\(\int (1+4 x-7 x^2)^2 (2+5 x+x^2) \sqrt {3+2 x+5 x^2} \, dx\) [3]

Optimal result

Integrand size = 35, antiderivative size = 166 \[ \int \left (1+4 x-7 x^2\right )^2 \left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2} \, dx=-\frac {2521723 (1+5 x) \sqrt {3+2 x+5 x^2}}{1250000}+\frac {198439 \left (3+2 x+5 x^2\right )^{3/2}}{750000}+\frac {1781669 x \left (3+2 x+5 x^2\right )^{3/2}}{250000}-\frac {77509 x^2 \left (3+2 x+5 x^2\right )^{3/2}}{25000}-\frac {25277 x^3 \left (3+2 x+5 x^2\right )^{3/2}}{3000}+\frac {989}{200} x^4 \left (3+2 x+5 x^2\right )^{3/2}+\frac {49}{40} x^5 \left (3+2 x+5 x^2\right )^{3/2}-\frac {17652061 \text {arcsinh}\left (\frac {1+5 x}{\sqrt {14}}\right )}{625000 \sqrt {5}} \] Output:

-2521723/1250000*(1+5*x)*(5*x^2+2*x+3)^(1/2)+198439/750000*(5*x^2+2*x+3)^( 
3/2)+1781669/250000*x*(5*x^2+2*x+3)^(3/2)-77509/25000*x^2*(5*x^2+2*x+3)^(3 
/2)-25277/3000*x^3*(5*x^2+2*x+3)^(3/2)+989/200*x^4*(5*x^2+2*x+3)^(3/2)+49/ 
40*x^5*(5*x^2+2*x+3)^(3/2)-17652061/3125000*arcsinh(1/14*(1+5*x)*14^(1/2)) 
*5^(1/2)
 

Mathematica [A] (verified)

Time = 0.84 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.54 \[ \int \left (1+4 x-7 x^2\right )^2 \left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2} \, dx=\frac {\sqrt {3+2 x+5 x^2} \left (-4588584+44333650 x+23531995 x^2+15583725 x^3-65693000 x^4-107112500 x^5+101906250 x^6+22968750 x^7\right )}{3750000}+\frac {17652061 \log \left (-1-5 x+\sqrt {5} \sqrt {3+2 x+5 x^2}\right )}{625000 \sqrt {5}} \] Input:

Integrate[(1 + 4*x - 7*x^2)^2*(2 + 5*x + x^2)*Sqrt[3 + 2*x + 5*x^2],x]
 

Output:

(Sqrt[3 + 2*x + 5*x^2]*(-4588584 + 44333650*x + 23531995*x^2 + 15583725*x^ 
3 - 65693000*x^4 - 107112500*x^5 + 101906250*x^6 + 22968750*x^7))/3750000 
+ (17652061*Log[-1 - 5*x + Sqrt[5]*Sqrt[3 + 2*x + 5*x^2]])/(625000*Sqrt[5] 
)
 

Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.19, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.371, Rules used = {2192, 2192, 27, 2192, 27, 2192, 27, 2192, 25, 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.

