\(\int \frac {1}{(1+2 x) (2+3 x+5 x^2)^3} \, dx\) [498]

Optimal result

Integrand size = 20, antiderivative size = 89 \[ \int \frac {1}{(1+2 x) \left (2+3 x+5 x^2\right )^3} \, dx=\frac {37+20 x}{434 \left (2+3 x+5 x^2\right )^2}+\frac {2 (2609+2290 x)}{47089 \left (2+3 x+5 x^2\right )}+\frac {125624 \arctan \left (\frac {3+10 x}{\sqrt {31}}\right )}{329623 \sqrt {31}}+\frac {32}{343} \log (1+2 x)-\frac {16}{343} \log \left (2+3 x+5 x^2\right ) \] Output:

1/434*(37+20*x)/(5*x^2+3*x+2)^2+2*(2609+2290*x)/(235445*x^2+141267*x+94178 
)+125624/10218313*arctan(1/31*(3+10*x)*31^(1/2))*31^(1/2)+32/343*ln(1+2*x) 
-16/343*ln(5*x^2+3*x+2)
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(1+2 x) \left (2+3 x+5 x^2\right )^3} \, dx=\frac {8 \left (\frac {217 \left (28901+53968 x+79660 x^2+45800 x^3\right )}{16 \left (2+3 x+5 x^2\right )^2}+15703 \sqrt {31} \arctan \left (\frac {3+10 x}{\sqrt {31}}\right )+119164 \log (1+2 x)-59582 \log \left (4 \left (2+3 x+5 x^2\right )\right )\right )}{10218313} \] Input:

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

Output:

(8*((217*(28901 + 53968*x + 79660*x^2 + 45800*x^3))/(16*(2 + 3*x + 5*x^2)^ 
2) + 15703*Sqrt[31]*ArcTan[(3 + 10*x)/Sqrt[31]] + 119164*Log[1 + 2*x] - 59 
582*Log[4*(2 + 3*x + 5*x^2)]))/10218313
 

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1165, 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.

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

\(\Big \downarrow \) 1165

\(\displaystyle \frac {1}{434} \int \frac {4 (30 x+77)}{(2 x+1) \left (5 x^2+3 x+2\right )^2}dx+\frac {20 x+37}{434 \left (5 x^2+3 x+2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{217} \int \frac {30 x+77}{(2 x+1) \left (5 x^2+3 x+2\right )^2}dx+\frac {20 x+37}{434 \left (5 x^2+3 x+2\right )^2}\)

\(\Big \downarrow \) 1235

\(\displaystyle \frac {2}{217} \left (\frac {1}{217} \int \frac {2 (2290 x+4989)}{(2 x+1) \left (5 x^2+3 x+2\right )}dx+\frac {2290 x+2609}{217 \left (5 x^2+3 x+2\right )}\right )+\frac {20 x+37}{434 \left (5 x^2+3 x+2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{217} \left (\frac {2}{217} \int \frac {2290 x+4989}{(2 x+1) \left (5 x^2+3 x+2\right )}dx+\frac {2290 x+2609}{217 \left (5 x^2+3 x+2\right )}\right )+\frac {20 x+37}{434 \left (5 x^2+3 x+2\right )^2}\)

\(\Big \downarrow \) 1200

\(\displaystyle \frac {2}{217} \left (\frac {2}{217} \int \left (\frac {4171-38440 x}{7 \left (5 x^2+3 x+2\right )}+\frac {15376}{7 (2 x+1)}\right )dx+\frac {2290 x+2609}{217 \left (5 x^2+3 x+2\right )}\right )+\frac {20 x+37}{434 \left (5 x^2+3 x+2\right )^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2}{217} \left (\frac {2}{217} \left (\frac {31406 \arctan \left (\frac {10 x+3}{\sqrt {31}}\right )}{7 \sqrt {31}}-\frac {3844}{7} \log \left (5 x^2+3 x+2\right )+\frac {7688}{7} \log (2 x+1)\right )+\frac {2290 x+2609}{217 \left (5 x^2+3 x+2\right )}\right )+\frac {20 x+37}{434 \left (5 x^2+3 x+2\right )^2}\)

Input:

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

Output:

(37 + 20*x)/(434*(2 + 3*x + 5*x^2)^2) + (2*((2609 + 2290*x)/(217*(2 + 3*x 
+ 5*x^2)) + (2*((31406*ArcTan[(3 + 10*x)/Sqrt[31]])/(7*Sqrt[31]) + (7688*L 
og[1 + 2*x])/7 - (3844*Log[2 + 3*x + 5*x^2])/7))/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 = 2.09 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.76

method	result	size
default	\(\frac {32 \ln \left (1+2 x \right )}{343}-\frac {25 \left (-\frac {6412}{961} x^{3}-\frac {55762}{4805} x^{2}-\frac {188888}{24025} x -\frac {202307}{48050}\right )}{343 \left (5 x^{2}+3 x +2\right )^{2}}-\frac {16 \ln \left (5 x^{2}+3 x +2\right )}{343}+\frac {125624 \arctan \left (\frac {\left (10 x +3\right ) \sqrt {31}}{31}\right ) \sqrt {31}}{10218313}\)	\(68\)
risch	\(\frac {\frac {22900}{47089} x^{3}+\frac {5690}{6727} x^{2}+\frac {26984}{47089} x +\frac {28901}{94178}}{\left (5 x^{2}+3 x +2\right )^{2}}+\frac {32 \ln \left (1+2 x \right )}{343}-\frac {16 \ln \left (24658420900 x^{2}[phone] x [phone]\right )}{343}+\frac {125624 \sqrt {31}\, \arctan \left (\frac {\left (157030 x +47109\right ) \sqrt {31}}{486793}\right )}{10218313}\)	\(68\)

Input:

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

Output:

32/343*ln(1+2*x)-25/343*(-6412/961*x^3-55762/4805*x^2-188888/24025*x-20230 
7/48050)/(5*x^2+3*x+2)^2-16/343*ln(5*x^2+3*x+2)+125624/10218313*arctan(1/3 
1*(10*x+3)*31^(1/2))*31^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.53 \[ \int \frac {1}{(1+2 x) \left (2+3 x+5 x^2\right )^3} \, dx=\frac {9938600 \, x^{3} + 251248 \, \sqrt {31} {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + 17286220 \, x^{2} - 953312 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) + 1906624 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \log \left (2 \, x + 1\right ) + 11711056 \, x + 6271517}{20436626 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} \] Input:

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

Output:

1/20436626*(9938600*x^3 + 251248*sqrt(31)*(25*x^4 + 30*x^3 + 29*x^2 + 12*x 
 + 4)*arctan(1/31*sqrt(31)*(10*x + 3)) + 17286220*x^2 - 953312*(25*x^4 + 3 
0*x^3 + 29*x^2 + 12*x + 4)*log(5*x^2 + 3*x + 2) + 1906624*(25*x^4 + 30*x^3 
 + 29*x^2 + 12*x + 4)*log(2*x + 1) + 11711056*x + 6271517)/(25*x^4 + 30*x^ 
3 + 29*x^2 + 12*x + 4)
 

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.01 \[ \int \frac {1}{(1+2 x) \left (2+3 x+5 x^2\right )^3} \, dx=\frac {45800 x^{3} + 79660 x^{2} + 53968 x + 28901}{2354450 x^{4} + 2825340 x^{3} + 2731162 x^{2} + 1130136 x + 376712} + \frac {32 \log {\left (x + \frac {1}{2} \right )}}{343} - \frac {16 \log {\left (x^{2} + \frac {3 x}{5} + \frac {2}{5} \right )}}{343} + \frac {125624 \sqrt {31} \operatorname {atan}{\left (\frac {10 \sqrt {31} x}{31} + \frac {3 \sqrt {31}}{31} \right )}}{10218313} \] Input:

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

Output:

(45800*x**3 + 79660*x**2 + 53968*x + 28901)/(2354450*x**4 + 2825340*x**3 + 
 2731162*x**2 + 1130136*x + 376712) + 32*log(x + 1/2)/343 - 16*log(x**2 + 
3*x/5 + 2/5)/343 + 125624*sqrt(31)*atan(10*sqrt(31)*x/31 + 3*sqrt(31)/31)/ 
10218313
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(1+2 x) \left (2+3 x+5 x^2\right )^3} \, dx=\frac {125624}{10218313} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {45800 \, x^{3} + 79660 \, x^{2} + 53968 \, x + 28901}{94178 \, {\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} - \frac {16}{343} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) + \frac {32}{343} \, \log \left (2 \, x + 1\right ) \] Input:

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

Output:

125624/10218313*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 1/94178*(45800 
*x^3 + 79660*x^2 + 53968*x + 28901)/(25*x^4 + 30*x^3 + 29*x^2 + 12*x + 4) 
- 16/343*log(5*x^2 + 3*x + 2) + 32/343*log(2*x + 1)
 

