Optimal. Leaf size=54 \[ \frac{2}{a c n \sqrt{a+b x^n}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{a^{3/2} c n} \]
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Rubi [A] time = 0.0332688, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {12, 266, 51, 63, 208} \[ \frac{2}{a c n \sqrt{a+b x^n}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{a^{3/2} c n} \]
Antiderivative was successfully verified.
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Rule 12
Rule 266
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{c x \left (a+b x^n\right )^{3/2}} \, dx &=\frac{\int \frac{1}{x \left (a+b x^n\right )^{3/2}} \, dx}{c}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{3/2}} \, dx,x,x^n\right )}{c n}\\ &=\frac{2}{a c n \sqrt{a+b x^n}}+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^n\right )}{a c n}\\ &=\frac{2}{a c 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 c n}\\ &=\frac{2}{a c n \sqrt{a+b x^n}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{a^{3/2} c n}\\ \end{align*}
Mathematica [C] time = 0.0140132, size = 40, normalized size = 0.74 \[ \frac{2 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{b x^n}{a}+1\right )}{a c n \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 42, normalized size = 0.8 \begin{align*}{\frac{1}{cn} \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 = 0.846647, size = 335, normalized size = 6.2 \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 c n x^{n} + a^{3} c 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 c n x^{n} + a^{3} c n}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 3.49617, size = 185, normalized size = 3.43 \begin{align*} \frac{\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}}}{c} \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}} c x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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