\(\displaystyle \int \left (-7 x^2+4 x+1\right )^2 \left (x^2+5 x+2\right ) \sqrt {5 x^2+2 x+3} \, dx\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{40} \int \sqrt {5 x^2+2 x+3} \left (6923 x^5-7935 x^4-3760 x^3+1800 x^2+840 x+80\right )dx+\frac {49}{40} \left (5 x^2+2 x+3\right )^{3/2} x^5\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{40} \left (\frac {1}{35} \int 14 \sqrt {5 x^2+2 x+3} \left (-25277 x^4-15334 x^3+4500 x^2+2100 x+200\right )dx+\frac {989}{5} \left (5 x^2+2 x+3\right )^{3/2} x^4\right )+\frac {49}{40} \left (5 x^2+2 x+3\right )^{3/2} x^5\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{40} \left (\frac {2}{5} \int \sqrt {5 x^2+2 x+3} \left (-25277 x^4-15334 x^3+4500 x^2+2100 x+200\right )dx+\frac {989}{5} \left (5 x^2+2 x+3\right )^{3/2} x^4\right )+\frac {49}{40} \left (5 x^2+2 x+3\right )^{3/2} x^5\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{40} \left (\frac {2}{5} \left (\frac {1}{30} \int 3 \sqrt {5 x^2+2 x+3} \left (-77509 x^3+120831 x^2+21000 x+2000\right )dx-\frac {25277}{30} x^3 \left (5 x^2+2 x+3\right )^{3/2}\right )+\frac {989}{5} \left (5 x^2+2 x+3\right )^{3/2} x^4\right )+\frac {49}{40} \left (5 x^2+2 x+3\right )^{3/2} x^5\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{40} \left (\frac {2}{5} \left (\frac {1}{10} \int \sqrt {5 x^2+2 x+3} \left (-77509 x^3+120831 x^2+21000 x+2000\right )dx-\frac {25277}{30} x^3 \left (5 x^2+2 x+3\right )^{3/2}\right )+\frac {989}{5} \left (5 x^2+2 x+3\right )^{3/2} x^4\right )+\frac {49}{40} \left (5 x^2+2 x+3\right )^{3/2} x^5\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{40} \left (\frac {2}{5} \left (\frac {1}{10} \left (\frac {1}{25} \int 2 \sqrt {5 x^2+2 x+3} \left (1781669 x^2+495027 x+25000\right )dx-\frac {77509}{25} x^2 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {25277}{30} x^3 \left (5 x^2+2 x+3\right )^{3/2}\right )+\frac {989}{5} \left (5 x^2+2 x+3\right )^{3/2} x^4\right )+\frac {49}{40} \left (5 x^2+2 x+3\right )^{3/2} x^5\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{40} \left (\frac {2}{5} \left (\frac {1}{10} \left (\frac {2}{25} \int \sqrt {5 x^2+2 x+3} \left (1781669 x^2+495027 x+25000\right )dx-\frac {77509}{25} x^2 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {25277}{30} x^3 \left (5 x^2+2 x+3\right )^{3/2}\right )+\frac {989}{5} \left (5 x^2+2 x+3\right )^{3/2} x^4\right )+\frac {49}{40} \left (5 x^2+2 x+3\right )^{3/2} x^5\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{40} \left (\frac {2}{5} \left (\frac {1}{10} \left (\frac {2}{25} \left (\frac {1}{20} \int -\left ((4845007-992195 x) \sqrt {5 x^2+2 x+3}\right )dx+\frac {1781669}{20} x \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {77509}{25} x^2 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {25277}{30} x^3 \left (5 x^2+2 x+3\right )^{3/2}\right )+\frac {989}{5} \left (5 x^2+2 x+3\right )^{3/2} x^4\right )+\frac {49}{40} \left (5 x^2+2 x+3\right )^{3/2} x^5\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{40} \left (\frac {2}{5} \left (\frac {1}{10} \left (\frac {2}{25} \left (\frac {1781669}{20} x \left (5 x^2+2 x+3\right )^{3/2}-\frac {1}{20} \int (4845007-992195 x) \sqrt {5 x^2+2 x+3}dx\right )-\frac {77509}{25} x^2 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {25277}{30} x^3 \left (5 x^2+2 x+3\right )^{3/2}\right )+\frac {989}{5} \left (5 x^2+2 x+3\right )^{3/2} x^4\right )+\frac {49}{40} \left (5 x^2+2 x+3\right )^{3/2} x^5\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {1}{40} \left (\frac {2}{5} \left (\frac {1}{10} \left (\frac {2}{25} \left (\frac {1}{20} \left (\frac {198439}{3} \left (5 x^2+2 x+3\right )^{3/2}-5043446 \int \sqrt {5 x^2+2 x+3}dx\right )+\frac {1781669}{20} x \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {77509}{25} x^2 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {25277}{30} x^3 \left (5 x^2+2 x+3\right )^{3/2}\right )+\frac {989}{5} \left (5 x^2+2 x+3\right )^{3/2} x^4\right )+\frac {49}{40} \left (5 x^2+2 x+3\right )^{3/2} x^5\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{40} \left (\frac {2}{5} \left (\frac {1}{10} \left (\frac {2}{25} \left (\frac {1}{20} \left (\frac {198439}{3} \left (5 x^2+2 x+3\right )^{3/2}-5043446 \left (\frac {7}{5} \int \frac {1}{\sqrt {5 x^2+2 x+3}}dx+\frac {1}{10} \sqrt {5 x^2+2 x+3} (5 x+1)\right )\right )+\frac {1781669}{20} x \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {77509}{25} x^2 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {25277}{30} x^3 \left (5 x^2+2 x+3\right )^{3/2}\right )+\frac {989}{5} \left (5 x^2+2 x+3\right )^{3/2} x^4\right )+\frac {49}{40} \left (5 x^2+2 x+3\right )^{3/2} x^5\)

