Optimal. Leaf size=105 \[ \frac {121 (10679-6744 x)}{8464 \sqrt {2 x^2-x+3}}+\frac {125}{16} x \sqrt {2 x^2-x+3}+\frac {3175}{64} \sqrt {2 x^2-x+3}-\frac {1331 (17-45 x)}{1104 \left (2 x^2-x+3\right )^{3/2}}-\frac {7495 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{128 \sqrt {2}} \]
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Rubi [A] time = 0.11, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1660, 1661, 640, 619, 215} \[ \frac {121 (10679-6744 x)}{8464 \sqrt {2 x^2-x+3}}+\frac {125}{16} x \sqrt {2 x^2-x+3}+\frac {3175}{64} \sqrt {2 x^2-x+3}-\frac {1331 (17-45 x)}{1104 \left (2 x^2-x+3\right )^{3/2}}-\frac {7495 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{128 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 215
Rule 619
Rule 640
Rule 1660
Rule 1661
Rubi steps
\begin {align*} \int \frac {\left (2+3 x+5 x^2\right )^3}{\left (3-x+2 x^2\right )^{5/2}} \, dx &=-\frac {1331 (17-45 x)}{1104 \left (3-x+2 x^2\right )^{3/2}}+\frac {2}{69} \int \frac {-\frac {91275}{64}-\frac {57201 x}{32}+\frac {66585 x^2}{16}+\frac {39675 x^3}{8}+\frac {8625 x^4}{4}}{\left (3-x+2 x^2\right )^{3/2}} \, dx\\ &=-\frac {1331 (17-45 x)}{1104 \left (3-x+2 x^2\right )^{3/2}}+\frac {121 (10679-6744 x)}{8464 \sqrt {3-x+2 x^2}}+\frac {4 \int \frac {\frac {1452105}{64}+\frac {277725 x}{8}+\frac {198375 x^2}{16}}{\sqrt {3-x+2 x^2}} \, dx}{1587}\\ &=-\frac {1331 (17-45 x)}{1104 \left (3-x+2 x^2\right )^{3/2}}+\frac {121 (10679-6744 x)}{8464 \sqrt {3-x+2 x^2}}+\frac {125}{16} x \sqrt {3-x+2 x^2}+\frac {\int \frac {\frac {214245}{4}+\frac {5038725 x}{32}}{\sqrt {3-x+2 x^2}} \, dx}{1587}\\ &=-\frac {1331 (17-45 x)}{1104 \left (3-x+2 x^2\right )^{3/2}}+\frac {121 (10679-6744 x)}{8464 \sqrt {3-x+2 x^2}}+\frac {3175}{64} \sqrt {3-x+2 x^2}+\frac {125}{16} x \sqrt {3-x+2 x^2}+\frac {7495}{128} \int \frac {1}{\sqrt {3-x+2 x^2}} \, dx\\ &=-\frac {1331 (17-45 x)}{1104 \left (3-x+2 x^2\right )^{3/2}}+\frac {121 (10679-6744 x)}{8464 \sqrt {3-x+2 x^2}}+\frac {3175}{64} \sqrt {3-x+2 x^2}+\frac {125}{16} x \sqrt {3-x+2 x^2}+\frac {7495 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+4 x\right )}{128 \sqrt {46}}\\ &=-\frac {1331 (17-45 x)}{1104 \left (3-x+2 x^2\right )^{3/2}}+\frac {121 (10679-6744 x)}{8464 \sqrt {3-x+2 x^2}}+\frac {3175}{64} \sqrt {3-x+2 x^2}+\frac {125}{16} x \sqrt {3-x+2 x^2}-\frac {7495 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{128 \sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 65, normalized size = 0.62 \[ \frac {3174000 x^5+16980900 x^4-29423976 x^3+101546529 x^2-62463282 x+89784565}{101568 \left (2 x^2-x+3\right )^{3/2}}+\frac {7495 \sinh ^{-1}\left (\frac {4 x-1}{\sqrt {23}}\right )}{128 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 122, normalized size = 1.16 \[ \frac {11894565 \, \sqrt {2} {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 8 \, {\left (3174000 \, x^{5} + 16980900 \, x^{4} - 29423976 \, x^{3} + 101546529 \, x^{2} - 62463282 \, x + 89784565\right )} \sqrt {2 \, x^{2} - x + 3}}{812544 \, {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 72, normalized size = 0.69 \[ -\frac {7495}{256} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) + \frac {3 \, {\left ({\left (4 \, {\left (13225 \, {\left (20 \, x + 107\right )} x - 2451998\right )} x + 33848843\right )} x - 20821094\right )} x + 89784565}{101568 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 180, normalized size = 1.71 \[ \frac {125 x^{5}}{4 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {2675 x^{4}}{16 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {7495 x^{3}}{192 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {222809 x^{2}}{256 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {281177 x}{2048 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {7495 x}{128 \sqrt {2 x^{2}-x +3}}+\frac {7495 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{256}-\frac {7495}{512 \sqrt {2 x^{2}-x +3}}+\frac {20961031}{24576 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {3391139 \left (4 x -1\right )}{203136 \sqrt {2 x^{2}-x +3}}-\frac {14081711 \left (4 x -1\right )}{565248 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.02, size = 219, normalized size = 2.09 \[ \frac {125 \, x^{5}}{4 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {2675 \, x^{4}}{16 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {7495}{203136} \, x {\left (\frac {284 \, x}{\sqrt {2 \, x^{2} - x + 3}} - \frac {3174 \, x^{2}}{{\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {71}{\sqrt {2 \, x^{2} - x + 3}} + \frac {805 \, x}{{\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {3243}{{\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}\right )} + \frac {7495}{256} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {532145}{101568} \, \sqrt {2 \, x^{2} - x + 3} - \frac {4515389 \, x}{50784 \, \sqrt {2 \, x^{2} - x + 3}} + \frac {7197 \, x^{2}}{8 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {396211}{50784 \, \sqrt {2 \, x^{2} - x + 3}} - \frac {269783 \, x}{1104 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {1002137}{1104 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
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 \[ \int \frac {{\left (5\,x^2+3\,x+2\right )}^3}{{\left (2\,x^2-x+3\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\left (5 x^{2} + 3 x + 2\right )^{3}}{\left (2 x^{2} - x + 3\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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