\(\int (5-x) (3+2 x)^{5/2} (2+5 x+3 x^2)^3 \, dx\) [820]

Optimal result

Integrand size = 27, antiderivative size = 105 \[ \int (5-x) (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^3 \, dx=\frac {1625}{896} (3+2 x)^{7/2}-\frac {7925 (3+2 x)^{9/2}}{1152}+\frac {1455}{128} (3+2 x)^{11/2}-\frac {17201 (3+2 x)^{13/2}}{1664}+\frac {2095}{384} (3+2 x)^{15/2}-\frac {207}{128} (3+2 x)^{17/2}+\frac {567 (3+2 x)^{19/2}}{2432}-\frac {9}{896} (3+2 x)^{21/2} \] Output:

1625/896*(3+2*x)^(7/2)-7925/1152*(3+2*x)^(9/2)+1455/128*(3+2*x)^(11/2)-172 
01/1664*(3+2*x)^(13/2)+2095/384*(3+2*x)^(15/2)-207/128*(3+2*x)^(17/2)+567/ 
2432*(3+2*x)^(19/2)-9/896*(3+2*x)^(21/2)
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.46 \[ \int (5-x) (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^3 \, dx=-\frac {(3+2 x)^{7/2} \left (-20346-160006 x-517293 x^2-871983 x^3-791700 x^4-339066 x^5-22113 x^6+20007 x^7\right )}{15561} \] Input:

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

Output:

-1/15561*((3 + 2*x)^(7/2)*(-20346 - 160006*x - 517293*x^2 - 871983*x^3 - 7 
91700*x^4 - 339066*x^5 - 22113*x^6 + 20007*x^7))
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 105, 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.

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

\(\Big \downarrow \) 1195

\(\displaystyle \int \left (-\frac {27}{128} (2 x+3)^{19/2}+\frac {567}{128} (2 x+3)^{17/2}-\frac {3519}{128} (2 x+3)^{15/2}+\frac {10475}{128} (2 x+3)^{13/2}-\frac {17201}{128} (2 x+3)^{11/2}+\frac {16005}{128} (2 x+3)^{9/2}-\frac {7925}{128} (2 x+3)^{7/2}+\frac {1625}{128} (2 x+3)^{5/2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {9}{896} (2 x+3)^{21/2}+\frac {567 (2 x+3)^{19/2}}{2432}-\frac {207}{128} (2 x+3)^{17/2}+\frac {2095}{384} (2 x+3)^{15/2}-\frac {17201 (2 x+3)^{13/2}}{1664}+\frac {1455}{128} (2 x+3)^{11/2}-\frac {7925 (2 x+3)^{9/2}}{1152}+\frac {1625}{896} (2 x+3)^{7/2}\)

Input:

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

Output:

(1625*(3 + 2*x)^(7/2))/896 - (7925*(3 + 2*x)^(9/2))/1152 + (1455*(3 + 2*x) 
^(11/2))/128 - (17201*(3 + 2*x)^(13/2))/1664 + (2095*(3 + 2*x)^(15/2))/384 
 - (207*(3 + 2*x)^(17/2))/128 + (567*(3 + 2*x)^(19/2))/2432 - (9*(3 + 2*x) 
^(21/2))/896
 

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 = 1.29 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.43