\(\Big \downarrow \) 1090

\(\displaystyle \frac {1}{40} \left (\frac {2}{5} \left (\frac {1}{10} \left (\frac {2}{25} \left (\frac {1}{20} \left (\frac {198439}{3} \left (5 x^2+2 x+3\right )^{3/2}-5043446 \left (\frac {1}{10} \sqrt {\frac {7}{10}} \int \frac {1}{\sqrt {\frac {1}{56} (10 x+2)^2+1}}d(10 x+2)+\frac {1}{10} \sqrt {5 x^2+2 x+3} (5 x+1)\right )\right )+\frac {1781669}{20} x \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {77509}{25} x^2 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {25277}{30} x^3 \left (5 x^2+2 x+3\right )^{3/2}\right )+\frac {989}{5} \left (5 x^2+2 x+3\right )^{3/2} x^4\right )+\frac {49}{40} \left (5 x^2+2 x+3\right )^{3/2} x^5\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {1}{40} \left (\frac {2}{5} \left (\frac {1}{10} \left (\frac {2}{25} \left (\frac {1}{20} \left (\frac {198439}{3} \left (5 x^2+2 x+3\right )^{3/2}-5043446 \left (\frac {7 \text {arcsinh}\left (\frac {10 x+2}{2 \sqrt {14}}\right )}{5 \sqrt {5}}+\frac {1}{10} \sqrt {5 x^2+2 x+3} (5 x+1)\right )\right )+\frac {1781669}{20} x \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {77509}{25} x^2 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {25277}{30} x^3 \left (5 x^2+2 x+3\right )^{3/2}\right )+\frac {989}{5} \left (5 x^2+2 x+3\right )^{3/2} x^4\right )+\frac {49}{40} \left (5 x^2+2 x+3\right )^{3/2} x^5\)

Input:

Int[(1 + 4*x - 7*x^2)^2*(2 + 5*x + x^2)*Sqrt[3 + 2*x + 5*x^2],x]
 

Output:

(49*x^5*(3 + 2*x + 5*x^2)^(3/2))/40 + ((989*x^4*(3 + 2*x + 5*x^2)^(3/2))/5 
 + (2*((-25277*x^3*(3 + 2*x + 5*x^2)^(3/2))/30 + ((-77509*x^2*(3 + 2*x + 5 
*x^2)^(3/2))/25 + (2*((1781669*x*(3 + 2*x + 5*x^2)^(3/2))/20 + ((198439*(3 
 + 2*x + 5*x^2)^(3/2))/3 - 5043446*(((1 + 5*x)*Sqrt[3 + 2*x + 5*x^2])/10 + 
 (7*ArcSinh[(2 + 10*x)/(2*Sqrt[14])])/(5*Sqrt[5])))/20))/25)/10))/5)/40
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 = 0.34 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.39

