rm e}^{n\ln \left ( x \right ) }}}{{b}^{2}n}}+{\frac{{a}^{2}\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{b}^{3}n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.997813, size = 61, normalized size = 1.33 \begin{align*} \frac{a^{2} \log \left (\frac{b x^{n} + a}{b}\right )}{b^{3} n} + \frac{b x^{2 \, n} - 2 \, a x^{n}}{2 \, b^{2} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.02448, size = 84, normalized size = 1.83 \begin{align*} \frac{b^{2} x^{2 \, n} - 2 \, a b x^{n} + 2 \, a^{2} \log \left (b x^{n} + a\right )}{2 \, b^{3} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 31.0822, size = 56, normalized size = 1.22 \begin{align*} \begin{cases} \frac{\log{\left (x \right )}}{a} & \text{for}\: b = 0 \wedge n = 0 \\\frac{x^{3 n}}{3 a n} & \text{for}\: b = 0 \\\frac{\log{\left (x \right )}}{a + b} & \text{for}\: n = 0 \\\frac{a^{2} \log{\left (\frac{a}{b} + x^{n} \right )}}{b^{3} n} - \frac{a x^{n}}{b^{2} n} + \frac{x^{2 n}}{2 b n} & \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{x^{3 \, n - 1}}{b x^{n} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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