Optimal. Leaf size=85 \[ -\frac {4 (691-13668 x)}{268203 \sqrt {3 x^2-x+2}}-\frac {2 (101-77 x)}{897 \left (3 x^2-x+2\right )^{3/2}}-\frac {8 \tanh ^{-1}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {3 x^2-x+2}}\right )}{169 \sqrt {13}} \]
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Rubi [A] time = 0.09, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {1646, 822, 12, 724, 206} \[ -\frac {4 (691-13668 x)}{268203 \sqrt {3 x^2-x+2}}-\frac {2 (101-77 x)}{897 \left (3 x^2-x+2\right )^{3/2}}-\frac {8 \tanh ^{-1}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {3 x^2-x+2}}\right )}{169 \sqrt {13}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 724
Rule 822
Rule 1646
Rubi steps
\begin {align*} \int \frac {1+3 x+4 x^2}{(1+2 x) \left (2-x+3 x^2\right )^{5/2}} \, dx &=-\frac {2 (101-77 x)}{897 \left (2-x+3 x^2\right )^{3/2}}+\frac {2}{69} \int \frac {\frac {223}{13}+\frac {308 x}{13}}{(1+2 x) \left (2-x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac {2 (101-77 x)}{897 \left (2-x+3 x^2\right )^{3/2}}-\frac {4 (691-13668 x)}{268203 \sqrt {2-x+3 x^2}}+\frac {4 \int \frac {3174}{13 (1+2 x) \sqrt {2-x+3 x^2}} \, dx}{20631}\\ &=-\frac {2 (101-77 x)}{897 \left (2-x+3 x^2\right )^{3/2}}-\frac {4 (691-13668 x)}{268203 \sqrt {2-x+3 x^2}}+\frac {8}{169} \int \frac {1}{(1+2 x) \sqrt {2-x+3 x^2}} \, dx\\ &=-\frac {2 (101-77 x)}{897 \left (2-x+3 x^2\right )^{3/2}}-\frac {4 (691-13668 x)}{268203 \sqrt {2-x+3 x^2}}-\frac {16}{169} \operatorname {Subst}\left (\int \frac {1}{52-x^2} \, dx,x,\frac {9-8 x}{\sqrt {2-x+3 x^2}}\right )\\ &=-\frac {2 (101-77 x)}{897 \left (2-x+3 x^2\right )^{3/2}}-\frac {4 (691-13668 x)}{268203 \sqrt {2-x+3 x^2}}-\frac {8 \tanh ^{-1}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {2-x+3 x^2}}\right )}{169 \sqrt {13}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 72, normalized size = 0.85 \[ \frac {2 \left (82008 x^3-31482 x^2+79077 x-32963\right )}{268203 \left (3 x^2-x+2\right )^{3/2}}-\frac {8 \tanh ^{-1}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {3 x^2-x+2}}\right )}{169 \sqrt {13}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 126, normalized size = 1.48 \[ \frac {2 \, {\left (3174 \, \sqrt {13} {\left (9 \, x^{4} - 6 \, x^{3} + 13 \, x^{2} - 4 \, x + 4\right )} \log \left (-\frac {4 \, \sqrt {13} \sqrt {3 \, x^{2} - x + 2} {\left (8 \, x - 9\right )} + 220 \, x^{2} - 196 \, x + 185}{4 \, x^{2} + 4 \, x + 1}\right ) + 13 \, {\left (82008 \, x^{3} - 31482 \, x^{2} + 79077 \, x - 32963\right )} \sqrt {3 \, x^{2} - x + 2}\right )}}{3486639 \, {\left (9 \, x^{4} - 6 \, x^{3} + 13 \, x^{2} - 4 \, x + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 101, normalized size = 1.19 \[ \frac {8}{2197} \, \sqrt {13} \log \left (-\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {13} - 2 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} - x + 2} \right |}}{2 \, {\left (2 \, \sqrt {3} x - \sqrt {13} + \sqrt {3} - 2 \, \sqrt {3 \, x^{2} - x + 2}\right )}}\right ) + \frac {2 \, {\left (3 \, {\left (6 \, {\left (4556 \, x - 1749\right )} x + 26359\right )} x - 32963\right )}}{268203 \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 158, normalized size = 1.86 \[ -\frac {8 \sqrt {13}\, \arctanh \left (\frac {2 \left (-4 x +\frac {9}{2}\right ) \sqrt {13}}{13 \sqrt {-16 x +12 \left (x +\frac {1}{2}\right )^{2}+5}}\right )}{2197}-\frac {2}{9 \left (3 x^{2}-x +2\right )^{\frac {3}{2}}}+\frac {\frac {10 x}{69}-\frac {5}{207}}{\left (3 x^{2}-x +2\right )^{\frac {3}{2}}}+\frac {\frac {80 x}{529}-\frac {40}{1587}}{\sqrt {3 x^{2}-x +2}}+\frac {1}{39 \left (-4 x +3 \left (x +\frac {1}{2}\right )^{2}+\frac {5}{4}\right )^{\frac {3}{2}}}+\frac {\frac {8 x}{299}-\frac {4}{897}}{\left (-4 x +3 \left (x +\frac {1}{2}\right )^{2}+\frac {5}{4}\right )^{\frac {3}{2}}}+\frac {\frac {4704 x}{89401}-\frac {784}{89401}}{\sqrt {-4 x +3 \left (x +\frac {1}{2}\right )^{2}+\frac {5}{4}}}+\frac {4}{169 \sqrt {-4 x +3 \left (x +\frac {1}{2}\right )^{2}+\frac {5}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 93, normalized size = 1.09 \[ \frac {8}{2197} \, \sqrt {13} \operatorname {arsinh}\left (\frac {8 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 1 \right |}} - \frac {9 \, \sqrt {23}}{23 \, {\left | 2 \, x + 1 \right |}}\right ) + \frac {18224 \, x}{89401 \, \sqrt {3 \, x^{2} - x + 2}} - \frac {2764}{268203 \, \sqrt {3 \, x^{2} - x + 2}} + \frac {154 \, x}{897 \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}}} - \frac {202}{897 \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}}} \]
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 \[ \int \frac {4\,x^2+3\,x+1}{\left (2\,x+1\right )\,{\left (3\,x^2-x+2\right )}^{5/2}} \,d x \]
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 \[ \int \frac {4 x^{2} + 3 x + 1}{\left (2 x + 1\right ) \left (3 x^{2} - x + 2\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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