Optimal. Leaf size=86 \[ \frac{2 (12839-3871 x)}{5589 \left (3 x^2-x+2\right )^{3/2}}+\frac{32}{27} \sqrt{3 x^2-x+2}-\frac{28 (42240 x+35809)}{128547 \sqrt{3 x^2-x+2}}-\frac{296 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{27 \sqrt{3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.112569, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1660, 640, 619, 215} \[ \frac{2 (12839-3871 x)}{5589 \left (3 x^2-x+2\right )^{3/2}}+\frac{32}{27} \sqrt{3 x^2-x+2}-\frac{28 (42240 x+35809)}{128547 \sqrt{3 x^2-x+2}}-\frac{296 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{27 \sqrt{3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1660
Rule 640
Rule 619
Rule 215
Rubi steps
\begin{align*} \int \frac{(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\left (2-x+3 x^2\right )^{5/2}} \, dx &=\frac{2 (12839-3871 x)}{5589 \left (2-x+3 x^2\right )^{3/2}}+\frac{2}{69} \int \frac{-\frac{4361}{81}+\frac{7682 x}{9}+\frac{2852 x^2}{3}+368 x^3}{\left (2-x+3 x^2\right )^{3/2}} \, dx\\ &=\frac{2 (12839-3871 x)}{5589 \left (2-x+3 x^2\right )^{3/2}}-\frac{28 (35809+42240 x)}{128547 \sqrt{2-x+3 x^2}}+\frac{4 \int \frac{\frac{37030}{9}+\frac{4232 x}{3}}{\sqrt{2-x+3 x^2}} \, dx}{1587}\\ &=\frac{2 (12839-3871 x)}{5589 \left (2-x+3 x^2\right )^{3/2}}-\frac{28 (35809+42240 x)}{128547 \sqrt{2-x+3 x^2}}+\frac{32}{27} \sqrt{2-x+3 x^2}+\frac{296}{27} \int \frac{1}{\sqrt{2-x+3 x^2}} \, dx\\ &=\frac{2 (12839-3871 x)}{5589 \left (2-x+3 x^2\right )^{3/2}}-\frac{28 (35809+42240 x)}{128547 \sqrt{2-x+3 x^2}}+\frac{32}{27} \sqrt{2-x+3 x^2}+\frac{296 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+6 x\right )}{27 \sqrt{69}}\\ &=\frac{2 (12839-3871 x)}{5589 \left (2-x+3 x^2\right )^{3/2}}-\frac{28 (35809+42240 x)}{128547 \sqrt{2-x+3 x^2}}+\frac{32}{27} \sqrt{2-x+3 x^2}-\frac{296 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{27 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0628087, size = 71, normalized size = 0.83 \[ \frac{2 \left (228528 x^4-743712 x^3+25890 x^2+78292 \sqrt{3} \left (3 x^2-x+2\right )^{3/2} \sinh ^{-1}\left (\frac{6 x-1}{\sqrt{23}}\right )-358377 x-134217\right )}{42849 \left (3 x^2-x+2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.056, size = 163, normalized size = 1.9 \begin{align*}{\frac{32\,{x}^{4}}{3} \left ( 3\,{x}^{2}-x+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{296\,{x}^{3}}{27} \left ( 3\,{x}^{2}-x+2 \right ) ^{-{\frac{3}{2}}}}+{\frac{8\,{x}^{2}}{27} \left ( 3\,{x}^{2}-x+2 \right ) ^{-{\frac{3}{2}}}}+{\frac{296\,\sqrt{3}}{81}{\it Arcsinh} \left ({\frac{6\,\sqrt{23}}{23} \left ( x-{\frac{1}{6}} \right ) } \right ) }-{\frac{461\,x}{81} \left ( 3\,{x}^{2}-x+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{296\,x}{27}{\frac{1}{\sqrt{3\,{x}^{2}-x+2}}}}+{\frac{-65264+391584\,x}{128547}{\frac{1}{\sqrt{3\,{x}^{2}-x+2}}}}+{\frac{-13763+82578\,x}{33534} \left ( 3\,{x}^{2}-x+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{1727}{1458} \left ( 3\,{x}^{2}-x+2 \right ) ^{-{\frac{3}{2}}}}-{\frac{148}{81}{\frac{1}{\sqrt{3\,{x}^{2}-x+2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.52883, size = 273, normalized size = 3.17 \begin{align*} \frac{32 \, x^{4}}{3 \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}}} + \frac{296}{42849} \, x{\left (\frac{426 \, x}{\sqrt{3 \, x^{2} - x + 2}} - \frac{4761 \, x^{2}}{{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}}} - \frac{71}{\sqrt{3 \, x^{2} - x + 2}} + \frac{805 \, x}{{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}}} - \frac{2162}{{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}}}\right )} + \frac{296}{81} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (6 \, x - 1\right )}\right ) - \frac{42032}{42849} \, \sqrt{3 \, x^{2} - x + 2} - \frac{47072 \, x}{42849 \, \sqrt{3 \, x^{2} - x + 2}} + \frac{52 \, x^{2}}{9 \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}}} - \frac{23104}{14283 \, \sqrt{3 \, x^{2} - x + 2}} - \frac{7742 \, x}{1863 \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}}} + \frac{1666}{1863 \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.14331, size = 325, normalized size = 3.78 \begin{align*} \frac{2 \,{\left (39146 \, \sqrt{3}{\left (9 \, x^{4} - 6 \, x^{3} + 13 \, x^{2} - 4 \, x + 4\right )} \log \left (-4 \, \sqrt{3} \sqrt{3 \, x^{2} - x + 2}{\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\right ) + 3 \,{\left (76176 \, x^{4} - 247904 \, x^{3} + 8630 \, x^{2} - 119459 \, x - 44739\right )} \sqrt{3 \, x^{2} - x + 2}\right )}}{42849 \,{\left (9 \, x^{4} - 6 \, x^{3} + 13 \, x^{2} - 4 \, x + 4\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{\left (2 x + 1\right )^{3} \left (4 x^{2} + 3 x + 1\right )}{\left (3 x^{2} - x + 2\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15634, size = 90, normalized size = 1.05 \begin{align*} -\frac{296}{81} \, \sqrt{3} \log \left (-2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} - x + 2}\right )} + 1\right ) + \frac{2 \,{\left ({\left (2 \,{\left (8 \,{\left (4761 \, x - 15494\right )} x + 4315\right )} x - 119459\right )} x - 44739\right )}}{14283 \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]