\(\int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^5 \sqrt {3-x+2 x^2}} \, dx\) [186]

Optimal result

Integrand size = 40, antiderivative size = 139 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^5 \sqrt {3-x+2 x^2}} \, dx=-\frac {3667 \sqrt {3-x+2 x^2}}{2304 (5+2 x)^4}+\frac {513097 \sqrt {3-x+2 x^2}}{497664 (5+2 x)^3}-\frac {16295969 \sqrt {3-x+2 x^2}}{71663616 (5+2 x)^2}+\frac {26800085 \sqrt {3-x+2 x^2}}{1719926784 (5+2 x)}+\frac {2053207 \text {arctanh}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {3-x+2 x^2}}\right )}{20639121408 \sqrt {2}} \] Output:

-3667/2304*(2*x^2-x+3)^(1/2)/(5+2*x)^4+513097/497664*(2*x^2-x+3)^(1/2)/(5+ 
2*x)^3-16295969/71663616*(2*x^2-x+3)^(1/2)/(5+2*x)^2+26800085*(2*x^2-x+3)^ 
(1/2)/(8599633920+3439853568*x)+2053207/41278242816*arctanh(1/24*(17-22*x) 
*2^(1/2)/(2*x^2-x+3)^(1/2))*2^(1/2)
 

Mathematica [A] (verified)

Time = 0.78 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.55 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^5 \sqrt {3-x+2 x^2}} \, dx=\frac {\frac {12 \sqrt {3-x+2 x^2} \left (-298655447-255525906 x+43592076 x^2+214400680 x^3\right )}{(5+2 x)^4}-2053207 \sqrt {2} \text {arctanh}\left (\frac {1}{6} \left (5+2 x-\sqrt {6-2 x+4 x^2}\right )\right )}{20639121408} \] Input:

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

Output:

((12*Sqrt[3 - x + 2*x^2]*(-298655447 - 255525906*x + 43592076*x^2 + 214400 
680*x^3))/(5 + 2*x)^4 - 2053207*Sqrt[2]*ArcTanh[(5 + 2*x - Sqrt[6 - 2*x + 
4*x^2])/6])/20639121408
 

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.11, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2181, 27, 2181, 2181, 27, 1228, 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.

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

\(\Big \downarrow \) 2181

\(\displaystyle -\frac {1}{288} \int \frac {-11520 x^3+31104 x^2-40668 x+37027}{16 (2 x+5)^4 \sqrt {2 x^2-x+3}}dx-\frac {3667 \sqrt {2 x^2-x+3}}{2304 (2 x+5)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {-11520 x^3+31104 x^2-40668 x+37027}{(2 x+5)^4 \sqrt {2 x^2-x+3}}dx}{4608}-\frac {3667 \sqrt {2 x^2-x+3}}{2304 (2 x+5)^4}\)

\(\Big \downarrow \) 2181

\(\displaystyle \frac {\frac {1}{216} \int \frac {1244160 x^2-2364856 x+2607829}{(2 x+5)^3 \sqrt {2 x^2-x+3}}dx+\frac {513097 \sqrt {2 x^2-x+3}}{108 (2 x+5)^3}}{4608}-\frac {3667 \sqrt {2 x^2-x+3}}{2304 (2 x+5)^4}\)

\(\Big \downarrow \) 2181

\(\displaystyle \frac {\frac {1}{216} \left (-\frac {1}{144} \int \frac {13 (1493165-1876588 x)}{(2 x+5)^2 \sqrt {2 x^2-x+3}}dx-\frac {16295969 \sqrt {2 x^2-x+3}}{72 (2 x+5)^2}\right )+\frac {513097 \sqrt {2 x^2-x+3}}{108 (2 x+5)^3}}{4608}-\frac {3667 \sqrt {2 x^2-x+3}}{2304 (2 x+5)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{216} \left (-\frac {13}{144} \int \frac {1493165-1876588 x}{(2 x+5)^2 \sqrt {2 x^2-x+3}}dx-\frac {16295969 \sqrt {2 x^2-x+3}}{72 (2 x+5)^2}\right )+\frac {513097 \sqrt {2 x^2-x+3}}{108 (2 x+5)^3}}{4608}-\frac {3667 \sqrt {2 x^2-x+3}}{2304 (2 x+5)^4}\)

