\(\int (5-x) (3+2 x)^{7/2} (2+5 x+3 x^2)^2 \, dx\) [811]

Optimal result
Mathematica [A] (verified)
Rubi [A] (verified)
Maple [A] (verified)
Fricas [A] (verification not implemented)
Sympy [B] (verification not implemented)
Maxima [A] (verification not implemented)
Giac [A] (verification not implemented)
Mupad [B] (verification not implemented)
Reduce [B] (verification not implemented)

Optimal result

Integrand size = 27, antiderivative size = 79 \[ \int (5-x) (3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^2 \, dx=\frac {325}{288} (3+2 x)^{9/2}-\frac {1065}{352} (3+2 x)^{11/2}+\frac {651}{208} (3+2 x)^{13/2}-\frac {359}{240} (3+2 x)^{15/2}+\frac {165}{544} (3+2 x)^{17/2}-\frac {9}{608} (3+2 x)^{19/2} \] Output:

325/288*(3+2*x)^(9/2)-1065/352*(3+2*x)^(11/2)+651/208*(3+2*x)^(13/2)-359/2 
40*(3+2*x)^(15/2)+165/544*(3+2*x)^(17/2)-9/608*(3+2*x)^(19/2)
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.48 \[ \int (5-x) (3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^2 \, dx=-\frac {(3+2 x)^{9/2} \left (-1670104-8846388 x-17037702 x^2-13495911 x^3-2702700 x^4+984555 x^5\right )}{2078505} \] Input:

Integrate[(5 - x)*(3 + 2*x)^(7/2)*(2 + 5*x + 3*x^2)^2,x]
 

Output:

-1/2078505*((3 + 2*x)^(9/2)*(-1670104 - 8846388*x - 17037702*x^2 - 1349591 
1*x^3 - 2702700*x^4 + 984555*x^5))
 

Rubi [A] (verified)

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.

\(\displaystyle \int (5-x) (2 x+3)^{7/2} \left (3 x^2+5 x+2\right )^2 \, dx\)

\(\Big \downarrow \) 1195

\(\displaystyle \int \left (-\frac {9}{32} (2 x+3)^{17/2}+\frac {165}{32} (2 x+3)^{15/2}-\frac {359}{16} (2 x+3)^{13/2}+\frac {651}{16} (2 x+3)^{11/2}-\frac {1065}{32} (2 x+3)^{9/2}+\frac {325}{32} (2 x+3)^{7/2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {9}{608} (2 x+3)^{19/2}+\frac {165}{544} (2 x+3)^{17/2}-\frac {359}{240} (2 x+3)^{15/2}+\frac {651}{208} (2 x+3)^{13/2}-\frac {1065}{352} (2 x+3)^{11/2}+\frac {325}{288} (2 x+3)^{9/2}\)

Input:

Int[(5 - x)*(3 + 2*x)^(7/2)*(2 + 5*x + 3*x^2)^2,x]
 

Output:

(325*(3 + 2*x)^(9/2))/288 - (1065*(3 + 2*x)^(11/2))/352 + (651*(3 + 2*x)^( 
13/2))/208 - (359*(3 + 2*x)^(15/2))/240 + (165*(3 + 2*x)^(17/2))/544 - (9* 
(3 + 2*x)^(19/2))/608
 

Defintions of rubi rules used

rule 1195
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]
 

rule 2009
Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 
Maple [A] (verified)

