\(\int \frac {2+5 x+x^2}{(1+4 x-7 x^2) \sqrt {3+2 x+5 x^2}} \, dx\) [17]

Optimal result

Integrand size = 35, antiderivative size = 164 \[ \int \frac {2+5 x+x^2}{\left (1+4 x-7 x^2\right ) \sqrt {3+2 x+5 x^2}} \, dx=-\frac {\text {arcsinh}\left (\frac {1+5 x}{\sqrt {14}}\right )}{7 \sqrt {5}}-\frac {3}{14} \sqrt {\frac {4091-1055 \sqrt {11}}{2794}} \text {arctanh}\left (\frac {23-\sqrt {11}+\left (17-5 \sqrt {11}\right ) x}{\sqrt {2 \left (125-17 \sqrt {11}\right )} \sqrt {3+2 x+5 x^2}}\right )+\frac {3}{14} \sqrt {\frac {4091+1055 \sqrt {11}}{2794}} \text {arctanh}\left (\frac {23+\sqrt {11}+\left (17+5 \sqrt {11}\right ) x}{\sqrt {2 \left (125+17 \sqrt {11}\right )} \sqrt {3+2 x+5 x^2}}\right ) \] Output:

-1/35*arcsinh(1/14*(1+5*x)*14^(1/2))*5^(1/2)-3/39116*(11430254-2947670*11^ 
(1/2))^(1/2)*arctanh((23-11^(1/2)+(17-5*11^(1/2))*x)/(250-34*11^(1/2))^(1/ 
2)/(5*x^2+2*x+3)^(1/2))+3/39116*(11430254+2947670*11^(1/2))^(1/2)*arctanh( 
(23+11^(1/2)+(17+5*11^(1/2))*x)/(250+34*11^(1/2))^(1/2)/(5*x^2+2*x+3)^(1/2 
))
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.47 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.29 \[ \int \frac {2+5 x+x^2}{\left (1+4 x-7 x^2\right ) \sqrt {3+2 x+5 x^2}} \, dx=\frac {\log \left (-1-5 x+\sqrt {5} \sqrt {3+2 x+5 x^2}\right )}{7 \sqrt {5}}+\frac {3}{14} \text {RootSum}\left [83-16 \sqrt {5} \text {$\#$1}-70 \text {$\#$1}^2+8 \sqrt {5} \text {$\#$1}^3+7 \text {$\#$1}^4\&,\frac {29 \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right )+10 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}-13 \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-4 \sqrt {5}-35 \text {$\#$1}+6 \sqrt {5} \text {$\#$1}^2+7 \text {$\#$1}^3}\&\right ] \] Input:

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

Output:

Log[-1 - 5*x + Sqrt[5]*Sqrt[3 + 2*x + 5*x^2]]/(7*Sqrt[5]) + (3*RootSum[83 
- 16*Sqrt[5]*#1 - 70*#1^2 + 8*Sqrt[5]*#1^3 + 7*#1^4 & , (29*Log[-(Sqrt[5]* 
x) + Sqrt[3 + 2*x + 5*x^2] - #1] + 10*Sqrt[5]*Log[-(Sqrt[5]*x) + Sqrt[3 + 
2*x + 5*x^2] - #1]*#1 - 13*Log[-(Sqrt[5]*x) + Sqrt[3 + 2*x + 5*x^2] - #1]* 
#1^2)/(-4*Sqrt[5] - 35*#1 + 6*Sqrt[5]*#1^2 + 7*#1^3) & ])/14
 

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.13, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {2143, 27, 1090, 222, 1365, 27, 1154, 219}

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.

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

\(\Big \downarrow \) 2143

\(\displaystyle -\frac {1}{7} \int \frac {1}{\sqrt {5 x^2+2 x+3}}dx-\frac {1}{7} \int -\frac {3 (13 x+5)}{\left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{7} \int \frac {13 x+5}{\left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}dx-\frac {1}{7} \int \frac {1}{\sqrt {5 x^2+2 x+3}}dx\)

\(\Big \downarrow \) 1090

\(\displaystyle \frac {3}{7} \int \frac {13 x+5}{\left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}dx-\frac {\int \frac {1}{\sqrt {\frac {1}{56} (10 x+2)^2+1}}d(10 x+2)}{14 \sqrt {70}}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {3}{7} \int \frac {13 x+5}{\left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}dx-\frac {\text {arcsinh}\left (\frac {10 x+2}{2 \sqrt {14}}\right )}{7 \sqrt {5}}\)

