Optimal. Leaf size=269 \[ -\frac {12280939-19536786 x}{2824232928 \left (3-x+2 x^2\right )^{3/2}}-\frac {1134826571-1504660754 x}{476353953856 \sqrt {3-x+2 x^2}}+\frac {4+65 x}{1364 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^2}+\frac {46386+86885 x}{1860496 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )}+\frac {35 \sqrt {\frac {1}{682} \left (2243059557247+2011748500000 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (2243059557247+2011748500000 \sqrt {2}\right )}} \left (1432939+2428746 \sqrt {2}+\left (6290431+3861685 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{1800960128}-\frac {35 \sqrt {\frac {1}{682} \left (-2243059557247+2011748500000 \sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (-2243059557247+2011748500000 \sqrt {2}\right )}} \left (1432939-2428746 \sqrt {2}+\left (6290431-3861685 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{1800960128} \]
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Rubi [A]
time = 0.38, antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps
used = 9, 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 {35 \sqrt {\frac {1}{682} \left (2243059557247+2011748500000 \sqrt {2}\right )} \text {ArcTan}\left (\frac {\sqrt {\frac {11}{31 \left (2243059557247+2011748500000 \sqrt {2}\right )}} \left (\left (6290431+3861685 \sqrt {2}\right ) x+2428746 \sqrt {2}+1432939\right )}{\sqrt {2 x^2-x+3}}\right )}{1800960128}-\frac {1134826571-1504660754 x}{476353953856 \sqrt {2 x^2-x+3}}+\frac {86885 x+46386}{1860496 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )}-\frac {12280939-19536786 x}{2824232928 \left (2 x^2-x+3\right )^{3/2}}+\frac {65 x+4}{1364 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )^2}-\frac {35 \sqrt {\frac {1}{682} \left (2011748500000 \sqrt {2}-2243059557247\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (2011748500000 \sqrt {2}-2243059557247\right )}} \left (\left (6290431-3861685 \sqrt {2}\right ) x-2428746 \sqrt {2}+1432939\right )}{\sqrt {2 x^2-x+3}}\right )}{1800960128} \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}{\left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )^3} \, dx &=\frac {4+65 x}{1364 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^2}-\frac {\int \frac {-5687+\frac {8635 x}{2}-8580 x^2}{\left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )^2} \, dx}{15004}\\ &=\frac {4+65 x}{1364 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^2}+\frac {46386+86885 x}{1860496 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )}-\frac {\int \frac {-\frac {27962737}{2}-\frac {34457291 x}{4}-42052340 x^2}{\left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )} \, dx}{112560008}\\ &=-\frac {12280939-19536786 x}{2824232928 \left (3-x+2 x^2\right )^{3/2}}+\frac {4+65 x}{1364 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^2}+\frac {46386+86885 x}{1860496 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )}-\frac {\int \frac {-\frac {119979599619}{4}-\frac {16689617967 x}{8}-65008655415 x^2}{\left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )} \, dx}{939763506792}\\ &=-\frac {12280939-19536786 x}{2824232928 \left (3-x+2 x^2\right )^{3/2}}-\frac {1134826571-1504660754 x}{476353953856 \sqrt {3-x+2 x^2}}+\frac {4+65 x}{1364 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^2}+\frac {46386+86885 x}{1860496 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )}-\frac {\int \frac {-\frac {107038717723245}{8}+\frac {333226839035475 x}{16}}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{2615361839402136}\\ &=-\frac {12280939-19536786 x}{2824232928 \left (3-x+2 x^2\right )^{3/2}}-\frac {1134826571-1504660754 x}{476353953856 \sqrt {3-x+2 x^2}}+\frac {4+65 x}{1364 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^2}+\frac {46386+86885 x}{1860496 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )}+\frac {\int \frac {\frac {8945577795}{16} \left (672997+263242 \sqrt {2}\right )+\frac {8945577795}{16} \left (146513-409755 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{57537960466846992 \sqrt {2}}-\frac {\int \frac {\frac {8945577795}{16} \left (672997-263242 \sqrt {2}\right )+\frac {8945577795}{16} \left (146513+409755 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{57537960466846992 \sqrt {2}}\\ &=-\frac {12280939-19536786 x}{2824232928 \left (3-x+2 x^2\right )^{3/2}}-\frac {1134826571-1504660754 x}{476353953856 \sqrt {3-x+2 