Optimal. Leaf size=112 \[ -\frac {1}{12} \sqrt {3 x^2+5 x+2} (2 x+3)^3+\frac {32}{27} \sqrt {3 x^2+5 x+2} (2 x+3)^2+\frac {5}{648} (1078 x+3261) \sqrt {3 x^2+5 x+2}+\frac {19405 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{1296 \sqrt {3}} \]
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Rubi [A] time = 0.06, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {832, 779, 621, 206} \begin {gather*} -\frac {1}{12} \sqrt {3 x^2+5 x+2} (2 x+3)^3+\frac {32}{27} \sqrt {3 x^2+5 x+2} (2 x+3)^2+\frac {5}{648} (1078 x+3261) \sqrt {3 x^2+5 x+2}+\frac {19405 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{1296 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 779
Rule 832
Rubi steps
\begin {align*} \int \frac {(5-x) (3+2 x)^3}{\sqrt {2+5 x+3 x^2}} \, dx &=-\frac {1}{12} (3+2 x)^3 \sqrt {2+5 x+3 x^2}+\frac {1}{12} \int \frac {(3+2 x)^2 \left (\frac {399}{2}+128 x\right )}{\sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {32}{27} (3+2 x)^2 \sqrt {2+5 x+3 x^2}-\frac {1}{12} (3+2 x)^3 \sqrt {2+5 x+3 x^2}+\frac {1}{108} \int \frac {(3+2 x) \left (\frac {6805}{2}+2695 x\right )}{\sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {32}{27} (3+2 x)^2 \sqrt {2+5 x+3 x^2}-\frac {1}{12} (3+2 x)^3 \sqrt {2+5 x+3 x^2}+\frac {5}{648} (3261+1078 x) \sqrt {2+5 x+3 x^2}+\frac {19405 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{1296}\\ &=\frac {32}{27} (3+2 x)^2 \sqrt {2+5 x+3 x^2}-\frac {1}{12} (3+2 x)^3 \sqrt {2+5 x+3 x^2}+\frac {5}{648} (3261+1078 x) \sqrt {2+5 x+3 x^2}+\frac {19405}{648} \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )\\ &=\frac {32}{27} (3+2 x)^2 \sqrt {2+5 x+3 x^2}-\frac {1}{12} (3+2 x)^3 \sqrt {2+5 x+3 x^2}+\frac {5}{648} (3261+1078 x) \sqrt {2+5 x+3 x^2}+\frac {19405 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{1296 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 67, normalized size = 0.60 \begin {gather*} \frac {19405 \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {9 x^2+15 x+6}}\right )-6 \sqrt {3 x^2+5 x+2} \left (432 x^3-1128 x^2-11690 x-21759\right )}{3888} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.87, size = 69, normalized size = 0.62 \begin {gather*} \frac {19405 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{648 \sqrt {3}}+\frac {1}{648} \sqrt {3 x^2+5 x+2} \left (-432 x^3+1128 x^2+11690 x+21759\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 68, normalized size = 0.61 \begin {gather*} -\frac {1}{648} \, {\left (432 \, x^{3} - 1128 \, x^{2} - 11690 \, x - 21759\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {19405}{7776} \, \sqrt {3} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 64, normalized size = 0.57 \begin {gather*} -\frac {1}{648} \, {\left (2 \, {\left (12 \, {\left (18 \, x - 47\right )} x - 5845\right )} x - 21759\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {19405}{3888} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 94, normalized size = 0.84 \begin {gather*} -\frac {2 \sqrt {3 x^{2}+5 x +2}\, x^{3}}{3}+\frac {47 \sqrt {3 x^{2}+5 x +2}\, x^{2}}{27}+\frac {5845 \sqrt {3 x^{2}+5 x +2}\, x}{324}+\frac {19405 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )}{3888}+\frac {7253 \sqrt {3 x^{2}+5 x +2}}{216} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 92, normalized size = 0.82 \begin {gather*} -\frac {2}{3} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x^{3} + \frac {47}{27} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x^{2} + \frac {5845}{324} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {19405}{3888} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac {7253}{216} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \end {gather*}
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 \begin {gather*} -\int \frac {{\left (2\,x+3\right )}^3\,\left (x-5\right )}{\sqrt {3\,x^2+5\,x+2}} \,d x \end {gather*}
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 \begin {gather*} - \int \left (- \frac {243 x}{\sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac {126 x^{2}}{\sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac {4 x^{3}}{\sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac {8 x^{4}}{\sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {135}{\sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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