\(\int \sqrt {3-x+2 x^2} (2+3 x+5 x^2)^3 \, dx\) [93]

Optimal result

Integrand size = 27, antiderivative size = 166 \[ \int \sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^3 \, dx=-\frac {6766097 (1-4 x) \sqrt {3-x+2 x^2}}{2097152}-\frac {22548119 \left (3-x+2 x^2\right )^{3/2}}{4587520}-\frac {9627393 x \left (3-x+2 x^2\right )^{3/2}}{1146880}+\frac {531681 x^2 \left (3-x+2 x^2\right )^{3/2}}{71680}+\frac {247435 x^3 \left (3-x+2 x^2\right )^{3/2}}{10752}+\frac {8825}{448} x^4 \left (3-x+2 x^2\right )^{3/2}+\frac {125}{16} x^5 \left (3-x+2 x^2\right )^{3/2}-\frac {155620231 \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )}{4194304 \sqrt {2}} \] Output:

-6766097/2097152*(1-4*x)*(2*x^2-x+3)^(1/2)-22548119/4587520*(2*x^2-x+3)^(3 
/2)-9627393/1146880*x*(2*x^2-x+3)^(3/2)+531681/71680*x^2*(2*x^2-x+3)^(3/2) 
+247435/10752*x^3*(2*x^2-x+3)^(3/2)+8825/448*x^4*(2*x^2-x+3)^(3/2)+125/16* 
x^5*(2*x^2-x+3)^(3/2)-155620231/8388608*arcsinh(1/23*(1-4*x)*23^(1/2))*2^( 
1/2)
 

Mathematica [A] (verified)

Time = 0.99 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.51 \[ \int \sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^3 \, dx=\frac {4 \sqrt {3-x+2 x^2} \left (-3957369321-1621307916 x+4583812128 x^2+9872163456 x^3+11212171264 x^4+10958233600 x^5+6955008000 x^6+3440640000 x^7\right )-16340124255 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{880803840} \] Input:

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

Output:

(4*Sqrt[3 - x + 2*x^2]*(-3957369321 - 1621307916*x + 4583812128*x^2 + 9872 
163456*x^3 + 11212171264*x^4 + 10958233600*x^5 + 6955008000*x^6 + 34406400 
00*x^7) - 16340124255*Sqrt[2]*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]])/8808 
03840
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.18, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.519, Rules used = {2192, 27, 2192, 27, 2192, 27, 2192, 27, 2192, 27, 1160, 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.