method	result	size
risch	\(\frac {\left (22968750 x^{7}[phone] x^{6}-107112500 x^{5}-65693000 x^{4}[phone] x^{3}[phone] x^{2}[phone] x -4588584\right ) \sqrt {5 x^{2}+2 x +3}}{3750000}-\frac {17652061 \sqrt {5}\, \operatorname {arcsinh}\left (\frac {5 \sqrt {14}\, \left (x +\frac {1}{5}\right )}{14}\right )}{3125000}\)	\(65\)
trager	\(\left (\frac {49}{8} x^{7}+\frac {1087}{40} x^{6}-\frac {8569}{300} x^{5}-\frac {65693}{3750} x^{4}+\frac {207783}{50000} x^{3}+\frac {4706399}{750000} x^{2}+\frac {886673}{75000} x -\frac {191191}{156250}\right ) \sqrt {5 x^{2}+2 x +3}+\frac {17652061 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x -\operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )+5 \sqrt {5 x^{2}+2 x +3}\right )}{3125000}\)	\(91\)
default	\(-\frac {2521723 \left (10 x +2\right ) \sqrt {5 x^{2}+2 x +3}}{2500000}-\frac {17652061 \sqrt {5}\, \operatorname {arcsinh}\left (\frac {5 \sqrt {14}\, \left (x +\frac {1}{5}\right )}{14}\right )}{3125000}+\frac {198439 \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}}}{750000}+\frac {1781669 x \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}}}{250000}-\frac {77509 x^{2} \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}}}{25000}-\frac {25277 x^{3} \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}}}{3000}+\frac {989 x^{4} \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}}}{200}+\frac {49 x^{5} \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}}}{40}\)	\(132\)

Input:

int((-7*x^2+4*x+1)^2*(x^2+5*x+2)*(5*x^2+2*x+3)^(1/2),x,method=_RETURNVERBO 
SE)
 

Output:

1/3750000*(22968750*x^7+101906250*x^6-107112500*x^5-65693000*x^4+15583725* 
x^3+23531995*x^2+44333650*x-4588584)*(5*x^2+2*x+3)^(1/2)-17652061/3125000* 
5^(1/2)*arcsinh(5/14*14^(1/2)*(x+1/5))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.52 \[ \int \left (1+4 x-7 x^2\right )^2 \left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2} \, dx=\frac {1}{3750000} \, {\left (22968750 \, x^{7} + 101906250 \, x^{6} - 107112500 \, x^{5} - 65693000 \, x^{4} + 15583725 \, x^{3} + 23531995 \, x^{2} + 44333650 \, x - 4588584\right )} \sqrt {5 \, x^{2} + 2 \, x + 3} + \frac {17652061}{6250000} \, \sqrt {5} \log \left (\sqrt {5} \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (5 \, x + 1\right )} - 25 \, x^{2} - 10 \, x - 8\right ) \] Input:

integrate((-7*x^2+4*x+1)^2*(x^2+5*x+2)*(5*x^2+2*x+3)^(1/2),x, algorithm="f 
ricas")
 

Output:

1/3750000*(22968750*x^7 + 101906250*x^6 - 107112500*x^5 - 65693000*x^4 + 1 
5583725*x^3 + 23531995*x^2 + 44333650*x - 4588584)*sqrt(5*x^2 + 2*x + 3) + 
 17652061/6250000*sqrt(5)*log(sqrt(5)*sqrt(5*x^2 + 2*x + 3)*(5*x + 1) - 25 
*x^2 - 10*x - 8)
 

Sympy [A] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.51 \[ \int \left (1+4 x-7 x^2\right )^2 \left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2} \, dx=\sqrt {5 x^{2} + 2 x + 3} \cdot \left (\frac {49 x^{7}}{8} + \frac {1087 x^{6}}{40} - \frac {8569 x^{5}}{300} - \frac {65693 x^{4}}{3750} + \frac {207783 x^{3}}{50000} + \frac {4706399 x^{2}}{750000} + \frac {886673 x}{75000} - \frac {191191}{156250}\right ) - \frac {17652061 \sqrt {5} \operatorname {asinh}{\left (\frac {5 \sqrt {14} \left (x + \frac {1}{5}\right )}{14} \right )}}{3125000} \] Input:

integrate((-7*x**2+4*x+1)**2*(x**2+5*x+2)*(5*x**2+2*x+3)**(1/2),x)
 

