Integrand size = 27, antiderivative size = 79 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^2}{\sqrt {3+2 x}} \, dx=\frac {325}{32} \sqrt {3+2 x}-\frac {355}{32} (3+2 x)^{3/2}+\frac {651}{80} (3+2 x)^{5/2}-\frac {359}{112} (3+2 x)^{7/2}+\frac {55}{96} (3+2 x)^{9/2}-\frac {9}{352} (3+2 x)^{11/2} \] Output:
325/32*(3+2*x)^(1/2)-355/32*(3+2*x)^(3/2)+651/80*(3+2*x)^(5/2)-359/112*(3+ 2*x)^(7/2)+55/96*(3+2*x)^(9/2)-9/352*(3+2*x)^(11/2)
Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.48 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^2}{\sqrt {3+2 x}} \, dx=-\frac {\sqrt {3+2 x} \left (-4344-6252 x-15354 x^2-12645 x^3-3500 x^4+945 x^5\right )}{1155} \] Input:
Integrate[((5 - x)*(2 + 5*x + 3*x^2)^2)/Sqrt[3 + 2*x],x]
Output:
-1/1155*(Sqrt[3 + 2*x]*(-4344 - 6252*x - 15354*x^2 - 12645*x^3 - 3500*x^4 + 945*x^5))
Time = 0.33 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1195, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(5-x) \left (3 x^2+5 x+2\right )^2}{\sqrt {2 x+3}} \, dx\) |
\(\Big \downarrow \) 1195 |
\(\displaystyle \int \left (-\frac {9}{32} (2 x+3)^{9/2}+\frac {165}{32} (2 x+3)^{7/2}-\frac {359}{16} (2 x+3)^{5/2}+\frac {651}{16} (2 x+3)^{3/2}-\frac {1065}{32} \sqrt {2 x+3}+\frac {325}{32 \sqrt {2 x+3}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {9}{352} (2 x+3)^{11/2}+\frac {55}{96} (2 x+3)^{9/2}-\frac {359}{112} (2 x+3)^{7/2}+\frac {651}{80} (2 x+3)^{5/2}-\frac {355}{32} (2 x+3)^{3/2}+\frac {325}{32} \sqrt {2 x+3}\) |
Input:
Int[((5 - x)*(2 + 5*x + 3*x^2)^2)/Sqrt[3 + 2*x],x]
Output:
(325*Sqrt[3 + 2*x])/32 - (355*(3 + 2*x)^(3/2))/32 + (651*(3 + 2*x)^(5/2))/ 80 - (359*(3 + 2*x)^(7/2))/112 + (55*(3 + 2*x)^(9/2))/96 - (9*(3 + 2*x)^(1 1/2))/352
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Time = 1.29 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.43
method | result | size |
trager | \(\left (-\frac {9}{11} x^{5}+\frac {100}{33} x^{4}+\frac {843}{77} x^{3}+\frac {5118}{385} x^{2}+\frac {2084}{385} x +\frac {1448}{385}\right ) \sqrt {2 x +3}\) | \(34\) |
gosper | \(-\frac {\left (945 x^{5}-3500 x^{4}-12645 x^{3}-15354 x^{2}-6252 x -4344\right ) \sqrt {2 x +3}}{1155}\) | \(35\) |
risch | \(-\frac {\left (945 x^{5}-3500 x^{4}-12645 x^{3}-15354 x^{2}-6252 x -4344\right ) \sqrt {2 x +3}}{1155}\) | \(35\) |
pseudoelliptic | \(-\frac {\left (945 x^{5}-3500 x^{4}-12645 x^{3}-15354 x^{2}-6252 x -4344\right ) \sqrt {2 x +3}}{1155}\) | \(35\) |
derivativedivides | \(\frac {325 \sqrt {2 x +3}}{32}-\frac {355 \left (2 x +3\right )^{\frac {3}{2}}}{32}+\frac {651 \left (2 x +3\right )^{\frac {5}{2}}}{80}-\frac {359 \left (2 x +3\right )^{\frac {7}{2}}}{112}+\frac {55 \left (2 x +3\right )^{\frac {9}{2}}}{96}-\frac {9 \left (2 x +3\right )^{\frac {11}{2}}}{352}\) | \(56\) |
