Optimal. Leaf size=37 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \left (c (d x)^m\right )^n}}{\sqrt{a}}\right )}{\sqrt{a} m n} \]
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Rubi [A] time = 0.166927, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {367, 12, 266, 63, 208} \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \left (c (d x)^m\right )^n}}{\sqrt{a}}\right )}{\sqrt{a} m n} \]
Antiderivative was successfully verified.
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Rule 367
Rule 12
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt{a+b \left (c (d x)^m\right )^n}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b (c x)^n}} \, dx,x,(d x)^m\right )}{m}\\ &=\frac{\operatorname{Subst}\left (\int \frac{c}{x \sqrt{a+b x^n}} \, dx,x,c (d x)^m\right )}{c m}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x^n}} \, dx,x,c (d x)^m\right )}{m}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\left (c (d x)^m\right )^n\right )}{m n}\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \left (c (d x)^m\right )^n}\right )}{b m n}\\ &=-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \left (c (d x)^m\right )^n}}{\sqrt{a}}\right )}{\sqrt{a} m n}\\ \end{align*}
Mathematica [A] time = 0.0658406, size = 37, normalized size = 1. \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \left (c (d x)^m\right )^n}}{\sqrt{a}}\right )}{\sqrt{a} m n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 32, normalized size = 0.9 \begin{align*} -2\,{\frac{1}{nm\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{a+b \left ( c \left ( dx \right ) ^{m} \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 (\left (d x\right )^{m} c\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 = 1.58939, size = 300, normalized size = 8.11 \begin{align*} \left [\frac{\log \left ({\left (b e^{\left (m n \log \left (d x\right ) + n \log \left (c\right )\right )} - 2 \, \sqrt{b e^{\left (m n \log \left (d x\right ) + n \log \left (c\right )\right )} + a} \sqrt{a} + 2 \, a\right )} e^{\left (-m n \log \left (d x\right ) - n \log \left (c\right )\right )}\right )}{\sqrt{a} m n}, \frac{2 \, \sqrt{-a} \arctan \left (\frac{\sqrt{b e^{\left (m n \log \left (d x\right ) + n \log \left (c\right )\right )} + a} \sqrt{-a}}{a}\right )}{a m 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 \left (d x\right )^{m}\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 (\left (d x\right )^{m} c\right )^{n} b + a} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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