Optimal. Leaf size=32 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right )}{\sqrt{a} n} \]
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Rubi [A] time = 0.031478, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {367, 12, 266, 63, 205} \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right )}{\sqrt{a} n} \]
Antiderivative was successfully verified.
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Rule 367
Rule 12
Rule 266
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt{-a+b (c x)^n}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{c}{x \sqrt{-a+b x^n}} \, dx,x,c x\right )}{c}\\ &=\operatorname{Subst}\left (\int \frac{1}{x \sqrt{-a+b x^n}} \, dx,x,c x\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{-a+b x}} \, dx,x,(c x)^n\right )}{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 )}{b n}\\ &=\frac{2 \tan ^{-1}\left (\frac{\sqrt{-a+b (c x)^n}}{\sqrt{a}}\right )}{\sqrt{a} n}\\ \end{align*}
Mathematica [A] time = 0.0202475, size = 32, normalized size = 1. \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right )}{\sqrt{a} n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 27, normalized size = 0.8 \begin{align*} 2\,{\frac{1}{n\sqrt{a}}\arctan \left ({\frac{\sqrt{-a+b \left ( cx \right ) ^{n}}}{\sqrt{a}}} \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}{\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 = 1.58977, size = 184, normalized size = 5.75 \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 )}{a n}, \frac{2 \, \arctan \left (\frac{\sqrt{\left (c x\right )^{n} b - a}}{\sqrt{a}}\right )}{\sqrt{a} 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{1}{x \sqrt{- a + b \left (c x\right )^{n}}}\, 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{1}{\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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