\(\Big \downarrow \) 1228

\(\displaystyle \frac {\frac {1}{216} \left (-\frac {13}{144} \left (\frac {157939}{24} \int \frac {1}{(2 x+5) \sqrt {2 x^2-x+3}}dx-\frac {2061545 \sqrt {2 x^2-x+3}}{12 (2 x+5)}\right )-\frac {16295969 \sqrt {2 x^2-x+3}}{72 (2 x+5)^2}\right )+\frac {513097 \sqrt {2 x^2-x+3}}{108 (2 x+5)^3}}{4608}-\frac {3667 \sqrt {2 x^2-x+3}}{2304 (2 x+5)^4}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {1}{216} \left (-\frac {13}{144} \left (-\frac {157939}{12} \int \frac {1}{288-\frac {(17-22 x)^2}{2 x^2-x+3}}d\frac {17-22 x}{\sqrt {2 x^2-x+3}}-\frac {2061545 \sqrt {2 x^2-x+3}}{12 (2 x+5)}\right )-\frac {16295969 \sqrt {2 x^2-x+3}}{72 (2 x+5)^2}\right )+\frac {513097 \sqrt {2 x^2-x+3}}{108 (2 x+5)^3}}{4608}-\frac {3667 \sqrt {2 x^2-x+3}}{2304 (2 x+5)^4}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {1}{216} \left (-\frac {13}{144} \left (-\frac {157939 \text {arctanh}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {2 x^2-x+3}}\right )}{144 \sqrt {2}}-\frac {2061545 \sqrt {2 x^2-x+3}}{12 (2 x+5)}\right )-\frac {16295969 \sqrt {2 x^2-x+3}}{72 (2 x+5)^2}\right )+\frac {513097 \sqrt {2 x^2-x+3}}{108 (2 x+5)^3}}{4608}-\frac {3667 \sqrt {2 x^2-x+3}}{2304 (2 x+5)^4}\)

Input:

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

Output:

(-3667*Sqrt[3 - x + 2*x^2])/(2304*(5 + 2*x)^4) + ((513097*Sqrt[3 - x + 2*x 
^2])/(108*(5 + 2*x)^3) + ((-16295969*Sqrt[3 - x + 2*x^2])/(72*(5 + 2*x)^2) 
 - (13*((-2061545*Sqrt[3 - x + 2*x^2])/(12*(5 + 2*x)) - (157939*ArcTanh[(1 
7 - 22*x)/(12*Sqrt[2]*Sqrt[3 - x + 2*x^2])])/(144*Sqrt[2])))/144)/216)/460 
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 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 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 1228

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + 
 b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e 
*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^ 
(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x 
] && EqQ[Simplify[m + 2*p + 3], 0]
 

rule 2181

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_ 
), x_Symbol] :> With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = Polynomi 
alRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + b*x + c*x^2) 
^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Simp[1/((m + 1)*(c*d^2 - 
b*d*e + a*e^2))   Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*ExpandToSum[(m 
+ 1)*(c*d^2 - b*d*e + a*e^2)*Qx + c*d*R*(m + 1) - b*e*R*(m + p + 2) - c*e*R 
*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, 
x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]
 

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.56

method	result	size
risch	\(\frac {428801360 x^{5}-127216528 x^{4}[phone] x^{3}-211008760 x^{2}-467922271 x -895966341}{1719926784 \left (5+2 x \right )^{4} \sqrt {2 x^{2}-x +3}}+\frac {2053207 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (\frac {17}{2}-11 x \right ) \sqrt {2}}{12 \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}\right )}{41278242816}\)	\(78\)
trager	\(\frac {\left (214400680 x^{3}[phone] x^{2}-255525906 x -298655447\right ) \sqrt {2 x^{2}-x +3}}{1719926784 \left (5+2 x \right )^{4}}+\frac {2053207 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {22 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x -17 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )-24 \sqrt {2 x^{2}-x +3}}{5+2 x}\right )}{41278242816}\)	\(88\)
default	\(-\frac {3667 \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}{36864 \left (x +\frac {5}{2}\right )^{4}}+\frac {513097 \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}{3981312 \left (x +\frac {5}{2}\right )^{3}}-\frac {16295969 \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}{286654464 \left (x +\frac {5}{2}\right )^{2}}+\frac {26800085 \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}{3439853568 \left (x +\frac {5}{2}\right )}+\frac {2053207 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (\frac {17}{2}-11 x \right ) \sqrt {2}}{12 \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}\right )}{41278242816}\)	\(116\)