method	result	size
gosper	\(-\frac {\left (20007 x^{7}-22113 x^{6}-339066 x^{5}-791700 x^{4}-871983 x^{3}-517293 x^{2}-160006 x -20346\right ) \left (2 x +3\right )^{\frac {7}{2}}}{15561}\)	\(45\)
pseudoelliptic	\(-\frac {\left (20007 x^{7}-22113 x^{6}-339066 x^{5}-791700 x^{4}-871983 x^{3}-517293 x^{2}-160006 x -20346\right ) \left (2 x +3\right )^{\frac {7}{2}}}{15561}\)	\(45\)
trager	\(\left (-\frac {72}{7} x^{10}-\frac {4644}{133} x^{9}+\frac {20754}{133} x^{8}+\frac {492151}{399} x^{7}+\frac {863233}{247} x^{6}+\frac {1387878}{247} x^{5}+\frac {12623654}{2223} x^{4}+\frac {19133449}{5187} x^{3}+\frac {2593299}{1729} x^{2}+\frac {602094}{1729} x +\frac {61038}{1729}\right ) \sqrt {2 x +3}\)	\(59\)
risch	\(-\frac {\left (160056 x^{10}+543348 x^{9}-2428218 x^{8}-19193889 x^{7}-54383679 x^{6}-87436314 x^{5}-88365578 x^{4}-57400347 x^{3}-23339691 x^{2}-5418846 x -549342\right ) \sqrt {2 x +3}}{15561}\)	\(60\)
derivativedivides	\(\frac {1625 \left (2 x +3\right )^{\frac {7}{2}}}{896}-\frac {7925 \left (2 x +3\right )^{\frac {9}{2}}}{1152}+\frac {1455 \left (2 x +3\right )^{\frac {11}{2}}}{128}-\frac {17201 \left (2 x +3\right )^{\frac {13}{2}}}{1664}+\frac {2095 \left (2 x +3\right )^{\frac {15}{2}}}{384}-\frac {207 \left (2 x +3\right )^{\frac {17}{2}}}{128}+\frac {567 \left (2 x +3\right )^{\frac {19}{2}}}{2432}-\frac {9 \left (2 x +3\right )^{\frac {21}{2}}}{896}\)	\(74\)
default	\(\frac {1625 \left (2 x +3\right )^{\frac {7}{2}}}{896}-\frac {7925 \left (2 x +3\right )^{\frac {9}{2}}}{1152}+\frac {1455 \left (2 x +3\right )^{\frac {11}{2}}}{128}-\frac {17201 \left (2 x +3\right )^{\frac {13}{2}}}{1664}+\frac {2095 \left (2 x +3\right )^{\frac {15}{2}}}{384}-\frac {207 \left (2 x +3\right )^{\frac {17}{2}}}{128}+\frac {567 \left (2 x +3\right )^{\frac {19}{2}}}{2432}-\frac {9 \left (2 x +3\right )^{\frac {21}{2}}}{896}\)	\(74\)
orering	\(\frac {\left (20007 x^{7}-22113 x^{6}-339066 x^{5}-791700 x^{4}-871983 x^{3}-517293 x^{2}-160006 x -20346\right ) \left (2 x +3\right )^{\frac {7}{2}} \left (5-x \right ) \left (3 x^{2}+5 x +2\right )^{3}}{15561 \left (-5+x \right ) \left (x +1\right )^{3} \left (3 x +2\right )^{3}}\)	\(79\)
meijerg	\(-\frac {9743085 \sqrt {3}\, \left (-\frac {4096 \sqrt {\pi }}{2297295}+\frac {4 \sqrt {\pi }\, \left (-\frac {1025024}{729} x^{8}-\frac {3587584}{729} x^{7}-\frac {1084160}{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}}}{2297295}\right )}{128 \sqrt {\pi }}-\frac {17878725 \sqrt {3}\, \left (\frac {2048 \sqrt {\pi }}{675675}-\frac {8 \sqrt {\pi }\, \left (\frac {256256}{729} x^{7}+\frac {305536}{243} x^{6}+\frac {31808}{27} 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}}}{675675}\right )}{128 \sqrt {\pi }}-\frac {14641965 \sqrt {3}\, \left (-\frac {256 \sqrt {\pi }}{45045}+\frac {2 \sqrt {\pi }\, \left (-\frac {39424}{243} x^{6}-\frac {1792}{3} x^{5}-\frac {47488}{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}}}{45045}\right )}{128 \sqrt {\pi }}-\frac {1585575 \sqrt {3}\, \left (\frac {128 \sqrt {\pi }}{10395}-\frac {8 \sqrt {\pi }\, \left (\frac {448}{27} x^{5}+\frac {5152}{81} x^{4}+\frac {1808}{27} x^{3}+\frac {8}{3} x^{2}-\frac {16}{3} x +16\right ) \sqrt {1+\frac {2 x}{3}}}{10395}\right )}{32 \sqrt {\pi }}-\frac {88695 \sqrt {3}\, \left (-\frac {32 \sqrt {\pi }}{945}+\frac {4 \sqrt {\pi }\, \left (-\frac {448}{81} x^{4}-\frac {608}{27} x^{3}-\frac {80}{3} x^{2}-\frac {8}{3} x +8\right ) \sqrt {1+\frac {2 x}{3}}}{945}\right )}{8 \sqrt {\pi }}-\frac {2025 \sqrt {3}\, \left (\frac {16 \sqrt {\pi }}{105}-\frac {8 \sqrt {\pi }\, \left (\frac {16}{27} x^{3}+\frac {8}{3} x^{2}+4 x +2\right ) \sqrt {1+\frac {2 x}{3}}}{105}\right )}{2 \sqrt {\pi }}+\frac {23914845 \sqrt {3}\, \left (-\frac {32768 \sqrt {\pi }}{43648605}+\frac {\sqrt {\pi }\, \left (-\frac {756760576}{19683} x^{10}-\frac {856334336}{6561} x^{9}-\frac {27529216}{243} x^{8}-\frac {292864}{729} x^{7}+\frac {157696}{243} x^{6}-\frac {28672}{27} x^{5}+\frac {143360}{81} x^{4}-\frac {81920}{27} x^{3}+\frac {16384}{3} x^{2}-\frac {32768}{3} x +32768\right ) \sqrt {1+\frac {2 x}{3}}}{43648605}\right )}{2048 \sqrt {\pi }}\)	\(399\)

