\(\int \frac {1}{(1+2 x)^2 (2+3 x+5 x^2)^4} \, dx\) [501]

Optimal result

Integrand size = 20, antiderivative size = 121 \[ \int \frac {1}{(1+2 x)^2 \left (2+3 x+5 x^2\right )^4} \, dx=-\frac {128}{2401 (1+2 x)}+\frac {43-270 x}{4557 \left (2+3 x+5 x^2\right )^3}+\frac {7117-27530 x}{329623 \left (2+3 x+5 x^2\right )^2}+\frac {2 (1739037-3736330 x)}{71528191 \left (2+3 x+5 x^2\right )}-\frac {116056984 \arctan \left (\frac {3+10 x}{\sqrt {31}}\right )}{500697337 \sqrt {31}}+\frac {2048 \log (1+2 x)}{16807}-\frac {1024 \log \left (2+3 x+5 x^2\right )}{16807} \] Output:

-128/(2401+4802*x)+1/4557*(43-270*x)/(5*x^2+3*x+2)^3+1/329623*(7117-27530* 
x)/(5*x^2+3*x+2)^2+2*(1739037-3736330*x)/(357640955*x^2+214584573*x+143056 
382)-116056984/15521617447*arctan(1/31*(3+10*x)*31^(1/2))*31^(1/2)+2048/16 
807*ln(1+2*x)-1024/16807*ln(5*x^2+3*x+2)
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.98 \[ \int \frac {1}{(1+2 x)^2 \left (2+3 x+5 x^2\right )^4} \, dx=\frac {8 \left (-\frac {310303056}{1+2 x}-\frac {10218313 (-43+270 x)}{8 \left (2+3 x+5 x^2\right )^3}-\frac {141267 (-7117+27530 x)}{8 \left (2+3 x+5 x^2\right )^2}-\frac {651 (-1739037+3736330 x)}{4 \left (2+3 x+5 x^2\right )}-43521369 \sqrt {31} \arctan \left (\frac {3+10 x}{\sqrt {31}}\right )[phone] \log (1+2 x)-354632064 \log \left (4 \left (2+3 x+5 x^2\right )\right )\right )}{46564852341} \] Input:

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

Output:

(8*(-310303056/(1 + 2*x) - (10218313*(-43 + 270*x))/(8*(2 + 3*x + 5*x^2)^3 
) - (141267*(-7117 + 27530*x))/(8*(2 + 3*x + 5*x^2)^2) - (651*(-1739037 + 
3736330*x))/(4*(2 + 3*x + 5*x^2)) - 43521369*Sqrt[31]*ArcTan[(3 + 10*x)/Sq 
rt[31]] + 709264128*Log[1 + 2*x] - 354632064*Log[4*(2 + 3*x + 5*x^2)]))/46 
564852341
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.30, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1165, 27, 1235, 27, 1235, 27, 1200, 2009}

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.

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

\(\Big \downarrow \) 1165

\(\displaystyle \frac {1}{651} \int \frac {6 (40 x+91)}{(2 x+1)^2 \left (5 x^2+3 x+2\right )^3}dx+\frac {20 x+37}{651 (2 x+1) \left (5 x^2+3 x+2\right )^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{217} \int \frac {40 x+91}{(2 x+1)^2 \left (5 x^2+3 x+2\right )^3}dx+\frac {20 x+37}{651 (2 x+1) \left (5 x^2+3 x+2\right )^3}\)

\(\Big \downarrow \) 1235

\(\displaystyle \frac {2}{217} \left (\frac {1}{434} \int \frac {2 (11280 x+16081)}{(2 x+1)^2 \left (5 x^2+3 x+2\right )^2}dx+\frac {2820 x+3047}{434 (2 x+1) \left (5 x^2+3 x+2\right )^2}\right )+\frac {20 x+37}{651 (2 x+1) \left (5 x^2+3 x+2\right )^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{217} \left (\frac {1}{217} \int \frac {11280 x+16081}{(2 x+1)^2 \left (5 x^2+3 x+2\right )^2}dx+\frac {2820 x+3047}{434 (2 x+1) \left (5 x^2+3 x+2\right )^2}\right )+\frac {20 x+37}{651 (2 x+1) \left (5 x^2+3 x+2\right )^3}\)

