Optimal. Leaf size=74 \[ \frac{a x^4 \sqrt{c-a^2 c x^2}}{4 \sqrt{1-a^2 x^2}}+\frac{x^3 \sqrt{c-a^2 c x^2}}{3 \sqrt{1-a^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.184429, antiderivative size = 74, 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 = {6153, 6150, 43} \[ \frac{a x^4 \sqrt{c-a^2 c x^2}}{4 \sqrt{1-a^2 x^2}}+\frac{x^3 \sqrt{c-a^2 c x^2}}{3 \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6153
Rule 6150
Rule 43
Rubi steps
\begin{align*} \int e^{\tanh ^{-1}(a x)} x^2 \sqrt{c-a^2 c x^2} \, dx &=\frac{\sqrt{c-a^2 c x^2} \int e^{\tanh ^{-1}(a x)} x^2 \sqrt{1-a^2 x^2} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\sqrt{c-a^2 c x^2} \int x^2 (1+a x) \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\sqrt{c-a^2 c x^2} \int \left (x^2+a x^3\right ) \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{x^3 \sqrt{c-a^2 c x^2}}{3 \sqrt{1-a^2 x^2}}+\frac{a x^4 \sqrt{c-a^2 c x^2}}{4 \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0231548, size = 42, normalized size = 0.57 \[ \frac{x^3 (3 a x+4) \sqrt{c-a^2 c x^2}}{12 \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.027, size = 37, normalized size = 0.5 \begin{align*}{\frac{{x}^{3} \left ( 3\,ax+4 \right ) }{12}\sqrt{-{a}^{2}c{x}^{2}+c}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.02855, size = 107, normalized size = 1.45 \begin{align*} -\frac{\sqrt{-a^{2} c x^{2} + c}{\left (3 \, a x^{4} + 4 \, x^{3}\right )} \sqrt{-a^{2} x^{2} + 1}}{12 \,{\left (a^{2} x^{2} - 1\right )}} \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^{2} \sqrt{- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )}{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}\, 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{-a^{2} c x^{2} + c}{\left (a x + 1\right )} x^{2}}{\sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]