Optimal. Leaf size=204 \[ -\frac {1}{36} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{9/2}+\frac {34}{99} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}+\frac {(390798 x+863825) \left (3 x^2+5 x+2\right )^{9/2}}{320760}+\frac {91087 (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}}{311040}-\frac {637609 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{22394880}+\frac {637609 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{214990848}-\frac {637609 (6 x+5) \sqrt {3 x^2+5 x+2}}{1719926784}+\frac {637609 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{3439853568 \sqrt {3}} \]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {832, 779, 612, 621, 206} \begin {gather*} -\frac {1}{36} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{9/2}+\frac {34}{99} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}+\frac {(390798 x+863825) \left (3 x^2+5 x+2\right )^{9/2}}{320760}+\frac {91087 (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}}{311040}-\frac {637609 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{22394880}+\frac {637609 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{214990848}-\frac {637609 (6 x+5) \sqrt {3 x^2+5 x+2}}{1719926784}+\frac {637609 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{3439853568 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 612
Rule 621
Rule 779
Rule 832
Rubi steps
\begin {align*} \int (5-x) (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2} \, dx &=-\frac {1}{36} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{9/2}+\frac {1}{36} \int (3+2 x)^2 \left (\frac {1239}{2}+408 x\right ) \left (2+5 x+3 x^2\right )^{7/2} \, dx\\ &=\frac {34}{99} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}-\frac {1}{36} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{9/2}+\frac {\int (3+2 x) \left (\frac {61053}{2}+21711 x\right ) \left (2+5 x+3 x^2\right )^{7/2} \, dx}{1188}\\ &=\frac {34}{99} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}-\frac {1}{36} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{9/2}+\frac {(863825+390798 x) \left (2+5 x+3 x^2\right )^{9/2}}{320760}+\frac {91087 \int \left (2+5 x+3 x^2\right )^{7/2} \, dx}{6480}\\ &=\frac {91087 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{311040}+\frac {34}{99} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}-\frac {1}{36} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{9/2}+\frac {(863825+390798 x) \left (2+5 x+3 x^2\right )^{9/2}}{320760}-\frac {637609 \int \left (2+5 x+3 x^2\right )^{5/2} \, dx}{622080}\\ &=-\frac {637609 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{22394880}+\frac {91087 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{311040}+\frac {34}{99} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}-\frac {1}{36} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{9/2}+\frac {(863825+390798 x) \left (2+5 x+3 x^2\right )^{9/2}}{320760}+\frac {637609 \int \left (2+5 x+3 x^2\right )^{3/2} \, dx}{8957952}\\ &=\frac {637609 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{214990848}-\frac {637609 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{22394880}+\frac {91087 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{311040}+\frac {34}{99} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}-\frac {1}{36} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{9/2}+\frac {(863825+390798 x) \left (2+5 x+3 x^2\right )^{9/2}}{320760}-\frac {637609 \int \sqrt {2+5 x+3 x^2} \, dx}{143327232}\\ &=-\frac {637609 (5+6 x) \sqrt {2+5 x+3 x^2}}{1719926784}+\frac {637609 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{214990848}-\frac {637609 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{22394880}+\frac {91087 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{311040}+\frac {34}{99} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}-\frac {1}{36} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{9/2}+\frac {(863825+390798 x) \left (2+5 x+3 x^2\right )^{9/2}}{320760}+\frac {637609 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{3439853568}\\ &=-\frac {637609 (5+6 x) \sqrt {2+5 x+3 x^2}}{1719926784}+\frac {637609 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{214990848}-\frac {637609 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{22394880}+\frac {91087 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{311040}+\frac {34}{99} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}-\frac {1}{36} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{9/2}+\frac {(863825+390798 x) \left (2+5 x+3 x^2\right )^{9/2}}{320760}+\frac {637609 \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{1719926784}\\ &=-\frac {637609 (5+6 x) \sqrt {2+5 x+3 x^2}}{1719926784}+\frac {637609 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{214990848}-\frac {637609 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{22394880}+\frac {91087 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{311040}+\frac {34}{99} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}-\frac {1}{36} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{9/2}+\frac {(863825+390798 