\(\Big \downarrow \) 1235

\(\displaystyle \frac {2}{217} \left (\frac {1}{217} \left (\frac {1}{217} \int \frac {2 (1207240 x+1453909)}{(2 x+1)^2 \left (5 x^2+3 x+2\right )}dx+\frac {603620 x+504757}{217 (2 x+1) \left (5 x^2+3 x+2\right )}\right )+\frac {2820 x+3047}{434 (2 x+1) \left (5 x^2+3 x+2\right )^2}\right )+\frac {20 x+37}{651 (2 x+1) \left (5 x^2+3 x+2\right )^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{217} \left (\frac {1}{217} \left (\frac {2}{217} \int \frac {1207240 x+1453909}{(2 x+1)^2 \left (5 x^2+3 x+2\right )}dx+\frac {603620 x+504757}{217 (2 x+1) \left (5 x^2+3 x+2\right )}\right )+\frac {2820 x+3047}{434 (2 x+1) \left (5 x^2+3 x+2\right )^2}\right )+\frac {20 x+37}{651 (2 x+1) \left (5 x^2+3 x+2\right )^3}\)

\(\Big \downarrow \) 1200

\(\displaystyle \frac {2}{217} \left (\frac {1}{217} \left (\frac {2}{217} \int \left (\frac {-76264960 x-37386611}{49 \left (5 x^2+3 x+2\right )}+\frac {30505984}{49 (2 x+1)}+\frac {3401156}{7 (2 x+1)^2}\right )dx+\frac {603620 x+504757}{217 (2 x+1) \left (5 x^2+3 x+2\right )}\right )+\frac {2820 x+3047}{434 (2 x+1) \left (5 x^2+3 x+2\right )^2}\right )+\frac {20 x+37}{651 (2 x+1) \left (5 x^2+3 x+2\right )^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2}{217} \left (\frac {1}{217} \left (\frac {2}{217} \left (-\frac {29014246 \arctan \left (\frac {10 x+3}{\sqrt {31}}\right )}{49 \sqrt {31}}-\frac {7626496}{49} \log \left (5 x^2+3 x+2\right )-\frac {1700578}{7 (2 x+1)}+\frac {15252992}{49} \log (2 x+1)\right )+\frac {603620 x+504757}{217 (2 x+1) \left (5 x^2+3 x+2\right )}\right )+\frac {2820 x+3047}{434 (2 x+1) \left (5 x^2+3 x+2\right )^2}\right )+\frac {20 x+37}{651 (2 x+1) \left (5 x^2+3 x+2\right )^3}\)

Input:

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

Output:

(37 + 20*x)/(651*(1 + 2*x)*(2 + 3*x + 5*x^2)^3) + (2*((3047 + 2820*x)/(434 
*(1 + 2*x)*(2 + 3*x + 5*x^2)^2) + ((504757 + 603620*x)/(217*(1 + 2*x)*(2 + 
 3*x + 5*x^2)) + (2*(-1700578/(7*(1 + 2*x)) - (29014246*ArcTan[(3 + 10*x)/ 
Sqrt[31]])/(49*Sqrt[31]) + (15252992*Log[1 + 2*x])/49 - (7626496*Log[2 + 3 
*x + 5*x^2])/49))/217)/217))/217
 

