\(\int (4+x-2 x^2)^3 \sqrt {2+3 x+5 x^2} \, dx\) [134]

Optimal result

Integrand size = 25, antiderivative size = 166 \[ \int \left (4+x-2 x^2\right )^3 \sqrt {2+3 x+5 x^2} \, dx=\frac {364734979 (3+10 x) \sqrt {2+3 x+5 x^2}}{160000000}+\frac {461470657 \left (2+3 x+5 x^2\right )^{3/2}}{84000000}-\frac {44194603 x \left (2+3 x+5 x^2\right )^{3/2}}{14000000}-\frac {938857 x^2 \left (2+3 x+5 x^2\right )^{3/2}}{350000}+\frac {25553 x^3 \left (2+3 x+5 x^2\right )^{3/2}}{21000}+\frac {159}{350} x^4 \left (2+3 x+5 x^2\right )^{3/2}-\frac {1}{5} x^5 \left (2+3 x+5 x^2\right )^{3/2}+\frac {11306784349 \text {arcsinh}\left (\frac {3+10 x}{\sqrt {31}}\right )}{320000000 \sqrt {5}} \] Output:

364734979/160000000*(3+10*x)*(5*x^2+3*x+2)^(1/2)[phone]/84000000*(5*x^2 
+3*x+2)^(3/2)-44194603/14000000*x*(5*x^2+3*x+2)^(3/2)-938857/350000*x^2*(5 
*x^2+3*x+2)^(3/2)+25553/21000*x^3*(5*x^2+3*x+2)^(3/2)+159/350*x^4*(5*x^2+3 
*x+2)^(3/2)-1/5*x^5*(5*x^2+3*x+2)^(3/2)[phone]/1600000000*arcsinh(1/3 
1*(3+10*x)*31^(1/2))*5^(1/2)
 

Mathematica [A] (verified)

Time = 1.02 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.54 \[ \int \left (4+x-2 x^2\right )^3 \sqrt {2+3 x+5 x^2} \, dx=\frac {\sqrt {2+3 x+5 x^2} \left (59895956237+110757414990 x+42447962840 x^2-71895645200 x^3-29746896000 x^4+23677600000 x^5+5616000000 x^6-3360000000 x^7\right )}{3360000000}-\frac {11306784349 \log \left (-3-10 x+2 \sqrt {5} \sqrt {2+3 x+5 x^2}\right )}{320000000 \sqrt {5}} \] Input:

Integrate[(4 + x - 2*x^2)^3*Sqrt[2 + 3*x + 5*x^2],x]
 

Output:

(Sqrt[2 + 3*x + 5*x^2]*(59895956237 + 110757414990*x + 42447962840*x^2 - 7 
1895645200*x^3 - 29746896000*x^4 + 23677600000*x^5 + 5616000000*x^6 - 3360 
000000*x^7))/3360000000 - (11306784349*Log[-3 - 10*x + 2*Sqrt[5]*Sqrt[2 + 
3*x + 5*x^2]])/(320000000*Sqrt[5])
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.18, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, 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.

