Optimal. Leaf size=223 \[ \frac {(3+10 x) \sqrt {3-x+2 x^2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {(3464+13665 x) \sqrt {3-x+2 x^2}}{84568 \left (2+3 x+5 x^2\right )}+\frac {\sqrt {\frac {1}{682} \left (112285869463+79399380740 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (112285869463+79399380740 \sqrt {2}\right )}} \left (509587+362788 \sqrt {2}+\left (1235163+872375 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{169136}-\frac {\sqrt {\frac {1}{682} \left (-112285869463+79399380740 \sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (-112285869463+79399380740 \sqrt {2}\right )}} \left (509587-362788 \sqrt {2}+\left (1235163-872375 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{169136} \]
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Rubi [A]
time = 0.30, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {985, 1074,
1049, 1043, 212, 210} \begin {gather*} \frac {\sqrt {\frac {1}{682} \left (112285869463+79399380740 \sqrt {2}\right )} \text {ArcTan}\left (\frac {\sqrt {\frac {11}{31 \left (112285869463+79399380740 \sqrt {2}\right )}} \left (\left (1235163+872375 \sqrt {2}\right ) x+362788 \sqrt {2}+509587\right )}{\sqrt {2 x^2-x+3}}\right )}{169136}+\frac {\sqrt {2 x^2-x+3} (10 x+3)}{62 \left (5 x^2+3 x+2\right )^2}+\frac {(13665 x+3464) \sqrt {2 x^2-x+3}}{84568 \left (5 x^2+3 x+2\right )}-\frac {\sqrt {\frac {1}{682} \left (79399380740 \sqrt {2}-112285869463\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (79399380740 \sqrt {2}-112285869463\right )}} \left (\left (1235163-872375 \sqrt {2}\right ) x-362788 \sqrt {2}+509587\right )}{\sqrt {2 x^2-x+3}}\right )}{169136} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 212
Rule 985
Rule 1043
Rule 1049
Rule 1074
Rubi steps
\begin {align*} \int \frac {\sqrt {3-x+2 x^2}}{\left (2+3 x+5 x^2\right )^3} \, dx &=\frac {(3+10 x) \sqrt {3-x+2 x^2}}{62 \left (2+3 x+5 x^2\right )^2}-\frac {1}{62} \int \frac {-\frac {183}{2}+31 x-40 x^2}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^2} \, dx\\ &=\frac {(3+10 x) \sqrt {3-x+2 x^2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {(3464+13665 x) \sqrt {3-x+2 x^2}}{84568 \left (2+3 x+5 x^2\right )}-\frac {\int \frac {-213004+\frac {358655 x}{4}}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{465124}\\ &=\frac {(3+10 x) \sqrt {3-x+2 x^2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {(3464+13665 x) \sqrt {3-x+2 x^2}}{84568 \left (2+3 x+5 x^2\right )}-\frac {\int \frac {\frac {121}{4} \left (110061-77456 \sqrt {2}\right )-\frac {121}{4} \left (44851-32605 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{10232728 \sqrt {2}}+\frac {\int \frac {\frac {121}{4} \left (110061+77456 \sqrt {2}\right )-\frac {121}{4} \left (44851+32605 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{10232728 \sqrt {2}}\\ &=\frac {(3+10 x) \sqrt {3-x+2 x^2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {(3464+13665 x) \sqrt {3-x+2 x^2}}{84568 \left (2+3 x+5 x^2\right )}-\frac {\left (11 \left (158798761480-112285869463 \sqrt {2}\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {453871}{16} \left (112285869463-79399380740 \sqrt {2}\right )-11 x^2} \, dx,x,\frac {\frac {121}{4} \left (509587-362788 \sqrt {2}\right )+\frac {121}{4} \left (1235163-872375 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2}}\right )}{123008}-\frac {\left (11 \left (158798761480+112285869463 \sqrt {2}\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {453871}{16} \left (112285869463+79399380740 \sqrt {2}\right )-11 x^2} \, dx,x,\frac {\frac {121}{4} \left (509587+362788 \sqrt {2}\right )+\frac {121}{4} \left (1235163+872375 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2}}\right )}{123008}\\ &=\frac {(3+10 x) \sqrt {3-x+2 x^2}}{62 \left (2+3 x+5 x^2\right )^2}+\frac {(3464+13665 x) \sqrt {3-x+2 x^2}}{84568 \left (2+3 x+5 x^2\right )}+\frac {\sqrt {\frac {1}{682} \left (112285869463+79399380740 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (112285869463+79399380740 \sqrt {2}\right )}} \left (509587+362788 \sqrt {2}+\left (1235163+872375 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{169136}-\frac {\sqrt {\frac {1}{682} \left (-112285869463+79399380740 \sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (-112285869463+79399380740 \sqrt {2}\right )}} \left (509587-362788 \sqrt {2}+\left (1235163-872375 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{169136}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.59, size = 392, normalized size = 1.76 \begin {gather*} \frac {\frac {661250 \sqrt {3-x+2 x^2} \left (11020+51362 x+58315 x^2+68325 x^3\right )}{\left (2+3 x+5 x^2\right )^2}+\text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {-537295920831 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+120146195680 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}-45923442075 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]-248 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {-2139373897 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+277937160 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}-228643025 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]}{55920590000} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(44342\) vs.