Time = 1.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.44

method result size
gosper \(-\frac {\left (984555 x^{5}-2702700 x^{4}-13495911 x^{3}-17037702 x^{2}-8846388 x -1670104\right ) \left (2 x +3\right )^{\frac {9}{2}}}{2078505}\) \(35\)
pseudoelliptic \(-\frac {\left (984555 x^{5}-2702700 x^{4}-13495911 x^{3}-17037702 x^{2}-8846388 x -1670104\right ) \left (2 x +3\right )^{\frac {9}{2}}}{2078505}\) \(35\)
trager \(\left (-\frac {144}{19} x^{9}-\frac {7968}{323} x^{8}+\frac {612424}{4845} x^{7}+\frac {19589192}{20995} x^{6}+\frac {577368069}{230945} x^{5}+\frac {1538030804}{415701} x^{4}+\frac {456297481}{138567} x^{3}+\frac {405735126}{230945} x^{2}+\frac {119699988}{230945} x +\frac {15030936}{230945}\right ) \sqrt {2 x +3}\) \(54\)
risch \(-\frac {\left (15752880 x^{9}+51274080 x^{8}-262729896 x^{7}-1939330008 x^{6}-5196312621 x^{5}-7690154020 x^{4}-6844462215 x^{3}-3651616134 x^{2}-1077299892 x -135278424\right ) \sqrt {2 x +3}}{2078505}\) \(55\)
derivativedivides \(\frac {325 \left (2 x +3\right )^{\frac {9}{2}}}{288}-\frac {1065 \left (2 x +3\right )^{\frac {11}{2}}}{352}+\frac {651 \left (2 x +3\right )^{\frac {13}{2}}}{208}-\frac {359 \left (2 x +3\right )^{\frac {15}{2}}}{240}+\frac {165 \left (2 x +3\right )^{\frac {17}{2}}}{544}-\frac {9 \left (2 x +3\right )^{\frac {19}{2}}}{608}\) \(56\)
default \(\frac {325 \left (2 x +3\right )^{\frac {9}{2}}}{288}-\frac {1065 \left (2 x +3\right )^{\frac {11}{2}}}{352}+\frac {651 \left (2 x +3\right )^{\frac {13}{2}}}{208}-\frac {359 \left (2 x +3\right )^{\frac {15}{2}}}{240}+\frac {165 \left (2 x +3\right )^{\frac {17}{2}}}{544}-\frac {9 \left (2 x +3\right )^{\frac {19}{2}}}{608}\) \(56\)
orering \(\frac {\left (984555 x^{5}-2702700 x^{4}-13495911 x^{3}-17037702 x^{2}-8846388 x -1670104\right ) \left (2 x +3\right )^{\frac {9}{2}} \left (5-x \right ) \left (3 x^{2}+5 x +2\right )^{2}}{2078505 \left (-5+x \right ) \left (3 x +2\right )^{2} \left (x +1\right )^{2}}\) \(69\)
meijerg \(\frac {10333575 \sqrt {3}\, \left (-\frac {4096 \sqrt {\pi }}{11486475}+\frac {16 \sqrt {\pi }\, \left (\frac {366080}{729} x^{8}+\frac {1903616}{729} x^{7}+\frac {1129216}{243} x^{6}+\frac {25856}{9} x^{5}+\frac {1120}{81} x^{4}-\frac {640}{27} x^{3}+\frac {128}{3} x^{2}-\frac {256}{3} x +256\right ) \sqrt {1+\frac {2 x}{3}}}{11486475}\right )}{512 \sqrt {\pi }}+\frac {25948755 \sqrt {3}\, \left (\frac {512 \sqrt {\pi }}{675675}-\frac {4 \sqrt {\pi }\, \left (-\frac {146432}{729} x^{7}-\frac {259072}{243} x^{6}-\frac {52736}{27} x^{5}-\frac {102400}{81} x^{4}-\frac {320}{27} x^{3}+\frac {64}{3} x^{2}-\frac {128}{3} x +128\right ) \sqrt {1+\frac {2 x}{3}}}{675675}\right )}{256 \sqrt {\pi }}+\frac {12629925 \sqrt {3}\, \left (-\frac {256 \sqrt {\pi }}{135135}+\frac {16 \sqrt {\pi }\, \left (\frac {1408}{81} x^{6}+\frac {2560}{27} x^{5}+\frac {14656}{81} x^{4}+\frac {3392}{27} x^{3}+\frac {8}{3} x^{2}-\frac {16}{3} x +16\right ) \sqrt {1+\frac {2 x}{3}}}{135135}\right )}{128 \sqrt {\pi }}+\frac {76545 \sqrt {3}\, \left (\frac {64 \sqrt {\pi }}{10395}-\frac {8 \sqrt {\pi }\, \left (-\frac {128}{27} x^{5}-\frac {2176}{81} x^{4}-\frac {1472}{27} x^{3}-\frac {128}{3} x^{2}-\frac {8}{3} x +8\right ) \sqrt {1+\frac {2 x}{3}}}{10395}\right )}{2 \sqrt {\pi }}+\frac {42525 \sqrt {3}\, \left (-\frac {32 \sqrt {\pi }}{945}+\frac {16 \sqrt {\pi }\, \left (\frac {32}{81} x^{4}+\frac {64}{27} x^{3}+\frac {16}{3} x^{2}+\frac {16}{3} x +2\right ) \sqrt {1+\frac {2 x}{3}}}{945}\right )}{8 \sqrt {\pi }}-\frac {18600435 \sqrt {3}\, \left (\frac {8192 \sqrt {\pi }}{43648605}-\frac {8 \sqrt {\pi }\, \left (-\frac {4978688}{2187} x^{9}-\frac {8493056}{729} x^{8}-\frac {14789632}{729} x^{7}-\frac {2951168}{243} x^{6}-\frac {896}{27} x^{5}+\frac {4480}{81} x^{4}-\frac {2560}{27} x^{3}+\frac {512}{3} x^{2}-\frac {1024}{3} x +1024\right ) \sqrt {1+\frac {2 x}{3}}}{43648605}\right )}{1024 \sqrt {\pi }}\) \(353\)