\(\displaystyle \int \left (-2 x^2+x+4\right )^3 \sqrt {5 x^2+3 x+2} \, dx\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{40} \int 4 \sqrt {5 x^2+3 x+2} \left (159 x^5+440 x^4-470 x^3-840 x^2+480 x+640\right )dx-\frac {1}{5} x^5 \left (5 x^2+3 x+2\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{10} \int \sqrt {5 x^2+3 x+2} \left (159 x^5+440 x^4-470 x^3-840 x^2+480 x+640\right )dx-\frac {1}{5} x^5 \left (5 x^2+3 x+2\right )^{3/2}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{10} \left (\frac {1}{35} \int \frac {1}{2} \sqrt {5 x^2+3 x+2} \left (25553 x^4-35444 x^3-58800 x^2+33600 x+44800\right )dx+\frac {159}{35} \left (5 x^2+3 x+2\right )^{3/2} x^4\right )-\frac {1}{5} x^5 \left (5 x^2+3 x+2\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{10} \left (\frac {1}{70} \int \sqrt {5 x^2+3 x+2} \left (25553 x^4-35444 x^3-58800 x^2+33600 x+44800\right )dx+\frac {159}{35} \left (5 x^2+3 x+2\right )^{3/2} x^4\right )-\frac {1}{5} x^5 \left (5 x^2+3 x+2\right )^{3/2}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{10} \left (\frac {1}{70} \left (\frac {1}{30} \int \frac {3}{2} \sqrt {5 x^2+3 x+2} \left (-938857 x^3-1278212 x^2+672000 x+896000\right )dx+\frac {25553}{30} \left (5 x^2+3 x+2\right )^{3/2} x^3\right )+\frac {159}{35} \left (5 x^2+3 x+2\right )^{3/2} x^4\right )-\frac {1}{5} x^5 \left (5 x^2+3 x+2\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{10} \left (\frac {1}{70} \left (\frac {1}{20} \int \sqrt {5 x^2+3 x+2} \left (-938857 x^3-1278212 x^2+672000 x+896000\right )dx+\frac {25553}{30} \left (5 x^2+3 x+2\right )^{3/2} x^3\right )+\frac {159}{35} \left (5 x^2+3 x+2\right )^{3/2} x^4\right )-\frac {1}{5} x^5 \left (5 x^2+3 x+2\right )^{3/2}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{10} \left (\frac {1}{70} \left (\frac {1}{20} \left (\frac {1}{25} \int \frac {1}{2} \left (-44194603 x^2+41110856 x+44800000\right ) \sqrt {5 x^2+3 x+2}dx-\frac {938857}{25} x^2 \left (5 x^2+3 x+2\right )^{3/2}\right )+\frac {25553}{30} \left (5 x^2+3 x+2\right )^{3/2} x^3\right )+\frac {159}{35} \left (5 x^2+3 x+2\right )^{3/2} x^4\right )-\frac {1}{5} x^5 \left (5 x^2+3 x+2\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{10} \left (\frac {1}{70} \left (\frac {1}{20} \left (\frac {1}{50} \int \left (-44194603 x^2+41110856 x+44800000\right ) \sqrt {5 x^2+3 x+2}dx-\frac {938857}{25} x^2 \left (5 x^2+3 x+2\right )^{3/2}\right )+\frac {25553}{30} \left (5 x^2+3 x+2\right )^{3/2} x^3\right )+\frac {159}{35} \left (5 x^2+3 x+2\right )^{3/2} x^4\right )-\frac {1}{5} x^5 \left (5 x^2+3 x+2\right )^{3/2}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{10} \left (\frac {1}{70} \left (\frac {1}{20} \left (\frac {1}{50} \left (\frac {1}{20} \int \frac {1}{2} (2307353285 x+1968778412) \sqrt {5 x^2+3 x+2}dx-\frac {44194603}{20} x \left (5 x^2+3 x+2\right )^{3/2}\right )-\frac {938857}{25} x^2 \left (5 x^2+3 x+2\right )^{3/2}\right )+\frac {25553}{30} \left (5 x^2+3 x+2\right )^{3/2} x^3\right )+\frac {159}{35} \left (5 x^2+3 x+2\right )^{3/2} x^4\right )-\frac {1}{5} x^5 \left (5 x^2+3 x+2\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{10} \left (\frac {1}{70} \left (\frac {1}{20} \left (\frac {1}{50} \left (\frac {1}{40} \int (2307353285 x+1968778412) \sqrt {5 x^2+3 x+2}dx-\frac {44194603}{20} x \left (5 x^2+3 x+2\right )^{3/2}\right )-\frac {938857}{25} x^2 \left (5 x^2+3 x+2\right )^{3/2}\right )+\frac {25553}{30} \left (5 x^2+3 x+2\right )^{3/2} x^3\right )+\frac {159}{35} \left (5 x^2+3 x+2\right )^{3/2} x^4\right )-\frac {1}{5} x^5 \left (5 x^2+3 x+2\right )^{3/2}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {1}{10} \left (\frac {1}{70} \left (\frac {1}{20} \left (\frac {1}{50} \left (\frac {1}{40} \left (\frac {2553144853}{2} \int \sqrt {5 x^2+3 x+2}dx+\frac {461470657}{3} \left (5 x^2+3 x+2\right )^{3/2}\right )-\frac {44194603}{20} x \left (5 x^2+3 x+2\right )^{3/2}\right )-\frac {938857}{25} x^2 \left (5 x^2+3 x+2\right )^{3/2}\right )+\frac {25553}{30} \left (5 x^2+3 x+2\right )^{3/2} x^3\right )+\frac {159}{35} \left (5 x^2+3 x+2\right )^{3/2} x^4\right )-\frac {1}{5} x^5 \left (5 x^2+3 x+2\right )^{3/2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{10} \left (\frac {1}{70} \left (\frac {1}{20} \left (\frac {1}{50} \left (\frac {1}{40} \left (\frac {2553144853}{2} \left (\frac {31}{40} \int \frac {1}{\sqrt {5 x^2+3 x+2}}dx+\frac {1}{20} \sqrt {5 x^2+3 x+2} (10 x+3)\right )+\frac {461470657}{3} \left (5 x^2+3 x+2\right )^{3/2}\right )-\frac {44194603}{20} x \left (5 x^2+3 x+2\right )^{3/2}\right )-\frac {938857}{25} x^2 \left (5 x^2+3 x+2\right )^{3/2}\right )+\frac {25553}{30} \left (5 x^2+3 x+2\right )^{3/2} x^3\right )+\frac {159}{35} \left (5 x^2+3 x+2\right )^{3/2} x^4\right )-\frac {1}{5} x^5 \left (5 x^2+3 x+2\right )^{3/2}\)