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 1165

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e) 
*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^ 
2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d 
+ e*x)^m*Simp[b*c*d*e*(2*p - m + 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p 
+ 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x, x]*(a + 
 b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p, -1] 
 && IntQuadraticQ[a, b, c, d, e, m, p, x]
 

rule 1200

Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* 
(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* 
x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In 
tegersQ[n]
 

rule 1235

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 
*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a 
+ b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] 
 + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^m 
*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 
 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* 
m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - 
f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] 
)
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 1.04 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.72

method	result	size
default	\(-\frac {128}{2401 \left (1+2 x \right )}+\frac {2048 \ln \left (1+2 x \right )}{16807}-\frac {125 \left (\frac {10461724}{29791} x^{5}+\frac {38423826}{148955} x^{4}+\frac {199128958}{744775} x^{3}-\frac {6944987}{3723875} x^{2}-\frac {410739}{744775} x -\frac {371196343}{11171625}\right )}{16807 \left (5 x^{2}+3 x +2\right )^{3}}-\frac {1024 \ln \left (5 x^{2}+3 x +2\right )}{16807}-\frac {116056984 \arctan \left (\frac {\left (10 x +3\right ) \sqrt {31}}{31}\right ) \sqrt {31}}{15521617447}\)	\(87\)
risch	\(\frac {-\frac {850289000}{71528191} x^{6}-\frac {1319253200}{71528191} x^{5}-\frac {1508473570}{71528191} x^{4}-\frac {929593024}{71528191} x^{3}-\frac {433131361}{71528191} x^{2}-\frac {304894531}{214584573} x -\frac {38489903}{214584573}}{\left (1+2 x \right ) \left (5 x^{2}+3 x +2\right )^{3}}-\frac {1024 \ln \left (100 x^{2}+60 x +40\right )}{16807}-\frac {116056984 \arctan \left (\frac {\left (10 x +3\right ) \sqrt {31}}{31}\right ) \sqrt {31}}{15521617447}+\frac {2048 \ln \left (1+2 x \right )}{16807}\)	\(90\)

Input:

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

Output:

-128/2401/(1+2*x)+2048/16807*ln(1+2*x)-125/16807*(10461724/29791*x^5+38423 
826/148955*x^4+199128958/744775*x^3-6944987/3723875*x^2-410739/744775*x-37 
1196343/11171625)/(5*x^2+3*x+2)^3-1024/16807*ln(5*x^2+3*x[phone]/155 
21617447*arctan(1/31*(10*x+3)*31^(1/2))*31^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 211 vs. \(2 (104) = 208\).

Time = 0.08 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.74 \[ \int \frac {1}{(1+2 x)^2 \left (2+3 x+5 x^2\right )^4} \, dx=-\frac {553538139000 \, x^{6} + 858833833200 \, x^{5} + 982016294070 \, x^{4} + 605165058624 \, x^{3} + 348170952 \, \sqrt {31} {\left (250 \, x^{7} + 575 \, x^{6} + 795 \, x^{5} + 699 \, x^{4} + 435 \, x^{3} + 186 \, x^{2} + 52 \, x + 8\right )} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + 281968516011 \, x^{2} + 2837056512 \, {\left (250 \, x^{7} + 575 \, x^{6} + 795 \, x^{5} + 699 \, x^{4} + 435 \, x^{3} + 186 \, x^{2} + 52 \, x + 8\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) - 5674113024 \, {\left (250 \, x^{7} + 575 \, x^{6} + 795 \, x^{5} + 699 \, x^{4} + 435 \, x^{3} + 186 \, x^{2} + 52 \, x + 8\right )} \log \left (2 \, x + 1\right ) + 66162113227 \, x + 8352308951}{46564852341 \, {\left (250 \, x^{7} + 575 \, x^{6} + 795 \, x^{5} + 699 \, x^{4} + 435 \, x^{3} + 186 \, x^{2} + 52 \, x + 8\right )}} \] Input:

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

Output:

-1/46564852341*(553538139000*x^6 + 858833833200*x^5 + 982016294070*x^4 + 6 
05165058624*x^3 + 348170952*sqrt(31)*(250*x^7 + 575*x^6 + 795*x^5 + 699*x^ 
4 + 435*x^3 + 186*x^2 + 52*x + 8)*arctan(1/31*sqrt(31)*(10*x + 3)) + 28196 
8516011*x^2 + 2837056512*(250*x^7 + 575*x^6 + 795*x^5 + 699*x^4 + 435*x^3 
+ 186*x^2 + 52*x + 8)*log(5*x^2 + 3*x + 2) - 5674113024*(250*x^7 + 575*x^6 
 + 795*x^5 + 699*x^4 + 435*x^3 + 186*x^2 + 52*x + 8)*log(2*x + 1) + 661621 
13227*x + 8352308951)/(250*x^7 + 575*x^6 + 795*x^5 + 699*x^4 + 435*x^3 + 1 
86*x^2 + 52*x + 8)
 

Sympy [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.01 \[ \int \frac {1}{(1+2 x)^2 \left (2+3 x+5 x^2\right )^4} \, dx=\frac {- 2550867000 x^{6} - 3957759600 x^{5} - 4525420710 x^{4} - 2788779072 x^{3} - 1299394083 x^{2} - 304894531 x - 38489903}{53646143250 x^{7} + 123386129475 x^{6} + 170594735535 x^{5} + 149994616527 x^{4} + 93344289255 x^{3} + 39912730578 x^{2} + 11158397796 x + 1716676584} + \frac {2048 \log {\left (x + \frac {1}{2} \right )}}{16807} - \frac {1024 \log {\left (x^{2} + \frac {3 x}{5} + \frac {2}{5} \right )}}{16807} - \frac {116056984 \sqrt {31} \operatorname {atan}{\left (\frac {10 \sqrt {31} x}{31} + \frac {3 \sqrt {31}}{31} \right )}}{15521617447} \] Input:

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

Output:

(-2550867000*x**6 - 3957759600*x**5 - 4525420710*x**4 - 2788779072*x**3 - 
1299394083*x**2 - 304894531*x - 38489903)/(53646143250*x**7 + 123386129475 
*x**6 + 170594735535*x**5 + 149994616527*x**4 + 93344289255*x**3 + 3991273 
0578*x**2 + 11158397796*x + 1716676584) + 2048*log(x + 1/2)/16807 - 1024*l 
og(x**2 + 3*x/5 + 2/5)/16807 - 116056984*sqrt(31)*atan(10*sqrt(31)*x/31 + 
3*sqrt(31)/31)/15521617447
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(1+2 x)^2 \left (2+3 x+5 x^2\right )^4} \, dx=-\frac {116056984}{15521617447} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) - \frac {2550867000 \, x^{6} + 3957759600 \, x^{5} + 4525420710 \, x^{4} + 2788779072 \, x^{3} + 1299394083 \, x^{2} + 304894531 \, x + 38489903}{214584573 \, {\left (250 \, x^{7} + 575 \, x^{6} + 795 \, x^{5} + 699 \, x^{4} + 435 \, x^{3} + 186 \, x^{2} + 52 \, x + 8\right )}} - \frac {1024}{16807} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) + \frac {2048}{16807} \, \log \left (2 \, x + 1\right ) \] Input:

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

Output:

-116056984/15521617447*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) - 1/21458 
4573*(2550867000*x^6 + 3957759600*x^5 + 4525420710*x^4 + 2788779072*x^3 + 
1299394083*x^2 + 304894531*x + 38489903)/(250*x^7 + 575*x^6 + 795*x^5 + 69 
9*x^4 + 435*x^3 + 186*x^2 + 52*x + 8) - 1024/16807*log(5*x^2 + 3*x + 2) + 
2048/16807*log(2*x + 1)
 

