Optimal. Leaf size=53 \[ \frac{2 \sqrt{b (c x)^n-a}}{n}-\frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right )}{n} \]
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Rubi [A] time = 0.0414853, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {367, 12, 266, 50, 63, 205} \[ \frac{2 \sqrt{b (c x)^n-a}}{n}-\frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right )}{n} \]
Antiderivative was successfully verified.
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Rule 367
Rule 12
Rule 266
Rule 50
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{\sqrt{-a+b (c x)^n}}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{c \sqrt{-a+b x^n}}{x} \, dx,x,c x\right )}{c}\\ &=\operatorname{Subst}\left (\int \frac{\sqrt{-a+b x^n}}{x} \, dx,x,c x\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{\sqrt{-a+b x}}{x} \, dx,x,(c x)^n\right )}{n}\\ &=\frac{2 \sqrt{-a+b (c x)^n}}{n}-\frac{a \operatorname{Subst}\left (\int \frac{1}{x \sqrt{-a+b x}} \, dx,x,(c x)^n\right )}{n}\\ &=\frac{2 \sqrt{-a+b (c x)^n}}{n}-\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{-a+b (c x)^n}\right )}{b n}\\ &=\frac{2 \sqrt{-a+b (c x)^n}}{n}-\frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{-a+b (c x)^n}}{\sqrt{a}}\right )}{n}\\ \end{align*}
Mathematica [A] time = 0.0223945, size = 51, normalized size = 0.96 \[ \frac{2 \sqrt{b (c x)^n-a}-2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right )}{n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 46, normalized size = 0.9 \begin{align*} -2\,{\frac{\sqrt{a}}{n}\arctan \left ({\frac{\sqrt{-a+b \left ( cx \right ) ^{n}}}{\sqrt{a}}} \right ) }+2\,{\frac{\sqrt{-a+b \left ( cx \right ) ^{n}}}{n}} \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{\sqrt{\left (c x\right )^{n} b - a}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60105, size = 243, normalized size = 4.58 \begin{align*} \left [\frac{\sqrt{-a} \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}}{n}, -\frac{2 \,{\left (\sqrt{a} \arctan \left (\frac{\sqrt{\left (c x\right )^{n} b - a}}{\sqrt{a}}\right ) - \sqrt{\left (c x\right )^{n} b - a}\right )}}{n}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- a + b \left (c x\right )^{n}}}{x}\, dx \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{\sqrt{\left (c x\right )^{n} b - a}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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