Optimal. Leaf size=50 \[ \frac{n x \left (a+b x^n\right )^{-1/n}}{a^2 (n+1)}+\frac{x \left (a+b x^n\right )^{-\frac{1}{n}-1}}{a (n+1)} \]
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Rubi [A] time = 0.0121668, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {192, 191} \[ \frac{n x \left (a+b x^n\right )^{-1/n}}{a^2 (n+1)}+\frac{x \left (a+b x^n\right )^{-\frac{1}{n}-1}}{a (n+1)} \]
Antiderivative was successfully verified.
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Rule 192
Rule 191
Rubi steps
\begin{align*} \int \left (a+b x^n\right )^{-2-\frac{1}{n}} \, dx &=\frac{x \left (a+b x^n\right )^{-1-\frac{1}{n}}}{a (1+n)}+\frac{n \int \left (a+b x^n\right )^{-1-\frac{1}{n}} \, dx}{a (1+n)}\\ &=\frac{x \left (a+b x^n\right )^{-1-\frac{1}{n}}}{a (1+n)}+\frac{n x \left (a+b x^n\right )^{-1/n}}{a^2 (1+n)}\\ \end{align*}
Mathematica [C] time = 0.0275574, size = 55, normalized size = 1.1 \[ \frac{x \left (a+b x^n\right )^{-1/n} \left (\frac{b x^n}{a}+1\right )^{\frac{1}{n}} \, _2F_1\left (2+\frac{1}{n},\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.093, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{x}^{n} \right ) ^{-2-{n}^{-1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{n} + a\right )}^{-\frac{1}{n} - 2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.3988, size = 139, normalized size = 2.78 \begin{align*} \frac{b^{2} n x x^{2 \, n} +{\left (2 \, a b n + a b\right )} x x^{n} +{\left (a^{2} n + a^{2}\right )} x}{{\left (a^{2} n + a^{2}\right )}{\left (b x^{n} + a\right )}^{\frac{2 \, n + 1}{n}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\left (b x^{n} + a\right )}^{-\frac{1}{n} - 2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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