Optimal. Leaf size=33 \[ \frac{x^{m+1} \sqrt{b-\frac{a}{x^2}}}{m \sqrt{a-b x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0314056, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {515, 23, 30} \[ \frac{x^{m+1} \sqrt{b-\frac{a}{x^2}}}{m \sqrt{a-b x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 515
Rule 23
Rule 30
Rubi steps
\begin{align*} \int \frac{\sqrt{b-\frac{a}{x^2}} x^m}{\sqrt{a-b x^2}} \, dx &=\frac{\left (\sqrt{b-\frac{a}{x^2}} x\right ) \int \frac{x^{-1+m} \sqrt{-a+b x^2}}{\sqrt{a-b x^2}} \, dx}{\sqrt{-a+b x^2}}\\ &=\frac{\left (\sqrt{b-\frac{a}{x^2}} x\right ) \int x^{-1+m} \, dx}{\sqrt{a-b x^2}}\\ &=\frac{\sqrt{b-\frac{a}{x^2}} x^{1+m}}{m \sqrt{a-b x^2}}\\ \end{align*}
Mathematica [A] time = 0.0174014, size = 33, normalized size = 1. \[ \frac{x^{m+1} \sqrt{b-\frac{a}{x^2}}}{m \sqrt{a-b x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.001, size = 35, normalized size = 1.1 \begin{align*}{\frac{{x}^{1+m}}{m}\sqrt{-{\frac{-b{x}^{2}+a}{{x}^{2}}}}{\frac{1}{\sqrt{-b{x}^{2}+a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [C] time = 1.15638, size = 11, normalized size = 0.33 \begin{align*} -\frac{i \, x^{m}}{m} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.51179, size = 85, normalized size = 2.58 \begin{align*} -\frac{\sqrt{-b x^{2} + a} x x^{m} \sqrt{\frac{b x^{2} - a}{x^{2}}}}{b m x^{2} - a m} \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{x^{m} \sqrt{- \frac{a}{x^{2}} + b}}{\sqrt{a - b x^{2}}}\, 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{b - \frac{a}{x^{2}}} x^{m}}{\sqrt{-b x^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]