\(\int (3-x+2 x^2)^{5/2} (2+3 x+5 x^2)^2 \, dx\) [108]

Optimal result

Integrand size = 27, antiderivative size = 170 \[ \int \left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )^2 \, dx=-\frac {4091815 (1-4 x) \sqrt {3-x+2 x^2}}{16777216}-\frac {177905 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{3145728}-\frac {1547 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{98304}+\frac {23225 \left (3-x+2 x^2\right )^{7/2}}{43008}+\frac {8467 x \left (3-x+2 x^2\right )^{7/2}}{4608}+\frac {305}{144} x^2 \left (3-x+2 x^2\right )^{7/2}+\frac {5}{4} x^3 \left (3-x+2 x^2\right )^{7/2}-\frac {94111745 \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )}{33554432 \sqrt {2}} \] Output:

-4091815/16777216*(1-4*x)*(2*x^2-x+3)^(1/2)-177905/3145728*(1-4*x)*(2*x^2- 
x+3)^(3/2)-1547/98304*(1-4*x)*(2*x^2-x+3)^(5/2)+23225/43008*(2*x^2-x+3)^(7 
/2)+8467/4608*x*(2*x^2-x+3)^(7/2)+305/144*x^2*(2*x^2-x+3)^(7/2)+5/4*x^3*(2 
*x^2-x+3)^(7/2)-94111745/67108864*arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)
 

Mathematica [A] (verified)

Time = 1.14 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.56 \[ \int \left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )^2 \, dx=\frac {4 \sqrt {3-x+2 x^2} \left (14824182519+39533249652 x+42992644128 x^2+77872272000 x^3+57147467776 x^4+75389820928 x^5+26401898496 x^6+44163137536 x^7+2055208960 x^8+10569646080 x^9\right )-5929039935 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{4227858432} \] Input:

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

Output:

(4*Sqrt[3 - x + 2*x^2]*(14824182519 + 39533249652*x + 42992644128*x^2 + 77 
872272000*x^3 + 57147467776*x^4 + 75389820928*x^5 + 26401898496*x^6 + 4416 
3137536*x^7 + 2055208960*x^8 + 10569646080*x^9) - 5929039935*Sqrt[2]*Log[1 
 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]])/4227858432
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.18, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {2192, 27, 2192, 27, 2192, 27, 1160, 1087, 1087, 1087, 1090, 222}

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.

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

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{20} \int \frac {5}{2} \left (2 x^2-x+3\right )^{5/2} \left (305 x^3+142 x^2+96 x+32\right )dx+\frac {5}{4} x^3 \left (2 x^2-x+3\right )^{7/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} \int \left (2 x^2-x+3\right )^{5/2} \left (305 x^3+142 x^2+96 x+32\right )dx+\frac {5}{4} x^3 \left (2 x^2-x+3\right )^{7/2}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{8} \left (\frac {1}{18} \int \frac {1}{2} \left (2 x^2-x+3\right )^{5/2} \left (8467 x^2-204 x+1152\right )dx+\frac {305}{18} x^2 \left (2 x^2-x+3\right )^{7/2}\right )+\frac {5}{4} x^3 \left (2 x^2-x+3\right )^{7/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} \left (\frac {1}{36} \int \left (2 x^2-x+3\right )^{5/2} \left (8467 x^2-204 x+1152\right )dx+\frac {305}{18} x^2 \left (2 x^2-x+3\right )^{7/2}\right )+\frac {5}{4} x^3 \left (2 x^2-x+3\right )^{7/2}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{8} \left (\frac {1}{36} \left (\frac {1}{16} \int -\frac {3}{2} (4646-23225 x) \left (2 x^2-x+3\right )^{5/2}dx+\frac {8467}{16} x \left (2 x^2-x+3\right )^{7/2}\right )+\frac {305}{18} x^2 \left (2 x^2-x+3\right )^{7/2}\right )+\frac {5}{4} x^3 \left (2 x^2-x+3\right )^{7/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} \left (\frac {1}{36} \left (\frac {8467}{16} x \left (2 x^2-x+3\right )^{7/2}-\frac {3}{32} \int (4646-23225 x) \left (2 x^2-x+3\right )^{5/2}dx\right )+\frac {305}{18} x^2 \left (2 x^2-x+3\right )^{7/2}\right )+\frac {5}{4} x^3 \left (2 x^2-x+3\right )^{7/2}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {1}{8} \left (\frac {1}{36} \left (\frac {8467}{16} x \left (2 x^2-x+3\right )^{7/2}-\frac {3}{32} \left (-\frac {4641}{4} \int \left (2 x^2-x+3\right )^{5/2}dx-\frac {23225}{14} \left (2 x^2-x+3\right )^{7/2}\right )\right )+\frac {305}{18} x^2 \left (2 x^2-x+3\right )^{7/2}\right )+\frac {5}{4} x^3 \left (2 x^2-x+3\right )^{7/2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{8} \left (\frac {1}{36} \left (\frac {8467}{16} x \left (2 x^2-x+3\right )^{7/2}-\frac {3}{32} \left (-\frac {4641}{4} \left (\frac {115}{48} \int \left (2 x^2-x+3\right )^{3/2}dx-\frac {1}{24} (1-4 x) \left (2 x^2-x+3\right )^{5/2}\right )-\frac {23225}{14} \left (2 x^2-x+3\right )^{7/2}\right )\right )+\frac {305}{18} x^2 \left (2 x^2-x+3\right )^{7/2}\right )+\frac {5}{4} x^3 \left (2 x^2-x+3\right )^{7/2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{8} \left (\frac {1}{36} \left (\frac {8467}{16} x \left (2 x^2-x+3\right )^{7/2}-\frac {3}{32} \left (-\frac {4641}{4} \left (\frac {115}{48} \left (\frac {69}{32} \int \sqrt {2 x^2-x+3}dx-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )-\frac {1}{24} (1-4 x) \left (2 x^2-x+3\right )^{5/2}\right )-\frac {23225}{14} \left (2 x^2-x+3\right )^{7/2}\right )\right )+\frac {305}{18} x^2 \left (2 x^2-x+3\right )^{7/2}\right )+\frac {5}{4} x^3 \left (2 x^2-x+3\right )^{7/2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{8} \left (\frac {1}{36} \left (\frac {8467}{16} x \left (2 x^2-x+3\right )^{7/2}-\frac {3}{32} \left (-\frac {4641}{4} \left (\frac {115}{48} \left (\frac {69}{32} \left (\frac {23}{16} \int \frac {1}{\sqrt {2 x^2-x+3}}dx-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )-\frac {1}{24} (1-4 x) \left (2 x^2-x+3\right )^{5/2}\right )-\frac {23225}{14} \left (2 x^2-x+3\right )^{7/2}\right )\right )+\frac {305}{18} x^2 \left (2 x^2-x+3\right )^{7/2}\right )+\frac {5}{4} x^3 \left (2 x^2-x+3\right )^{7/2}\)

