Optimal. Leaf size=246 \[ -\frac {4353943-6508666 x}{941410976 \sqrt {3-x+2 x^2}}+\frac {4+65 x}{1364 \sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^2}+\frac {5 (7318+17315 x)}{1860496 \sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )}+\frac {3 \sqrt {\frac {1}{682} \left (13874275807943+9819738650000 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (13874275807943+9819738650000 \sqrt {2}\right )}} \left (5538393+4123702 \sqrt {2}+\left (13785797+9662095 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{81861824}-\frac {3 \sqrt {\frac {1}{682} \left (-13874275807943+9819738650000 \sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (-13874275807943+9819738650000 \sqrt {2}\right )}} \left (5538393-4123702 \sqrt {2}+\left (13785797-9662095 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{81861824} \]
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Rubi [A]
time = 0.34, antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps
used = 8, 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 {3 \sqrt {\frac {1}{682} \left (13874275807943+9819738650000 \sqrt {2}\right )} \text {ArcTan}\left (\frac {\sqrt {\frac {11}{31 \left (13874275807943+9819738650000 \sqrt {2}\right )}} \left (\left (13785797+9662095 \sqrt {2}\right ) x+4123702 \sqrt {2}+5538393\right )}{\sqrt {2 x^2-x+3}}\right )}{81861824}-\frac {4353943-6508666 x}{941410976 \sqrt {2 x^2-x+3}}+\frac {5 (17315 x+7318)}{1860496 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}+\frac {65 x+4}{1364 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )^2}-\frac {3 \sqrt {\frac {1}{682} \left (9819738650000 \sqrt {2}-13874275807943\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (9819738650000 \sqrt {2}-13874275807943\right )}} \left (\left (13785797-9662095 \sqrt {2}\right ) x-4123702 \sqrt {2}+5538393\right )}{\sqrt {2 x^2-x+3}}\right )}{81861824} \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 )^{3/2} \left (2+3 x+5 x^2\right )^3} \, dx &=\frac {4+65 x}{1364 \sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^2}-\frac {\int \frac {-5731+\frac {7557 x}{2}-5720 x^2}{\left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^2} \, dx}{15004}\\ &=\frac {4+65 x}{1364 \sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^2}+\frac {5 (7318+17315 x)}{1860496 \sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )}-\frac {\int \frac {-\frac {29276797}{2}+\frac {3439425 x}{4}-20951150 x^2}{\left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )} \, dx}{112560008}\\ &=-\frac {4353943-6508666 x}{941410976 \sqrt {3-x+2 x^2}}+\frac {4+65 x}{1364 \sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^2}+\frac {5 (7318+17315 x)}{1860496 \sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )}-\frac {\int \frac {-\frac {38923847853}{4}+\frac {36293395215 x}{8}}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{313254502264}\\ &=-\frac {4353943-6508666 x}{941410976 \sqrt {3-x+2 x^2}}+\frac {4+65 x}{1364 \sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^2}+\frac {5 (7318+17315 x)}{1860496 \sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )}-\frac {\int \frac {\frac {1010229}{8} \left (1242839-847654 \sqrt {2}\right )-\frac {1010229}{8} \left (452469-395185 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{6891599049808 \sqrt {2}}+\frac {\int \frac {\frac {1010229}{8} \left (1242839+847654 \sqrt {2}\right )-\frac {1010229}{8} \left (452469+395185 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )} \, dx}{6891599049808 \sqrt {2}}\\ &=-\frac {4353943-6508666 x}{941410976 \sqrt {3-x+2 x^2}}+\frac {4+65 x}{1364 \sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^2}+\frac {5 (7318+17315 x)}{1860496 \sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )}-\frac {\left (2277 \left (19639477300000-13874275807943 \sqrt {2}\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {31637441605671}{64} \left (13874275807943-9819738650000 \sqrt {2}\right )-11 x^2} \, dx,x,\frac {\frac {1010229}{8} \left (5538393-4123702 \sqrt {2}\right )+\frac {1010229}{8} \left (13785797-9662095 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2}}\right )}{984064}-\frac {\left (2277 \left (19639477300000+13874275807943 \sqrt {2}\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {31637441605671}{64} \left (13874275807943+9819738650000 \sqrt {2}\right )-11 x^2} \, dx,x,\frac {\frac {1010229}{8} \left (5538393+4123702 \sqrt {2}\right )+\frac {1010229}{8} \left (13785797+9662095 \sqrt {2}\right ) x}{\sqrt {3-x+2 x^2}}\right )}{984064}\\ &=-\frac {4353943-6508666 x}{941410976 \sqrt {3-x+2 x^2}}+\frac {4+65 x}{1364 \sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^2}+\frac {5 (7318+17315 x)}{1860496 \sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )}+\frac {3 \sqrt {\frac {1}{682} \left (13874275807943+9819738650000 \sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (13874275807943+9819738650000 \sqrt {2}\right )}} \left (5538393+4123702 \sqrt {2}+\left (13785797+9662095 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{81861824}-\frac {3 \sqrt {\frac {1}{682} \left (-13874275807943+9819738650000 \sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {11}{31 \left (-13874275807943+9819738650000 \sqrt {2}\right )}} \left (5538393-4123702 \sqrt {2}+\left (13785797-9662095 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )}{81861824}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 1.12, size = 607, normalized size = 2.47 \begin {gather*} \frac {\frac {4 \left (22374044+161806828 x+175833195 x^2+277167774 x^3+86411405 x^4+162716650 x^5\right )}{\sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^2}-176824 \sqrt {2} \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {-491 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+208 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+5 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]+124 \sqrt {2} \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {7194481 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )-3798456 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+575915 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]-7 \sqrt {2} \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {143178771 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )-105962920 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+6180225 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ]}{3765643904} \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. \(18980\) vs.
\(2(190)=380\).
time = 0.86, size = 18981, normalized size = 77.16
method | result | size |
trager | \(\text {Expression too large to display}\) | \(493\) |
risch | \(\frac {162716650 x^{5}+86411405 x^{4}+277167774 x^{3}+175833195 x^{2}+161806828 x +22374044}{941410976 \left (5 x^{2}+3 x +2\right )^{2} \sqrt {2 x^{2}-x +3}}+\frac {3 \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 (372457261 \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}}+526930410 \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}}+553009243226 \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}-759830139398 \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 )}{1730722683008 \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}}}\) | \(736\) |
default | \(\text {Expression too large to display}\) | \(18981\) |
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. 2263 vs.
\(2 (190) = 380\).
time = 2.74, size = 2263, normalized size = 9.20 \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 {3}{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 )}^{3/2}\,{\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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