default | \(\frac {325 \sqrt {2 x +3}}{32}-\frac {355 \left (2 x +3\right )^{\frac {3}{2}}}{32}+\frac {651 \left (2 x +3\right )^{\frac {5}{2}}}{80}-\frac {359 \left (2 x +3\right )^{\frac {7}{2}}}{112}+\frac {55 \left (2 x +3\right )^{\frac {9}{2}}}{96}-\frac {9 \left (2 x +3\right )^{\frac {11}{2}}}{352}\) | \(56\) |
orering | \(\frac {\left (945 x^{5}-3500 x^{4}-12645 x^{3}-15354 x^{2}-6252 x -4344\right ) \sqrt {2 x +3}\, \left (5-x \right ) \left (3 x^{2}+5 x +2\right )^{2}}{1155 \left (-5+x \right ) \left (3 x +2\right )^{2} \left (x +1\right )^{2}}\) | \(69\) |
meijerg | \(\frac {10 \sqrt {3}\, \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {1+\frac {2 x}{3}}\right )}{\sqrt {\pi }}+\frac {1485 \sqrt {3}\, \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (\frac {8}{3} x^{2}-\frac {16}{3} x +16\right ) \sqrt {1+\frac {2 x}{3}}}{15}\right )}{8 \sqrt {\pi }}+\frac {72 \sqrt {3}\, \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-\frac {8 x}{3}+8\right ) \sqrt {1+\frac {2 x}{3}}}{6}\right )}{\sqrt {\pi }}+\frac {1215 \sqrt {3}\, \left (-\frac {256 \sqrt {\pi }}{315}+\frac {\sqrt {\pi }\, \left (\frac {1120}{81} x^{4}-\frac {640}{27} x^{3}+\frac {128}{3} x^{2}-\frac {256}{3} x +256\right ) \sqrt {1+\frac {2 x}{3}}}{315}\right )}{32 \sqrt {\pi }}+\frac {3051 \sqrt {3}\, \left (\frac {32 \sqrt {\pi }}{35}-\frac {\sqrt {\pi }\, \left (-\frac {320}{27} x^{3}+\frac {64}{3} x^{2}-\frac {128}{3} x +128\right ) \sqrt {1+\frac {2 x}{3}}}{140}\right )}{16 \sqrt {\pi }}-\frac {2187 \sqrt {3}\, \left (\frac {512 \sqrt {\pi }}{693}-\frac {\sqrt {\pi }\, \left (-\frac {896}{27} x^{5}+\frac {4480}{81} x^{4}-\frac {2560}{27} x^{3}+\frac {512}{3} x^{2}-\frac {1024}{3} x +1024\right ) \sqrt {1+\frac {2 x}{3}}}{1386}\right )}{64 \sqrt {\pi }}\) | \(233\) |
Input:
int((5-x)*(3*x^2+5*x+2)^2/(2*x+3)^(1/2),x,method=_RETURNVERBOSE)
Output:
(-9/11*x^5+100/33*x^4+843/77*x^3+5118/385*x^2+2084/385*x+1448/385)*(2*x+3) ^(1/2)
Time = 0.07 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.43 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^2}{\sqrt {3+2 x}} \, dx=-\frac {1}{1155} \, {\left (945 \, x^{5} - 3500 \, x^{4} - 12645 \, x^{3} - 15354 \, x^{2} - 6252 \, x - 4344\right )} \sqrt {2 \, x + 3} \] Input:
integrate((5-x)*(3*x^2+5*x+2)^2/(3+2*x)^(1/2),x, algorithm="fricas")
Output:
-1/1155*(945*x^5 - 3500*x^4 - 12645*x^3 - 15354*x^2 - 6252*x - 4344)*sqrt( 2*x + 3)
Time = 1.02 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.89 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^2}{\sqrt {3+2 x}} \, dx=- \frac {9 \left (2 x + 3\right )^{\frac {11}{2}}}{352} + \frac {55 \left (2 x + 3\right )^{\frac {9}{2}}}{96} - \frac {359 \left (2 x + 3\right )^{\frac {7}{2}}}{112} + \frac {651 \left (2 x + 3\right )^{\frac {5}{2}}}{80} - \frac {355 \left (2 x + 3\right )^{\frac {3}{2}}}{32} + \frac {325 \sqrt {2 x + 3}}{32} \] Input:
