Optimal. Leaf size=40 \[ \frac{x^2 \sqrt{a+\frac{b}{x^4}}}{a^2}-\frac{x^2}{2 a \sqrt{a+\frac{b}{x^4}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0109285, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {273, 264} \[ \frac{x^2 \sqrt{a+\frac{b}{x^4}}}{a^2}-\frac{x^2}{2 a \sqrt{a+\frac{b}{x^4}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 273
Rule 264
Rubi steps
\begin{align*} \int \frac{x}{\left (a+\frac{b}{x^4}\right )^{3/2}} \, dx &=-\frac{x^2}{2 a \sqrt{a+\frac{b}{x^4}}}+\frac{2 \int \frac{x}{\sqrt{a+\frac{b}{x^4}}} \, dx}{a}\\ &=-\frac{x^2}{2 a \sqrt{a+\frac{b}{x^4}}}+\frac{\sqrt{a+\frac{b}{x^4}} x^2}{a^2}\\ \end{align*}
Mathematica [A] time = 0.0154149, size = 30, normalized size = 0.75 \[ \frac{a x^4+2 b}{2 a^2 x^2 \sqrt{a+\frac{b}{x^4}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.004, size = 38, normalized size = 1. \begin{align*}{\frac{ \left ( a{x}^{4}+b \right ) \left ( a{x}^{4}+2\,b \right ) }{2\,{a}^{2}{x}^{6}} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.957835, size = 49, normalized size = 1.22 \begin{align*} \frac{\sqrt{a + \frac{b}{x^{4}}} x^{2}}{2 \, a^{2}} + \frac{b}{2 \, \sqrt{a + \frac{b}{x^{4}}} a^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.5176, size = 85, normalized size = 2.12 \begin{align*} \frac{{\left (a x^{6} + 2 \, b x^{2}\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{2 \,{\left (a^{3} x^{4} + a^{2} b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.48567, size = 42, normalized size = 1.05 \begin{align*} \frac{x^{4}}{2 a \sqrt{b} \sqrt{\frac{a x^{4}}{b} + 1}} + \frac{\sqrt{b}}{a^{2} \sqrt{\frac{a x^{4}}{b} + 1}} \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{x}{{\left (a + \frac{b}{x^{4}}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]