Input:

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

Output:

-1/15561*(20007*x^7-22113*x^6-339066*x^5-791700*x^4-871983*x^3-517293*x^2- 
160006*x-20346)*(2*x+3)^(7/2)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.56 \[ \int (5-x) (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^3 \, dx=-\frac {1}{15561} \, {\left (160056 \, x^{10} + 543348 \, x^{9} - 2428218 \, x^{8} - 19193889 \, x^{7} - 54383679 \, x^{6} - 87436314 \, x^{5} - 88365578 \, x^{4} - 57400347 \, x^{3} - 23339691 \, x^{2} - 5418846 \, x - 549342\right )} \sqrt {2 \, x + 3} \] Input:

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

Output:

-1/15561*(160056*x^10 + 543348*x^9 - 2428218*x^8 - 19193889*x^7 - 54383679 
*x^6 - 87436314*x^5 - 88365578*x^4 - 57400347*x^3 - 23339691*x^2 - 5418846 
*x - 549342)*sqrt(2*x + 3)
                                                                                    
                                                                                    
 

Sympy [A] (verification not implemented)

Time = 1.28 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int (5-x) (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^3 \, dx=- \frac {9 \left (2 x + 3\right )^{\frac {21}{2}}}{896} + \frac {567 \left (2 x + 3\right )^{\frac {19}{2}}}{2432} - \frac {207 \left (2 x + 3\right )^{\frac {17}{2}}}{128} + \frac {2095 \left (2 x + 3\right )^{\frac {15}{2}}}{384} - \frac {17201 \left (2 x + 3\right )^{\frac {13}{2}}}{1664} + \frac {1455 \left (2 x + 3\right )^{\frac {11}{2}}}{128} - \frac {7925 \left (2 x + 3\right )^{\frac {9}{2}}}{1152} + \frac {1625 \left (2 x + 3\right )^{\frac {7}{2}}}{896} \] Input:

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

Output:

