Optimal. Leaf size=38 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x^{2-n}-b x^2}}\right )}{\sqrt{b} n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0226047, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {1979, 2008, 203} \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x^{2-n}-b x^2}}\right )}{\sqrt{b} n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1979
Rule 2008
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{x^{2-n} \left (a-b x^n\right )}} \, dx &=\int \frac{1}{\sqrt{-b x^2+a x^{2-n}}} \, dx\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{1+b x^2} \, dx,x,\frac{x}{\sqrt{-b x^2+a x^{2-n}}}\right )}{n}\\ &=\frac{2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{-b x^2+a x^{2-n}}}\right )}{\sqrt{b} n}\\ \end{align*}
Mathematica [B] time = 0.0428816, size = 78, normalized size = 2.05 \[ \frac{2 \sqrt{a} x^{1-\frac{n}{2}} \sqrt{1-\frac{b x^n}{a}} \sin ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a}}\right )}{\sqrt{b} n \sqrt{x^{2-n} \left (a-b x^n\right )}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.586, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt{{x}^{2-n} \left ( a-b{x}^{n} \right ) }}}\, 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{1}{\sqrt{-{\left (b x^{n} - a\right )} x^{-n + 2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.00405, size = 219, normalized size = 5.76 \begin{align*} \left [-\frac{\sqrt{-b} \log \left (-\frac{2 \, b x x^{n} - a x - 2 \, \sqrt{-b} x^{n} \sqrt{-\frac{b x^{2} x^{n} - a x^{2}}{x^{n}}}}{x}\right )}{b n}, -\frac{2 \, \arctan \left (\frac{\sqrt{-\frac{b x^{2} x^{n} - a x^{2}}{x^{n}}}}{\sqrt{b} x}\right )}{\sqrt{b} n}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{1}{\sqrt{-{\left (b x^{n} - a\right )} x^{-n + 2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]