\(\Big \downarrow \) 1090

\(\displaystyle \frac {1}{8} \left (\frac {1}{36} \left (\frac {8467}{16} x \left (2 x^2-x+3\right )^{7/2}-\frac {3}{32} \left (-\frac {4641}{4} \left (\frac {115}{48} \left (\frac {69}{32} \left (\frac {1}{16} \sqrt {\frac {23}{2}} \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )-\frac {1}{24} (1-4 x) \left (2 x^2-x+3\right )^{5/2}\right )-\frac {23225}{14} \left (2 x^2-x+3\right )^{7/2}\right )\right )+\frac {305}{18} x^2 \left (2 x^2-x+3\right )^{7/2}\right )+\frac {5}{4} x^3 \left (2 x^2-x+3\right )^{7/2}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {1}{8} \left (\frac {1}{36} \left (\frac {8467}{16} x \left (2 x^2-x+3\right )^{7/2}-\frac {3}{32} \left (-\frac {4641}{4} \left (\frac {115}{48} \left (\frac {69}{32} \left (\frac {23 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{16 \sqrt {2}}-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )-\frac {1}{16} (1-4 x) \left (2 x^2-x+3\right )^{3/2}\right )-\frac {1}{24} (1-4 x) \left (2 x^2-x+3\right )^{5/2}\right )-\frac {23225}{14} \left (2 x^2-x+3\right )^{7/2}\right )\right )+\frac {305}{18} x^2 \left (2 x^2-x+3\right )^{7/2}\right )+\frac {5}{4} x^3 \left (2 x^2-x+3\right )^{7/2}\)

Input:

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

Output:

(5*x^3*(3 - x + 2*x^2)^(7/2))/4 + ((305*x^2*(3 - x + 2*x^2)^(7/2))/18 + (( 
8467*x*(3 - x + 2*x^2)^(7/2))/16 - (3*((-23225*(3 - x + 2*x^2)^(7/2))/14 - 
 (4641*(-1/24*((1 - 4*x)*(3 - x + 2*x^2)^(5/2)) + (115*(-1/16*((1 - 4*x)*( 
3 - x + 2*x^2)^(3/2)) + (69*(-1/8*((1 - 4*x)*Sqrt[3 - x + 2*x^2]) + (23*Ar 
cSinh[(-1 + 4*x)/Sqrt[23]])/(16*Sqrt[2])))/32))/48))/4))/32)/36)/8
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt 
[a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
 

rule 1087

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* 
p + 1)))   Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && 
GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
 

rule 1090

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/(2*c*(-4* 
(c/(b^2 - 4*a*c)))^p)   Subst[Int[Simp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, 
b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]
 

rule 1160

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol 
] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b 
*e)/(2*c)   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] 
 && NeQ[p, -1]
 

rule 2192

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = 
Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x + 
 c*x^2)^(p + 1)/(c*(q + 2*p + 1))), x] + Simp[1/(c*(q + 2*p + 1))   Int[(a 
+ b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b 
*e*(q + p)*x^(q - 1) - c*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c 
, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]
 