x^2}}+\frac {4+65 x}{1364 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^2}+\frac {46386+86885 x}{1860496 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )}-\frac {\left (21384825 \left (4023497000000-2243059557247 \sqrt {2}\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {2480724224678308922775}{256} \left (2243059557247-2011748500000 \sqrt {2}\right )-11 x^2} \, dx,x,\frac {\frac {8945577795}{16} \left (1432939-2428746 \sqrt {2}\right )+\frac {8945577795}{16} \left (6290431-3861685 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2}}\right )}{3936256}-\frac {\left (21384825 \left (4023497000000+2243059557247 \sqrt {2}\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {2480724224678308922775}{256} \left (2243059557247+2011748500000 \sqrt {2}\right )-11 x^2} \, dx,x,\frac {\frac {8945577795}{16} \left (1432939+2428746 \sqrt {2}\right )+\frac {8945577795}{16} \left (6290431+3861685 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2}}\right )}{3936256}\\ &=-\frac {12280939-19536786 x}{2824232928 \left (3-x+2 x^2\right )^{3/2}}-\frac {1134826571-1504660754 x}{476353953856 \sqrt {3-x+2 x^2}}+\frac {4+65 x}{1364 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^2}+\frac {46386+86885 x}{1860496 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )}+\frac {35 \sqrt {\frac {1}{682} \left (2243059557247+2011748500000 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (2243059557247+2011748500000 \sqrt {2}\right )}} \left (1432939+2428746 \sqrt {2}+\left (6290431+3861685 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{1800960128}-\frac {35 \sqrt {\frac {1}{682} \left (-2243059557247+2011748500000 \sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (-2243059557247+2011748500000 \sqrt {2}\right )}} \left (1432939-2428746 \sqrt {2}+\left (6290431-3861685 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{1800960128}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 1.20, size = 605, normalized size = 2.25 \begin {gather*} \frac {\frac {4 \sqrt {3-x+2 x^2} \left (9739335532+218659985088 x+178650961091 x^2+519223213785 x^3+174241614961 x^4+592923725931 x^5-12234606480 x^6+225699113100 x^7\right )}{\left (6+7 x+16 x^2+x^3+10 x^4\right )^2}-2976 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {-26154346 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+37230166 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}-1193705 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]-24401712 \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {-3647 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+3172 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}-485 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]+15 \sqrt {2} \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {-9138129081 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+16445754136 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+1004412885 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]}{5716247446272} \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. \(19013\) vs.
\(2(209)=418\).
time = 0.82, size = 19014, normalized size = 70.68
method | result | size |
trager | \(\text {Expression too large to display}\) | \(510\) |
risch | \(\frac {225699113100 x^{7}-12234606480 x^{6}+592923725931 x^{5}+174241614961 x^{4}+519223213785 x^{3}+178650961091 x^{2}+218659985088 x +9739335532}{1429061861568 \left (2 x^{2}-x +3\right )^{\frac {3}{2}} \left (5 x^{2}+3 x +2\right )^{2}}+\frac {35 \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 (159009303 \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}}+226212430 \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}}+287037935998 \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}-254173077554 \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 )}{38075899026176 \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}}}\) | \(746\) |
default | \(\text {Expression too large to display}\) | \(19014\) |
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. 2343 vs.
\(2 (209) = 418\).
time = 6.36, size = 2343, normalized size = 8.71 \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}{\left (2 x^{2} - x + 3\right )^{\frac {5}{2}} \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}{{\left (2\,x^2-x+3\right )}^{5/2}\,{\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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