Optimal. Leaf size=53 \[ \frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{a^{3/2} n}-\frac{x^{-n} \sqrt{a+b x^n}}{a n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0253995, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {266, 51, 63, 208} \[ \frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{a^{3/2} n}-\frac{x^{-n} \sqrt{a+b x^n}}{a n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{x^{-1-n}}{\sqrt{a+b x^n}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,x^n\right )}{n}\\ &=-\frac{x^{-n} \sqrt{a+b x^n}}{a n}-\frac{b \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^n\right )}{2 a n}\\ &=-\frac{x^{-n} \sqrt{a+b x^n}}{a n}-\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^n}\right )}{a n}\\ &=-\frac{x^{-n} \sqrt{a+b x^n}}{a n}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{a^{3/2} n}\\ \end{align*}
Mathematica [A] time = 0.0617062, size = 67, normalized size = 1.26 \[ \frac{2 b \sqrt{a+b x^n} \left (\frac{\tanh ^{-1}\left (\sqrt{\frac{b x^n}{a}+1}\right )}{2 \sqrt{\frac{b x^n}{a}+1}}-\frac{a x^{-n}}{2 b}\right )}{a^2 n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.083, size = 0, normalized size = 0. \begin{align*} \int{{x}^{-1-n}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \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{x^{-n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.06722, size = 265, normalized size = 5. \begin{align*} \left [\frac{\sqrt{a} b x^{n} \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} a}{2 \, a^{2} n x^{n}}, -\frac{\sqrt{-a} b x^{n} \arctan \left (\frac{\sqrt{b x^{n} + a} \sqrt{-a}}{a}\right ) + \sqrt{b x^{n} + a} a}{a^{2} n x^{n}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 171.165, size = 49, normalized size = 0.92 \begin{align*} - \frac{\sqrt{b} x^{- \frac{n}{2}} \sqrt{\frac{a x^{- n}}{b} + 1}}{a n} + \frac{b \operatorname{asinh}{\left (\frac{\sqrt{a} x^{- \frac{n}{2}}}{\sqrt{b}} \right )}}{a^{\frac{3}{2}} n} \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{x^{-n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]