Maple [A] (verified)

Time = 2.44 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.44

method	result	size
risch	\(\frac {\left (10569646080 x^{9}[phone] x^{8}[phone] x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {2 x^{2}-x +3}}{1056964608}+\frac {94111745 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{67108864}\)	\(75\)
trager	\(\left (10 x^{9}+\frac {35}{18} x^{8}+\frac {24067}{576} x^{7}+\frac {134287}{5376} x^{6}+\frac {9202859}{129024} x^{5}+\frac {3986291}{73728} x^{4}+\frac {202792375}{2752512} x^{3}+\frac {63977149}{1572864} x^{2}+\frac {3294437471}{88080384} x +\frac {1647131391}{117440512}\right ) \sqrt {2 x^{2}-x +3}-\frac {94111745 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )+4 \sqrt {2 x^{2}-x +3}\right )}{67108864}\)	\(99\)
default	\(\frac {1547 \left (4 x -1\right ) \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{98304}+\frac {177905 \left (4 x -1\right ) \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{3145728}+\frac {4091815 \left (4 x -1\right ) \sqrt {2 x^{2}-x +3}}{16777216}+\frac {94111745 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{67108864}+\frac {23225 \left (2 x^{2}-x +3\right )^{\frac {7}{2}}}{43008}+\frac {8467 x \left (2 x^{2}-x +3\right )^{\frac {7}{2}}}{4608}+\frac {305 x^{2} \left (2 x^{2}-x +3\right )^{\frac {7}{2}}}{144}+\frac {5 x^{3} \left (2 x^{2}-x +3\right )^{\frac {7}{2}}}{4}\)	\(136\)

Input:

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

Output:

1/1056964608*(10569646080*x^9+2055208960*x^8+44163137536*x^7+26401898496*x 
^6+75389820928*x^5+57147467776*x^4+77872272000*x^3+42992644128*x^2+3953324 
9652*x[phone])*(2*x^2-x+3)^(1/2)[phone]/67108864*2^(1/2)*arcsinh(4/ 
23*23^(1/2)*(x-1/4))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.58 \[ \int \left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )^2 \, dx=\frac {1}{1056964608} \, {\left (10569646080 \, x^{9} + 2055208960 \, x^{8} + 44163137536 \, x^{7} + 26401898496 \, x^{6} + 75389820928 \, x^{5} + 57147467776 \, x^{4} + 77872272000 \, x^{3} + 42992644128 \, x^{2} + 39533249652 \, x + 14824182519\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {94111745}{134217728} \, \sqrt {2} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \] Input:

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

Output:

1/1056964608*(10569646080*x^9 + 2055208960*x^8 + 44163137536*x^7 + 2640189 
8496*x^6 + 75389820928*x^5 + 57147467776*x^4 + 77872272000*x^3 + 429926441 
28*x^2 + 39533249652*x + 14824182519)*sqrt(2*x^2 - x + 3) + 94111745/13421 
7728*sqrt(2)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x 
- 25)
 

Sympy [A] (verification not implemented)

Time = 0.60 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.56 \[ \int \left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )^2 \, dx=\sqrt {2 x^{2} - x + 3} \cdot \left (10 x^{9} + \frac {35 x^{8}}{18} + \frac {24067 x^{7}}{576} + \frac {134287 x^{6}}{5376} + \frac {9202859 x^{5}}{129024} + \frac {3986291 x^{4}}{73728} + \frac {202792375 x^{3}}{2752512} + \frac {63977149 x^{2}}{1572864} + \frac {3294437471 x}{88080384} + \frac {1647131391}{117440512}\right ) + \frac {94111745 \sqrt {2} \operatorname {asinh}{\left (\frac {4 \sqrt {23} \left (x - \frac {1}{4}\right )}{23} \right )}}{67108864} \] Input:

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

Output:

sqrt(2*x**2 - x + 3)*(10*x**9 + 35*x**8/18 + 24067*x**7/576 + 134287*x**6/ 
5376 + 9202859*x**5/129024 + 3986291*x**4/73728 + 202792375*x**3/2752512 + 
 63977149*x**2/1572864 + 3294437471*x/88080384 + 1647131391/117440512) + 9 
4111745*sqrt(2)*asinh(4*sqrt(23)*(x - 1/4)/23)/67108864
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.98 \[ \int \left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )^2 \, dx=\frac {5}{4} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {7}{2}} x^{3} + \frac {305}{144} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {7}{2}} x^{2} + \frac {8467}{4608} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {7}{2}} x + \frac {23225}{43008} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {7}{2}} + \frac {1547}{24576} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x - \frac {1547}{98304} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {177905}{786432} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {177905}{3145728} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {4091815}{4194304} \, \sqrt {2 \, x^{2} - x + 3} x + \frac {94111745}{67108864} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {4091815}{16777216} \, \sqrt {2 \, x^{2} - x + 3} \] Input:

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

Output:

5/4*(2*x^2 - x + 3)^(7/2)*x^3 + 305/144*(2*x^2 - x + 3)^(7/2)*x^2 + 8467/4 
608*(2*x^2 - x + 3)^(7/2)*x + 23225/43008*(2*x^2 - x + 3)^(7/2) + 1547/245 
76*(2*x^2 - x + 3)^(5/2)*x - 1547/98304*(2*x^2 - x + 3)^(5/2) + 177905/786 
432*(2*x^2 - x + 3)^(3/2)*x - 177905/3145728*(2*x^2 - x + 3)^(3/2) + 40918 
15/4194304*sqrt(2*x^2 - x + 3)*x + 94111745/67108864*sqrt(2)*arcsinh(1/23* 
sqrt(23)*(4*x - 1)) - 4091815/16777216*sqrt(2*x^2 - x + 3)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.55 \[ \int \left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )^2 \, dx=\frac {1}{1056964608} \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (4 \, {\left (8 \, {\left (28 \, {\left (160 \, {\left (36 \, x + 7\right )} x + 24067\right )} x + 402861\right )} x + 9202859\right )} x + 27904037\right )} x + 608377125\right )} x + 1343520129\right )} x + 9883312413\right )} x + 14824182519\right )} \sqrt {2 \, x^{2} - x + 3} - \frac {94111745}{67108864} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) \] Input:

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

Output:

1/1056964608*(4*(8*(4*(16*(4*(8*(28*(160*(36*x + 7)*x + 24067)*x + 402861) 
*x + 9202859)*x + 27904037)*x + 608377125)*x + 1343520129)*x + 9883312413) 
*x + 14824182519)*sqrt(2*x^2 - x + 3) - 94111745/67108864*sqrt(2)*log(-2*s 
qrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1)
 

Mupad [F(-1)]

Timed out. \[ \int \left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )^2 \, dx=\int {\left (2\,x^2-x+3\right )}^{5/2}\,{\left (5\,x^2+3\,x+2\right )}^2 \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.09 \[ \int \left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )^2 \, dx=10 \sqrt {2 x^{2}-x +3}\, x^{9}+\frac {35 \sqrt {2 x^{2}-x +3}\, x^{8}}{18}+\frac {24067 \sqrt {2 x^{2}-x +3}\, x^{7}}{576}+\frac {134287 \sqrt {2 x^{2}-x +3}\, x^{6}}{5376}+\frac {9202859 \sqrt {2 x^{2}-x +3}\, x^{5}}{129024}+\frac {3986291 \sqrt {2 x^{2}-x +3}\, x^{4}}{73728}+\frac {202792375 \sqrt {2 x^{2}-x +3}\, x^{3}}{2752512}+\frac {63977149 \sqrt {2 x^{2}-x +3}\, x^{2}}{1572864}+\frac {3294437471 \sqrt {2 x^{2}-x +3}\, x}{88080384}+\frac {1647131391 \sqrt {2 x^{2}-x +3}}{117440512}+\frac {94111745 \sqrt {2}\, \mathrm {log}\left (\frac {2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+4 x -1}{\sqrt {23}}\right )}{67108864} \] Input:

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

Output:

(42278584320*sqrt(2*x**2 - x + 3)*x**9 + 8220835840*sqrt(2*x**2 - x + 3)*x 
**8 + 176652550144*sqrt(2*x**2 - x + 3)*x**7 + 105607593984*sqrt(2*x**2 - 
x + 3)*x**6 + 301559283712*sqrt(2*x**2 - x + 3)*x**5 + 228589871104*sqrt(2 
*x**2 - x + 3)*x**4 + 311489088000*sqrt(2*x**2 - x + 3)*x**3 + 17197057651 
2*sqrt(2*x**2 - x + 3)*x**2 + 158132998608*sqrt(2*x**2 - x + 3)*x + 592967 
30076*sqrt(2*x**2 - x + 3) + 5929039935*sqrt(2)*log((2*sqrt(2*x**2 - x + 3 
)*sqrt(2) + 4*x - 1)/sqrt(23)))/4227858432