-9*(2*x + 3)**(21/2)/896 + 567*(2*x + 3)**(19/2)/2432 - 207*(2*x + 3)**(17 
/2)/128 + 2095*(2*x + 3)**(15/2)/384 - 17201*(2*x + 3)**(13/2)/1664 + 1455 
*(2*x + 3)**(11/2)/128 - 7925*(2*x + 3)**(9/2)/1152 + 1625*(2*x + 3)**(7/2 
)/896
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.70 \[ \int (5-x) (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^3 \, dx=-\frac {9}{896} \, {\left (2 \, x + 3\right )}^{\frac {21}{2}} + \frac {567}{2432} \, {\left (2 \, x + 3\right )}^{\frac {19}{2}} - \frac {207}{128} \, {\left (2 \, x + 3\right )}^{\frac {17}{2}} + \frac {2095}{384} \, {\left (2 \, x + 3\right )}^{\frac {15}{2}} - \frac {17201}{1664} \, {\left (2 \, x + 3\right )}^{\frac {13}{2}} + \frac {1455}{128} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} - \frac {7925}{1152} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} + \frac {1625}{896} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} \] Input:

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

Output:

-9/896*(2*x + 3)^(21/2) + 567/2432*(2*x + 3)^(19/2) - 207/128*(2*x + 3)^(1 
7/2) + 2095/384*(2*x + 3)^(15/2) - 17201/1664*(2*x + 3)^(13/2) + 1455/128* 
(2*x + 3)^(11/2) - 7925/1152*(2*x + 3)^(9/2) + 1625/896*(2*x + 3)^(7/2)
 

Giac [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.70 \[ \int (5-x) (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^3 \, dx=-\frac {9}{896} \, {\left (2 \, x + 3\right )}^{\frac {21}{2}} + \frac {567}{2432} \, {\left (2 \, x + 3\right )}^{\frac {19}{2}} - \frac {207}{128} \, {\left (2 \, x + 3\right )}^{\frac {17}{2}} + \frac {2095}{384} \, {\left (2 \, x + 3\right )}^{\frac {15}{2}} - \frac {17201}{1664} \, {\left (2 \, x + 3\right )}^{\frac {13}{2}} + \frac {1455}{128} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} - \frac {7925}{1152} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} + \frac {1625}{896} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} \] Input:

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

Output:

-9/896*(2*x + 3)^(21/2) + 567/2432*(2*x + 3)^(19/2) - 207/128*(2*x + 3)^(1 
7/2) + 2095/384*(2*x + 3)^(15/2) - 17201/1664*(2*x + 3)^(13/2) + 1455/128* 
(2*x + 3)^(11/2) - 7925/1152*(2*x + 3)^(9/2) + 1625/896*(2*x + 3)^(7/2)
 

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.70 \[ \int (5-x) (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^3 \, dx=\frac {1625\,{\left (2\,x+3\right )}^{7/2}}{896}-\frac {7925\,{\left (2\,x+3\right )}^{9/2}}{1152}+\frac {1455\,{\left (2\,x+3\right )}^{11/2}}{128}-\frac {17201\,{\left (2\,x+3\right )}^{13/2}}{1664}+\frac {2095\,{\left (2\,x+3\right )}^{15/2}}{384}-\frac {207\,{\left (2\,x+3\right )}^{17/2}}{128}+\frac {567\,{\left (2\,x+3\right )}^{19/2}}{2432}-\frac {9\,{\left (2\,x+3\right )}^{21/2}}{896} \] Input:

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

Output:

(1625*(2*x + 3)^(7/2))/896 - (7925*(2*x + 3)^(9/2))/1152 + (1455*(2*x + 3) 
^(11/2))/128 - (17201*(2*x + 3)^(13/2))/1664 + (2095*(2*x + 3)^(15/2))/384 
 - (207*(2*x + 3)^(17/2))/128 + (567*(2*x + 3)^(19/2))/2432 - (9*(2*x + 3) 
^(21/2))/896
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.55 \[ \int (5-x) (3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^3 \, dx=\frac {\sqrt {2 x +3}\, \left (-160056 x^{10}-543348 x^{9}+2428218 x^{8}[phone] x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}+5418846 x +549342\right )}{15561} \] Input:

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

Output:

(sqrt(2*x + 3)*( - 160056*x**10 - 543348*x**9 + 2428218*x**8 + 19193889*x* 
*7 + 54383679*x**6 + 87436314*x**5 + 88365578*x**4 + 57400347*x**3 + 23339 
691*x**2 + 5418846*x + 549342))/15561