Output:

sqrt(5*x**2 + 2*x + 3)*(49*x**7/8 + 1087*x**6/40 - 8569*x**5/300 - 65693*x 
**4/3750 + 207783*x**3/50000 + 4706399*x**2/750000 + 886673*x/75000 - 1911 
91/156250) - 17652061*sqrt(5)*asinh(5*sqrt(14)*(x + 1/5)/14)/3125000
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.86 \[ \int \left (1+4 x-7 x^2\right )^2 \left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2} \, dx=\frac {49}{40} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} x^{5} + \frac {989}{200} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} x^{4} - \frac {25277}{3000} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} x^{3} - \frac {77509}{25000} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} x^{2} + \frac {1781669}{250000} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} x + \frac {198439}{750000} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} - \frac {2521723}{250000} \, \sqrt {5 \, x^{2} + 2 \, x + 3} x - \frac {17652061}{3125000} \, \sqrt {5} \operatorname {arsinh}\left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) - \frac {2521723}{1250000} \, \sqrt {5 \, x^{2} + 2 \, x + 3} \] Input:

integrate((-7*x^2+4*x+1)^2*(x^2+5*x+2)*(5*x^2+2*x+3)^(1/2),x, algorithm="m 
axima")
 

Output:

49/40*(5*x^2 + 2*x + 3)^(3/2)*x^5 + 989/200*(5*x^2 + 2*x + 3)^(3/2)*x^4 - 
25277/3000*(5*x^2 + 2*x + 3)^(3/2)*x^3 - 77509/25000*(5*x^2 + 2*x + 3)^(3/ 
2)*x^2 + 1781669/250000*(5*x^2 + 2*x + 3)^(3/2)*x + 198439/750000*(5*x^2 + 
 2*x + 3)^(3/2) - 2521723/250000*sqrt(5*x^2 + 2*x + 3)*x - 17652061/312500 
0*sqrt(5)*arcsinh(1/14*sqrt(14)*(5*x + 1)) - 2521723/1250000*sqrt(5*x^2 + 
2*x + 3)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.49 \[ \int \left (1+4 x-7 x^2\right )^2 \left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2} \, dx=\frac {1}{3750000} \, {\left (5 \, {\left ({\left (5 \, {\left (10 \, {\left (25 \, {\left (15 \, {\left (245 \, x + 1087\right )} x - 17138\right )} x - 262772\right )} x + 623349\right )} x + 4706399\right )} x + 8866730\right )} x - 4588584\right )} \sqrt {5 \, x^{2} + 2 \, x + 3} + \frac {17652061}{3125000} \, \sqrt {5} \log \left (-\sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )} - 1\right ) \] Input:

integrate((-7*x^2+4*x+1)^2*(x^2+5*x+2)*(5*x^2+2*x+3)^(1/2),x, algorithm="g 
iac")
 

Output:

1/3750000*(5*((5*(10*(25*(15*(245*x + 1087)*x - 17138)*x - 262772)*x + 623 
349)*x + 4706399)*x + 8866730)*x - 4588584)*sqrt(5*x^2 + 2*x + 3) + 176520 
61/3125000*sqrt(5)*log(-sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3)) - 1)
 

Mupad [B] (verification not implemented)

