Optimal. Leaf size=195 \[ -\frac {3730507 \left (2 x^2-x+3\right )^{5/2}}{11943936 (2 x+5)^3}+\frac {158527 \left (2 x^2-x+3\right )^{5/2}}{165888 (2 x+5)^4}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{2880 (2 x+5)^5}+\frac {(44773976 x+246012435) \left (2 x^2-x+3\right )^{3/2}}{95551488 (2 x+5)^2}-\frac {(1028823716 x+5658774871) \sqrt {2 x^2-x+3}}{127401984 (2 x+5)}+\frac {70991525167 \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {2 x^2-x+3}}\right )}{1528823808 \sqrt {2}}-\frac {23775 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{512 \sqrt {2}} \]
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Rubi [A] time = 0.26, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {1650, 812, 843, 619, 215, 724, 206} \[ -\frac {3730507 \left (2 x^2-x+3\right )^{5/2}}{11943936 (2 x+5)^3}+\frac {158527 \left (2 x^2-x+3\right )^{5/2}}{165888 (2 x+5)^4}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{2880 (2 x+5)^5}+\frac {(44773976 x+246012435) \left (2 x^2-x+3\right )^{3/2}}{95551488 (2 x+5)^2}-\frac {(1028823716 x+5658774871) \sqrt {2 x^2-x+3}}{127401984 (2 x+5)}+\frac {70991525167 \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {2 x^2-x+3}}\right )}{1528823808 \sqrt {2}}-\frac {23775 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{512 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 215
Rule 619
Rule 724
Rule 812
Rule 843
Rule 1650
Rubi steps
\begin {align*} \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^6} \, dx &=-\frac {3667 \left (3-x+2 x^2\right )^{5/2}}{2880 (5+2 x)^5}-\frac {1}{360} \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (\frac {60035}{16}-6615 x+2430 x^2-900 x^3\right )}{(5+2 x)^5} \, dx\\ &=-\frac {3667 \left (3-x+2 x^2\right )^{5/2}}{2880 (5+2 x)^5}+\frac {158527 \left (3-x+2 x^2\right )^{5/2}}{165888 (5+2 x)^4}+\frac {\int \frac {\left (3-x+2 x^2\right )^{3/2} \left (\frac {8114455}{16}-\frac {3488315 x}{4}+129600 x^2\right )}{(5+2 x)^4} \, dx}{103680}\\ &=-\frac {3667 \left (3-x+2 x^2\right )^{5/2}}{2880 (5+2 x)^5}+\frac {158527 \left (3-x+2 x^2\right )^{5/2}}{165888 (5+2 x)^4}-\frac {3730507 \left (3-x+2 x^2\right )^{5/2}}{11943936 (5+2 x)^3}-\frac {\int \frac {\left (\frac {332138325}{16}-\frac {83951205 x}{2}\right ) \left (3-x+2 x^2\right )^{3/2}}{(5+2 x)^3} \, dx}{22394880}\\ &=\frac {(246012435+44773976 x) \left (3-x+2 x^2\right )^{3/2}}{95551488 (5+2 x)^2}-\frac {3667 \left (3-x+2 x^2\right )^{5/2}}{2880 (5+2 x)^5}+\frac {158527 \left (3-x+2 x^2\right )^{5/2}}{165888 (5+2 x)^4}-\frac {3730507 \left (3-x+2 x^2\right )^{5/2}}{11943936 (5+2 x)^3}+\frac {\int \frac {\left (\frac {7719844365}{4}-3858088935 x\right ) \sqrt {3-x+2 x^2}}{(5+2 x)^2} \, dx}{238878720}\\ &=-\frac {(5658774871+1028823716 x) \sqrt {3-x+2 x^2}}{127401984 (5+2 x)}+\frac {(246012435+44773976 x) \left (3-x+2 x^2\right )^{3/2}}{95551488 (5+2 x)^2}-\frac {3667 \left (3-x+2 x^2\right )^{5/2}}{2880 (5+2 x)^5}+\frac {158527 \left (3-x+2 x^2\right )^{5/2}}{165888 (5+2 x)^4}-\frac {3730507 \left (3-x+2 x^2\right )^{5/2}}{11943936 (5+2 x)^3}-\frac {\int \frac {\frac {177475757505}{2}-177479424000 x}{(5+2 x) \sqrt {3-x+2 x^2}} \, dx}{1911029760}\\ &=-\frac {(5658774871+1028823716 x) \sqrt {3-x+2 x^2}}{127401984 (5+2 x)}+\frac {(246012435+44773976 x) \left (3-x+2 x^2\right )^{3/2}}{95551488 (5+2 x)^2}-\frac {3667 \left (3-x+2 x^2\right )^{5/2}}{2880 (5+2 x)^5}+\frac {158527 \left (3-x+2 x^2\right )^{5/2}}{165888 (5+2 x)^4}-\frac {3730507 \left (3-x+2 x^2\right )^{5/2}}{11943936 (5+2 x)^3}+\frac {23775}{512} \int \frac {1}{\sqrt {3-x+2 x^2}} \, dx-\frac {70991525167 \int \frac {1}{(5+2 x) \sqrt {3-x+2 x^2}} \, dx}{254803968}\\ &=-\frac {(5658774871+1028823716 x) \sqrt {3-x+2 x^2}}{127401984 (5+2 x)}+\frac {(246012435+44773976 x) \left (3-x+2 x^2\right )^{3/2}}{95551488 (5+2 x)^2}-\frac {3667 \left (3-x+2 x^2\right )^{5/2}}{2880 (5+2 x)^5}+\frac {158527 \left (3-x+2 x^2\right )^{5/2}}{165888 (5+2 x)^4}-\frac {3730507 \left (3-x+2 x^2\right )^{5/2}}{11943936 (5+2 x)^3}+\frac {70991525167 \operatorname {Subst}\left (\int \frac {1}{288-x^2} \, dx,x,\frac {17-22 x}{\sqrt {3-x+2 x^2}}\right )}{127401984}+\frac {23775 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+4 x\right )}{512 \sqrt {46}}\\ &=-\frac {(5658774871+1028823716 x) \sqrt {3-x+2 x^2}}{127401984 (5+2 x)}+\frac {(246012435+44773976 x) \left (3-x+2 x^2\right )^{3/2}}{95551488 (5+2 x)^2}-\frac {3667 \left (3-x+2 x^2\right )^{5/2}}{2880 (5+2 x)^5}+\frac {158527 \left (3-x+2 x^2\right )^{5/2}}{165888 (5+2 x)^4}-\frac {3730507 \left (3-x+2 x^2\right )^{5/2}}{11943936 (5+2 x)^3}-\frac {23775 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{512 \sqrt {2}}+\frac {70991525167 \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {3-x+2 x^2}}\right )}{1528823808 \sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 108, normalized size = 0.55 \[ \frac {354957625835 \sqrt {2} \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {4 x^2-2 x+6}}\right )+\frac {24 \sqrt {2 x^2-x+3} \left (1592524800 x^6-30496849920 x^5-1023534029552 x^4-7117092892448 x^3-21590439797064 x^2-30872393829992 x-17093312738327\right )}{(2 x+5)^5}-354958848000 \sqrt {2} \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{15288238080} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.03, size = 213, normalized size = 1.09 \[ \frac {354958848000 \, \sqrt {2} {\left (32 \, x^{5} + 400 \, x^{4} + 2000 \, x^{3} + 5000 \, x^{2} + 6250 \, x + 3125\right )} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 354957625835 \, \sqrt {2} {\left (32 \, x^{5} + 400 \, x^{4} + 2000 \, x^{3} + 5000 \, x^{2} + 6250 \, x + 3125\right )} \log \left (\frac {24 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (22 \, x - 17\right )} - 1060 \, x^{2} + 1036 \, x - 1153}{4 \, x^{2} + 20 \, x + 25}\right ) + 48 \, {\left (1592524800 \, x^{6} - 30496849920 \, x^{5} - 1023534029552 \, x^{4} - 7117092892448 \, x^{3} - 21590439797064 \, x^{2} - 30872393829992 \, x - 17093312738327\right )} \sqrt {2 \, x^{2} - x + 3}}{30576476160 \, {\left (32 \, x^{5} + 400 \, x^{4} + 2000 \, x^{3} + 5000 \, x^{2} + 6250 \, x + 3125\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.32, size = 406, normalized size = 2.08 \[ \frac {1}{256} \, \sqrt {2 \, x^{2} - x + 3} {\left (20 \, x - 633\right )} - \frac {23775}{1024} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) + \frac {70991525167}{3057647616} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x + \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) - \frac {70991525167}{3057647616} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x - 11 \, \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) - \frac {\sqrt {2} {\left (8281387393360 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{9} + 275661428628240 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{8} + 1560382703345760 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{7} + 4938646760855520 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{6} - 9673562837036232 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{5} - 30647310393849000 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{4} + 70060241036847960 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{3} - 97730658088823880 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{2} + 