Optimal. Leaf size=44 \[ -\frac{\tan ^{-1}\left (\frac{b-2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2-c x^4}}\right )}{2 \sqrt{c}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0382176, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {1107, 621, 204} \[ -\frac{\tan ^{-1}\left (\frac{b-2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2-c x^4}}\right )}{2 \sqrt{c}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1107
Rule 621
Rule 204
Rubi steps
\begin{align*} \int \frac{x}{\sqrt{a+b x^2-c x^4}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x-c x^2}} \, dx,x,x^2\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{-4 c-x^2} \, dx,x,\frac{b-2 c x^2}{\sqrt{a+b x^2-c x^4}}\right )\\ &=-\frac{\tan ^{-1}\left (\frac{b-2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2-c x^4}}\right )}{2 \sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.0056679, size = 44, normalized size = 1. \[ -\frac{\tan ^{-1}\left (\frac{b-2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2-c x^4}}\right )}{2 \sqrt{c}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.158, size = 36, normalized size = 0.8 \begin{align*}{\frac{1}{2}\arctan \left ({\sqrt{c} \left ({x}^{2}-{\frac{b}{2\,c}} \right ){\frac{1}{\sqrt{-c{x}^{4}+b{x}^{2}+a}}}} \right ){\frac{1}{\sqrt{c}}}} \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 = 1.53858, size = 288, normalized size = 6.55 \begin{align*} \left [-\frac{\sqrt{-c} \log \left (8 \, c^{2} x^{4} - 8 \, b c x^{2} + b^{2} - 4 \, \sqrt{-c x^{4} + b x^{2} + a}{\left (2 \, c x^{2} - b\right )} \sqrt{-c} - 4 \, a c\right )}{4 \, c}, -\frac{\arctan \left (\frac{\sqrt{-c x^{4} + b x^{2} + a}{\left (2 \, c x^{2} - b\right )} \sqrt{c}}{2 \,{\left (c^{2} x^{4} - b c x^{2} - a c\right )}}\right )}{2 \, \sqrt{c}}\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}{\sqrt{a + b x^{2} - c x^{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.2244, size = 61, normalized size = 1.39 \begin{align*} -\frac{\log \left ({\left | 2 \,{\left (\sqrt{-c} x^{2} - \sqrt{-c x^{4} + b x^{2} + a}\right )} \sqrt{-c} + b \right |}\right )}{2 \, \sqrt{-c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]