Optimal. Leaf size=52 \[ \frac{2}{a n \sqrt{a+b (c x)^n}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right )}{a^{3/2} n} \]
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Rubi [A] time = 0.046182, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {367, 12, 266, 51, 63, 208} \[ \frac{2}{a n \sqrt{a+b (c x)^n}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right )}{a^{3/2} n} \]
Antiderivative was successfully verified.
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Rule 367
Rule 12
Rule 266
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x \left (a+b (c x)^n\right )^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{c}{x \left (a+b x^n\right )^{3/2}} \, dx,x,c x\right )}{c}\\ &=\operatorname{Subst}\left (\int \frac{1}{x \left (a+b x^n\right )^{3/2}} \, dx,x,c x\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{3/2}} \, dx,x,(c x)^n\right )}{n}\\ &=\frac{2}{a n \sqrt{a+b (c x)^n}}+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,(c x)^n\right )}{a n}\\ &=\frac{2}{a n \sqrt{a+b (c x)^n}}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b (c x)^n}\right )}{a b n}\\ &=\frac{2}{a n \sqrt{a+b (c x)^n}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right )}{a^{3/2} n}\\ \end{align*}
Mathematica [C] time = 0.0286098, size = 41, normalized size = 0.79 \[ \frac{2 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{b (c x)^n}{a}+1\right )}{a n \sqrt{a+b (c x)^n}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 43, normalized size = 0.8 \begin{align*}{\frac{1}{n} \left ( -2\,{\frac{1}{{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{a+b \left ( cx \right ) ^{n}}}{\sqrt{a}}} \right ) }+2\,{\frac{1}{a\sqrt{a+b \left ( cx \right ) ^{n}}}} \right ) } \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{1}{{\left (\left (c x\right )^{n} b + a\right )}^{\frac{3}{2}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78772, size = 378, normalized size = 7.27 \begin{align*} \left [\frac{{\left (\left (c x\right )^{n} \sqrt{a} b + a^{\frac{3}{2}}\right )} \log \left (\frac{\left (c x\right )^{n} b - 2 \, \sqrt{\left (c x\right )^{n} b + a} \sqrt{a} + 2 \, a}{\left (c x\right )^{n}}\right ) + 2 \, \sqrt{\left (c x\right )^{n} b + a} a}{\left (c x\right )^{n} a^{2} b n + a^{3} n}, \frac{2 \,{\left ({\left (\left (c x\right )^{n} \sqrt{-a} b + \sqrt{-a} a\right )} \arctan \left (\frac{\sqrt{\left (c x\right )^{n} b + a} \sqrt{-a}}{a}\right ) + \sqrt{\left (c x\right )^{n} b + a} a\right )}}{\left (c x\right )^{n} a^{2} b n + a^{3} n}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.01044, size = 48, normalized size = 0.92 \begin{align*} \frac{2}{a n \sqrt{a + b \left (c x\right )^{n}}} + \frac{2 \operatorname{atan}{\left (\frac{\sqrt{a + b \left (c x\right )^{n}}}{\sqrt{- a}} \right )}}{a n \sqrt{- a}} \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 (\left (c x\right )^{n} b + a\right )}^{\frac{3}{2}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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