30180638363071845 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} - 7096913381268319\right )}}{1274019840 \, {\left (2 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{2} + 10 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} - 11\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 225, normalized size = 1.15 \[ \frac {23775 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{1024}+\frac {70991525167 \sqrt {2}\, \arctanh \left (\frac {\left (-11 x +\frac {17}{2}\right ) \sqrt {2}}{12 \sqrt {-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}}}\right )}{3057647616}-\frac {70991525167 \sqrt {-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}}}{9172942848}-\frac {70991525167 \left (-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}\right )^{\frac {3}{2}}}{495338913792}-\frac {3667 \left (-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}\right )^{\frac {5}{2}}}{92160 \left (x +\frac {5}{2}\right )^{5}}+\frac {158527 \left (-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}\right )^{\frac {5}{2}}}{2654208 \left (x +\frac {5}{2}\right )^{4}}-\frac {3730507 \left (-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}\right )^{\frac {5}{2}}}{95551488 \left (x +\frac {5}{2}\right )^{3}}+\frac {134077495 \left (-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}\right )^{\frac {5}{2}}}{6879707136 \left (x +\frac {5}{2}\right )^{2}}+\frac {4698578717 \left (4 x -1\right ) \left (-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}\right )^{\frac {3}{2}}}{495338913792}-\frac {4698578717 \left (-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}\right )^{\frac {5}{2}}}{247669456896 \left (x +\frac {5}{2}\right )}+\frac {3086715581 \left (4 x -1\right ) \sqrt {-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}}}{9172942848} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 251, normalized size = 1.29 \[ -\frac {134077495}{3439853568} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {3667 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{2880 \, {\left (32 \, x^{5} + 400 \, x^{4} + 2000 \, x^{3} + 5000 \, x^{2} + 6250 \, x + 3125\right )}} + \frac {158527 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{165888 \, {\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )}} - \frac {3730507 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{11943936 \, {\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )}} + \frac {134077495 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{1719926784 \, {\left (4 \, x^{2} + 20 \, x + 25\right )}} + \frac {3086715581}{2293235712} \, \sqrt {2 \, x^{2} - x + 3} x + \frac {23775}{1024} \, \sqrt {2} \operatorname {arsinh}\left (\frac {4}{23} \, \sqrt {23} x - \frac {1}{23} \, \sqrt {23}\right ) - \frac {70991525167}{3057647616} \, \sqrt {2} \operatorname {arsinh}\left (\frac {22 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 5 \right |}} - \frac {17 \, \sqrt {23}}{23 \, {\left | 2 \, x + 5 \right |}}\right ) - \frac {6173186729}{764411904} \, \sqrt {2 \, x^{2} - x + 3} - \frac {4698578717 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{6879707136 \, {\left (2 \, x + 5\right )}} \]
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 \[ \int \frac {{\left (2\,x^2-x+3\right )}^{3/2}\,\left (5\,x^4-x^3+3\,x^2+x+2\right )}{{\left (2\,x+5\right )}^6} \,d x \]
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 \[ \int \frac {\left (2 x^{2} - x + 3\right )^{\frac {3}{2}} \left (5 x^{4} - x^{3} + 3 x^{2} + x + 2\right )}{\left (2 x + 5\right )^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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