Optimal. Leaf size=49 \[ \frac{2 \sqrt{a+b (c x)^n}}{n}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right )}{n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0415735, antiderivative size = 49, 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, 50, 63, 208} \[ \frac{2 \sqrt{a+b (c x)^n}}{n}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right )}{n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 367
Rule 12
Rule 266
Rule 50
Rule 63
Rule 208
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} \tanh ^{-1}\left (\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right )}{n}\\ \end{align*}
Mathematica [A] time = 0.0214182, size = 47, normalized size = 0.96 \[ \frac{2 \sqrt{a+b (c x)^n}-2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right )}{n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 40, normalized size = 0.8 \begin{align*}{\frac{1}{n} \left ( 2\,\sqrt{a+b \left ( cx \right ) ^{n}}-2\,\sqrt{a}{\it Artanh} \left ({\frac{\sqrt{a+b \left ( cx \right ) ^{n}}}{\sqrt{a}}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.7595, size = 244, normalized size = 4.98 \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}}{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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]