x) \left (2+5 x+3 x^2\right )^{9/2}}{320760}+\frac {637609 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{3439853568 \sqrt {3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.15, size = 163, normalized size = 0.80 \begin {gather*} \frac {1}{36} \left (-(2 x+3)^3 \left (3 x^2+5 x+2\right )^{9/2}+\frac {136}{11} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}+\frac {(390798 x+863825) \left (3 x^2+5 x+2\right )^{9/2}}{8910}+\frac {91087 \left (35 \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 (4478976 x^7+26127360 x^6+64800000 x^5+88560000 x^4+72023472 x^3+34858680 x^2+9298342 x+1054785\right )\right )}{1433272320}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 1.17, size = 109, normalized size = 0.53 \begin {gather*} \frac {637609 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{1719926784 \sqrt {3}}+\frac {\sqrt {3 x^2+5 x+2} \left (-1702727516160 x^{11}-8487838679040 x^{10}+15591566278656 x^9+235832896880640 x^8+866110416795648 x^7+1766184385305600 x^6+2298912734198016 x^5+1992318117275520 x^4+1149328734822000 x^3+425035984788120 x^2+91318722047870 x+8675936123685\right )}{94595973120} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 108, normalized size = 0.53 \begin {gather*} -\frac {1}{94595973120} \, {\left (1702727516160 \, x^{11} + 8487838679040 \, x^{10} - 15591566278656 \, x^{9} - 235832896880640 \, x^{8} - 866110416795648 \, x^{7} - 1766184385305600 \, x^{6} - 2298912734198016 \, x^{5} - 1992318117275520 \, x^{4} - 1149328734822000 \, x^{3} - 425035984788120 \, x^{2} - 91318722047870 \, x - 8675936123685\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {637609}{20639121408} \, \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 = 0.33, size = 104, normalized size = 0.51 \begin {gather*} -\frac {1}{94595973120} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (8 \, {\left (6 \, {\left (36 \, {\left (2 \, {\left (48 \, {\left (54 \, {\left (20 \, {\left (66 \, x + 329\right )} x - 12087\right )} x - 9872495\right )} x - 1740351757\right )} x - 7097898925\right )} x - 332597328443\right )} x - 1729442810135\right )} x - 7981449547375\right )} x - 17709832699505\right )} x - 45659361023935\right )} x - 8675936123685\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {637609}{10319560704} \, \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 = 170, normalized size = 0.83 \begin {gather*} -\frac {2 \left (3 x^{2}+5 x +2\right )^{\frac {9}{2}} x^{3}}{9}+\frac {37 \left (3 x^{2}+5 x +2\right )^{\frac {9}{2}} x^{2}}{99}+\frac {22807 \left (3 x^{2}+5 x +2\right )^{\frac {9}{2}} x}{5940}+\frac {637609 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )}{10319560704}+\frac {91087 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{311040}-\frac {637609 \left (6 x +5\right ) \sqrt {3 x^{2}+5 x +2}}{1719926784}+\frac {637609 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{214990848}-\frac {637609 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{22394880}+\frac {322939 \left (3 x^{2}+5 x +2\right )^{\frac {9}{2}}}{64152} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.26, size = 208, normalized size = 1.02 \begin {gather*} -\frac {2}{9} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}} x^{3} + \frac {37}{99} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}} x^{2} + \frac {22807}{5940} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}} x + \frac {322939}{64152} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}} + \frac {91087}{51840} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x + \frac {91087}{62208} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} - \frac {637609}{3732480} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x - \frac {637609}{4478976} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} + \frac {637609}{35831808} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {3188045}{214990848} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {637609}{286654464} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {637609}{10319560704} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac {3188045}{1719926784} \, \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 = -0.00 \begin {gather*} -\int {\left (2\,x+3\right )}^3\,\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{7/2} \,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 (- 10044 x \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 40698 x^{2} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 93965 x^{3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 135392 x^{4} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 124716 x^{5} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 71336 x^{6} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 22247 x^{7} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 1710 x^{8} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 972 x^{9} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int 216 x^{10} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int \left (- 1080 \sqrt {3 x^{2} + 5 x + 2}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________