Optimal. Leaf size=48 \[ -\frac{a^2}{b^3 n \left (a+b x^n\right )}-\frac{2 a \log \left (a+b x^n\right )}{b^3 n}+\frac{x^n}{b^2 n} \]
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Rubi [A] time = 0.0315231, antiderivative size = 48, 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} \[ -\frac{a^2}{b^3 n \left (a+b x^n\right )}-\frac{2 a \log \left (a+b x^n\right )}{b^3 n}+\frac{x^n}{b^2 n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{-1+3 n}}{\left (a+b x^n\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{(a+b x)^2} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{b^2}+\frac{a^2}{b^2 (a+b x)^2}-\frac{2 a}{b^2 (a+b x)}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac{x^n}{b^2 n}-\frac{a^2}{b^3 n \left (a+b x^n\right )}-\frac{2 a \log \left (a+b x^n\right )}{b^3 n}\\ \end{align*}
Mathematica [A] time = 0.0363026, size = 38, normalized size = 0.79 \[ \frac{-\frac{a^2}{a+b x^n}-2 a \log \left (a+b x^n\right )+b x^n}{b^3 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 59, normalized size = 1.2 \begin{align*}{\frac{1}{a+b{{\rm e}^{n\ln \left ( x \right ) }}} \left ({\frac{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{bn}}-2\,{\frac{{a}^{2}}{{b}^{3}n}} \right ) }-2\,{\frac{a\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 = 1.00195, size = 82, normalized size = 1.71 \begin{align*} \frac{b^{2} x^{2 \, n} + a b x^{n} - a^{2}}{b^{4} n x^{n} + a b^{3} n} - \frac{2 \, a \log \left (\frac{b x^{n} + a}{b}\right )}{b^{3} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.04351, size = 119, normalized size = 2.48 \begin{align*} \frac{b^{2} x^{2 \, n} + a b x^{n} - a^{2} - 2 \,{\left (a b x^{n} + a^{2}\right )} \log \left (b x^{n} + a\right )}{b^{4} n x^{n} + a b^{3} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 93.1684, size = 133, normalized size = 2.77 \begin{align*} \begin{cases} \frac{\log{\left (x \right )}}{a^{2}} & \text{for}\: b = 0 \wedge n = 0 \\\frac{x^{3 n}}{3 a^{2} n} & \text{for}\: b = 0 \\\frac{\log{\left (x \right )}}{\left (a + b\right )^{2}} & \text{for}\: n = 0 \\- \frac{2 a^{2} \log{\left (\frac{a}{b} + x^{n} \right )}}{a b^{3} n + b^{4} n x^{n}} - \frac{2 a b x^{n} \log{\left (\frac{a}{b} + x^{n} \right )}}{a b^{3} n + b^{4} n x^{n}} + \frac{2 a b x^{n}}{a b^{3} n + b^{4} n x^{n}} + \frac{b^{2} x^{2 n}}{a b^{3} n + b^{4} n x^{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}}{{\left (b x^{n} + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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