\(\Big \downarrow \) 1090

\(\displaystyle \frac {1}{10} \left (\frac {1}{70} \left (\frac {1}{20} \left (\frac {1}{50} \left (\frac {1}{40} \left (\frac {2553144853}{2} \left (\frac {1}{40} \sqrt {\frac {31}{5}} \int \frac {1}{\sqrt {\frac {1}{31} (10 x+3)^2+1}}d(10 x+3)+\frac {1}{20} \sqrt {5 x^2+3 x+2} (10 x+3)\right )+\frac {461470657}{3} \left (5 x^2+3 x+2\right )^{3/2}\right )-\frac {44194603}{20} x \left (5 x^2+3 x+2\right )^{3/2}\right )-\frac {938857}{25} x^2 \left (5 x^2+3 x+2\right )^{3/2}\right )+\frac {25553}{30} \left (5 x^2+3 x+2\right )^{3/2} x^3\right )+\frac {159}{35} \left (5 x^2+3 x+2\right )^{3/2} x^4\right )-\frac {1}{5} x^5 \left (5 x^2+3 x+2\right )^{3/2}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {1}{10} \left (\frac {1}{70} \left (\frac {1}{20} \left (\frac {1}{50} \left (\frac {1}{40} \left (\frac {2553144853}{2} \left (\frac {31 \text {arcsinh}\left (\frac {10 x+3}{\sqrt {31}}\right )}{40 \sqrt {5}}+\frac {1}{20} \sqrt {5 x^2+3 x+2} (10 x+3)\right )+\frac {461470657}{3} \left (5 x^2+3 x+2\right )^{3/2}\right )-\frac {44194603}{20} x \left (5 x^2+3 x+2\right )^{3/2}\right )-\frac {938857}{25} x^2 \left (5 x^2+3 x+2\right )^{3/2}\right )+\frac {25553}{30} \left (5 x^2+3 x+2\right )^{3/2} x^3\right )+\frac {159}{35} \left (5 x^2+3 x+2\right )^{3/2} x^4\right )-\frac {1}{5} x^5 \left (5 x^2+3 x+2\right )^{3/2}\)

Input:

Int[(4 + x - 2*x^2)^3*Sqrt[2 + 3*x + 5*x^2],x]
 

Output:

-1/5*(x^5*(2 + 3*x + 5*x^2)^(3/2)) + ((159*x^4*(2 + 3*x + 5*x^2)^(3/2))/35 
 + ((25553*x^3*(2 + 3*x + 5*x^2)^(3/2))/30 + ((-938857*x^2*(2 + 3*x + 5*x^ 
2)^(3/2))/25 + ((-44194603*x*(2 + 3*x + 5*x^2)^(3/2))/20 + ((461470657*(2 
+ 3*x + 5*x^2)^(3/2))/3 + (2553144853*(((3 + 10*x)*Sqrt[2 + 3*x + 5*x^2])/ 
20 + (31*ArcSinh[(3 + 10*x)/Sqrt[31]])/(40*Sqrt[5])))/2)/40)/50)/20)/70)/1 
0
 

