Optimal. Leaf size=87 \[ -\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{4 a^{5/2} n}+\frac{3 b x^{-n} \sqrt{a+b x^n}}{4 a^2 n}-\frac{x^{-2 n} \sqrt{a+b x^n}}{2 a n} \]
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Rubi [A] time = 0.0357873, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {266, 51, 63, 208} \[ -\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{4 a^{5/2} n}+\frac{3 b x^{-n} \sqrt{a+b x^n}}{4 a^2 n}-\frac{x^{-2 n} \sqrt{a+b x^n}}{2 a n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{x^{-1-2 n}}{\sqrt{a+b x^n}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{a+b x}} \, dx,x,x^n\right )}{n}\\ &=-\frac{x^{-2 n} \sqrt{a+b x^n}}{2 a n}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,x^n\right )}{4 a n}\\ &=-\frac{x^{-2 n} \sqrt{a+b x^n}}{2 a n}+\frac{3 b x^{-n} \sqrt{a+b x^n}}{4 a^2 n}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^n\right )}{8 a^2 n}\\ &=-\frac{x^{-2 n} \sqrt{a+b x^n}}{2 a n}+\frac{3 b x^{-n} \sqrt{a+b x^n}}{4 a^2 n}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^n}\right )}{4 a^2 n}\\ &=-\frac{x^{-2 n} \sqrt{a+b x^n}}{2 a n}+\frac{3 b x^{-n} \sqrt{a+b x^n}}{4 a^2 n}-\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{4 a^{5/2} n}\\ \end{align*}
Mathematica [C] time = 0.0096511, size = 40, normalized size = 0.46 \[ -\frac{2 b^2 \sqrt{a+b x^n} \, _2F_1\left (\frac{1}{2},3;\frac{3}{2};\frac{b x^n}{a}+1\right )}{a^3 n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.086, size = 0, normalized size = 0. \begin{align*} \int{{x}^{-1-2\,n}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, 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 \frac{x^{-2 \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.06533, size = 350, normalized size = 4.02 \begin{align*} \left [\frac{3 \, \sqrt{a} b^{2} x^{2 \, n} \log \left (\frac{b x^{n} - 2 \, \sqrt{b x^{n} + a} \sqrt{a} + 2 \, a}{x^{n}}\right ) + 2 \,{\left (3 \, a b x^{n} - 2 \, a^{2}\right )} \sqrt{b x^{n} + a}}{8 \, a^{3} n x^{2 \, n}}, \frac{3 \, \sqrt{-a} b^{2} x^{2 \, n} \arctan \left (\frac{\sqrt{b x^{n} + a} \sqrt{-a}}{a}\right ) +{\left (3 \, a b x^{n} - 2 \, a^{2}\right )} \sqrt{b x^{n} + a}}{4 \, a^{3} n x^{2 \, n}}\right ] \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 \frac{x^{-2 \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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