Optimal. Leaf size=169 \[ -\frac {1172725 (17-22 x) \sqrt {3-x+2 x^2}}{330225942528 (5+2 x)^2}-\frac {3667 \left (3-x+2 x^2\right )^{3/2}}{3456 (5+2 x)^6}+\frac {92239 \left (3-x+2 x^2\right )^{3/2}}{138240 (5+2 x)^5}-\frac {5703277 \left (3-x+2 x^2\right )^{3/2}}{39813120 (5+2 x)^4}+\frac {87677717 \left (3-x+2 x^2\right )^{3/2}}{8599633920 (5+2 x)^3}-\frac {26972675 \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {3-x+2 x^2}}\right )}{3962711310336 \sqrt {2}} \]
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Rubi [A]
time = 0.13, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1664, 820, 734,
738, 212} \begin {gather*} \frac {87677717 \left (2 x^2-x+3\right )^{3/2}}{8599633920 (2 x+5)^3}-\frac {5703277 \left (2 x^2-x+3\right )^{3/2}}{39813120 (2 x+5)^4}+\frac {92239 \left (2 x^2-x+3\right )^{3/2}}{138240 (2 x+5)^5}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{3456 (2 x+5)^6}-\frac {1172725 (17-22 x) \sqrt {2 x^2-x+3}}{330225942528 (2 x+5)^2}-\frac {26972675 \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {2 x^2-x+3}}\right )}{3962711310336 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 734
Rule 738
Rule 820
Rule 1664
Rubi steps
\begin {align*} \int \frac {\sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^7} \, dx &=-\frac {3667 \left (3-x+2 x^2\right )^{3/2}}{3456 (5+2 x)^6}-\frac {1}{432} \int \frac {\sqrt {3-x+2 x^2} \left (\frac {61041}{16}-\frac {20751 x}{4}+2916 x^2-1080 x^3\right )}{(5+2 x)^6} \, dx\\ &=-\frac {3667 \left (3-x+2 x^2\right )^{3/2}}{3456 (5+2 x)^6}+\frac {92239 \left (3-x+2 x^2\right )^{3/2}}{138240 (5+2 x)^5}+\frac {\int \frac {\sqrt {3-x+2 x^2} \left (\frac {8057313}{16}-\frac {1191609 x}{2}+194400 x^2\right )}{(5+2 x)^5} \, dx}{155520}\\ &=-\frac {3667 \left (3-x+2 x^2\right )^{3/2}}{3456 (5+2 x)^6}+\frac {92239 \left (3-x+2 x^2\right )^{3/2}}{138240 (5+2 x)^5}-\frac {5703277 \left (3-x+2 x^2\right )^{3/2}}{39813120 (5+2 x)^4}-\frac {\int \frac {\left (\frac {182650383}{16}-\frac {60644907 x}{4}\right ) \sqrt {3-x+2 x^2}}{(5+2 x)^4} \, dx}{44789760}\\ &=-\frac {3667 \left (3-x+2 x^2\right )^{3/2}}{3456 (5+2 x)^6}+\frac {92239 \left (3-x+2 x^2\right )^{3/2}}{138240 (5+2 x)^5}-\frac {5703277 \left (3-x+2 x^2\right )^{3/2}}{39813120 (5+2 x)^4}+\frac {87677717 \left (3-x+2 x^2\right )^{3/2}}{8599633920 (5+2 x)^3}+\frac {1172725 \int \frac {\sqrt {3-x+2 x^2}}{(5+2 x)^3} \, dx}{1146617856}\\ &=-\frac {1172725 (17-22 x) \sqrt {3-x+2 x^2}}{330225942528 (5+2 x)^2}-\frac {3667 \left (3-x+2 x^2\right )^{3/2}}{3456 (5+2 x)^6}+\frac {92239 \left (3-x+2 x^2\right )^{3/2}}{138240 (5+2 x)^5}-\frac {5703277 \left (3-x+2 x^2\right )^{3/2}}{39813120 (5+2 x)^4}+\frac {87677717 \left (3-x+2 x^2\right )^{3/2}}{8599633920 (5+2 x)^3}+\frac {26972675 \int \frac {1}{(5+2 x) \sqrt {3-x+2 x^2}} \, dx}{660451885056}\\ &=-\frac {1172725 (17-22 x) \sqrt {3-x+2 x^2}}{330225942528 (5+2 x)^2}-\frac {3667 \left (3-x+2 x^2\right )^{3/2}}{3456 (5+2 x)^6}+\frac {92239 \left (3-x+2 x^2\right )^{3/2}}{138240 (5+2 x)^5}-\frac {5703277 \left (3-x+2 x^2\right )^{3/2}}{39813120 (5+2 x)^4}+\frac {87677717 \left (3-x+2 x^2\right )^{3/2}}{8599633920 (5+2 x)^3}-\frac {26972675 \text {Subst}\left (\int \frac {1}{288-x^2} \, dx,x,\frac {17-22 x}{\sqrt {3-x+2 x^2}}\right )}{330225942528}\\ &=-\frac {1172725 (17-22 x) \sqrt {3-x+2 x^2}}{330225942528 (5+2 x)^2}-\frac {3667 \left (3-x+2 x^2\right )^{3/2}}{3456 (5+2 x)^6}+\frac {92239 \left (3-x+2 x^2\right )^{3/2}}{138240 (5+2 x)^5}-\frac {5703277 \left (3-x+2 x^2\right )^{3/2}}{39813120 (5+2 x)^4}+\frac {87677717 \left (3-x+2 x^2\right )^{3/2}}{8599633920 (5+2 x)^3}-\frac {26972675 \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {3-x+2 x^2}}\right )}{3962711310336 \sqrt {2}}\\ \end {align*}