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 = 2.58 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.39

method	result	size
risch	\(-\frac {\left (3360000000 x^{7}-5616000000 x^{6}-23677600000 x^{5}[phone] x^{4}[phone] x^{3}-42447962840 x^{2}-110757414990 x -59895956237\right ) \sqrt {5 x^{2}+3 x +2}}{3360000000}+\frac {11306784349 \sqrt {5}\, \operatorname {arcsinh}\left (\frac {10 \sqrt {31}\, \left (x +\frac {3}{10}\right )}{31}\right )}{1600000000}\)	\(65\)
trager	\(\left (-x^{7}+\frac {117}{70} x^{6}+\frac {29597}{4200} x^{5}-\frac {619727}{70000} x^{4}-\frac {179739113}{8400000} x^{3}+\frac {1061199071}{84000000} x^{2}+\frac {3691913833}{112000000} x +\frac {59895956237}{3360000000}\right ) \sqrt {5 x^{2}+3 x +2}-\frac {11306784349 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (-10 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {5 x^{2}+3 x +2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )\right )}{1600000000}\)	\(91\)
default	\(\frac {364734979 \left (10 x +3\right ) \sqrt {5 x^{2}+3 x +2}}{160000000}+\frac {11306784349 \sqrt {5}\, \operatorname {arcsinh}\left (\frac {10 \sqrt {31}\, \left (x +\frac {3}{10}\right )}{31}\right )}{1600000000}+\frac {461470657 \left (5 x^{2}+3 x +2\right )^{\frac {3}{2}}}{84000000}-\frac {44194603 x \left (5 x^{2}+3 x +2\right )^{\frac {3}{2}}}{14000000}-\frac {938857 x^{2} \left (5 x^{2}+3 x +2\right )^{\frac {3}{2}}}{350000}+\frac {25553 x^{3} \left (5 x^{2}+3 x +2\right )^{\frac {3}{2}}}{21000}+\frac {159 x^{4} \left (5 x^{2}+3 x +2\right )^{\frac {3}{2}}}{350}-\frac {x^{5} \left (5 x^{2}+3 x +2\right )^{\frac {3}{2}}}{5}\)	\(132\)

Input:

int((-2*x^2+x+4)^3*(5*x^2+3*x+2)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/3360000000*(3360000000*x^7-5616000000*x^6-23677600000*x^5+29746896000*x 
^4+71895645200*x^3-42447962840*x^2-110757414990*x-59895956237)*(5*x^2+3*x+ 
2)^(1/2)[phone]/1600000000*5^(1/2)*arcsinh(10/31*31^(1/2)*(x+3/10))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.53 \[ \int \left (4+x-2 x^2\right )^3 \sqrt {2+3 x+5 x^2} \, dx=-\frac {1}{3360000000} \, {\left (3360000000 \, x^{7} - 5616000000 \, x^{6} - 23677600000 \, x^{5} + 29746896000 \, x^{4} + 71895645200 \, x^{3} - 42447962840 \, x^{2} - 110757414990 \, x - 59895956237\right )} \sqrt {5 \, x^{2} + 3 \, x + 2} + \frac {11306784349}{3200000000} \, \sqrt {5} \log \left (-4 \, \sqrt {5} \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (10 \, x + 3\right )} - 200 \, x^{2} - 120 \, x - 49\right ) \] Input:

integrate((-2*x^2+x+4)^3*(5*x^2+3*x+2)^(1/2),x, algorithm="fricas")
 

Output:

-1/3360000000*(3360000000*x^7 - 5616000000*x^6 - 23677600000*x^5 + 2974689 
6000*x^4 + 71895645200*x^3 - 42447962840*x^2 - 110757414990*x - 5989595623 
7)*sqrt(5*x^2 + 3*x + 2) + 11306784349/3200000000*sqrt(5)*log(-4*sqrt(5)*s 
qrt(5*x^2 + 3*x + 2)*(10*x + 3) - 200*x^2 - 120*x - 49)
 

Sympy [A] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.49 \[ \int \left (4+x-2 x^2\right )^3 \sqrt {2+3 x+5 x^2} \, dx=\sqrt {5 x^{2} + 3 x + 2} \left (- x^{7} + \frac {117 x^{6}}{70} + \frac {29597 x^{5}}{4200} - \frac {619727 x^{4}}{70000} - \frac {179739113 x^{3}}{8400000} + \frac {1061199071 x^{2}}{84000000} + \frac {3691913833 x}{112000000} + \frac {59895956237}{3360000000}\right ) + \frac {11306784349 \sqrt {5} \operatorname {asinh}{\left (\frac {10 \sqrt {31} \left (x + \frac {3}{10}\right )}{31} \right )}}{1600000000} \] Input:

integrate((-2*x**2+x+4)**3*(5*x**2+3*x+2)**(1/2),x)
 

