Optimal. Leaf size=112 \[ \frac {32}{27} (3+2 x)^2 \sqrt {2+5 x+3 x^2}-\frac {1}{12} (3+2 x)^3 \sqrt {2+5 x+3 x^2}+\frac {5}{648} (3261+1078 x) \sqrt {2+5 x+3 x^2}+\frac {19405 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{1296 \sqrt {3}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, 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 = {846, 793, 635,
212} \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.
[In]
[Out]
Rule 212
Rule 635
Rule 793
Rule 846
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} \text {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*}
________________________________________________________________________________________
Mathematica [A]
time = 0.28, size = 66, normalized size = 0.59 \begin {gather*} \frac {-3 \sqrt {2+5 x+3 x^2} \left (-21759-11690 x-1128 x^2+432 x^3\right )+19405 \sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{1944} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.08, size = 94, normalized size = 0.84
method | result | size |
risch | \(-\frac {\left (432 x^{3}-1128 x^{2}-11690 x -21759\right ) \sqrt {3 x^{2}+5 x +2}}{648}+\frac {19405 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{3888}\) | \(60\) |
trager | \(\left (-\frac {2}{3} x^{3}+\frac {47}{27} x^{2}+\frac {5845}{324} x +\frac {7253}{216}\right ) \sqrt {3 x^{2}+5 x +2}-\frac {19405 \RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (-6 \RootOf \left (\textit {\_Z}^{2}-3\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-3\right )+6 \sqrt {3 x^{2}+5 x +2}\right )}{3888}\) | \(71\) |
default | \(-\frac {2 x^{3} \sqrt {3 x^{2}+5 x +2}}{3}+\frac {47 x^{2} \sqrt {3 x^{2}+5 x +2}}{27}+\frac {5845 x \sqrt {3 x^{2}+5 x +2}}{324}+\frac {7253 \sqrt {3 x^{2}+5 x +2}}{216}+\frac {19405 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{3888}\) | \(94\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.52, 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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 3.32, 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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.82, 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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________