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

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{16} \int \frac {1}{2} \sqrt {2 x^2-x+3} \left (8825 x^5+5370 x^4+6624 x^3+3648 x^2+1152 x+256\right )dx+\frac {125}{16} \left (2 x^2-x+3\right )^{3/2} x^5\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{32} \int \sqrt {2 x^2-x+3} \left (8825 x^5+5370 x^4+6624 x^3+3648 x^2+1152 x+256\right )dx+\frac {125}{16} \left (2 x^2-x+3\right )^{3/2} x^5\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{32} \left (\frac {1}{14} \int \frac {1}{2} \sqrt {2 x^2-x+3} \left (247435 x^4-26328 x^3+102144 x^2+32256 x+7168\right )dx+\frac {8825}{14} \left (2 x^2-x+3\right )^{3/2} x^4\right )+\frac {125}{16} \left (2 x^2-x+3\right )^{3/2} x^5\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{32} \left (\frac {1}{28} \int \sqrt {2 x^2-x+3} \left (247435 x^4-26328 x^3+102144 x^2+32256 x+7168\right )dx+\frac {8825}{14} \left (2 x^2-x+3\right )^{3/2} x^4\right )+\frac {125}{16} \left (2 x^2-x+3\right )^{3/2} x^5\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{32} \left (\frac {1}{28} \left (\frac {1}{12} \int \frac {3}{2} \sqrt {2 x^2-x+3} \left (531681 x^3-667458 x^2+258048 x+57344\right )dx+\frac {247435}{12} \left (2 x^2-x+3\right )^{3/2} x^3\right )+\frac {8825}{14} \left (2 x^2-x+3\right )^{3/2} x^4\right )+\frac {125}{16} \left (2 x^2-x+3\right )^{3/2} x^5\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{32} \left (\frac {1}{28} \left (\frac {1}{8} \int \sqrt {2 x^2-x+3} \left (531681 x^3-667458 x^2+258048 x+57344\right )dx+\frac {247435}{12} \left (2 x^2-x+3\right )^{3/2} x^3\right )+\frac {8825}{14} \left (2 x^2-x+3\right )^{3/2} x^4\right )+\frac {125}{16} \left (2 x^2-x+3\right )^{3/2} x^5\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{32} \left (\frac {1}{28} \left (\frac {1}{8} \left (\frac {1}{10} \int \frac {1}{2} \left (-9627393 x^2-1219212 x+1146880\right ) \sqrt {2 x^2-x+3}dx+\frac {531681}{10} \left (2 x^2-x+3\right )^{3/2} x^2\right )+\frac {247435}{12} \left (2 x^2-x+3\right )^{3/2} x^3\right )+\frac {8825}{14} \left (2 x^2-x+3\right )^{3/2} x^4\right )+\frac {125}{16} \left (2 x^2-x+3\right )^{3/2} x^5\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{32} \left (\frac {1}{28} \left (\frac {1}{8} \left (\frac {1}{20} \int \left (-9627393 x^2-1219212 x+1146880\right ) \sqrt {2 x^2-x+3}dx+\frac {531681}{10} \left (2 x^2-x+3\right )^{3/2} x^2\right )+\frac {247435}{12} \left (2 x^2-x+3\right )^{3/2} x^3\right )+\frac {8825}{14} \left (2 x^2-x+3\right )^{3/2} x^4\right )+\frac {125}{16} \left (2 x^2-x+3\right )^{3/2} x^5\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{32} \left (\frac {1}{28} \left (\frac {1}{8} \left (\frac {1}{20} \left (\frac {1}{8} \int \frac {1}{2} (76114438-67644357 x) \sqrt {2 x^2-x+3}dx-\frac {9627393}{8} x \left (2 x^2-x+3\right )^{3/2}\right )+\frac {531681}{10} \left (2 x^2-x+3\right )^{3/2} x^2\right )+\frac {247435}{12} \left (2 x^2-x+3\right )^{3/2} x^3\right )+\frac {8825}{14} \left (2 x^2-x+3\right )^{3/2} x^4\right )+\frac {125}{16} \left (2 x^2-x+3\right )^{3/2} x^5\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{32} \left (\frac {1}{28} \left (\frac {1}{8} \left (\frac {1}{20} \left (\frac {1}{16} \int (76114438-67644357 x) \sqrt {2 x^2-x+3}dx-\frac {9627393}{8} x \left (2 x^2-x+3\right )^{3/2}\right )+\frac {531681}{10} \left (2 x^2-x+3\right )^{3/2} x^2\right )+\frac {247435}{12} \left (2 x^2-x+3\right )^{3/2} x^3\right )+\frac {8825}{14} \left (2 x^2-x+3\right )^{3/2} x^4\right )+\frac {125}{16} \left (2 x^2-x+3\right )^{3/2} x^5\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {1}{32} \left (\frac {1}{28} \left (\frac {1}{8} \left (\frac {1}{20} \left (\frac {1}{16} \left (\frac {236813395}{4} \int \sqrt {2 x^2-x+3}dx-\frac {22548119}{2} \left (2 x^2-x+3\right )^{3/2}\right )-\frac {9627393}{8} x \left (2 x^2-x+3\right )^{3/2}\right )+\frac {531681}{10} \left (2 x^2-x+3\right )^{3/2} x^2\right )+\frac {247435}{12} \left (2 x^2-x+3\right )^{3/2} x^3\right )+\frac {8825}{14} \left (2 x^2-x+3\right )^{3/2} x^4\right )+\frac {125}{16} \left (2 x^2-x+3\right )^{3/2} x^5\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{32} \left (\frac {1}{28} \left (\frac {1}{8} \left (\frac {1}{20} \left (\frac {1}{16} \left (\frac {236813395}{4} \left (\frac {23}{16} \int \frac {1}{\sqrt {2 x^2-x+3}}dx-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )-\frac {22548119}{2} \left (2 x^2-x+3\right )^{3/2}\right )-\frac {9627393}{8} x \left (2 x^2-x+3\right )^{3/2}\right )+\frac {531681}{10} \left (2 x^2-x+3\right )^{3/2} x^2\right )+\frac {247435}{12} \left (2 x^2-x+3\right )^{3/2} x^3\right )+\frac {8825}{14} \left (2 x^2-x+3\right )^{3/2} x^4\right )+\frac {125}{16} \left (2 x^2-x+3\right )^{3/2} x^5\)