Output:

sqrt(5*x**2 + 3*x + 2)*(-x**7 + 117*x**6/70 + 29597*x**5/4200 - 619727*x** 
4/70000 - 179739113*x**3/8400000 + 1061199071*x**2/84000000 + 3691913833*x 
/112000000 + 59895956237/3360000000) + 11306784349*sqrt(5)*asinh(10*sqrt(3 
1)*(x + 3/10)/31)/1600000000
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.86 \[ \int \left (4+x-2 x^2\right )^3 \sqrt {2+3 x+5 x^2} \, dx=-\frac {1}{5} \, {\left (5 \, x^{2} + 3 \, x + 2\right )}^{\frac {3}{2}} x^{5} + \frac {159}{350} \, {\left (5 \, x^{2} + 3 \, x + 2\right )}^{\frac {3}{2}} x^{4} + \frac {25553}{21000} \, {\left (5 \, x^{2} + 3 \, x + 2\right )}^{\frac {3}{2}} x^{3} - \frac {938857}{350000} \, {\left (5 \, x^{2} + 3 \, x + 2\right )}^{\frac {3}{2}} x^{2} - \frac {44194603}{14000000} \, {\left (5 \, x^{2} + 3 \, x + 2\right )}^{\frac {3}{2}} x + \frac {461470657}{84000000} \, {\left (5 \, x^{2} + 3 \, x + 2\right )}^{\frac {3}{2}} + \frac {364734979}{16000000} \, \sqrt {5 \, x^{2} + 3 \, x + 2} x + \frac {11306784349}{1600000000} \, \sqrt {5} \operatorname {arsinh}\left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {1094204937}{160000000} \, \sqrt {5 \, x^{2} + 3 \, x + 2} \] Input:

integrate((-2*x^2+x+4)^3*(5*x^2+3*x+2)^(1/2),x, algorithm="maxima")
 

Output:

-1/5*(5*x^2 + 3*x + 2)^(3/2)*x^5 + 159/350*(5*x^2 + 3*x + 2)^(3/2)*x^4 + 2 
5553/21000*(5*x^2 + 3*x + 2)^(3/2)*x^3 - 938857/350000*(5*x^2 + 3*x + 2)^( 
3/2)*x^2 - 44194603/14000000*(5*x^2 + 3*x + 2)^(3/2)*x + 461470657/8400000 
0*(5*x^2 + 3*x + 2)^(3/2) + 364734979/16000000*sqrt(5*x^2 + 3*x + 2)*x + 1 
1306784349/1600000000*sqrt(5)*arcsinh(1/31*sqrt(31)*(10*x + 3)) + 10942049 
37/160000000*sqrt(5*x^2 + 3*x + 2)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.50 \[ \int \left (4+x-2 x^2\right )^3 \sqrt {2+3 x+5 x^2} \, dx=-\frac {1}{3360000000} \, {\left (10 \, {\left (4 \, {\left (10 \, {\left (40 \, {\left (50 \, {\left (60 \, {\left (70 \, x - 117\right )} x - 29597\right )} x + 1859181\right )} x + 179739113\right )} x - 1061199071\right )} x - 11075741499\right )} x - 59895956237\right )} \sqrt {5 \, x^{2} + 3 \, x + 2} - \frac {11306784349}{1600000000} \, \sqrt {5} \log \left (-2 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )} - 3\right ) \] Input:

integrate((-2*x^2+x+4)^3*(5*x^2+3*x+2)^(1/2),x, algorithm="giac")
 

Output:

-1/3360000000*(10*(4*(10*(40*(50*(60*(70*x - 117)*x - 29597)*x + 1859181)* 
x + 179739113)*x - 1061199071)*x - 11075741499)*x - 59895956237)*sqrt(5*x^ 
2 + 3*x + 2) - 11306784349/1600000000*sqrt(5)*log(-2*sqrt(5)*(sqrt(5)*x - 
sqrt(5*x^2 + 3*x + 2)) - 3)
 

Mupad [B] (verification not implemented)