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Mathematica [A]
time = 0.77, size = 86, normalized size = 0.51 \begin {gather*} \frac {\frac {12 \sqrt {3-x+2 x^2} \left (-219337079305+27245373694 x+158340720344 x^2+397498825328 x^3+12256250416 x^4+271409942624 x^5\right )}{(5+2 x)^6}+134863375 \sqrt {2} \tanh ^{-1}\left (\frac {1}{6} \left (5+2 x-\sqrt {6-2 x+4 x^2}\right )\right )}{19813556551680} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 195, normalized size = 1.15
method | result | size |
risch | \(\frac {542819885248 x^{7}-246897441792 x^{6}+1596971228112 x^{5}-44048633392 x^{4}+1088646503028 x^{3}+9102628728 x^{2}+301073200387 x -658011237915}{1651129712640 \left (5+2 x \right )^{6} \sqrt {2 x^{2}-x +3}}-\frac {26972675 \sqrt {2}\, \arctanh \left (\frac {\left (\frac {17}{2}-11 x \right ) \sqrt {2}}{12 \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}\right )}{7925422620672}\) | \(88\) |
trager | \(\frac {\left (271409942624 x^{5}+12256250416 x^{4}+397498825328 x^{3}+158340720344 x^{2}+27245373694 x -219337079305\right ) \sqrt {2 x^{2}-x +3}}{1651129712640 \left (5+2 x \right )^{6}}-\frac {26972675 \RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {22 \RootOf \left (\textit {\_Z}^{2}-2\right ) x -17 \RootOf \left (\textit {\_Z}^{2}-2\right )-24 \sqrt {2 x^{2}-x +3}}{5+2 x}\right )}{7925422620672}\) | \(98\) |
default | \(\frac {92239 \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {3}{2}}}{4423680 \left (x +\frac {5}{2}\right )^{5}}-\frac {5703277 \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {3}{2}}}{637009920 \left (x +\frac {5}{2}\right )^{4}}+\frac {87677717 \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {3}{2}}}{68797071360 \left (x +\frac {5}{2}\right )^{3}}-\frac {1172725 \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {3}{2}}}{330225942528 \left (x +\frac {5}{2}\right )^{2}}-\frac {12899975 \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {3}{2}}}{11888133931008 \left (x +\frac {5}{2}\right )}+\frac {12899975 \left (4 x -1\right ) \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}{23776267862016}-\frac {3667 \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {3}{2}}}{221184 \left (x +\frac {5}{2}\right )^{6}}-\frac {26972675 \sqrt {2}\, \arctanh \left (\frac {\left (\frac {17}{2}-11 x \right ) \sqrt {2}}{12 \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}\right )}{7925422620672}+\frac {26972675 \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}{23776267862016}\) | \(195\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 250, normalized size = 1.48 \begin {gather*} \frac {26972675}{7925422620672} \, \sqrt {2} \operatorname {arsinh}\left (\frac {22 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 5 \right |}} - \frac {17 \, \sqrt {23}}{23 \, {\left | 2 \, x + 5 \right |}}\right ) + \frac {1172725}{165112971264} \, \sqrt {2 \, x^{2} - x + 3} - \frac {3667 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{3456 \, {\left (64 \, x^{6} + 960 \, x^{5} + 6000 \, x^{4} + 20000 \, x^{3} + 37500 \, x^{2} + 37500 \, x + 15625\right )}} + \frac {92239 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{138240 \, {\left (32 \, x^{5} + 400 \, x^{4} + 2000 \, x^{3} + 5000 \, x^{2} + 6250 \, x + 3125\right )}} - \frac {5703277 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{39813120 \, {\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )}} + \frac {87677717 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{8599633920 \, {\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )}} - \frac {1172725 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{82556485632 \, {\left (4 \, x^{2} + 20 \, x + 25\right )}} - \frac {12899975 \, \sqrt {2 \, x^{2} - x + 3}}{330225942528 \, {\left (2 \, x + 5\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 156, normalized size = 0.92 \begin {gather*} \frac {134863375 \, \sqrt {2} {\left (64 \, x^{6} + 960 \, x^{5} + 6000 \, x^{4} + 20000 \, x^{3} + 37500 \, x^{2} + 37500 \, x + 15625\right )} \log \left (-\frac {24 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (22 \, x - 17\right )} + 1060 \, x^{2} - 1036 \, x + 1153}{4 \, x^{2} + 20 \, x + 25}\right ) + 48 \, {\left (271409942624 \, x^{5} + 12256250416 \, x^{4} + 397498825328 \, x^{3} + 158340720344 \, x^{2} + 27245373694 \, x - 219337079305\right )} \sqrt {2 \, x^{2} - x + 3}}{79254226206720 \, {\left (64 \, x^{6} + 960 \, x^{5} + 6000 \, x^{4} + 20000 \, x^{3} + 37500 \, x^{2} + 37500 \, x + 15625\right )}} \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} \cdot \left (5 x^{4} - x^{3} + 3 x^{2} + x + 2\right )}{\left (2 x + 5\right )^{7}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 405 vs.
\(2 (139) = 278\).
time = 3.24, size = 405, normalized size = 2.40 \begin {gather*} -\frac {26972675}{7925422620672} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x + \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) + \frac {26972675}{7925422620672} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x - 11 \, \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) + \frac {\sqrt {2} {\left (16506981498400 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{11} + 389429252643040 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{10} + 2263923918689840 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{9} + 11663651054548560 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{8} + 902212326134736 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{7} - 84192729519861840 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{6} - 4317200555009448 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{5} + 351543414066518760 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{4} - 376787166452923830 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{3} + 356306707647610982 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{2} - 82348353128195465 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 15499394004553969\right )}}{3302259425280 \, {\left (2 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{2} + 10 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} - 11\right )}^{6}} \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 = -0.01 \begin {gather*} \int \frac {\sqrt {2\,x^2-x+3}\,\left (5\,x^4-x^3+3\,x^2+x+2\right )}{{\left (2\,x+5\right )}^7} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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