Optimal. Leaf size=21 \[ -\frac{\sqrt{a+c x^4}}{2 a x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0046456, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {4, 264} \[ -\frac{\sqrt{a+c x^4}}{2 a x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4
Rule 264
Rubi steps
\begin{align*} \int \frac{1}{x^3 \sqrt{a+(2+2 b-2 (1+b)) x^2+c x^4}} \, dx &=\int \frac{1}{x^3 \sqrt{a+c x^4}} \, dx\\ &=-\frac{\sqrt{a+c x^4}}{2 a x^2}\\ \end{align*}
Mathematica [A] time = 0.0040815, size = 21, normalized size = 1. \[ -\frac{\sqrt{a+c x^4}}{2 a x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.045, size = 18, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,a{x}^{2}}\sqrt{c{x}^{4}+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.944374, size = 23, normalized size = 1.1 \begin{align*} -\frac{\sqrt{c x^{4} + a}}{2 \, a x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.4529, size = 41, normalized size = 1.95 \begin{align*} -\frac{\sqrt{c x^{4} + a}}{2 \, a x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.616605, size = 20, normalized size = 0.95 \begin{align*} - \frac{\sqrt{c} \sqrt{\frac{a}{c x^{4}} + 1}}{2 a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.1882, size = 19, normalized size = 0.9 \begin{align*} -\frac{\sqrt{c + \frac{a}{x^{4}}}}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]