integrate((5-x)*(3*x**2+5*x+2)**2/(3+2*x)**(1/2),x)
Output:
-9*(2*x + 3)**(11/2)/352 + 55*(2*x + 3)**(9/2)/96 - 359*(2*x + 3)**(7/2)/1 12 + 651*(2*x + 3)**(5/2)/80 - 355*(2*x + 3)**(3/2)/32 + 325*sqrt(2*x + 3) /32
Time = 0.04 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.70 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^2}{\sqrt {3+2 x}} \, dx=-\frac {9}{352} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} + \frac {55}{96} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} - \frac {359}{112} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} + \frac {651}{80} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} - \frac {355}{32} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} + \frac {325}{32} \, \sqrt {2 \, x + 3} \] Input:
integrate((5-x)*(3*x^2+5*x+2)^2/(3+2*x)^(1/2),x, algorithm="maxima")
Output:
-9/352*(2*x + 3)^(11/2) + 55/96*(2*x + 3)^(9/2) - 359/112*(2*x + 3)^(7/2) + 651/80*(2*x + 3)^(5/2) - 355/32*(2*x + 3)^(3/2) + 325/32*sqrt(2*x + 3)
Time = 0.20 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.70 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^2}{\sqrt {3+2 x}} \, dx=-\frac {9}{352} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} + \frac {55}{96} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} - \frac {359}{112} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} + \frac {651}{80} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} - \frac {355}{32} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} + \frac {325}{32} \, \sqrt {2 \, x + 3} \] Input:
integrate((5-x)*(3*x^2+5*x+2)^2/(3+2*x)^(1/2),x, algorithm="giac")
Output:
-9/352*(2*x + 3)^(11/2) + 55/96*(2*x + 3)^(9/2) - 359/112*(2*x + 3)^(7/2) + 651/80*(2*x + 3)^(5/2) - 355/32*(2*x + 3)^(3/2) + 325/32*sqrt(2*x + 3)
Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.70 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^2}{\sqrt {3+2 x}} \, dx=\frac {325\,\sqrt {2\,x+3}}{32}-\frac {355\,{\left (2\,x+3\right )}^{3/2}}{32}+\frac {651\,{\left (2\,x+3\right )}^{5/2}}{80}-\frac {359\,{\left (2\,x+3\right )}^{7/2}}{112}+\frac {55\,{\left (2\,x+3\right )}^{9/2}}{96}-\frac {9\,{\left (2\,x+3\right )}^{11/2}}{352} \] Input:
int(-((x - 5)*(5*x + 3*x^2 + 2)^2)/(2*x + 3)^(1/2),x)
Output:
(325*(2*x + 3)^(1/2))/32 - (355*(2*x + 3)^(3/2))/32 + (651*(2*x + 3)^(5/2) )/80 - (359*(2*x + 3)^(7/2))/112 + (55*(2*x + 3)^(9/2))/96 - (9*(2*x + 3)^ (11/2))/352
Time = 0.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.42 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^2}{\sqrt {3+2 x}} \, dx=\frac {\sqrt {2 x +3}\, \left (-945 x^{5}+3500 x^{4}+12645 x^{3}+15354 x^{2}+6252 x +4344\right )}{1155} \] Input:
int((5-x)*(3*x^2+5*x+2)^2/(3+2*x)^(1/2),x)
Output:
(sqrt(2*x + 3)*( - 945*x**5 + 3500*x**4 + 12645*x**3 + 15354*x**2 + 6252*x + 4344))/1155