Optimal. Leaf size=45 \[ \frac {2 \sqrt {a+b x^n}}{n}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {266, 50, 63, 208} \[ \frac {2 \sqrt {a+b x^n}}{n}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^n}}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,x^n\right )}{n}\\ &=\frac {2 \sqrt {a+b x^n}}{n}+\frac {a \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^n\right )}{n}\\ &=\frac {2 \sqrt {a+b x^n}}{n}+\frac {(2 a) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^n}\right )}{b n}\\ &=\frac {2 \sqrt {a+b x^n}}{n}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 43, normalized size = 0.96 \[ \frac {2 \sqrt {a+b x^n}-2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.58, size = 91, normalized size = 2.02 \[ \left [\frac {\sqrt {a} \log \left (\frac {b x^{n} - 2 \, \sqrt {b x^{n} + a} \sqrt {a} + 2 \, a}{x^{n}}\right ) + 2 \, \sqrt {b x^{n} + a}}{n}, \frac {2 \, {\left (\sqrt {-a} \arctan \left (\frac {\sqrt {b x^{n} + a} \sqrt {-a}}{a}\right ) + \sqrt {b x^{n} + a}\right )}}{n}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {b x^{n} + a}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 36, normalized size = 0.80 \[ \frac {-2 \sqrt {a}\, \arctanh \left (\frac {\sqrt {b \,x^{n}+a}}{\sqrt {a}}\right )+2 \sqrt {b \,x^{n}+a}}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.19, size = 54, normalized size = 1.20 \[ \frac {\sqrt {a} \log \left (\frac {\sqrt {b x^{n} + a} - \sqrt {a}}{\sqrt {b x^{n} + a} + \sqrt {a}}\right )}{n} + \frac {2 \, \sqrt {b x^{n} + a}}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {a+b\,x^n}}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 1.71, size = 76, normalized size = 1.69 \[ - \frac {2 \sqrt {a} \operatorname {asinh}{\left (\frac {\sqrt {a} x^{- \frac {n}{2}}}{\sqrt {b}} \right )}}{n} + \frac {2 a x^{- \frac {n}{2}}}{\sqrt {b} n \sqrt {\frac {a x^{- n}}{b} + 1}} + \frac {2 \sqrt {b} x^{\frac {n}{2}}}{n \sqrt {\frac {a x^{- n}}{b} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________