\(\Big \downarrow \) 1090

\(\displaystyle \frac {1}{32} \left (\frac {1}{28} \left (\frac {1}{8} \left (\frac {1}{20} \left (\frac {1}{16} \left (\frac {236813395}{4} \left (\frac {1}{16} \sqrt {\frac {23}{2}} \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )-\frac {22548119}{2} \left (2 x^2-x+3\right )^{3/2}\right )-\frac {9627393}{8} x \left (2 x^2-x+3\right )^{3/2}\right )+\frac {531681}{10} \left (2 x^2-x+3\right )^{3/2} x^2\right )+\frac {247435}{12} \left (2 x^2-x+3\right )^{3/2} x^3\right )+\frac {8825}{14} \left (2 x^2-x+3\right )^{3/2} x^4\right )+\frac {125}{16} \left (2 x^2-x+3\right )^{3/2} x^5\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {1}{32} \left (\frac {1}{28} \left (\frac {1}{8} \left (\frac {1}{20} \left (\frac {1}{16} \left (\frac {236813395}{4} \left (\frac {23 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{16 \sqrt {2}}-\frac {1}{8} (1-4 x) \sqrt {2 x^2-x+3}\right )-\frac {22548119}{2} \left (2 x^2-x+3\right )^{3/2}\right )-\frac {9627393}{8} x \left (2 x^2-x+3\right )^{3/2}\right )+\frac {531681}{10} \left (2 x^2-x+3\right )^{3/2} x^2\right )+\frac {247435}{12} \left (2 x^2-x+3\right )^{3/2} x^3\right )+\frac {8825}{14} \left (2 x^2-x+3\right )^{3/2} x^4\right )+\frac {125}{16} \left (2 x^2-x+3\right )^{3/2} x^5\)

Input:

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

Output:

(125*x^5*(3 - x + 2*x^2)^(3/2))/16 + ((8825*x^4*(3 - x + 2*x^2)^(3/2))/14 
+ ((247435*x^3*(3 - x + 2*x^2)^(3/2))/12 + ((531681*x^2*(3 - x + 2*x^2)^(3 
/2))/10 + ((-9627393*x*(3 - x + 2*x^2)^(3/2))/8 + ((-22548119*(3 - x + 2*x 
^2)^(3/2))/2 + (236813395*(-1/8*((1 - 4*x)*Sqrt[3 - x + 2*x^2]) + (23*ArcS 
inh[(-1 + 4*x)/Sqrt[23]])/(16*Sqrt[2])))/4)/16)/20)/8)/28)/32
 

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 = 2.41 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.39

method	result	size
risch	\(\frac {\left (3440640000 x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}-1621307916 x -3957369321\right ) \sqrt {2 x^{2}-x +3}}{220200960}+\frac {155620231 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{8388608}\)	\(65\)
trager	\(\left (\frac {125}{8} x^{7}+\frac {7075}{224} x^{6}+\frac {267535}{5376} x^{5}+\frac {782099}{15360} x^{4}+\frac {25708759}{573440} x^{3}+\frac {6821149}{327680} x^{2}-\frac {135108993}{18350080} x -\frac {1319123107}{73400320}\right ) \sqrt {2 x^{2}-x +3}-\frac {155620231 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )+4 \sqrt {2 x^{2}-x +3}\right )}{8388608}\)	\(89\)
default	\(\frac {6766097 \left (4 x -1\right ) \sqrt {2 x^{2}-x +3}}{2097152}+\frac {155620231 \sqrt {2}\, \operatorname {arcsinh}\left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{8388608}-\frac {22548119 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{4587520}-\frac {9627393 x \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{1146880}+\frac {531681 x^{2} \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{71680}+\frac {247435 x^{3} \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{10752}+\frac {8825 x^{4} \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{448}+\frac {125 x^{5} \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{16}\)	\(132\)

Input:

int((2*x^2-x+3)^(1/2)*(5*x^2+3*x+2)^3,x,method=_RETURNVERBOSE)
 

Output:

1/220200960*(3440640000*x^7+6955008000*x^6+10958233600*x^5+11212171264*x^4 
[phone]*x^3+4583812128*x^2-1621307916*x-3957369321)*(2*x^2-x+3)^(1/2)+ 
155620231/8388608*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.53 \[ \int \sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^3 \, dx=\frac {1}{220200960} \, {\left (3440640000 \, x^{7} + 6955008000 \, x^{6} + 10958233600 \, x^{5} + 11212171264 \, x^{4} + 9872163456 \, x^{3} + 4583812128 \, x^{2} - 1621307916 \, x - 3957369321\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {155620231}{16777216} \, \sqrt {2} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \] Input:

integrate((2*x^2-x+3)^(1/2)*(5*x^2+3*x+2)^3,x, algorithm="fricas")
 

Output:

1/220200960*(3440640000*x^7 + 6955008000*x^6 + 10958233600*x^5 + 112121712 
64*x^4 + 9872163456*x^3 + 4583812128*x^2 - 1621307916*x - 3957369321)*sqrt 
(2*x^2 - x + 3) + 155620231/16777216*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 = 0.41 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.50 \[ \int \sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^3 \, dx=\sqrt {2 x^{2} - x + 3} \cdot \left (\frac {125 x^{7}}{8} + \frac {7075 x^{6}}{224} + \frac {267535 x^{5}}{5376} + \frac {782099 x^{4}}{15360} + \frac {25708759 x^{3}}{573440} + \frac {6821149 x^{2}}{327680} - \frac {135108993 x}{18350080} - \frac {1319123107}{73400320}\right ) + \frac {155620231 \sqrt {2} \operatorname {asinh}{\left (\frac {4 \sqrt {23} \left (x - \frac {1}{4}\right )}{23} \right )}}{8388608} \] Input:

integrate((2*x**2-x+3)**(1/2)*(5*x**2+3*x+2)**3,x)
 

Output:

sqrt(2*x**2 - x + 3)*(125*x**7/8 + 7075*x**6/224 + 267535*x**5/5376 + 7820 
99*x**4/15360 + 25708759*x**3/573440 + 6821149*x**2/327680 - 135108993*x/1 
8350080 - 1319123107/73400320) + 155620231*sqrt(2)*asinh(4*sqrt(23)*(x - 1 
/4)/23)/8388608
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.86 \[ \int \sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^3 \, dx=\frac {125}{16} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{5} + \frac {8825}{448} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{4} + \frac {247435}{10752} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + \frac {531681}{71680} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} - \frac {9627393}{1146880} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {22548119}{4587520} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {6766097}{524288} \, \sqrt {2 \, x^{2} - x + 3} x + \frac {155620231}{8388608} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {6766097}{2097152} \, \sqrt {2 \, x^{2} - x + 3} \] Input:

integrate((2*x^2-x+3)^(1/2)*(5*x^2+3*x+2)^3,x, algorithm="maxima")
 

Output:

125/16*(2*x^2 - x + 3)^(3/2)*x^5 + 8825/448*(2*x^2 - x + 3)^(3/2)*x^4 + 24 
7435/10752*(2*x^2 - x + 3)^(3/2)*x^3 + 531681/71680*(2*x^2 - x + 3)^(3/2)* 
x^2 - 9627393/1146880*(2*x^2 - x + 3)^(3/2)*x - 22548119/4587520*(2*x^2 - 
x + 3)^(3/2) + 6766097/524288*sqrt(2*x^2 - x + 3)*x + 155620231/8388608*sq 
rt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 6766097/2097152*sqrt(2*x^2 - x + 
3)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.50 \[ \int \sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^3 \, dx=\frac {1}{220200960} \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (100 \, {\left (120 \, {\left (140 \, x + 283\right )} x + 53507\right )} x + 5474693\right )} x + 77126277\right )} x + 143244129\right )} x - 405326979\right )} x - 3957369321\right )} \sqrt {2 \, x^{2} - x + 3} - \frac {155620231}{8388608} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) \] Input:

integrate((2*x^2-x+3)^(1/2)*(5*x^2+3*x+2)^3,x, algorithm="giac")
 

Output:

1/220200960*(4*(8*(4*(16*(100*(120*(140*x + 283)*x + 53507)*x + 5474693)*x 
 + 77126277)*x + 143244129)*x - 405326979)*x - 3957369321)*sqrt(2*x^2 - x 
+ 3) - 155620231/8388608*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - 
x + 3)) + 1)
 

