Optimal. Leaf size=84 \[ \frac {121 (21193-12828 x)}{33856 \left (2 x^2-x+3\right )}-\frac {1331 (17-45 x)}{1472 \left (2 x^2-x+3\right )^2}+\frac {825}{32} \log \left (2 x^2-x+3\right )+\frac {125 x}{8}+\frac {165099 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{8464 \sqrt {23}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1660, 1657, 634, 618, 204, 628} \[ \frac {121 (21193-12828 x)}{33856 \left (2 x^2-x+3\right )}-\frac {1331 (17-45 x)}{1472 \left (2 x^2-x+3\right )^2}+\frac {825}{32} \log \left (2 x^2-x+3\right )+\frac {125 x}{8}+\frac {165099 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{8464 \sqrt {23}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 618
Rule 628
Rule 634
Rule 1657
Rule 1660
Rubi steps
\begin {align*} \int \frac {\left (2+3 x+5 x^2\right )^3}{\left (3-x+2 x^2\right )^3} \, dx &=-\frac {1331 (17-45 x)}{1472 \left (3-x+2 x^2\right )^2}+\frac {1}{46} \int \frac {-\frac {40885}{32}-\frac {19067 x}{8}+\frac {22195 x^2}{4}+\frac {13225 x^3}{2}+2875 x^4}{\left (3-x+2 x^2\right )^2} \, dx\\ &=-\frac {1331 (17-45 x)}{1472 \left (3-x+2 x^2\right )^2}+\frac {121 (21193-12828 x)}{33856 \left (3-x+2 x^2\right )}+\frac {\int \frac {\frac {23997}{2}+92575 x+\frac {66125 x^2}{2}}{3-x+2 x^2} \, dx}{1058}\\ &=-\frac {1331 (17-45 x)}{1472 \left (3-x+2 x^2\right )^2}+\frac {121 (21193-12828 x)}{33856 \left (3-x+2 x^2\right )}+\frac {\int \left (\frac {66125}{4}-\frac {33 (4557-13225 x)}{4 \left (3-x+2 x^2\right )}\right ) \, dx}{1058}\\ &=\frac {125 x}{8}-\frac {1331 (17-45 x)}{1472 \left (3-x+2 x^2\right )^2}+\frac {121 (21193-12828 x)}{33856 \left (3-x+2 x^2\right )}-\frac {33 \int \frac {4557-13225 x}{3-x+2 x^2} \, dx}{4232}\\ &=\frac {125 x}{8}-\frac {1331 (17-45 x)}{1472 \left (3-x+2 x^2\right )^2}+\frac {121 (21193-12828 x)}{33856 \left (3-x+2 x^2\right )}-\frac {165099 \int \frac {1}{3-x+2 x^2} \, dx}{16928}+\frac {825}{32} \int \frac {-1+4 x}{3-x+2 x^2} \, dx\\ &=\frac {125 x}{8}-\frac {1331 (17-45 x)}{1472 \left (3-x+2 x^2\right )^2}+\frac {121 (21193-12828 x)}{33856 \left (3-x+2 x^2\right )}+\frac {825}{32} \log \left (3-x+2 x^2\right )+\frac {165099 \operatorname {Subst}\left (\int \frac {1}{-23-x^2} \, dx,x,-1+4 x\right )}{8464}\\ &=\frac {125 x}{8}-\frac {1331 (17-45 x)}{1472 \left (3-x+2 x^2\right )^2}+\frac {121 (21193-12828 x)}{33856 \left (3-x+2 x^2\right )}+\frac {165099 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{8464 \sqrt {23}}+\frac {825}{32} \log \left (3-x+2 x^2\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 84, normalized size = 1.00 \[ -\frac {121 (12828 x-21193)}{33856 \left (2 x^2-x+3\right )}+\frac {1331 (45 x-17)}{1472 \left (2 x^2-x+3\right )^2}+\frac {825}{32} \log \left (2 x^2-x+3\right )+\frac {125 x}{8}-\frac {165099 \tan ^{-1}\left (\frac {4 x-1}{\sqrt {23}}\right )}{8464 \sqrt {23}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.82, size = 118, normalized size = 1.40 \[ \frac {24334000 \, x^{5} - 24334000 \, x^{4} + 43385176 \, x^{3} - 330198 \, \sqrt {23} {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + 40329281 \, x^{2} + 10037775 \, {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \log \left (2 \, x^{2} - x + 3\right ) - 12446818 \, x + 82485337}{389344 \, {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 62, normalized size = 0.74 \[ -\frac {165099}{194672} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {125}{8} \, x - \frac {121 \, {\left (12828 \, x^{3} - 27607 \, x^{2} + 24146 \, x - 29639\right )}}{16928 \, {\left (2 \, x^{2} - x + 3\right )}^{2}} + \frac {825}{32} \, \log \left (2 \, x^{2} - x + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 63, normalized size = 0.75 \[ \frac {125 x}{8}-\frac {165099 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{194672}+\frac {825 \ln \left (2 x^{2}-x +3\right )}{32}+\frac {-\frac {388047}{4232} x^{3}+\frac {3340447}{16928} x^{2}-\frac {1460833}{8464} x +\frac {3586319}{16928}}{\left (2 x^{2}-x +3\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.96, size = 72, normalized size = 0.86 \[ -\frac {165099}{194672} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {125}{8} \, x - \frac {121 \, {\left (12828 \, x^{3} - 27607 \, x^{2} + 24146 \, x - 29639\right )}}{16928 \, {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} + \frac {825}{32} \, \log \left (2 \, x^{2} - x + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.05, size = 72, normalized size = 0.86 \[ \frac {125\,x}{8}+\frac {825\,\ln \left (2\,x^2-x+3\right )}{32}-\frac {165099\,\sqrt {23}\,\mathrm {atan}\left (\frac {4\,\sqrt {23}\,x}{23}-\frac {\sqrt {23}}{23}\right )}{194672}-\frac {\frac {388047\,x^3}{16928}-\frac {3340447\,x^2}{67712}+\frac {1460833\,x}{33856}-\frac {3586319}{67712}}{x^4-x^3+\frac {13\,x^2}{4}-\frac {3\,x}{2}+\frac {9}{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.23, size = 82, normalized size = 0.98 \[ \frac {125 x}{8} + \frac {- 1552188 x^{3} + 3340447 x^{2} - 2921666 x + 3586319}{67712 x^{4} - 67712 x^{3} + 220064 x^{2} - 101568 x + 152352} + \frac {825 \log {\left (x^{2} - \frac {x}{2} + \frac {3}{2} \right )}}{32} - \frac {165099 \sqrt {23} \operatorname {atan}{\left (\frac {4 \sqrt {23} x}{23} - \frac {\sqrt {23}}{23} \right )}}{194672} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________