\(\Big \downarrow \) 1365

\(\displaystyle \frac {3}{7} \left (\frac {1}{11} \left (143-61 \sqrt {11}\right ) \int \frac {1}{2 \left (-7 x-\sqrt {11}+2\right ) \sqrt {5 x^2+2 x+3}}dx+\frac {1}{11} \left (143+61 \sqrt {11}\right ) \int \frac {1}{2 \left (-7 x+\sqrt {11}+2\right ) \sqrt {5 x^2+2 x+3}}dx\right )-\frac {\text {arcsinh}\left (\frac {10 x+2}{2 \sqrt {14}}\right )}{7 \sqrt {5}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{7} \left (\frac {1}{22} \left (143-61 \sqrt {11}\right ) \int \frac {1}{\left (-7 x-\sqrt {11}+2\right ) \sqrt {5 x^2+2 x+3}}dx+\frac {1}{22} \left (143+61 \sqrt {11}\right ) \int \frac {1}{\left (-7 x+\sqrt {11}+2\right ) \sqrt {5 x^2+2 x+3}}dx\right )-\frac {\text {arcsinh}\left (\frac {10 x+2}{2 \sqrt {14}}\right )}{7 \sqrt {5}}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {3}{7} \left (-\frac {1}{11} \left (143-61 \sqrt {11}\right ) \int \frac {1}{8 \left (125-17 \sqrt {11}\right )-\frac {4 \left (\left (17-5 \sqrt {11}\right ) x-\sqrt {11}+23\right )^2}{5 x^2+2 x+3}}d\left (-\frac {2 \left (\left (17-5 \sqrt {11}\right ) x-\sqrt {11}+23\right )}{\sqrt {5 x^2+2 x+3}}\right )-\frac {1}{11} \left (143+61 \sqrt {11}\right ) \int \frac {1}{8 \left (125+17 \sqrt {11}\right )-\frac {4 \left (\left (17+5 \sqrt {11}\right ) x+\sqrt {11}+23\right )^2}{5 x^2+2 x+3}}d\left (-\frac {2 \left (\left (17+5 \sqrt {11}\right ) x+\sqrt {11}+23\right )}{\sqrt {5 x^2+2 x+3}}\right )\right )-\frac {\text {arcsinh}\left (\frac {10 x+2}{2 \sqrt {14}}\right )}{7 \sqrt {5}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {3}{7} \left (\frac {\left (143-61 \sqrt {11}\right ) \text {arctanh}\left (\frac {\left (17-5 \sqrt {11}\right ) x-\sqrt {11}+23}{\sqrt {2 \left (125-17 \sqrt {11}\right )} \sqrt {5 x^2+2 x+3}}\right )}{22 \sqrt {2 \left (125-17 \sqrt {11}\right )}}+\frac {\left (143+61 \sqrt {11}\right ) \text {arctanh}\left (\frac {\left (17+5 \sqrt {11}\right ) x+\sqrt {11}+23}{\sqrt {2 \left (125+17 \sqrt {11}\right )} \sqrt {5 x^2+2 x+3}}\right )}{22 \sqrt {2 \left (125+17 \sqrt {11}\right )}}\right )-\frac {\text {arcsinh}\left (\frac {10 x+2}{2 \sqrt {14}}\right )}{7 \sqrt {5}}\)

Input:

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

Output:

-1/7*ArcSinh[(2 + 10*x)/(2*Sqrt[14])]/Sqrt[5] + (3*(((143 - 61*Sqrt[11])*A 
rcTanh[(23 - Sqrt[11] + (17 - 5*Sqrt[11])*x)/(Sqrt[2*(125 - 17*Sqrt[11])]* 
Sqrt[3 + 2*x + 5*x^2])])/(22*Sqrt[2*(125 - 17*Sqrt[11])]) + ((143 + 61*Sqr 
t[11])*ArcTanh[(23 + Sqrt[11] + (17 + 5*Sqrt[11])*x)/(Sqrt[2*(125 + 17*Sqr 
t[11])]*Sqrt[3 + 2*x + 5*x^2])])/(22*Sqrt[2*(125 + 17*Sqrt[11])])))/7
 

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

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 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 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1365

Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + ( 
e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Sim 
p[(2*c*g - h*(b - q))/q   Int[1/((b - q + 2*c*x)*Sqrt[d + e*x + f*x^2]), x] 
, x] - Simp[(2*c*g - h*(b + q))/q   Int[1/((b + q + 2*c*x)*Sqrt[d + e*x + f 
*x^2]), x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0 
] && NeQ[e^2 - 4*d*f, 0] && PosQ[b^2 - 4*a*c]
 

rule 2143

Int[(Px_)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_ 
.)*(x_)^2]), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C 
 = Coeff[Px, x, 2]}, Simp[C/c   Int[1/Sqrt[d + e*x + f*x^2], x], x] + Simp[ 
1/c   Int[(A*c - a*C + (B*c - b*C)*x)/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x 
^2]), x], x]] /; FreeQ[{a, b, c, d, e, f}, x] && PolyQ[Px, x, 2]
 

Maple [A] (verified)

Time = 1.09 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.24

method	result	size
default	\(-\frac {\sqrt {5}\, \operatorname {arcsinh}\left (\frac {5 \sqrt {14}\, \left (x +\frac {1}{5}\right )}{14}\right )}{35}+\frac {3 \left (-61+13 \sqrt {11}\right ) \sqrt {11}\, \operatorname {arctanh}\left (\frac {250-34 \sqrt {11}+\frac {49 \left (\frac {34}{7}-\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}+\frac {\sqrt {11}}{7}\right )}{2}}{\sqrt {250-34 \sqrt {11}}\, \sqrt {245 \left (x -\frac {2}{7}+\frac {\sqrt {11}}{7}\right )^{2}+49 \left (\frac {34}{7}-\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}+\frac {\sqrt {11}}{7}\right )+250-34 \sqrt {11}}}\right )}{154 \sqrt {250-34 \sqrt {11}}}+\frac {3 \left (61+13 \sqrt {11}\right ) \sqrt {11}\, \operatorname {arctanh}\left (\frac {250+34 \sqrt {11}+\frac {49 \left (\frac {34}{7}+\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}-\frac {\sqrt {11}}{7}\right )}{2}}{\sqrt {250+34 \sqrt {11}}\, \sqrt {245 \left (x -\frac {2}{7}-\frac {\sqrt {11}}{7}\right )^{2}+49 \left (\frac {34}{7}+\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}-\frac {\sqrt {11}}{7}\right )+250+34 \sqrt {11}}}\right )}{154 \sqrt {250+34 \sqrt {11}}}\)	\(204\)
trager	\(\frac {\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 )}{35}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}[phone] \operatorname {RootOf}\left (24095456 \textit {\_Z}^{4}-3240072 \textit {\_Z}^{2}+29241\right )^{2}-51436143\right ) \ln \left (\frac {9459586880128 \operatorname {RootOf}\left (\textit {\_Z}^{2}[phone] \operatorname {RootOf}\left (24095456 \textit {\_Z}^{4}-3240072 \textit {\_Z}^{2}+29241\right )^{2}-51436143\right ) \operatorname {RootOf}\left (24095456 \textit {\_Z}^{4}-3240072 \textit {\_Z}^{2}+29241\right )^{4} x -1347145103304 \operatorname {RootOf}\left (24095456 \textit {\_Z}^{4}-3240072 \textit {\_Z}^{2}+29241\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}[phone] \operatorname {RootOf}\left (24095456 \textit {\_Z}^{4}-3240072 \textit {\_Z}^{2}+29241\right )^{2}-51436143\right ) x [phone] \sqrt {5 x^{2}+2 x +3}\, \operatorname {RootOf}\left (24095456 \textit {\_Z}^{4}-3240072 \textit {\_Z}^{2}+29241\right )^{2}-866938620240 \operatorname {RootOf}\left (24095456 \textit {\_Z}^{4}-3240072 \textit {\_Z}^{2}+29241\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}[phone] \operatorname {RootOf}\left (24095456 \textit {\_Z}^{4}-3240072 \textit {\_Z}^{2}+29241\right )^{2}-51436143\right )[phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}[phone] \operatorname {RootOf}\left (24095456 \textit {\_Z}^{4}-3240072 \textit {\_Z}^{2}+29241\right )^{2}-51436143\right ) x [phone] \sqrt {5 x^{2}+2 x +3}[phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}[phone] \operatorname {RootOf}\left (24095456 \textit {\_Z}^{4}-3240072 \textit {\_Z}^{2}+29241\right )^{2}-51436143\right )}{547624 x \operatorname {RootOf}\left (24095456 \textit {\_Z}^{4}-3240072 \textit {\_Z}^{2}+29241\right )^{2}-17829 x +9495}\right )}{19558}-\operatorname {RootOf}\left (24095456 \textit {\_Z}^{4}-3240072 \textit {\_Z}^{2}+29241\right ) \ln \left (-\frac {16819145472867584 x \operatorname {RootOf}\left (24095456 \textit {\_Z}^{4}-3240072 \textit {\_Z}^{2}+29241\right )^{5}-2128055608136304 \operatorname {RootOf}\left (24095456 \textit {\_Z}^{4}-3240072 \textit {\_Z}^{2}+29241\right )^{3} x [phone] \sqrt {5 x^{2}+2 x +3}\, \operatorname {RootOf}\left (24095456 \textit {\_Z}^{4}-3240072 \textit {\_Z}^{2}+29241\right )^{2}+1541416866786720 \operatorname {RootOf}\left (24095456 \textit {\_Z}^{4}-3240072 \textit {\_Z}^{2}+29241\right )^{3}[phone] \operatorname {RootOf}\left (24095456 \textit {\_Z}^{4}-3240072 \textit {\_Z}^{2}+29241\right ) x -10498126497165 \sqrt {5 x^{2}+2 x +3}-183024937404450 \operatorname {RootOf}\left (24095456 \textit {\_Z}^{4}-3240072 \textit {\_Z}^{2}+29241\right )}{547624 x \operatorname {RootOf}\left (24095456 \textit {\_Z}^{4}-3240072 \textit {\_Z}^{2}+29241\right )^{2}-55809 x -9495}\right )\)	\(483\)