\(2(171)=342\).
time = 1.00, size = 44343, normalized size = 198.85
method | result | size |
trager | \(\text {Expression too large to display}\) | \(483\) |
risch | \(\frac {\left (68325 x^{3}+58315 x^{2}+51362 x +11020\right ) \sqrt {2 x^{2}-x +3}}{84568 \left (5 x^{2}+3 x +2\right )^{2}}+\frac {\sqrt {\frac {8 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}\, \sqrt {2}\, \left (33504619 \sqrt {2}\, \sqrt {-8866+6820 \sqrt {2}}\, \arctan \left (\frac {\sqrt {-775687+549362 \sqrt {2}}\, \sqrt {-23 \left (8+3 \sqrt {2}\right ) \left (-\frac {23 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+24 \sqrt {2}-41\right )}\, \left (\frac {6485 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {10368 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+22379 \sqrt {2}+32016\right ) \left (\sqrt {2}-1+x \right ) \left (8+3 \sqrt {2}\right )}{11692487 \left (\frac {23 \left (\sqrt {2}-1+x \right )^{4}}{\left (\sqrt {2}+1-x \right )^{4}}+\frac {82 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+23\right ) \left (\sqrt {2}+1-x \right )}\right ) \sqrt {-775687+549362 \sqrt {2}}+47385010 \sqrt {-8866+6820 \sqrt {2}}\, \arctan \left (\frac {\sqrt {-775687+549362 \sqrt {2}}\, \sqrt {-23 \left (8+3 \sqrt {2}\right ) \left (-\frac {23 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+24 \sqrt {2}-41\right )}\, \left (\frac {6485 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {10368 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+22379 \sqrt {2}+32016\right ) \left (\sqrt {2}-1+x \right ) \left (8+3 \sqrt {2}\right )}{11692487 \left (\frac {23 \left (\sqrt {2}-1+x \right )^{4}}{\left (\sqrt {2}+1-x \right )^{4}}+\frac {82 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+23\right ) \left (\sqrt {2}+1-x \right )}\right ) \sqrt {-775687+549362 \sqrt {2}}+49124007834 \arctanh \left (\frac {31 \sqrt {\frac {8 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}}{2 \sqrt {-8866+6820 \sqrt {2}}}\right ) \sqrt {2}-69208569562 \arctanh \left (\frac {31 \sqrt {\frac {8 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}}{2 \sqrt {-8866+6820 \sqrt {2}}}\right )\right )}{3575873312 \sqrt {\frac {\frac {8 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}{\left (1+\frac {\sqrt {2}-1+x}{\sqrt {2}+1-x}\right )^{2}}}\, \left (1+\frac {\sqrt {2}-1+x}{\sqrt {2}+1-x}\right ) \left (8+3 \sqrt {2}\right ) \sqrt {-8866+6820 \sqrt {2}}}\) | \(726\) |
default | \(\text {Expression too large to display}\) | \(44343\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2182 vs.
\(2 (171) = 342\).
time = 5.79, size = 2182, normalized size = 9.78 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {2 x^{2} - x + 3}}{\left (5 x^{2} + 3 x + 2\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {2\,x^2-x+3}}{{\left (5\,x^2+3\,x+2\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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