Mupad [B] (verification not implemented)

Time = 17.12 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.13 \[ \int \sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^3 \, dx=\frac {531681\,x^2\,{\left (2\,x^2-x+3\right )}^{3/2}}{71680}+\frac {247435\,x^3\,{\left (2\,x^2-x+3\right )}^{3/2}}{10752}+\frac {8825\,x^4\,{\left (2\,x^2-x+3\right )}^{3/2}}{448}+\frac {125\,x^5\,{\left (2\,x^2-x+3\right )}^{3/2}}{16}+\frac {875316037\,\sqrt {2}\,\ln \left (\sqrt {2\,x^2-x+3}+\frac {\sqrt {2}\,\left (2\,x-\frac {1}{2}\right )}{2}\right )}{36700160}+\frac {38057219\,\left (\frac {x}{2}-\frac {1}{8}\right )\,\sqrt {2\,x^2-x+3}}{1146880}-\frac {22548119\,\sqrt {2\,x^2-x+3}\,\left (32\,x^2-4\,x+45\right )}{73400320}-\frac {9627393\,x\,{\left (2\,x^2-x+3\right )}^{3/2}}{1146880}-\frac {1555820211\,\sqrt {2}\,\ln \left (2\,\sqrt {2\,x^2-x+3}+\frac {\sqrt {2}\,\left (4\,x-1\right )}{2}\right )}{293601280} \] Input:

int((2*x^2 - x + 3)^(1/2)*(3*x + 5*x^2 + 2)^3,x)
 

Output:

(531681*x^2*(2*x^2 - x + 3)^(3/2))/71680 + (247435*x^3*(2*x^2 - x + 3)^(3/ 
2))/10752 + (8825*x^4*(2*x^2 - x + 3)^(3/2))/448 + (125*x^5*(2*x^2 - x + 3 
)^(3/2))/16 + (875316037*2^(1/2)*log((2*x^2 - x + 3)^(1/2) + (2^(1/2)*(2*x 
 - 1/2))/2))/36700160 + (38057219*(x/2 - 1/8)*(2*x^2 - x + 3)^(1/2))/11468 
80 - (22548119*(2*x^2 - x + 3)^(1/2)*(32*x^2 - 4*x + 45))/73400320 - (9627 
393*x*(2*x^2 - x + 3)^(3/2))/1146880 - (1555820211*2^(1/2)*log(2*(2*x^2 - 
x + 3)^(1/2) + (2^(1/2)*(4*x - 1))/2))/293601280
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.93 \[ \int \sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^3 \, dx=\frac {125 \sqrt {2 x^{2}-x +3}\, x^{7}}{8}+\frac {7075 \sqrt {2 x^{2}-x +3}\, x^{6}}{224}+\frac {267535 \sqrt {2 x^{2}-x +3}\, x^{5}}{5376}+\frac {782099 \sqrt {2 x^{2}-x +3}\, x^{4}}{15360}+\frac {25708759 \sqrt {2 x^{2}-x +3}\, x^{3}}{573440}+\frac {6821149 \sqrt {2 x^{2}-x +3}\, x^{2}}{327680}-\frac {135108993 \sqrt {2 x^{2}-x +3}\, x}{18350080}-\frac {1319123107 \sqrt {2 x^{2}-x +3}}{73400320}+\frac {155620231 \sqrt {2}\, \mathrm {log}\left (\frac {2 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+4 x -1}{\sqrt {23}}\right )}{8388608} \] Input:

int((2*x^2-x+3)^(1/2)*(5*x^2+3*x+2)^3,x)
 

Output:

(13762560000*sqrt(2*x**2 - x + 3)*x**7 + 27820032000*sqrt(2*x**2 - x + 3)* 
x**6 + 43832934400*sqrt(2*x**2 - x + 3)*x**5 + 44848685056*sqrt(2*x**2 - x 
 + 3)*x**4 + 39488653824*sqrt(2*x**2 - x + 3)*x**3 + 18335248512*sqrt(2*x* 
*2 - x + 3)*x**2 - 6485231664*sqrt(2*x**2 - x + 3)*x - 15829477284*sqrt(2* 
x**2 - x + 3) + 16340124255*sqrt(2)*log((2*sqrt(2*x**2 - x + 3)*sqrt(2) + 
4*x - 1)/sqrt(23)))/880803840