Optimal. Leaf size=94 \[ \frac{13-6 x}{506 \left (2 x^2-x+3\right )}-\frac{13}{968} \log \left (2 x^2-x+3\right )+\frac{13}{968} \log \left (5 x^2+3 x+2\right )+\frac{241 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{11132 \sqrt{23}}+\frac{69 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{484 \sqrt{31}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0884198, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {974, 1072, 634, 618, 204, 628} \[ \frac{13-6 x}{506 \left (2 x^2-x+3\right )}-\frac{13}{968} \log \left (2 x^2-x+3\right )+\frac{13}{968} \log \left (5 x^2+3 x+2\right )+\frac{241 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{11132 \sqrt{23}}+\frac{69 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{484 \sqrt{31}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 974
Rule 1072
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{\left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )} \, dx &=\frac{13-6 x}{506 \left (3-x+2 x^2\right )}-\frac{\int \frac{-1892-1067 x+330 x^2}{\left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )} \, dx}{5566}\\ &=\frac{13-6 x}{506 \left (3-x+2 x^2\right )}-\frac{\int \frac{-3509+72358 x}{3-x+2 x^2} \, dx}{1346972}-\frac{\int \frac{-150282-180895 x}{2+3 x+5 x^2} \, dx}{1346972}\\ &=\frac{13-6 x}{506 \left (3-x+2 x^2\right )}-\frac{241 \int \frac{1}{3-x+2 x^2} \, dx}{22264}-\frac{13}{968} \int \frac{-1+4 x}{3-x+2 x^2} \, dx+\frac{13}{968} \int \frac{3+10 x}{2+3 x+5 x^2} \, dx+\frac{69}{968} \int \frac{1}{2+3 x+5 x^2} \, dx\\ &=\frac{13-6 x}{506 \left (3-x+2 x^2\right )}-\frac{13}{968} \log \left (3-x+2 x^2\right )+\frac{13}{968} \log \left (2+3 x+5 x^2\right )+\frac{241 \operatorname{Subst}\left (\int \frac{1}{-23-x^2} \, dx,x,-1+4 x\right )}{11132}-\frac{69}{484} \operatorname{Subst}\left (\int \frac{1}{-31-x^2} \, dx,x,3+10 x\right )\\ &=\frac{13-6 x}{506 \left (3-x+2 x^2\right )}+\frac{241 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{11132 \sqrt{23}}+\frac{69 \tan ^{-1}\left (\frac{3+10 x}{\sqrt{31}}\right )}{484 \sqrt{31}}-\frac{13}{968} \log \left (3-x+2 x^2\right )+\frac{13}{968} \log \left (2+3 x+5 x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0552043, size = 94, normalized size = 1. \[ \frac{13-6 x}{506 \left (2 x^2-x+3\right )}-\frac{13}{968} \log \left (2 x^2-x+3\right )+\frac{13}{968} \log \left (5 x^2+3 x+2\right )-\frac{241 \tan ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{11132 \sqrt{23}}+\frac{69 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{484 \sqrt{31}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.049, size = 77, normalized size = 0.8 \begin{align*}{\frac{13\,\ln \left ( 5\,{x}^{2}+3\,x+2 \right ) }{968}}+{\frac{69\,\sqrt{31}}{15004}\arctan \left ({\frac{ \left ( 3+10\,x \right ) \sqrt{31}}{31}} \right ) }-{\frac{1}{484} \left ({\frac{66\,x}{23}}-{\frac{143}{23}} \right ) \left ({x}^{2}-{\frac{x}{2}}+{\frac{3}{2}} \right ) ^{-1}}-{\frac{13\,\ln \left ( 2\,{x}^{2}-x+3 \right ) }{968}}-{\frac{241\,\sqrt{23}}{256036}\arctan \left ({\frac{ \left ( -1+4\,x \right ) \sqrt{23}}{23}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.44483, size = 105, normalized size = 1.12 \begin{align*} \frac{69}{15004} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) - \frac{241}{256036} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{6 \, x - 13}{506 \,{\left (2 \, x^{2} - x + 3\right )}} + \frac{13}{968} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) - \frac{13}{968} \, \log \left (2 \, x^{2} - x + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.00196, size = 363, normalized size = 3.86 \begin{align*} \frac{73002 \, \sqrt{31}{\left (2 \, x^{2} - x + 3\right )} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) - 14942 \, \sqrt{23}{\left (2 \, x^{2} - x + 3\right )} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + 213187 \,{\left (2 \, x^{2} - x + 3\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) - 213187 \,{\left (2 \, x^{2} - x + 3\right )} \log \left (2 \, x^{2} - x + 3\right ) - 188232 \, x + 407836}{15874232 \,{\left (2 \, x^{2} - x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.306932, size = 102, normalized size = 1.09 \begin{align*} - \frac{6 x - 13}{1012 x^{2} - 506 x + 1518} - \frac{13 \log{\left (x^{2} - \frac{x}{2} + \frac{3}{2} \right )}}{968} + \frac{13 \log{\left (x^{2} + \frac{3 x}{5} + \frac{2}{5} \right )}}{968} - \frac{241 \sqrt{23} \operatorname{atan}{\left (\frac{4 \sqrt{23} x}{23} - \frac{\sqrt{23}}{23} \right )}}{256036} + \frac{69 \sqrt{31} \operatorname{atan}{\left (\frac{10 \sqrt{31} x}{31} + \frac{3 \sqrt{31}}{31} \right )}}{15004} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.13347, size = 105, normalized size = 1.12 \begin{align*} \frac{69}{15004} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) - \frac{241}{256036} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{6 \, x - 13}{506 \,{\left (2 \, x^{2} - x + 3\right )}} + \frac{13}{968} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) - \frac{13}{968} \, \log \left (2 \, x^{2} - x + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]