Optimal. Leaf size=108 \[ \frac {1}{270} (161-30 x) \left (3 x^2+5 x+2\right )^{5/2}+\frac {839 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{2592}-\frac {839 (6 x+5) \sqrt {3 x^2+5 x+2}}{20736}+\frac {839 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{41472 \sqrt {3}} \]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {779, 612, 621, 206} \begin {gather*} \frac {1}{270} (161-30 x) \left (3 x^2+5 x+2\right )^{5/2}+\frac {839 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{2592}-\frac {839 (6 x+5) \sqrt {3 x^2+5 x+2}}{20736}+\frac {839 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{41472 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 612
Rule 621
Rule 779
Rubi steps
\begin {align*} \int (5-x) (3+2 x) \left (2+5 x+3 x^2\right )^{3/2} \, dx &=\frac {1}{270} (161-30 x) \left (2+5 x+3 x^2\right )^{5/2}+\frac {839}{108} \int \left (2+5 x+3 x^2\right )^{3/2} \, dx\\ &=\frac {839 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{2592}+\frac {1}{270} (161-30 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac {839 \int \sqrt {2+5 x+3 x^2} \, dx}{1728}\\ &=-\frac {839 (5+6 x) \sqrt {2+5 x+3 x^2}}{20736}+\frac {839 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{2592}+\frac {1}{270} (161-30 x) \left (2+5 x+3 x^2\right )^{5/2}+\frac {839 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{41472}\\ &=-\frac {839 (5+6 x) \sqrt {2+5 x+3 x^2}}{20736}+\frac {839 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{2592}+\frac {1}{270} (161-30 x) \left (2+5 x+3 x^2\right )^{5/2}+\frac {839 \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{20736}\\ &=-\frac {839 (5+6 x) \sqrt {2+5 x+3 x^2}}{20736}+\frac {839 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{2592}+\frac {1}{270} (161-30 x) \left (2+5 x+3 x^2\right )^{5/2}+\frac {839 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{41472 \sqrt {3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 77, normalized size = 0.71 \begin {gather*} \frac {4195 \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 (103680 x^5-210816 x^4-2032560 x^3-3567288 x^2-2406950 x-561921\right )}{622080} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.61, size = 79, normalized size = 0.73 \begin {gather*} \frac {839 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{20736 \sqrt {3}}+\frac {\sqrt {3 x^2+5 x+2} \left (-103680 x^5+210816 x^4+2032560 x^3+3567288 x^2+2406950 x+561921\right )}{103680} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.40, size = 78, normalized size = 0.72 \begin {gather*} -\frac {1}{103680} \, {\left (103680 \, x^{5} - 210816 \, x^{4} - 2032560 \, x^{3} - 3567288 \, x^{2} - 2406950 \, x - 561921\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {839}{248832} \, \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 = 0.24, size = 74, normalized size = 0.69 \begin {gather*} -\frac {1}{103680} \, {\left (2 \, {\left (12 \, {\left (18 \, {\left (8 \, {\left (30 \, x - 61\right )} x - 4705\right )} x - 148637\right )} x - 1203475\right )} x - 561921\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {839}{124416} \, \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 = 0.05, size = 98, normalized size = 0.91 \begin {gather*} -\frac {\left (3 x^{2}+5 x +2\right )^{\frac {5}{2}} x}{9}+\frac {839 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )}{124416}+\frac {161 \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{270}+\frac {839 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{2592}-\frac {839 \left (6 x +5\right ) \sqrt {3 x^{2}+5 x +2}}{20736} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.35, size = 116, normalized size = 1.07 \begin {gather*} -\frac {1}{9} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x + \frac {161}{270} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} + \frac {839}{432} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {4195}{2592} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {839}{3456} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {839}{124416} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac {4195}{20736} \, \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 \left (2\,x+3\right )\,\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2} \,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 (- 89 x \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 76 x^{2} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 11 x^{3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 6 x^{4} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int \left (- 30 \sqrt {3 x^{2} + 5 x + 2}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________