Optimal. Leaf size=48 \[ \frac{2}{a n \sqrt{a+b x^n}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{a^{3/2} n} \]
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Rubi [A] time = 0.026104, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 51, 63, 208} \[ \frac{2}{a n \sqrt{a+b x^n}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{a^{3/2} n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x \left (a+b x^n\right )^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{3/2}} \, dx,x,x^n\right )}{n}\\ &=\frac{2}{a n \sqrt{a+b x^n}}+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^n\right )}{a n}\\ &=\frac{2}{a n \sqrt{a+b x^n}}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^n}\right )}{a b n}\\ &=\frac{2}{a n \sqrt{a+b x^n}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{a^{3/2} n}\\ \end{align*}
Mathematica [C] time = 0.0083244, size = 37, normalized size = 0.77 \[ \frac{2 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{b x^n}{a}+1\right )}{a n \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 39, normalized size = 0.8 \begin{align*}{\frac{1}{n} \left ( -2\,{\frac{1}{{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{a+b{x}^{n}}}{\sqrt{a}}} \right ) }+2\,{\frac{1}{a\sqrt{a+b{x}^{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 = 1.05994, size = 324, normalized size = 6.75 \begin{align*} \left [\frac{{\left (\sqrt{a} b x^{n} + a^{\frac{3}{2}}\right )} \log \left (\frac{b x^{n} - 2 \, \sqrt{b x^{n} + a} \sqrt{a} + 2 \, a}{x^{n}}\right ) + 2 \, \sqrt{b x^{n} + a} a}{a^{2} b n x^{n} + a^{3} n}, \frac{2 \,{\left ({\left (\sqrt{-a} b x^{n} + \sqrt{-a} a\right )} \arctan \left (\frac{\sqrt{b x^{n} + a} \sqrt{-a}}{a}\right ) + \sqrt{b x^{n} + a} a\right )}}{a^{2} b n x^{n} + a^{3} n}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.50453, size = 184, normalized size = 3.83 \begin{align*} \frac{2 a^{3} \sqrt{1 + \frac{b x^{n}}{a}}}{a^{\frac{9}{2}} n + a^{\frac{7}{2}} b n x^{n}} + \frac{a^{3} \log{\left (\frac{b x^{n}}{a} \right )}}{a^{\frac{9}{2}} n + a^{\frac{7}{2}} b n x^{n}} - \frac{2 a^{3} \log{\left (\sqrt{1 + \frac{b x^{n}}{a}} + 1 \right )}}{a^{\frac{9}{2}} n + a^{\frac{7}{2}} b n x^{n}} + \frac{a^{2} b x^{n} \log{\left (\frac{b x^{n}}{a} \right )}}{a^{\frac{9}{2}} n + a^{\frac{7}{2}} b n x^{n}} - \frac{2 a^{2} b x^{n} \log{\left (\sqrt{1 + \frac{b x^{n}}{a}} + 1 \right )}}{a^{\frac{9}{2}} n + a^{\frac{7}{2}} b n x^{n}} \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{1}{{\left (b x^{n} + a\right )}^{\frac{3}{2}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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