Optimal. Leaf size=135 \[ \frac {2 (3693 x+2363)}{151593 \left (3 x^2-x+2\right )^{3/2}}-\frac {144 \sqrt {3 x^2-x+2}}{28561 (2 x+1)}-\frac {8 \sqrt {3 x^2-x+2}}{2197 (2 x+1)^2}+\frac {12 (103526 x+25771)}{15108769 \sqrt {3 x^2-x+2}}-\frac {2084 \tanh ^{-1}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {3 x^2-x+2}}\right )}{28561 \sqrt {13}} \]
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Rubi [A] time = 0.21, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {1646, 1650, 806, 724, 206} \[ \frac {2 (3693 x+2363)}{151593 \left (3 x^2-x+2\right )^{3/2}}-\frac {144 \sqrt {3 x^2-x+2}}{28561 (2 x+1)}-\frac {8 \sqrt {3 x^2-x+2}}{2197 (2 x+1)^2}+\frac {12 (103526 x+25771)}{15108769 \sqrt {3 x^2-x+2}}-\frac {2084 \tanh ^{-1}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {3 x^2-x+2}}\right )}{28561 \sqrt {13}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 724
Rule 806
Rule 1646
Rule 1650
Rubi steps
\begin {align*} \int \frac {1+3 x+4 x^2}{(1+2 x)^3 \left (2-x+3 x^2\right )^{5/2}} \, dx &=\frac {2 (2363+3693 x)}{151593 \left (2-x+3 x^2\right )^{3/2}}+\frac {2}{69} \int \frac {\frac {32433}{2197}+\frac {106830 x}{2197}+\frac {160116 x^2}{2197}+\frac {59088 x^3}{2197}}{(1+2 x)^3 \left (2-x+3 x^2\right )^{3/2}} \, dx\\ &=\frac {2 (2363+3693 x)}{151593 \left (2-x+3 x^2\right )^{3/2}}+\frac {12 (25771+103526 x)}{15108769 \sqrt {2-x+3 x^2}}+\frac {4 \int \frac {\frac {1434648}{28561}+\frac {3345396 x}{28561}+\frac {3097824 x^2}{28561}}{(1+2 x)^3 \sqrt {2-x+3 x^2}} \, dx}{1587}\\ &=\frac {2 (2363+3693 x)}{151593 \left (2-x+3 x^2\right )^{3/2}}+\frac {12 (25771+103526 x)}{15108769 \sqrt {2-x+3 x^2}}-\frac {8 \sqrt {2-x+3 x^2}}{2197 (1+2 x)^2}-\frac {2 \int \frac {-\frac {2167842}{2197}-\frac {2850252 x}{2197}}{(1+2 x)^2 \sqrt {2-x+3 x^2}} \, dx}{20631}\\ &=\frac {2 (2363+3693 x)}{151593 \left (2-x+3 x^2\right )^{3/2}}+\frac {12 (25771+103526 x)}{15108769 \sqrt {2-x+3 x^2}}-\frac {8 \sqrt {2-x+3 x^2}}{2197 (1+2 x)^2}-\frac {144 \sqrt {2-x+3 x^2}}{28561 (1+2 x)}+\frac {2084 \int \frac {1}{(1+2 x) \sqrt {2-x+3 x^2}} \, dx}{28561}\\ &=\frac {2 (2363+3693 x)}{151593 \left (2-x+3 x^2\right )^{3/2}}+\frac {12 (25771+103526 x)}{15108769 \sqrt {2-x+3 x^2}}-\frac {8 \sqrt {2-x+3 x^2}}{2197 (1+2 x)^2}-\frac {144 \sqrt {2-x+3 x^2}}{28561 (1+2 x)}-\frac {4168 \operatorname {Subst}\left (\int \frac {1}{52-x^2} \, dx,x,\frac {9-8 x}{\sqrt {2-x+3 x^2}}\right )}{28561}\\ &=\frac {2 (2363+3693 x)}{151593 \left (2-x+3 x^2\right )^{3/2}}+\frac {12 (25771+103526 x)}{15108769 \sqrt {2-x+3 x^2}}-\frac {8 \sqrt {2-x+3 x^2}}{2197 (1+2 x)^2}-\frac {144 \sqrt {2-x+3 x^2}}{28561 (1+2 x)}-\frac {2084 \tanh ^{-1}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {2-x+3 x^2}}\right )}{28561 \sqrt {13}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 89, normalized size = 0.66 \[ \frac {2 \left (20304864 x^5+20074356 x^4+19381992 x^3+21890266 x^2+10777477 x+847141\right )}{45326307 (2 x+1)^2 \left (3 x^2-x+2\right )^{3/2}}-\frac {2084 \tanh ^{-1}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {3 x^2-x+2}}\right )}{28561 \sqrt {13}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 156, normalized size = 1.16 \[ \frac {2 \, {\left (826827 \, \sqrt {13} {\left (36 \, x^{6} + 12 \, x^{5} + 37 \, x^{4} + 30 \, x^{3} + 13 \, x^{2} + 12 \, x + 4\right )} \log \left (-\frac {4 \, \sqrt {13} \sqrt {3 \, x^{2} - x + 2} {\left (8 \, x - 9\right )} + 220 \, x^{2} - 196 \, x + 185}{4 \, x^{2} + 4 \, x + 1}\right ) + 13 \, {\left (20304864 \, x^{5} + 20074356 \, x^{4} + 19381992 \, x^{3} + 21890266 \, x^{2} + 10777477 \, x + 847141\right )} \sqrt {3 \, x^{2} - x + 2}\right )}}{589241991 \, {\left (36 \, x^{6} + 12 \, x^{5} + 37 \, x^{4} + 30 \, x^{3} + 13 \, x^{2} + 12 \, x + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.35, size = 233, normalized size = 1.73 \[ \frac {2084}{371293} \, \sqrt {13} \log \left (-\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {13} - 2 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} - x + 2} \right |}}{2 \, {\left (2 \, \sqrt {3} x - \sqrt {13} + \sqrt {3} - 2 \, \sqrt {3 \, x^{2} - x + 2}\right )}}\right ) + \frac {2 \, {\left (3 \, {\left (6 \, {\left (310578 \, x - 26213\right )} x + 1455755\right )} x + 1634293\right )}}{45326307 \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}}} - \frac {8 \, {\left (66 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - x + 2}\right )}^{3} + 21 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - x + 2}\right )}^{2} - 1015 \, \sqrt {3} x + 431 \, \sqrt {3} + 1015 \, \sqrt {3 \, x^{2} - x + 2}\right )}}{28561 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - x + 2}\right )}^{2} + 2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - x + 2}\right )} - 5\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 148, normalized size = 1.10 \[ -\frac {2084 \sqrt {13}\, \arctanh \left (\frac {2 \left (-4 x +\frac {9}{2}\right ) \sqrt {13}}{13 \sqrt {-16 x +12 \left (x +\frac {1}{2}\right )^{2}+5}}\right )}{371293}+\frac {521}{13182 \left (-4 x +3 \left (x +\frac {1}{2}\right )^{2}+\frac {5}{4}\right )^{\frac {3}{2}}}+\frac {\frac {1772 x}{50531}-\frac {886}{151593}}{\left (-4 x +3 \left (x +\frac {1}{2}\right )^{2}+\frac {5}{4}\right )^{\frac {3}{2}}}+\frac {\frac {1128048 x}{15108769}-\frac {188008}{15108769}}{\sqrt {-4 x +3 \left (x +\frac {1}{2}\right )^{2}+\frac {5}{4}}}+\frac {1042}{28561 \sqrt {-4 x +3 \left (x +\frac {1}{2}\right )^{2}+\frac {5}{4}}}-\frac {1}{338 \left (x +\frac {1}{2}\right ) \left (-4 x +3 \left (x +\frac {1}{2}\right )^{2}+\frac {5}{4}\right )^{\frac {3}{2}}}-\frac {1}{104 \left (x +\frac {1}{2}\right )^{2} \left (-4 x +3 \left (x +\frac {1}{2}\right )^{2}+\frac {5}{4}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 174, normalized size = 1.29 \[ \frac {2084}{371293} \, \sqrt {13} \operatorname {arsinh}\left (\frac {8 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 1 \right |}} - \frac {9 \, \sqrt {23}}{23 \, {\left | 2 \, x + 1 \right |}}\right ) + \frac {1128048 \, x}{15108769 \, \sqrt {3 \, x^{2} - x + 2}} + \frac {363210}{15108769 \, \sqrt {3 \, x^{2} - x + 2}} + \frac {1772 \, x}{50531 \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}}} - \frac {1}{26 \, {\left (4 \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} x^{2} + 4 \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} x + {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}}\right )}} - \frac {1}{169 \, {\left (2 \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} x + {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}}\right )}} + \frac {10211}{303186 \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}}} \]
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 \[ \int \frac {4\,x^2+3\,x+1}{{\left (2\,x+1\right )}^3\,{\left (3\,x^2-x+2\right )}^{5/2}} \,d x \]
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 \[ \int \frac {4 x^{2} + 3 x + 1}{\left (2 x + 1\right )^{3} \left (3 x^{2} - x + 2\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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