Time = 16.54 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.13 \[ \int \left (4+x-2 x^2\right )^3 \sqrt {2+3 x+5 x^2} \, dx=\frac {25553\,x^3\,{\left (5\,x^2+3\,x+2\right )}^{3/2}}{21000}-\frac {938857\,x^2\,{\left (5\,x^2+3\,x+2\right )}^{3/2}}{350000}+\frac {159\,x^4\,{\left (5\,x^2+3\,x+2\right )}^{3/2}}{350}-\frac {x^5\,{\left (5\,x^2+3\,x+2\right )}^{3/2}}{5}+\frac {15258032693\,\sqrt {5}\,\ln \left (\sqrt {5\,x^2+3\,x+2}+\frac {\sqrt {5}\,\left (5\,x+\frac {3}{2}\right )}{5}\right )}{1400000000}+\frac {492194603\,\left (\frac {x}{2}+\frac {3}{20}\right )\,\sqrt {5\,x^2+3\,x+2}}{7000000}+\frac {461470657\,\sqrt {5\,x^2+3\,x+2}\,\left (200\,x^2+30\,x+53\right )}{3360000000}-\frac {44194603\,x\,{\left (5\,x^2+3\,x+2\right )}^{3/2}}{14000000}-\frac {42916771101\,\sqrt {5}\,\ln \left (2\,\sqrt {5\,x^2+3\,x+2}+\frac {\sqrt {5}\,\left (10\,x+3\right )}{5}\right )}{11200000000} \] Input:

int((x - 2*x^2 + 4)^3*(3*x + 5*x^2 + 2)^(1/2),x)
 

Output:

(25553*x^3*(3*x + 5*x^2 + 2)^(3/2))/21000 - (938857*x^2*(3*x + 5*x^2 + 2)^ 
(3/2))/350000 + (159*x^4*(3*x + 5*x^2 + 2)^(3/2))/350 - (x^5*(3*x + 5*x^2 
+ 2)^(3/2))/5 + (15258032693*5^(1/2)*log((3*x + 5*x^2 + 2)^(1/2) + (5^(1/2 
)*(5*x + 3/2))/5))/1400000000 + (492194603*(x/2 + 3/20)*(3*x + 5*x^2 + 2)^ 
(1/2))/7000000 + (461470657*(3*x + 5*x^2 + 2)^(1/2)*(30*x + 200*x^2 + 53)) 
/3360000000 - (44194603*x*(3*x + 5*x^2 + 2)^(3/2))/14000000 - (42916771101 
*5^(1/2)*log(2*(3*x + 5*x^2 + 2)^(1/2) + (5^(1/2)*(10*x + 3))/5))/11200000 
000
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.93 \[ \int \left (4+x-2 x^2\right )^3 \sqrt {2+3 x+5 x^2} \, dx=-\sqrt {5 x^{2}+3 x +2}\, x^{7}+\frac {117 \sqrt {5 x^{2}+3 x +2}\, x^{6}}{70}+\frac {29597 \sqrt {5 x^{2}+3 x +2}\, x^{5}}{4200}-\frac {619727 \sqrt {5 x^{2}+3 x +2}\, x^{4}}{70000}-\frac {179739113 \sqrt {5 x^{2}+3 x +2}\, x^{3}}{8400000}+\frac {1061199071 \sqrt {5 x^{2}+3 x +2}\, x^{2}}{84000000}+\frac {3691913833 \sqrt {5 x^{2}+3 x +2}\, x}{112000000}+\frac {59895956237 \sqrt {5 x^{2}+3 x +2}}{3360000000}+\frac {11306784349 \sqrt {5}\, \mathrm {log}\left (\frac {2 \sqrt {5 x^{2}+3 x +2}\, \sqrt {5}+10 x +3}{\sqrt {31}}\right )}{1600000000} \] Input:

int((-2*x^2+x+4)^3*(5*x^2+3*x+2)^(1/2),x)
 

Output:

( - 33600000000*sqrt(5*x**2 + 3*x + 2)*x**7 + 56160000000*sqrt(5*x**2 + 3* 
x + 2)*x**6 + 236776000000*sqrt(5*x**2 + 3*x + 2)*x**5 - 297468960000*sqrt 
(5*x**2 + 3*x + 2)*x**4 - 718956452000*sqrt(5*x**2 + 3*x + 2)*x**3 + 42447 
9628400*sqrt(5*x**2 + 3*x + 2)*x**2 + 1107574149900*sqrt(5*x**2 + 3*x + 2) 
*x + 598959562370*sqrt(5*x**2 + 3*x + 2) + 237442471329*sqrt(5)*log((2*sqr 
t(5*x**2 + 3*x + 2)*sqrt(5) + 10*x + 3)/sqrt(31)))/33600000000