Optimal. Leaf size=223 \[ \frac {(4+65 x) \sqrt {3-x+2 x^2}}{1364 \left (2+3 x+5 x^2\right )^2}+\frac {(26794+86265 x) \sqrt {3-x+2 x^2}}{1860496 \left (2+3 x+5 x^2\right )}+\frac {25 \sqrt {\frac {1}{682} \left (6414867847+4536374600 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (6414867847+4536374600 \sqrt {2}\right )}} \left (123161+85754 \sqrt {2}+\left (294669+208915 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{3720992}-\frac {25 \sqrt {\frac {1}{682} \left (-6414867847+4536374600 \sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (-6414867847+4536374600 \sqrt {2}\right )}} \left (123161-85754 \sqrt {2}+\left (294669-208915 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{3720992} \]
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Rubi [A]
time = 0.31, 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 = {988, 1074,
1049, 1043, 212, 210} \begin {gather*} \frac {25 \sqrt {\frac {1}{682} \left (6414867847+4536374600 \sqrt {2}\right )} \text {ArcTan}\left (\frac {\sqrt {\frac {11}{31 \left (6414867847+4536374600 \sqrt {2}\right )}} \left (\left (294669+208915 \sqrt {2}\right ) x+85754 \sqrt {2}+123161\right )}{\sqrt {2 x^2-x+3}}\right )}{3720992}+\frac {\sqrt {2 x^2-x+3} (65 x+4)}{1364 \left (5 x^2+3 x+2\right )^2}+\frac {(86265 x+26794) \sqrt {2 x^2-x+3}}{1860496 \left (5 x^2+3 x+2\right )}-\frac {25 \sqrt {\frac {1}{682} \left (4536374600 \sqrt {2}-6414867847\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (4536374600 \sqrt {2}-6414867847\right )}} \left (\left (294669-208915 \sqrt {2}\right ) x-85754 \sqrt {2}+123161\right )}{\sqrt {2 x^2-x+3}}\right )}{3720992} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 212
Rule 988
Rule 1043
Rule 1049
Rule 1074
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^3} \, dx &=\frac {(4+65 x) \sqrt {3-x+2 x^2}}{1364 \left (2+3 x+5 x^2\right )^2}-\frac {\int \frac {-5775+\frac {6479 x}{2}-2860 x^2}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^2} \, dx}{15004}\\ &=\frac {(4+65 x) \sqrt {3-x+2 x^2}}{1364 \left (2+3 x+5 x^2\right )^2}+\frac {(26794+86265 x) \sqrt {3-x+2 x^2}}{1860496 \left (2+3 x+5 x^2\right )}-\frac {\int \frac {-\frac {28220225}{2}+\frac {22521125 x}{4}}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{112560008}\\ &=\frac {(4+65 x) \sqrt {3-x+2 x^2}}{1364 \left (2+3 x+5 x^2\right )^2}+\frac {(26794+86265 x) \sqrt {3-x+2 x^2}}{1860496 \left (2+3 x+5 x^2\right )}-\frac {\int \frac {\frac {33275}{4} \left (26103-18658 \sqrt {2}\right )-\frac {33275}{4} \left (11213-7445 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{2476320176 \sqrt {2}}+\frac {\int \frac {\frac {33275}{4} \left (26103+18658 \sqrt {2}\right )-\frac {33275}{4} \left (11213+7445 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{2476320176 \sqrt {2}}\\ &=\frac {(4+65 x) \sqrt {3-x+2 x^2}}{1364 \left (2+3 x+5 x^2\right )^2}+\frac {(26794+86265 x) \sqrt {3-x+2 x^2}}{1860496 \left (2+3 x+5 x^2\right )}-\frac {\left (6875 \left (9072749200-6414867847 \sqrt {2}\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {34323994375}{16} \left (6414867847-4536374600 \sqrt {2}\right )-11 x^2} \, dx,x,\frac {\frac {33275}{4} \left (123161-85754 \sqrt {2}\right )+\frac {33275}{4} \left (294669-208915 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2}}\right )}{246016}-\frac {\left (6875 \left (9072749200+6414867847 \sqrt {2}\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {34323994375}{16} \left (6414867847+4536374600 \sqrt {2}\right )-11 x^2} \, dx,x,\frac {\frac {33275}{4} \left (123161+85754 \sqrt {2}\right )+\frac {33275}{4} \left (294669+208915 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2}}\right )}{246016}\\ &=\frac {(4+65 x) \sqrt {3-x+2 x^2}}{1364 \left (2+3 x+5 x^2\right )^2}+\frac {(26794+86265 x) \sqrt {3-x+2 x^2}}{1860496 \left (2+3 x+5 x^2\right )}+\frac {25 \sqrt {\frac {1}{682} \left (6414867847+4536374600 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (6414867847+4536374600 \sqrt {2}\right )}} \left (123161+85754 \sqrt {2}+\left (294669+208915 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{3720992}-\frac {25 \sqrt {\frac {1}{682} \left (-6414867847+4536374600 \sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (-6414867847+4536374600 \sqrt {2}\right )}} \left (123161-85754 \sqrt {2}+\left (294669-208915 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{3720992}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.63, size = 396, normalized size = 1.78 \begin {gather*} \frac {\sqrt {3-x+2 x^2} \left (59044+341572 x+392765 x^2+431325 x^3\right )}{1860496 \left (2+3 x+5 x^2\right )^2}+\frac {3 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {-42330420383 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+11629301740 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}-2992879225 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]}{49210119200}-\frac {16 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {-720397 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+129160 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}-65525 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]}{4509725} \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. \(13039\) vs.
\(2(171)=342\).
time = 0.65, size = 13040, normalized size = 58.48
method | result | size |
trager | \(\text {Expression too large to display}\) | \(483\) |
risch | \(\frac {\left (431325 x^{3}+392765 x^{2}+341572 x +59044\right ) \sqrt {2 x^{2}-x +3}}{1860496 \left (5 x^{2}+3 x +2\right )^{2}}+\frac {25 \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 (11325170 \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}}+8008997 \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}}+11668925202 \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}-16645371446 \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 )}{78669212864 \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 | \(\text {Expression too large to display}\) | \(13040\) |
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. 2183 vs.
\(2 (171) = 342\).
time = 2.86, size = 2183, normalized size = 9.79 \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 {1}{\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 {1}{\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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