Input:

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

Output:

1/1719926784*(428801360*x^5-127216528*x^4+88558152*x^3-211008760*x^2-46792 
2271*x-895966341)/(5+2*x)^4/(2*x^2-x+3)^(1/2)+2053207/41278242816*2^(1/2)* 
arctanh(1/12*(17/2-11*x)*2^(1/2)/(2*(x+5/2)^2-11*x-19/2)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.90 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^5 \sqrt {3-x+2 x^2}} \, dx=\frac {2053207 \, \sqrt {2} {\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )} \log \left (\frac {24 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (22 \, x - 17\right )} - 1060 \, x^{2} + 1036 \, x - 1153}{4 \, x^{2} + 20 \, x + 25}\right ) + 48 \, {\left (214400680 \, x^{3} + 43592076 \, x^{2} - 255525906 \, x - 298655447\right )} \sqrt {2 \, x^{2} - x + 3}}{82556485632 \, {\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )}} \] Input:

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

Output:

1/82556485632*(2053207*sqrt(2)*(16*x^4 + 160*x^3 + 600*x^2 + 1000*x + 625) 
*log((24*sqrt(2)*sqrt(2*x^2 - x + 3)*(22*x - 17) - 1060*x^2 + 1036*x - 115 
3)/(4*x^2 + 20*x + 25)) + 48*(214400680*x^3 + 43592076*x^2 - 255525906*x - 
 298655447)*sqrt(2*x^2 - x + 3))/(16*x^4 + 160*x^3 + 600*x^2 + 1000*x + 62 
5)
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.07 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^5 \sqrt {3-x+2 x^2}} \, dx=-\frac {2053207}{41278242816} \, \sqrt {2} \operatorname {arsinh}\left (\frac {22 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 5 \right |}} - \frac {17 \, \sqrt {23}}{23 \, {\left | 2 \, x + 5 \right |}}\right ) - \frac {3667 \, \sqrt {2 \, x^{2} - x + 3}}{2304 \, {\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )}} + \frac {513097 \, \sqrt {2 \, x^{2} - x + 3}}{497664 \, {\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )}} - \frac {16295969 \, \sqrt {2 \, x^{2} - x + 3}}{71663616 \, {\left (4 \, x^{2} + 20 \, x + 25\right )}} + \frac {26800085 \, \sqrt {2 \, x^{2} - x + 3}}{1719926784 \, {\left (2 \, x + 5\right )}} \] Input:

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

Output:

-2053207/41278242816*sqrt(2)*arcsinh(22/23*sqrt(23)*x/abs(2*x + 5) - 17/23 
*sqrt(23)/abs(2*x + 5)) - 3667/2304*sqrt(2*x^2 - x + 3)/(16*x^4 + 160*x^3 
+ 600*x^2 + 1000*x + 625) + 513097/497664*sqrt(2*x^2 - x + 3)/(8*x^3 + 60* 
x^2 + 150*x + 125) - 16295969/71663616*sqrt(2*x^2 - x + 3)/(4*x^2 + 20*x + 
 25) + 26800085/1719926784*sqrt(2*x^2 - x + 3)/(2*x + 5)
 

Giac [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.18 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^5 \sqrt {3-x+2 x^2}} \, dx=\frac {1}{41278242816} \, \sqrt {2} {\left (12 \, {\left (\frac {24 \, {\left (\frac {144 \, {\left (\frac {513097}{\mathrm {sgn}\left (\frac {1}{2 \, x + 5}\right )} - \frac {792072}{{\left (2 \, x + 5\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 5}\right )}\right )}}{2 \, x + 5} - \frac {16295969}{\mathrm {sgn}\left (\frac {1}{2 \, x + 5}\right )}\right )}}{2 \, x + 5} + \frac {26800085}{\mathrm {sgn}\left (\frac {1}{2 \, x + 5}\right )}\right )} \sqrt {-\frac {11}{2 \, x + 5} + \frac {36}{{\left (2 \, x + 5\right )}^{2}} + 1} + \frac {2053207 \, \log \left (12 \, \sqrt {-\frac {11}{2 \, x + 5} + \frac {36}{{\left (2 \, x + 5\right )}^{2}} + 1} + \frac {72}{2 \, x + 5} - 11\right )}{\mathrm {sgn}\left (\frac {1}{2 \, x + 5}\right )} - 321601020 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 5}\right )\right )} \] Input:

