Optimal. Leaf size=55 \[ \frac {2-51 x}{18 \sqrt {3 x^2+2}}+\frac {8}{9} \sqrt {3 x^2+2}+\frac {10 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1814, 641, 215} \[ \frac {2-51 x}{18 \sqrt {3 x^2+2}}+\frac {8}{9} \sqrt {3 x^2+2}+\frac {10 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 215
Rule 641
Rule 1814
Rubi steps
\begin {align*} \int \frac {(1+2 x) \left (1+3 x+4 x^2\right )}{\left (2+3 x^2\right )^{3/2}} \, dx &=\frac {2-51 x}{18 \sqrt {2+3 x^2}}-\frac {1}{2} \int \frac {-\frac {20}{3}-\frac {16 x}{3}}{\sqrt {2+3 x^2}} \, dx\\ &=\frac {2-51 x}{18 \sqrt {2+3 x^2}}+\frac {8}{9} \sqrt {2+3 x^2}+\frac {10}{3} \int \frac {1}{\sqrt {2+3 x^2}} \, dx\\ &=\frac {2-51 x}{18 \sqrt {2+3 x^2}}+\frac {8}{9} \sqrt {2+3 x^2}+\frac {10 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 48, normalized size = 0.87 \[ \frac {48 x^2+20 \sqrt {9 x^2+6} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )-51 x+34}{18 \sqrt {3 x^2+2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.76, size = 67, normalized size = 1.22 \[ \frac {10 \, \sqrt {3} {\left (3 \, x^{2} + 2\right )} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + {\left (48 \, x^{2} - 51 \, x + 34\right )} \sqrt {3 \, x^{2} + 2}}{18 \, {\left (3 \, x^{2} + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.23, size = 44, normalized size = 0.80 \[ -\frac {10}{9} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) + \frac {3 \, {\left (16 \, x - 17\right )} x + 34}{18 \, \sqrt {3 \, x^{2} + 2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 51, normalized size = 0.93 \[ \frac {8 x^{2}}{3 \sqrt {3 x^{2}+2}}-\frac {17 x}{6 \sqrt {3 x^{2}+2}}+\frac {10 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{9}+\frac {17}{9 \sqrt {3 x^{2}+2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.96, size = 50, normalized size = 0.91 \[ \frac {8 \, x^{2}}{3 \, \sqrt {3 \, x^{2} + 2}} + \frac {10}{9} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) - \frac {17 \, x}{6 \, \sqrt {3 \, x^{2} + 2}} + \frac {17}{9 \, \sqrt {3 \, x^{2} + 2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.04, size = 100, normalized size = 1.82 \[ \frac {8\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{9}+\frac {10\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{9}+\frac {\sqrt {3}\,\sqrt {6}\,\left (-6+\sqrt {6}\,51{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{648\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {\sqrt {3}\,\sqrt {6}\,\left (6+\sqrt {6}\,51{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{648\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 15.96, size = 114, normalized size = 2.07 \[ \frac {30 \sqrt {3} x^{2} \operatorname {asinh}{\left (\frac {\sqrt {6} x}{2} \right )}}{27 x^{2} + 18} + \frac {8 x^{2}}{3 \sqrt {3 x^{2} + 2}} - \frac {30 x \sqrt {3 x^{2} + 2}}{27 x^{2} + 18} + \frac {x}{2 \sqrt {3 x^{2} + 2}} + \frac {20 \sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {6} x}{2} \right )}}{27 x^{2} + 18} + \frac {17}{9 \sqrt {3 x^{2} + 2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________