Input:

int((5-x)*(2*x+3)^(7/2)*(3*x^2+5*x+2)^2,x,method=_RETURNVERBOSE)
 

Output:

-1/2078505*(984555*x^5-2702700*x^4-13495911*x^3-17037702*x^2-8846388*x-167 
0104)*(2*x+3)^(9/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.68 \[ \int (5-x) (3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^2 \, dx=-\frac {1}{2078505} \, {\left (15752880 \, x^{9} + 51274080 \, x^{8} - 262729896 \, x^{7} - 1939330008 \, x^{6} - 5196312621 \, x^{5} - 7690154020 \, x^{4} - 6844462215 \, x^{3} - 3651616134 \, x^{2} - 1077299892 \, x - 135278424\right )} \sqrt {2 \, x + 3} \] Input:

integrate((5-x)*(3+2*x)^(7/2)*(3*x^2+5*x+2)^2,x, algorithm="fricas")
 

Output:

-1/2078505*(15752880*x^9 + 51274080*x^8 - 262729896*x^7 - 1939330008*x^6 - 
 5196312621*x^5 - 7690154020*x^4 - 6844462215*x^3 - 3651616134*x^2 - 10772 
99892*x - 135278424)*sqrt(2*x + 3)
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 146 vs. \(2 (70) = 140\).

Time = 0.52 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.85 \[ \int (5-x) (3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^2 \, dx=- \frac {144 x^{9} \sqrt {2 x + 3}}{19} - \frac {7968 x^{8} \sqrt {2 x + 3}}{323} + \frac {612424 x^{7} \sqrt {2 x + 3}}{4845} + \frac {19589192 x^{6} \sqrt {2 x + 3}}{20995} + \frac {577368069 x^{5} \sqrt {2 x + 3}}{230945} + \frac {1538030804 x^{4} \sqrt {2 x + 3}}{415701} + \frac {456297481 x^{3} \sqrt {2 x + 3}}{138567} + \frac {405735126 x^{2} \sqrt {2 x + 3}}{230945} + \frac {119699988 x \sqrt {2 x + 3}}{230945} + \frac {15030936 \sqrt {2 x + 3}}{230945} \] Input:

integrate((5-x)*(3+2*x)**(7/2)*(3*x**2+5*x+2)**2,x)
 

Output:

-144*x**9*sqrt(2*x + 3)/19 - 7968*x**8*sqrt(2*x + 3)/323 + 612424*x**7*sqr 
t(2*x + 3)/4845 + 19589192*x**6*sqrt(2*x + 3)/20995 + 577368069*x**5*sqrt( 
2*x + 3)/230945 + 1538030804*x**4*sqrt(2*x + 3)/415701 + 456297481*x**3*sq 
rt(2*x + 3)/138567 + 405735126*x**2*sqrt(2*x + 3)/230945 + 119699988*x*sqr 
t(2*x + 3)/230945 + 15030936*sqrt(2*x + 3)/230945
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.70 \[ \int (5-x) (3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^2 \, dx=-\frac {9}{608} \, {\left (2 \, x + 3\right )}^{\frac {19}{2}} + \frac {165}{544} \, {\left (2 \, x + 3\right )}^{\frac {17}{2}} - \frac {359}{240} \, {\left (2 \, x + 3\right )}^{\frac {15}{2}} + \frac {651}{208} \, {\left (2 \, x + 3\right )}^{\frac {13}{2}} - \frac {1065}{352} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} + \frac {325}{288} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} \] Input:

integrate((5-x)*(3+2*x)^(7/2)*(3*x^2+5*x+2)^2,x, algorithm="maxima")
 

Output:

-9/608*(2*x + 3)^(19/2) + 165/544*(2*x + 3)^(17/2) - 359/240*(2*x + 3)^(15 
/2) + 651/208*(2*x + 3)^(13/2) - 1065/352*(2*x + 3)^(11/2) + 325/288*(2*x 
+ 3)^(9/2)
 

Giac [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.70 \[ \int (5-x) (3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^2 \, dx=-\frac {9}{608} \, {\left (2 \, x + 3\right )}^{\frac {19}{2}} + \frac {165}{544} \, {\left (2 \, x + 3\right )}^{\frac {17}{2}} - \frac {359}{240} \, {\left (2 \, x + 3\right )}^{\frac {15}{2}} + \frac {651}{208} \, {\left (2 \, x + 3\right )}^{\frac {13}{2}} - \frac {1065}{352} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} + \frac {325}{288} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} \] Input:

integrate((5-x)*(3+2*x)^(7/2)*(3*x^2+5*x+2)^2,x, algorithm="giac")
 

Output:

-9/608*(2*x + 3)^(19/2) + 165/544*(2*x + 3)^(17/2) - 359/240*(2*x + 3)^(15 
/2) + 651/208*(2*x + 3)^(13/2) - 1065/352*(2*x + 3)^(11/2) + 325/288*(2*x 
+ 3)^(9/2)
 

Mupad [B] (verification not implemented)

Time = 12.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.70 \[ \int (5-x) (3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^2 \, dx=\frac {325\,{\left (2\,x+3\right )}^{9/2}}{288}-\frac {1065\,{\left (2\,x+3\right )}^{11/2}}{352}+\frac {651\,{\left (2\,x+3\right )}^{13/2}}{208}-\frac {359\,{\left (2\,x+3\right )}^{15/2}}{240}+\frac {165\,{\left (2\,x+3\right )}^{17/2}}{544}-\frac {9\,{\left (2\,x+3\right )}^{19/2}}{608} \] Input:

int(-(2*x + 3)^(7/2)*(x - 5)*(5*x + 3*x^2 + 2)^2,x)
 

Output:

(325*(2*x + 3)^(9/2))/288 - (1065*(2*x + 3)^(11/2))/352 + (651*(2*x + 3)^( 
13/2))/208 - (359*(2*x + 3)^(15/2))/240 + (165*(2*x + 3)^(17/2))/544 - (9* 
(2*x + 3)^(19/2))/608
 

Reduce [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.67 \[ \int (5-x) (3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^2 \, dx=\frac {\sqrt {2 x +3}\, \left (-15752880 x^{9}-51274080 x^{8}+262729896 x^{7}+1939330008 x^{6}+5196312621 x^{5}+7690154020 x^{4}+6844462215 x^{3}+3651616134 x^{2}+1077299892 x +135278424\right )}{2078505} \] Input:

int((5-x)*(3+2*x)^(7/2)*(3*x^2+5*x+2)^2,x)
 

Output:

(sqrt(2*x + 3)*( - 15752880*x**9 - 51274080*x**8 + 262729896*x**7 + 193933 
0008*x**6 + 5196312621*x**5 + 7690154020*x**4 + 6844462215*x**3 + 36516161 
34*x**2 + 1077299892*x + 135278424))/2078505