Optimal. Leaf size=85 \[ \frac {16688 (10 x+3)}{148955 \left (5 x^2+3 x+2\right )}+\frac {11 (12060 x+4579)}{120125 \left (5 x^2+3 x+2\right )^2}+\frac {121 (69 x+61)}{11625 \left (5 x^2+3 x+2\right )^3}+\frac {66752 \tan ^{-1}\left (\frac {10 x+3}{\sqrt {31}}\right )}{29791 \sqrt {31}} \]
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Rubi [A] time = 0.06, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1660, 12, 614, 618, 204} \begin {gather*} \frac {16688 (10 x+3)}{148955 \left (5 x^2+3 x+2\right )}+\frac {11 (12060 x+4579)}{120125 \left (5 x^2+3 x+2\right )^2}+\frac {121 (69 x+61)}{11625 \left (5 x^2+3 x+2\right )^3}+\frac {66752 \tan ^{-1}\left (\frac {10 x+3}{\sqrt {31}}\right )}{29791 \sqrt {31}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 204
Rule 614
Rule 618
Rule 1660
Rubi steps
\begin {align*} \int \frac {\left (3-x+2 x^2\right )^2}{\left (2+3 x+5 x^2\right )^4} \, dx &=\frac {121 (61+69 x)}{11625 \left (2+3 x+5 x^2\right )^3}+\frac {1}{93} \int \frac {\frac {77178}{125}-\frac {2976 x}{25}+\frac {372 x^2}{5}}{\left (2+3 x+5 x^2\right )^3} \, dx\\ &=\frac {121 (61+69 x)}{11625 \left (2+3 x+5 x^2\right )^3}+\frac {11 (4579+12060 x)}{120125 \left (2+3 x+5 x^2\right )^2}+\frac {\int \frac {100128}{5 \left (2+3 x+5 x^2\right )^2} \, dx}{5766}\\ &=\frac {121 (61+69 x)}{11625 \left (2+3 x+5 x^2\right )^3}+\frac {11 (4579+12060 x)}{120125 \left (2+3 x+5 x^2\right )^2}+\frac {16688 \int \frac {1}{\left (2+3 x+5 x^2\right )^2} \, dx}{4805}\\ &=\frac {121 (61+69 x)}{11625 \left (2+3 x+5 x^2\right )^3}+\frac {11 (4579+12060 x)}{120125 \left (2+3 x+5 x^2\right )^2}+\frac {16688 (3+10 x)}{148955 \left (2+3 x+5 x^2\right )}+\frac {33376 \int \frac {1}{2+3 x+5 x^2} \, dx}{29791}\\ &=\frac {121 (61+69 x)}{11625 \left (2+3 x+5 x^2\right )^3}+\frac {11 (4579+12060 x)}{120125 \left (2+3 x+5 x^2\right )^2}+\frac {16688 (3+10 x)}{148955 \left (2+3 x+5 x^2\right )}-\frac {66752 \operatorname {Subst}\left (\int \frac {1}{-31-x^2} \, dx,x,3+10 x\right )}{29791}\\ &=\frac {121 (61+69 x)}{11625 \left (2+3 x+5 x^2\right )^3}+\frac {11 (4579+12060 x)}{120125 \left (2+3 x+5 x^2\right )^2}+\frac {16688 (3+10 x)}{148955 \left (2+3 x+5 x^2\right )}+\frac {66752 \tan ^{-1}\left (\frac {3+10 x}{\sqrt {31}}\right )}{29791 \sqrt {31}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 63, normalized size = 0.74 \begin {gather*} \frac {12516000 x^5+18774000 x^4+21491796 x^3+12780597 x^2+5674908 x+1259239}{446865 \left (5 x^2+3 x+2\right )^3}+\frac {66752 \tan ^{-1}\left (\frac {10 x+3}{\sqrt {31}}\right )}{29791 \sqrt {31}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (3-x+2 x^2\right )^2}{\left (2+3 x+5 x^2\right )^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.38, size = 105, normalized size = 1.24 \begin {gather*} \frac {387996000 \, x^{5} + 581994000 \, x^{4} + 666245676 \, x^{3} + 1001280 \, \sqrt {31} {\left (125 \, x^{6} + 225 \, x^{5} + 285 \, x^{4} + 207 \, x^{3} + 114 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + 396198507 \, x^{2} + 175922148 \, x + 39036409}{13852815 \, {\left (125 \, x^{6} + 225 \, x^{5} + 285 \, x^{4} + 207 \, x^{3} + 114 \, x^{2} + 36 \, x + 8\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 56, normalized size = 0.66 \begin {gather*} \frac {66752}{923521} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {12516000 \, x^{5} + 18774000 \, x^{4} + 21491796 \, x^{3} + 12780597 \, x^{2} + 5674908 \, x + 1259239}{446865 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 57, normalized size = 0.67 \begin {gather*} \frac {66752 \sqrt {31}\, \arctan \left (\frac {\left (10 x +3\right ) \sqrt {31}}{31}\right )}{923521}+\frac {\frac {834400}{29791} x^{5}+\frac {1251600}{29791} x^{4}+\frac {7163932}{148955} x^{3}+\frac {4260199}{148955} x^{2}+\frac {1891636}{148955} x +\frac {1259239}{446865}}{\left (5 x^{2}+3 x +2\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 76, normalized size = 0.89 \begin {gather*} \frac {66752}{923521} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {12516000 \, x^{5} + 18774000 \, x^{4} + 21491796 \, x^{3} + 12780597 \, x^{2} + 5674908 \, x + 1259239}{446865 \, {\left (125 \, x^{6} + 225 \, x^{5} + 285 \, x^{4} + 207 \, x^{3} + 114 \, x^{2} + 36 \, x + 8\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.47, size = 75, normalized size = 0.88 \begin {gather*} \frac {66752\,\sqrt {31}\,\mathrm {atan}\left (\frac {10\,\sqrt {31}\,x}{31}+\frac {3\,\sqrt {31}}{31}\right )}{923521}+\frac {\frac {33376\,x^5}{148955}+\frac {50064\,x^4}{148955}+\frac {7163932\,x^3}{18619375}+\frac {4260199\,x^2}{18619375}+\frac {1891636\,x}{18619375}+\frac {1259239}{55858125}}{x^6+\frac {9\,x^5}{5}+\frac {57\,x^4}{25}+\frac {207\,x^3}{125}+\frac {114\,x^2}{125}+\frac {36\,x}{125}+\frac {8}{125}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.23, size = 83, normalized size = 0.98 \begin {gather*} \frac {12516000 x^{5} + 18774000 x^{4} + 21491796 x^{3} + 12780597 x^{2} + 5674908 x + 1259239}{55858125 x^{6} + 100544625 x^{5} + 127356525 x^{4} + 92501055 x^{3} + 50942610 x^{2} + 16087140 x + 3574920} + \frac {66752 \sqrt {31} \operatorname {atan}{\left (\frac {10 \sqrt {31} x}{31} + \frac {3 \sqrt {31}}{31} \right )}}{923521} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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