Time = 19.06 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.13 \[ \int \left (1+4 x-7 x^2\right )^2 \left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2} \, dx=\frac {989\,x^4\,{\left (5\,x^2+2\,x+3\right )}^{3/2}}{200}-\frac {25277\,x^3\,{\left (5\,x^2+2\,x+3\right )}^{3/2}}{3000}-\frac {77509\,x^2\,{\left (5\,x^2+2\,x+3\right )}^{3/2}}{25000}+\frac {49\,x^5\,{\left (5\,x^2+2\,x+3\right )}^{3/2}}{40}-\frac {33915049\,\sqrt {5}\,\ln \left (\sqrt {5\,x^2+2\,x+3}+\frac {\sqrt {5}\,\left (5\,x+1\right )}{5}\right )}{6250000}-\frac {4845007\,\left (\frac {x}{2}+\frac {1}{10}\right )\,\sqrt {5\,x^2+2\,x+3}}{250000}+\frac {198439\,\sqrt {5\,x^2+2\,x+3}\,\left (200\,x^2+20\,x+108\right )}{30000000}+\frac {1781669\,x\,{\left (5\,x^2+2\,x+3\right )}^{3/2}}{250000}-\frac {1389073\,\sqrt {5}\,\ln \left (2\,\sqrt {5\,x^2+2\,x+3}+\frac {\sqrt {5}\,\left (10\,x+2\right )}{5}\right )}{6250000} \] Input:

int((5*x + x^2 + 2)*(2*x + 5*x^2 + 3)^(1/2)*(4*x - 7*x^2 + 1)^2,x)
                                                                                    
                                                                                    
 

Output:

(989*x^4*(2*x + 5*x^2 + 3)^(3/2))/200 - (25277*x^3*(2*x + 5*x^2 + 3)^(3/2) 
)/3000 - (77509*x^2*(2*x + 5*x^2 + 3)^(3/2))/25000 + (49*x^5*(2*x + 5*x^2 
+ 3)^(3/2))/40 - (33915049*5^(1/2)*log((2*x + 5*x^2 + 3)^(1/2) + (5^(1/2)* 
(5*x + 1))/5))/6250000 - (4845007*(x/2 + 1/10)*(2*x + 5*x^2 + 3)^(1/2))/25 
0000 + (198439*(2*x + 5*x^2 + 3)^(1/2)*(20*x + 200*x^2 + 108))/30000000 + 
(1781669*x*(2*x + 5*x^2 + 3)^(3/2))/250000 - (1389073*5^(1/2)*log(2*(2*x + 
 5*x^2 + 3)^(1/2) + (5^(1/2)*(10*x + 2))/5))/6250000
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.92 \[ \int \left (1+4 x-7 x^2\right )^2 \left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2} \, dx=\frac {49 \sqrt {5 x^{2}+2 x +3}\, x^{7}}{8}+\frac {1087 \sqrt {5 x^{2}+2 x +3}\, x^{6}}{40}-\frac {8569 \sqrt {5 x^{2}+2 x +3}\, x^{5}}{300}-\frac {65693 \sqrt {5 x^{2}+2 x +3}\, x^{4}}{3750}+\frac {207783 \sqrt {5 x^{2}+2 x +3}\, x^{3}}{50000}+\frac {4706399 \sqrt {5 x^{2}+2 x +3}\, x^{2}}{750000}+\frac {886673 \sqrt {5 x^{2}+2 x +3}\, x}{75000}-\frac {191191 \sqrt {5 x^{2}+2 x +3}}{156250}-\frac {17652061 \sqrt {5}\, \mathrm {log}\left (\frac {\sqrt {5 x^{2}+2 x +3}\, \sqrt {5}+5 x +1}{\sqrt {14}}\right )}{3125000} \] Input:

int((-7*x^2+4*x+1)^2*(x^2+5*x+2)*(5*x^2+2*x+3)^(1/2),x)
 

Output:

(114843750*sqrt(5*x**2 + 2*x + 3)*x**7 + 509531250*sqrt(5*x**2 + 2*x + 3)* 
x**6 - 535562500*sqrt(5*x**2 + 2*x + 3)*x**5 - 328465000*sqrt(5*x**2 + 2*x 
 + 3)*x**4 + 77918625*sqrt(5*x**2 + 2*x + 3)*x**3 + 117659975*sqrt(5*x**2 
+ 2*x + 3)*x**2 + 221668250*sqrt(5*x**2 + 2*x + 3)*x - 22942920*sqrt(5*x** 
2 + 2*x + 3) - 105912366*sqrt(5)*log((sqrt(5*x**2 + 2*x + 3)*sqrt(5) + 5*x 
 + 1)/sqrt(14)))/18750000