Optimal. Leaf size=85 \[ \frac {917 x+1191}{9936 \left (2 x^2-x+3\right )^{3/2}}-\frac {146729 x+335337}{1371168 \sqrt {2 x^2-x+3}}-\frac {3667 \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {2 x^2-x+3}}\right )}{31104 \sqrt {2}} \]
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Rubi [A] time = 0.13, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1646, 12, 724, 206} \[ \frac {917 x+1191}{9936 \left (2 x^2-x+3\right )^{3/2}}-\frac {146729 x+335337}{1371168 \sqrt {2 x^2-x+3}}-\frac {3667 \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {2 x^2-x+3}}\right )}{31104 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 724
Rule 1646
Rubi steps
\begin {align*} \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x) \left (3-x+2 x^2\right )^{5/2}} \, dx &=\frac {1191+917 x}{9936 \left (3-x+2 x^2\right )^{3/2}}+\frac {2}{69} \int \frac {-\frac {1877}{576}+\frac {695 x}{18}+\frac {345 x^2}{4}}{(5+2 x) \left (3-x+2 x^2\right )^{3/2}} \, dx\\ &=\frac {1191+917 x}{9936 \left (3-x+2 x^2\right )^{3/2}}-\frac {335337+146729 x}{1371168 \sqrt {3-x+2 x^2}}+\frac {4 \int \frac {1939843}{6912 (5+2 x) \sqrt {3-x+2 x^2}} \, dx}{1587}\\ &=\frac {1191+917 x}{9936 \left (3-x+2 x^2\right )^{3/2}}-\frac {335337+146729 x}{1371168 \sqrt {3-x+2 x^2}}+\frac {3667 \int \frac {1}{(5+2 x) \sqrt {3-x+2 x^2}} \, dx}{5184}\\ &=\frac {1191+917 x}{9936 \left (3-x+2 x^2\right )^{3/2}}-\frac {335337+146729 x}{1371168 \sqrt {3-x+2 x^2}}-\frac {3667 \operatorname {Subst}\left (\int \frac {1}{288-x^2} \, dx,x,\frac {17-22 x}{\sqrt {3-x+2 x^2}}\right )}{2592}\\ &=\frac {1191+917 x}{9936 \left (3-x+2 x^2\right )^{3/2}}-\frac {335337+146729 x}{1371168 \sqrt {3-x+2 x^2}}-\frac {3667 \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {3-x+2 x^2}}\right )}{31104 \sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.47, size = 80, normalized size = 0.94 \[ \frac {-3667 \log \left (12 \sqrt {4 x^2-2 x+6}-22 x+17\right )-\frac {12 \sqrt {2} \left (293458 x^3+523945 x^2-21696 x+841653\right )}{529 \left (2 x^2-x+3\right )^{3/2}}+3667 \log (2 x+5)}{31104 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 126, normalized size = 1.48 \[ \frac {1939843 \, \sqrt {2} {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \log \left (-\frac {24 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (22 \, x - 17\right )} + 1060 \, x^{2} - 1036 \, x + 1153}{4 \, x^{2} + 20 \, x + 25}\right ) - 48 \, {\left (293458 \, x^{3} + 523945 \, x^{2} - 21696 \, x + 841653\right )} \sqrt {2 \, x^{2} - x + 3}}{65816064 \, {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 92, normalized size = 1.08 \[ -\frac {3667}{62208} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x + \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) + \frac {3667}{62208} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x - 11 \, \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) - \frac {{\left ({\left (293458 \, x + 523945\right )} x - 21696\right )} x + 841653}{1371168 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 190, normalized size = 2.24 \[ -\frac {5 x^{2}}{4 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {59 x}{32 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {3667 \sqrt {2}\, \arctanh \left (\frac {\left (-11 x +\frac {17}{2}\right ) \sqrt {2}}{12 \sqrt {-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}}}\right )}{62208}-\frac {1597}{384 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {3817 \left (4 x -1\right )}{2944 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {3817 \left (4 x -1\right )}{4232 \sqrt {2 x^{2}-x +3}}+\frac {3667}{1728 \left (-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}\right )^{\frac {3}{2}}}+\frac {\frac {40337 x}{9936}-\frac {40337}{39744}}{\left (-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}\right )^{\frac {3}{2}}}+\frac {\frac {4800103 x}{1371168}-\frac {4800103}{5484672}}{\sqrt {-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}}}+\frac {3667}{10368 \sqrt {-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 110, normalized size = 1.29 \[ \frac {3667}{62208} \, \sqrt {2} \operatorname {arsinh}\left (\frac {22 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 5 \right |}} - \frac {17 \, \sqrt {23}}{23 \, {\left | 2 \, x + 5 \right |}}\right ) - \frac {146729 \, x}{1371168 \, \sqrt {2 \, x^{2} - x + 3}} - \frac {5 \, x^{2}}{4 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {173881}{457056 \, \sqrt {2 \, x^{2} - x + 3}} + \frac {7127 \, x}{9936 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {5813}{3312 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
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 \[ \int \frac {5\,x^4-x^3+3\,x^2+x+2}{\left (2\,x+5\right )\,{\left (2\,x^2-x+3\right )}^{5/2}} \,d x \]
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 \[ \int \frac {5 x^{4} - x^{3} + 3 x^{2} + x + 2}{\left (2 x + 5\right ) \left (2 x^{2} - x + 3\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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