Input:

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

Output:

-1/35*5^(1/2)*arcsinh(5/14*14^(1/2)*(x+1/5))+3/154*(-61+13*11^(1/2))*11^(1 
/2)/(250-34*11^(1/2))^(1/2)*arctanh(49/2*(500/49-68/49*11^(1/2)+(34/7-10/7 
*11^(1/2))*(x-2/7+1/7*11^(1/2)))/(250-34*11^(1/2))^(1/2)/(245*(x-2/7+1/7*1 
1^(1/2))^2+49*(34/7-10/7*11^(1/2))*(x-2/7+1/7*11^(1/2))+250-34*11^(1/2))^( 
1/2))+3/154*(61+13*11^(1/2))*11^(1/2)/(250+34*11^(1/2))^(1/2)*arctanh(49/2 
*(500/49+68/49*11^(1/2)+(34/7+10/7*11^(1/2))*(x-2/7-1/7*11^(1/2)))/(250+34 
*11^(1/2))^(1/2)/(245*(x-2/7-1/7*11^(1/2))^2+49*(34/7+10/7*11^(1/2))*(x-2/ 
7-1/7*11^(1/2))+250+34*11^(1/2))^(1/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 289 vs. \(2 (114) = 228\).

Time = 0.10 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.76 \[ \int \frac {2+5 x+x^2}{\left (1+4 x-7 x^2\right ) \sqrt {3+2 x+5 x^2}} \, dx=-\frac {3}{28} \, \sqrt {\frac {1}{2794}} \sqrt {1055 \, \sqrt {11} + 4091} \log \left (\frac {3 \, {\left (2 \, \sqrt {\frac {1}{2794}} \sqrt {5 \, x^{2} + 2 \, x + 3} \sqrt {1055 \, \sqrt {11} + 4091} {\left (172 \, \sqrt {11} - 715\right )} + 133 \, \sqrt {11} {\left (x + 3\right )} + 399 \, x - 665\right )}}{x}\right ) + \frac {3}{28} \, \sqrt {\frac {1}{2794}} \sqrt {1055 \, \sqrt {11} + 4091} \log \left (-\frac {3 \, {\left (2 \, \sqrt {\frac {1}{2794}} \sqrt {5 \, x^{2} + 2 \, x + 3} \sqrt {1055 \, \sqrt {11} + 4091} {\left (172 \, \sqrt {11} - 715\right )} - 133 \, \sqrt {11} {\left (x + 3\right )} - 399 \, x + 665\right )}}{x}\right ) + \frac {1}{70} \, \sqrt {5} \log \left (\sqrt {5} \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (5 \, x + 1\right )} - 25 \, x^{2} - 10 \, x - 8\right ) - \frac {1}{28} \, \sqrt {-\frac {9495}{2794} \, \sqrt {11} + \frac {36819}{2794}} \log \left (-\frac {2 \, \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (172 \, \sqrt {11} + 715\right )} \sqrt {-\frac {9495}{2794} \, \sqrt {11} + \frac {36819}{2794}} + 399 \, \sqrt {11} {\left (x + 3\right )} - 1197 \, x + 1995}{x}\right ) + \frac {1}{28} \, \sqrt {-\frac {9495}{2794} \, \sqrt {11} + \frac {36819}{2794}} \log \left (\frac {2 \, \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (172 \, \sqrt {11} + 715\right )} \sqrt {-\frac {9495}{2794} \, \sqrt {11} + \frac {36819}{2794}} - 399 \, \sqrt {11} {\left (x + 3\right )} + 1197 \, x - 1995}{x}\right ) \] Input:

