Optimal. Leaf size=76 \[ \frac{x^3 \sqrt{c-\frac{c}{a^2 x^2}}}{3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x^2 \sqrt{c-\frac{c}{a^2 x^2}}}{2 a \sqrt{1-\frac{1}{a^2 x^2}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.260666, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {6197, 6193, 43} \[ \frac{x^3 \sqrt{c-\frac{c}{a^2 x^2}}}{3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x^2 \sqrt{c-\frac{c}{a^2 x^2}}}{2 a \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6197
Rule 6193
Rule 43
Rubi steps
\begin{align*} \int e^{\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} x^2 \, dx &=\frac{\sqrt{c-\frac{c}{a^2 x^2}} \int e^{\coth ^{-1}(a x)} \sqrt{1-\frac{1}{a^2 x^2}} x^2 \, dx}{\sqrt{1-\frac{1}{a^2 x^2}}}\\ &=\frac{\sqrt{c-\frac{c}{a^2 x^2}} \int x (1+a x) \, dx}{a \sqrt{1-\frac{1}{a^2 x^2}}}\\ &=\frac{\sqrt{c-\frac{c}{a^2 x^2}} \int \left (x+a x^2\right ) \, dx}{a \sqrt{1-\frac{1}{a^2 x^2}}}\\ &=\frac{\sqrt{c-\frac{c}{a^2 x^2}} x^2}{2 a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{c-\frac{c}{a^2 x^2}} x^3}{3 \sqrt{1-\frac{1}{a^2 x^2}}}\\ \end{align*}
Mathematica [A] time = 0.024884, size = 45, normalized size = 0.59 \[ \frac{x^2 (2 a x+3) \sqrt{c-\frac{c}{a^2 x^2}}}{6 a \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.125, size = 53, normalized size = 0.7 \begin{align*}{\frac{ \left ( 2\,ax+3 \right ){x}^{3}}{6\,ax+6}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}}}{\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}} \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{c - \frac{c}{a^{2} x^{2}}} x^{2}}{\sqrt{\frac{a x - 1}{a x + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.57269, size = 53, normalized size = 0.7 \begin{align*} \frac{{\left (2 \, a x^{3} + 3 \, x^{2}\right )} \sqrt{a^{2} c}}{6 \, a^{2}} \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{\sqrt{c - \frac{c}{a^{2} x^{2}}} x^{2}}{\sqrt{\frac{a x - 1}{a x + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]