Giac [A] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.76 \[ \int \frac {1}{(1+2 x) \left (2+3 x+5 x^2\right )^3} \, dx=\frac {125624}{10218313} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {45800 \, x^{3} + 79660 \, x^{2} + 53968 \, x + 28901}{94178 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}^{2}} - \frac {16}{343} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) + \frac {32}{343} \, \log \left ({\left | 2 \, x + 1 \right |}\right ) \] Input:

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

Output:

125624/10218313*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 1/94178*(45800 
*x^3 + 79660*x^2 + 53968*x + 28901)/(5*x^2 + 3*x + 2)^2 - 16/343*log(5*x^2 
 + 3*x + 2) + 32/343*log(abs(2*x + 1))
 

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.92 \[ \int \frac {1}{(1+2 x) \left (2+3 x+5 x^2\right )^3} \, dx=\frac {32\,\ln \left (x+\frac {1}{2}\right )}{343}-\ln \left (x+\frac {3}{10}-\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (\frac {16}{343}+\frac {\sqrt {31}\,62812{}\mathrm {i}}{10218313}\right )+\ln \left (x+\frac {3}{10}+\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (-\frac {16}{343}+\frac {\sqrt {31}\,62812{}\mathrm {i}}{10218313}\right )+\frac {\frac {916\,x^3}{47089}+\frac {1138\,x^2}{33635}+\frac {26984\,x}{1177225}+\frac {28901}{2354450}}{x^4+\frac {6\,x^3}{5}+\frac {29\,x^2}{25}+\frac {12\,x}{25}+\frac {4}{25}} \] Input:

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

Output:

(32*log(x + 1/2))/343 - log(x - (31^(1/2)*1i)/10 + 3/10)*((31^(1/2)*62812i 
)/10218313 + 16/343) + log(x + (31^(1/2)*1i)/10 + 3/10)*((31^(1/2)*62812i) 
/10218313 - 16/343) + ((26984*x)/1177225 + (1138*x^2)/33635 + (916*x^3)/47 
089 + 28901/2354450)/((12*x)/25 + (29*x^2)/25 + (6*x^3)/5 + x^4 + 4/25)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 248, normalized size of antiderivative = 2.79 \[ \int \frac {1}{(1+2 x) \left (2+3 x+5 x^2\right )^3} \, dx=\frac {18843600 \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right ) x^{4}[phone] \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right ) x^{3}[phone] \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right ) x^{2}+9044928 \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right ) x +3014976 \sqrt {31}\, \mathit {atan} \left (\frac {10 x +3}{\sqrt {31}}\right )-71498400 \,\mathrm {log}\left (5 x^{2}+3 x +2\right ) x^{4}-85798080 \,\mathrm {log}\left (5 x^{2}+3 x +2\right ) x^{3}-82938144 \,\mathrm {log}\left (5 x^{2}+3 x +2\right ) x^{2}-34319232 \,\mathrm {log}\left (5 x^{2}+3 x +2\right ) x -11439744 \,\mathrm {log}\left (5 x^{2}+3 x +2\right )[phone] \,\mathrm {log}\left (2 x +1\right ) x^{4}[phone] \,\mathrm {log}\left (2 x +1\right ) x^{3}[phone] \,\mathrm {log}\left (2 x +1\right ) x^{2}[phone] \,\mathrm {log}\left (2 x +1\right ) x [phone] \,\mathrm {log}\left (2 x +1\right )-24846500 x^{4}[phone] x^{2}[phone] x [phone]}{1532746950 x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]} \] Input:

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

Output:

(18843600*sqrt(31)*atan((10*x + 3)/sqrt(31))*x**4 + 22612320*sqrt(31)*atan 
((10*x + 3)/sqrt(31))*x**3 + 21858576*sqrt(31)*atan((10*x + 3)/sqrt(31))*x 
**2 + 9044928*sqrt(31)*atan((10*x + 3)/sqrt(31))*x + 3014976*sqrt(31)*atan 
((10*x + 3)/sqrt(31)) - 71498400*log(5*x**2 + 3*x + 2)*x**4 - 85798080*log 
(5*x**2 + 3*x + 2)*x**3 - 82938144*log(5*x**2 + 3*x + 2)*x**2 - 34319232*l 
og(5*x**2 + 3*x + 2)*x - 11439744*log(5*x**2 + 3*x + 2) + 142996800*log(2* 
x + 1)*x**4 + 171596160*log(2*x + 1)*x**3 + 165876288*log(2*x + 1)*x**2 + 
68638464*log(2*x + 1)*x + 22879488*log(2*x + 1) - 24846500*x**4 + 23036720 
*x**2 + 23206848*x + 14839111)/(61309878*(25*x**4 + 30*x**3 + 29*x**2 + 12 
*x + 4))