Optimal. Leaf size=87 \[ -\frac {1}{9} \sqrt {3 x^2+5 x+2} (2 x+3)^2+\frac {1}{54} (194 x+699) \sqrt {3 x^2+5 x+2}+\frac {1147 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{108 \sqrt {3}} \]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 4, 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}{9} \sqrt {3 x^2+5 x+2} (2 x+3)^2+\frac {1}{54} (194 x+699) \sqrt {3 x^2+5 x+2}+\frac {1147 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{108 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 621
Rule 779
Rule 832
Rubi steps
\begin {align*} \int \frac {(5-x) (3+2 x)^2}{\sqrt {2+5 x+3 x^2}} \, dx &=-\frac {1}{9} (3+2 x)^2 \sqrt {2+5 x+3 x^2}+\frac {1}{9} \int \frac {(3+2 x) \left (\frac {301}{2}+97 x\right )}{\sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {1}{9} (3+2 x)^2 \sqrt {2+5 x+3 x^2}+\frac {1}{54} (699+194 x) \sqrt {2+5 x+3 x^2}+\frac {1147}{108} \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {1}{9} (3+2 x)^2 \sqrt {2+5 x+3 x^2}+\frac {1}{54} (699+194 x) \sqrt {2+5 x+3 x^2}+\frac {1147}{54} \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )\\ &=-\frac {1}{9} (3+2 x)^2 \sqrt {2+5 x+3 x^2}+\frac {1}{54} (699+194 x) \sqrt {2+5 x+3 x^2}+\frac {1147 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{108 \sqrt {3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 62, normalized size = 0.71 \begin {gather*} \frac {1}{324} \left (1147 \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 (24 x^2-122 x-645\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.62, size = 64, normalized size = 0.74 \begin {gather*} \frac {1}{54} \sqrt {3 x^2+5 x+2} \left (-24 x^2+122 x+645\right )+\frac {1147 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{54 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.40, size = 63, normalized size = 0.72 \begin {gather*} -\frac {1}{54} \, {\left (24 \, x^{2} - 122 \, x - 645\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {1147}{648} \, \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.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.23, size = 59, normalized size = 0.68 \begin {gather*} -\frac {1}{54} \, {\left (2 \, {\left (12 \, x - 61\right )} x - 645\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {1147}{324} \, \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.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 77, normalized size = 0.89 \begin {gather*} -\frac {4 \sqrt {3 x^{2}+5 x +2}\, x^{2}}{9}+\frac {61 \sqrt {3 x^{2}+5 x +2}\, x}{27}+\frac {1147 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )}{324}+\frac {215 \sqrt {3 x^{2}+5 x +2}}{18} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.24, size = 75, normalized size = 0.86 \begin {gather*} -\frac {4}{9} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x^{2} + \frac {61}{27} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {1147}{324} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac {215}{18} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {{\left (2\,x+3\right )}^2\,\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.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {51 x}{\sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac {8 x^{2}}{\sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac {4 x^{3}}{\sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {45}{\sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________