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

Output:

-3/28*sqrt(1/2794)*sqrt(1055*sqrt(11) + 4091)*log(3*(2*sqrt(1/2794)*sqrt(5 
*x^2 + 2*x + 3)*sqrt(1055*sqrt(11) + 4091)*(172*sqrt(11) - 715) + 133*sqrt 
(11)*(x + 3) + 399*x - 665)/x) + 3/28*sqrt(1/2794)*sqrt(1055*sqrt(11) + 40 
91)*log(-3*(2*sqrt(1/2794)*sqrt(5*x^2 + 2*x + 3)*sqrt(1055*sqrt(11) + 4091 
)*(172*sqrt(11) - 715) - 133*sqrt(11)*(x + 3) - 399*x + 665)/x) + 1/70*sqr 
t(5)*log(sqrt(5)*sqrt(5*x^2 + 2*x + 3)*(5*x + 1) - 25*x^2 - 10*x - 8) - 1/ 
28*sqrt(-9495/2794*sqrt(11) + 36819/2794)*log(-(2*sqrt(5*x^2 + 2*x + 3)*(1 
72*sqrt(11) + 715)*sqrt(-9495/2794*sqrt(11) + 36819/2794) + 399*sqrt(11)*( 
x + 3) - 1197*x + 1995)/x) + 1/28*sqrt(-9495/2794*sqrt(11) + 36819/2794)*l 
og((2*sqrt(5*x^2 + 2*x + 3)*(172*sqrt(11) + 715)*sqrt(-9495/2794*sqrt(11) 
+ 36819/2794) - 399*sqrt(11)*(x + 3) + 1197*x - 1995)/x)
 

Sympy [F]

\[ \int \frac {2+5 x+x^2}{\left (1+4 x-7 x^2\right ) \sqrt {3+2 x+5 x^2}} \, dx=- \int \frac {5 x}{7 x^{2} \sqrt {5 x^{2} + 2 x + 3} - 4 x \sqrt {5 x^{2} + 2 x + 3} - \sqrt {5 x^{2} + 2 x + 3}}\, dx - \int \frac {x^{2}}{7 x^{2} \sqrt {5 x^{2} + 2 x + 3} - 4 x \sqrt {5 x^{2} + 2 x + 3} - \sqrt {5 x^{2} + 2 x + 3}}\, dx - \int \frac {2}{7 x^{2} \sqrt {5 x^{2} + 2 x + 3} - 4 x \sqrt {5 x^{2} + 2 x + 3} - \sqrt {5 x^{2} + 2 x + 3}}\, dx \] Input:

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

Output:

-Integral(5*x/(7*x**2*sqrt(5*x**2 + 2*x + 3) - 4*x*sqrt(5*x**2 + 2*x + 3) 
- sqrt(5*x**2 + 2*x + 3)), x) - Integral(x**2/(7*x**2*sqrt(5*x**2 + 2*x + 
3) - 4*x*sqrt(5*x**2 + 2*x + 3) - sqrt(5*x**2 + 2*x + 3)), x) - Integral(2 
/(7*x**2*sqrt(5*x**2 + 2*x + 3) - 4*x*sqrt(5*x**2 + 2*x + 3) - sqrt(5*x**2 
 + 2*x + 3)), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 465 vs. \(2 (114) = 228\).