Giac [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.04 \[ \int \frac {1}{(1+2 x)^2 \left (2+3 x+5 x^2\right )^4} \, dx=-\frac {116056984}{15521617447} \, \sqrt {31} \arctan \left (-\frac {1}{31} \, \sqrt {31} {\left (\frac {7}{2 \, x + 1} - 2\right )}\right ) - \frac {128}{2401 \, {\left (2 \, x + 1\right )}} - \frac {8 \, {\left (\frac {3841449975}{2 \, x + 1} - \frac {8833663680}{{\left (2 \, x + 1\right )}^{2}} + \frac {7499779568}{{\left (2 \, x + 1\right )}^{3}} - \frac {7050406230}{{\left (2 \, x + 1\right )}^{4}} + \frac {1291725897}{{\left (2 \, x + 1\right )}^{5}} - 2009265250\right )}}{1502092011 \, {\left (\frac {4}{2 \, x + 1} - \frac {7}{{\left (2 \, x + 1\right )}^{2}} - 5\right )}^{3}} - \frac {1024}{16807} \, \log \left (-\frac {4}{2 \, x + 1} + \frac {7}{{\left (2 \, x + 1\right )}^{2}} + 5\right ) \] Input:

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

Output:

-116056984/15521617447*sqrt(31)*arctan(-1/31*sqrt(31)*(7/(2*x + 1) - 2)) - 
 128/2401/(2*x + 1) - 8/1502092011*(3841449975/(2*x + 1) - 8833663680/(2*x 
 + 1)^2 + 7499779568/(2*x + 1)^3 - 7050406230/(2*x + 1)^4 + 1291725897/(2* 
x + 1)^5 - 2009265250)/(4/(2*x + 1) - 7/(2*x + 1)^2 - 5)^3 - 1024/16807*lo 
g(-4/(2*x + 1) + 7/(2*x + 1)^2 + 5)
 

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.93 \[ \int \frac {1}{(1+2 x)^2 \left (2+3 x+5 x^2\right )^4} \, dx=\frac {2048\,\ln \left (x+\frac {1}{2}\right )}{16807}+\ln \left (x+\frac {3}{10}-\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (-\frac {1024}{16807}+\frac {\sqrt {31}\,58028492{}\mathrm {i}}{15521617447}\right )-\ln \left (x+\frac {3}{10}+\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (\frac {1024}{16807}+\frac {\sqrt {31}\,58028492{}\mathrm {i}}{15521617447}\right )-\frac {\frac {3401156\,x^6}{71528191}+\frac {26385064\,x^5}{357640955}+\frac {150847357\,x^4}{1788204775}+\frac {464796512\,x^3}{8941023875}+\frac {433131361\,x^2}{17882047750}+\frac {304894531\,x}{53646143250}+\frac {38489903}{53646143250}}{x^7+\frac {23\,x^6}{10}+\frac {159\,x^5}{50}+\frac {699\,x^4}{250}+\frac {87\,x^3}{50}+\frac {93\,x^2}{125}+\frac {26\,x}{125}+\frac {4}{125}} \] Input:

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

Output:

(2048*log(x + 1/2))/16807 + log(x - (31^(1/2)*1i)/10 + 3/10)*((31^(1/2)*58 
028492i)/15521617447 - 1024/16807) - log(x + (31^(1/2)*1i)/10 + 3/10)*((31 
^(1/2)*58028492i)/15521617447 + 1024/16807) - ((304894531*x)/53646143250 + 
 (433131361*x^2)/17882047750 + (464796512*x^3)/8941023875 + (150847357*x^4 
)/1788204775 + (26385064*x^5)/357640955 + (3401156*x^6)/71528191 + 3848990 
3/53646143250)/((26*x)/125 + (93*x^2)/125 + (87*x^3)/50 + (699*x^4)/250 + 
(159*x^5)/50 + (23*x^6)/10 + x^7 + 4/125)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 413, normalized size of antiderivative = 3.41 \[ \int \frac {1}{(1+2 x)^2 \left (2+3 x+5 x^2\right )^4} \, dx=\frac {-14970901393-370369275101 x -2366952114093 x^{2}-522018398208 \,\mathrm {log}\left (5 x^{2}+3 x +2\right )[phone] \,\mathrm {log}\left (2 x +1\right )-6366305857320 \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right ) x^{5}-5597544395304 \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right ) x^{4}-3483450374760 \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right ) x^{3}-1489475332656 \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right ) x^{2}-416412458592 \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right ) x -2001982974000 \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right ) x^{7}-2150665343400 x^{5}-4604560840200 \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right ) x^{6}-51875578321920 \,\mathrm {log}\left (5 x^{2}+3 x +2\right ) x^{5}-4287232729752 x^{3}[phone] x^{7}-16313074944000 \,\mathrm {log}\left (5 x^{2}+3 x +2\right ) x^{7}[phone] \,\mathrm {log}\left (2 x +1\right ) x^{7}-7109448397170 x^{4}-37520072371200 \,\mathrm {log}\left (5 x^{2}+3 x +2\right ) x^{6}[phone] \,\mathrm {log}\left (2 x +1\right ) x^{6}-64063455168 \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right )-45611357543424 \,\mathrm {log}\left (5 x^{2}+3 x +2\right ) x^{4}-28384750402560 \,\mathrm {log}\left (5 x^{2}+3 x +2\right ) x^{3}-12136927758336 \,\mathrm {log}\left (5 x^{2}+3 x +2\right ) x^{2}-3393119588352 \,\mathrm {log}\left (5 x^{2}+3 x +2\right ) x [phone] \,\mathrm {log}\left (2 x +1\right ) x^{4}[phone] \,\mathrm {log}\left (2 x +1\right ) x^{3}[phone] \,\mathrm {log}\left (2 x +1\right ) x^{2}[phone] \,\mathrm {log}\left (2 x +1\right ) x [phone] \,\mathrm {log}\left (2 x +1\right ) x^{5}}{267747900960750 x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]} \] Input:

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

Output:

( - 2001982974000*sqrt(31)*atan((10*x + 3)/sqrt(31))*x**7 - 4604560840200* 
sqrt(31)*atan((10*x + 3)/sqrt(31))*x**6 - 6366305857320*sqrt(31)*atan((10* 
x + 3)/sqrt(31))*x**5 - 5597544395304*sqrt(31)*atan((10*x + 3)/sqrt(31))*x 
**4 - 3483450374760*sqrt(31)*atan((10*x + 3)/sqrt(31))*x**3 - 148947533265 
6*sqrt(31)*atan((10*x + 3)/sqrt(31))*x**2 - 416412458592*sqrt(31)*atan((10 
*x + 3)/sqrt(31))*x - 64063455168*sqrt(31)*atan((10*x + 3)/sqrt(31)) - 163 
13074944000*log(5*x**2 + 3*x + 2)*x**7 - 37520072371200*log(5*x**2 + 3*x + 
 2)*x**6 - 51875578321920*log(5*x**2 + 3*x + 2)*x**5 - 45611357543424*log( 
5*x**2 + 3*x + 2)*x**4 - 28384750402560*log(5*x**2 + 3*x + 2)*x**3 - 12136 
927758336*log(5*x**2 + 3*x + 2)*x**2 - 3393119588352*log(5*x**2 + 3*x + 2) 
*x - 522018398208*log(5*x**2 + 3*x + 2) + 32626149888000*log(2*x + 1)*x**7 
 + 75040144742400*log(2*x + 1)*x**6 + 103751156643840*log(2*x + 1)*x**5 + 
91222715086848*log(2*x + 1)*x**4 + 56769500805120*log(2*x + 1)*x**3 + 2427 
3855516672*log(2*x + 1)*x**2 + 6786239176704*log(2*x + 1)*x + 104403679641 
6*log(2*x + 1) + 5535381390000*x**7 - 2150665343400*x**5 - 7109448397170*x 
**4 - 4287232729752*x**3 - 2366952114093*x**2 - 370369275101*x - 149709013 
93)/(1070991603843*(250*x**7 + 575*x**6 + 795*x**5 + 699*x**4 + 435*x**3 + 
 186*x**2 + 52*x + 8))