Optimal. Leaf size=63 \[ \frac {4 x}{25}+\frac {121 (61+69 x)}{3875 \left (2+3 x+5 x^2\right )}+\frac {41932 \tan ^{-1}\left (\frac {3+10 x}{\sqrt {31}}\right )}{3875 \sqrt {31}}-\frac {22}{125} \log \left (2+3 x+5 x^2\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1674, 1671,
648, 632, 210, 642} \begin {gather*} \frac {41932 \text {ArcTan}\left (\frac {10 x+3}{\sqrt {31}}\right )}{3875 \sqrt {31}}+\frac {121 (69 x+61)}{3875 \left (5 x^2+3 x+2\right )}-\frac {22}{125} \log \left (5 x^2+3 x+2\right )+\frac {4 x}{25} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 1671
Rule 1674
Rubi steps
\begin {align*} \int \frac {\left (3-x+2 x^2\right )^2}{\left (2+3 x+5 x^2\right )^2} \, dx &=\frac {121 (61+69 x)}{3875 \left (2+3 x+5 x^2\right )}+\frac {1}{31} \int \frac {\frac {4032}{25}-\frac {992 x}{25}+\frac {124 x^2}{5}}{2+3 x+5 x^2} \, dx\\ &=\frac {121 (61+69 x)}{3875 \left (2+3 x+5 x^2\right )}+\frac {1}{31} \int \left (\frac {124}{25}+\frac {44 (86-31 x)}{25 \left (2+3 x+5 x^2\right )}\right ) \, dx\\ &=\frac {4 x}{25}+\frac {121 (61+69 x)}{3875 \left (2+3 x+5 x^2\right )}+\frac {44}{775} \int \frac {86-31 x}{2+3 x+5 x^2} \, dx\\ &=\frac {4 x}{25}+\frac {121 (61+69 x)}{3875 \left (2+3 x+5 x^2\right )}-\frac {22}{125} \int \frac {3+10 x}{2+3 x+5 x^2} \, dx+\frac {20966 \int \frac {1}{2+3 x+5 x^2} \, dx}{3875}\\ &=\frac {4 x}{25}+\frac {121 (61+69 x)}{3875 \left (2+3 x+5 x^2\right )}-\frac {22}{125} \log \left (2+3 x+5 x^2\right )-\frac {41932 \text {Subst}\left (\int \frac {1}{-31-x^2} \, dx,x,3+10 x\right )}{3875}\\ &=\frac {4 x}{25}+\frac {121 (61+69 x)}{3875 \left (2+3 x+5 x^2\right )}+\frac {41932 \tan ^{-1}\left (\frac {3+10 x}{\sqrt {31}}\right )}{3875 \sqrt {31}}-\frac {22}{125} \log \left (2+3 x+5 x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 59, normalized size = 0.94 \begin {gather*} \frac {19220 x+\frac {3751 (61+69 x)}{2+3 x+5 x^2}+41932 \sqrt {31} \tan ^{-1}\left (\frac {3+10 x}{\sqrt {31}}\right )-21142 \log \left (2+3 x+5 x^2\right )}{120125} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 51, normalized size = 0.81
method | result | size |
risch | \(\frac {4 x}{25}+\frac {\frac {8349 x}{19375}+\frac {7381}{19375}}{x^{2}+\frac {3}{5} x +\frac {2}{5}}-\frac {22 \ln \left (100 x^{2}+60 x +40\right )}{125}+\frac {41932 \arctan \left (\frac {\left (3+10 x \right ) \sqrt {31}}{31}\right ) \sqrt {31}}{120125}\) | \(50\) |
default | \(\frac {4 x}{25}-\frac {11 \left (-\frac {759 x}{775}-\frac {671}{775}\right )}{25 \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right )}-\frac {22 \ln \left (5 x^{2}+3 x +2\right )}{125}+\frac {41932 \arctan \left (\frac {\left (3+10 x \right ) \sqrt {31}}{31}\right ) \sqrt {31}}{120125}\) | \(51\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 52, normalized size = 0.83 \begin {gather*} \frac {41932}{120125} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {4}{25} \, x + \frac {121 \, {\left (69 \, x + 61\right )}}{3875 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}} - \frac {22}{125} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.64, size = 78, normalized size = 1.24 \begin {gather*} \frac {96100 \, x^{3} + 41932 \, \sqrt {31} {\left (5 \, x^{2} + 3 \, x + 2\right )} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + 57660 \, x^{2} - 21142 \, {\left (5 \, x^{2} + 3 \, x + 2\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) + 297259 \, x + 228811}{120125 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 65, normalized size = 1.03 \begin {gather*} \frac {4 x}{25} + \frac {8349 x + 7381}{19375 x^{2} + 11625 x + 7750} - \frac {22 \log {\left (x^{2} + \frac {3 x}{5} + \frac {2}{5} \right )}}{125} + \frac {41932 \sqrt {31} \operatorname {atan}{\left (\frac {10 \sqrt {31} x}{31} + \frac {3 \sqrt {31}}{31} \right )}}{120125} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.02, size = 52, normalized size = 0.83 \begin {gather*} \frac {41932}{120125} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {4}{25} \, x + \frac {121 \, {\left (69 \, x + 61\right )}}{3875 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}} - \frac {22}{125} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 51, normalized size = 0.81 \begin {gather*} \frac {4\,x}{25}-\frac {22\,\ln \left (5\,x^2+3\,x+2\right )}{125}+\frac {\frac {8349\,x}{19375}+\frac {7381}{19375}}{x^2+\frac {3\,x}{5}+\frac {2}{5}}+\frac {41932\,\sqrt {31}\,\mathrm {atan}\left (\frac {10\,\sqrt {31}\,x}{31}+\frac {3\,\sqrt {31}}{31}\right )}{120125} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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