Optimal. Leaf size=56 \[ \frac{149}{16} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )-\frac{1}{8} \left (37-6 x^2\right ) \sqrt{x^4+5 x^2+3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0449077, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {1251, 779, 621, 206} \[ \frac{149}{16} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )-\frac{1}{8} \left (37-6 x^2\right ) \sqrt{x^4+5 x^2+3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1251
Rule 779
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{x^3 \left (2+3 x^2\right )}{\sqrt{3+5 x^2+x^4}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x (2+3 x)}{\sqrt{3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac{1}{8} \left (37-6 x^2\right ) \sqrt{3+5 x^2+x^4}+\frac{149}{16} \operatorname{Subst}\left (\int \frac{1}{\sqrt{3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac{1}{8} \left (37-6 x^2\right ) \sqrt{3+5 x^2+x^4}+\frac{149}{8} \operatorname{Subst}\left (\int \frac{1}{4-x^2} \, dx,x,\frac{5+2 x^2}{\sqrt{3+5 x^2+x^4}}\right )\\ &=-\frac{1}{8} \left (37-6 x^2\right ) \sqrt{3+5 x^2+x^4}+\frac{149}{16} \tanh ^{-1}\left (\frac{5+2 x^2}{2 \sqrt{3+5 x^2+x^4}}\right )\\ \end{align*}
Mathematica [A] time = 0.0157702, size = 56, normalized size = 1. \[ \frac{1}{16} \left (2 \sqrt{x^4+5 x^2+3} \left (6 x^2-37\right )+149 \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.011, size = 53, normalized size = 1. \begin{align*}{\frac{3\,{x}^{2}}{4}\sqrt{{x}^{4}+5\,{x}^{2}+3}}-{\frac{37}{8}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{149}{16}\ln \left ({\frac{5}{2}}+{x}^{2}+\sqrt{{x}^{4}+5\,{x}^{2}+3} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.968043, size = 76, normalized size = 1.36 \begin{align*} \frac{3}{4} \, \sqrt{x^{4} + 5 \, x^{2} + 3} x^{2} - \frac{37}{8} \, \sqrt{x^{4} + 5 \, x^{2} + 3} + \frac{149}{16} \, \log \left (2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} + 5\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.28019, size = 123, normalized size = 2.2 \begin{align*} \frac{1}{8} \, \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (6 \, x^{2} - 37\right )} - \frac{149}{16} \, \log \left (-2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} - 5\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^{3} \left (3 x^{2} + 2\right )}{\sqrt{x^{4} + 5 x^{2} + 3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.11393, size = 62, normalized size = 1.11 \begin{align*} \frac{1}{8} \, \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (6 \, x^{2} - 37\right )} - \frac{149}{16} \, \log \left (2 \, x^{2} - 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} + 5\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]