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

Output:

1/41278242816*sqrt(2)*(12*(24*(144*(513097/sgn(1/(2*x + 5)) - 792072/((2*x 
 + 5)*sgn(1/(2*x + 5))))/(2*x + 5) - 16295969/sgn(1/(2*x + 5)))/(2*x + 5) 
+ 26800085/sgn(1/(2*x + 5)))*sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 20 
53207*log(12*sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 72/(2*x + 5) - 11) 
/sgn(1/(2*x + 5)) - 321601020*sgn(1/(2*x + 5)))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 278, normalized size of antiderivative = 2.00 \[ \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^5 \sqrt {3-x+2 x^2}} \, dx=\frac {5145616320 \sqrt {2 x^{2}-x +3}\, x^{3}[phone] \sqrt {2 x^{2}-x +3}\, x^{2}-6132621744 \sqrt {2 x^{2}-x +3}\, x -7167730728 \sqrt {2 x^{2}-x +3}[phone] \sqrt {2}\, \mathrm {log}\left (-12 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+22 x -17\right ) x^{4}[phone] \sqrt {2}\, \mathrm {log}\left (-12 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+22 x -17\right ) x^{3}[phone] \sqrt {2}\, \mathrm {log}\left (-12 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+22 x -17\right ) x^{2}[phone] \sqrt {2}\, \mathrm {log}\left (-12 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+22 x -17\right ) x [phone] \sqrt {2}\, \mathrm {log}\left (-12 \sqrt {2 x^{2}-x +3}\, \sqrt {2}+22 x -17\right )-32851312 \sqrt {2}\, \mathrm {log}\left (2 x +5\right ) x^{4}-328513120 \sqrt {2}\, \mathrm {log}\left (2 x +5\right ) x^{3}-1231924200 \sqrt {2}\, \mathrm {log}\left (2 x +5\right ) x^{2}-2053207000 \sqrt {2}\, \mathrm {log}\left (2 x +5\right ) x -1283254375 \sqrt {2}\, \mathrm {log}\left (2 x +5\right )}{660451885056 x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]} \] Input:

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

Output:

(5145616320*sqrt(2*x**2 - x + 3)*x**3 + 1046209824*sqrt(2*x**2 - x + 3)*x* 
*2 - 6132621744*sqrt(2*x**2 - x + 3)*x - 7167730728*sqrt(2*x**2 - x + 3) + 
 32851312*sqrt(2)*log( - 12*sqrt(2*x**2 - x + 3)*sqrt(2) + 22*x - 17)*x**4 
 + 328513120*sqrt(2)*log( - 12*sqrt(2*x**2 - x + 3)*sqrt(2) + 22*x - 17)*x 
**3 + 1231924200*sqrt(2)*log( - 12*sqrt(2*x**2 - x + 3)*sqrt(2) + 22*x - 1 
7)*x**2 + 2053207000*sqrt(2)*log( - 12*sqrt(2*x**2 - x + 3)*sqrt(2) + 22*x 
 - 17)*x + 1283254375*sqrt(2)*log( - 12*sqrt(2*x**2 - x + 3)*sqrt(2) + 22* 
x - 17) - 32851312*sqrt(2)*log(2*x + 5)*x**4 - 328513120*sqrt(2)*log(2*x + 
 5)*x**3 - 1231924200*sqrt(2)*log(2*x + 5)*x**2 - 2053207000*sqrt(2)*log(2 
*x + 5)*x - 1283254375*sqrt(2)*log(2*x + 5))/(41278242816*(16*x**4 + 160*x 
**3 + 600*x**2 + 1000*x + 625))