Time = 0.16 (sec) , antiderivative size = 465, normalized size of antiderivative = 2.84 \[ \int \frac {2+5 x+x^2}{\left (1+4 x-7 x^2\right ) \sqrt {3+2 x+5 x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/10780*sqrt(11)*(28*sqrt(11)*sqrt(5)*arcsinh(5/14*sqrt(7)*sqrt(2)*x + 1/ 
14*sqrt(7)*sqrt(2)) - 1365*sqrt(11)*sqrt(2)*arcsinh(5/7*sqrt(11)*sqrt(7)*s 
qrt(2)*x/abs(14*x - 2*sqrt(11) - 4) + 17/7*sqrt(7)*sqrt(2)*x/abs(14*x - 2* 
sqrt(11) - 4) + 1/7*sqrt(11)*sqrt(7)*sqrt(2)/abs(14*x - 2*sqrt(11) - 4) + 
23/7*sqrt(7)*sqrt(2)/abs(14*x - 2*sqrt(11) - 4))/sqrt(17*sqrt(11) + 125) + 
 390*sqrt(11)*arcsinh(5/7*sqrt(11)*sqrt(7)*sqrt(2)*x/abs(14*x + 2*sqrt(11) 
 - 4) - 17/7*sqrt(7)*sqrt(2)*x/abs(14*x + 2*sqrt(11) - 4) + 1/7*sqrt(11)*s 
qrt(7)*sqrt(2)/abs(14*x + 2*sqrt(11) - 4) - 23/7*sqrt(7)*sqrt(2)/abs(14*x 
+ 2*sqrt(11) - 4))/sqrt(-34/49*sqrt(11) + 250/49) - 6405*sqrt(2)*arcsinh(5 
/7*sqrt(11)*sqrt(7)*sqrt(2)*x/abs(14*x - 2*sqrt(11) - 4) + 17/7*sqrt(7)*sq 
rt(2)*x/abs(14*x - 2*sqrt(11) - 4) + 1/7*sqrt(11)*sqrt(7)*sqrt(2)/abs(14*x 
 - 2*sqrt(11) - 4) + 23/7*sqrt(7)*sqrt(2)/abs(14*x - 2*sqrt(11) - 4))/sqrt 
(17*sqrt(11) + 125) - 1830*arcsinh(5/7*sqrt(11)*sqrt(7)*sqrt(2)*x/abs(14*x 
 + 2*sqrt(11) - 4) - 17/7*sqrt(7)*sqrt(2)*x/abs(14*x + 2*sqrt(11) - 4) + 1 
/7*sqrt(11)*sqrt(7)*sqrt(2)/abs(14*x + 2*sqrt(11) - 4) - 23/7*sqrt(7)*sqrt 
(2)/abs(14*x + 2*sqrt(11) - 4))/sqrt(-34/49*sqrt(11) + 250/49))
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.76 \[ \int \frac {2+5 x+x^2}{\left (1+4 x-7 x^2\right ) \sqrt {3+2 x+5 x^2}} \, dx=\frac {1}{35} \, \sqrt {5} \log \left (-5 \, \sqrt {5} x - \sqrt {5} + 5 \, \sqrt {5 \, x^{2} + 2 \, x + 3}\right ) + 0.353184817631429 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 2 \, x + 3} + 4.41924736459000\right ) - 0.0986339689905714 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 2 \, x + 3} + 1.25295163054000\right ) - 0.353184817631429 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 2 \, x + 3} - 1.02258038113000\right ) + 0.0986339689905714 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 2 \, x + 3} - 2.09411235400000\right ) \] Input:

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

Output:

1/35*sqrt(5)*log(-5*sqrt(5)*x - sqrt(5) + 5*sqrt(5*x^2 + 2*x + 3)) + 0.353 
184817631429*log(-sqrt(5)*x + sqrt(5*x^2 + 2*x + 3) + 4.41924736459000) - 
0.0986339689905714*log(-sqrt(5)*x + sqrt(5*x^2 + 2*x + 3) + 1.252951630540 
00) - 0.353184817631429*log(-sqrt(5)*x + sqrt(5*x^2 + 2*x + 3) - 1.0225803 
8113000) + 0.0986339689905714*log(-sqrt(5)*x + sqrt(5*x^2 + 2*x + 3) - 2.0 
9411235400000)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {2+5 x+x^2}{\left (1+4 x-7 x^2\right ) \sqrt {3+2 x+5 x^2}} \, dx=\int \frac {x^2+5\,x+2}{\sqrt {5\,x^2+2\,x+3}\,\left (-7\,x^2+4\,x+1\right )} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {2+5 x+x^2}{\left (1+4 x-7 x^2\right ) \sqrt {3+2 x+5 x^2}} \, dx=\int \frac {x^{2}+5 x +2}{\left (-7 x^{2}+4 x +1\right ) \sqrt {5 x^{2}+2 x +3}}d x \] Input:

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

Output:

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