Integrand size = 20, antiderivative size = 110 \[ \int \frac {1}{(1+2 x) \left (2+3 x+5 x^2\right )^4} \, dx=\frac {37+20 x}{651 \left (2+3 x+5 x^2\right )^3}+\frac {4 (1983+1805 x)}{141267 \left (2+3 x+5 x^2\right )^2}+\frac {4 (180133+203230 x)}{10218313 \left (2+3 x+5 x^2\right )}+\frac {19007376 \arctan \left (\frac {3+10 x}{\sqrt {31}}\right )}{71528191 \sqrt {31}}+\frac {128 \log (1+2 x)}{2401}-\frac {64 \log \left (2+3 x+5 x^2\right )}{2401} \]
1/651*(37+20*x)/(5*x^2+3*x+2)^3+4/141267*(1983+1805*x)/(5*x^2+3*x+2)^2+4/1 0218313*(180133+203230*x)/(5*x^2+3*x+2)+128/2401*ln(1+2*x)-64/2401*ln(5*x^ 2+3*x+2)+19007376/2217373921*arctan(1/31*(3+10*x)*31^(1/2))*31^(1/2)
Time = 0.08 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.80 \[ \int \frac {1}{(1+2 x) \left (2+3 x+5 x^2\right )^4} \, dx=\frac {16 \left (\frac {217 \left (13831165+44933184 x+105257844 x^2+143405620 x^3+127202700 x^4+60969000 x^5\right )}{16 \left (2+3 x+5 x^2\right )^3}+3563883 \sqrt {31} \arctan \left (\frac {3+10 x}{\sqrt {31}}\right )+22164504 \log (1+2 x)-11082252 \log \left (4 \left (2+3 x+5 x^2\right )\right )\right )}{6652121763} \]
(16*((217*(13831165 + 44933184*x + 105257844*x^2 + 143405620*x^3 + 1272027 00*x^4 + 60969000*x^5))/(16*(2 + 3*x + 5*x^2)^3) + 3563883*Sqrt[31]*ArcTan [(3 + 10*x)/Sqrt[31]] + 22164504*Log[1 + 2*x] - 11082252*Log[4*(2 + 3*x + 5*x^2)]))/6652121763
Time = 0.33 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.14, 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.
| \(\displaystyle \int \frac {1}{(2 x+1) \left (5 x^2+3 x+2\right )^4} \, dx\) |
| \(\Big \downarrow \) 1165 |
| \(\displaystyle \frac {1}{651} \int \frac {8 (25 x+59)}{(2 x+1) \left (5 x^2+3 x+2\right )^3}dx+\frac {20 x+37}{651 \left (5 x^2+3 x+2\right )^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {8}{651} \int \frac {25 x+59}{(2 x+1) \left (5 x^2+3 x+2\right )^3}dx+\frac {20 x+37}{651 \left (5 x^2+3 x+2\right )^3}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {8}{651} \left (\frac {1}{434} \int \frac {3 (3610 x+5649)}{(2 x+1) \left (5 x^2+3 x+2\right )^2}dx+\frac {1805 x+1983}{434 \left (5 x^2+3 x+2\right )^2}\right )+\frac {20 x+37}{651 \left (5 x^2+3 x+2\right )^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {8}{651} \left (\frac {3}{434} \int \frac {3610 x+5649}{(2 x+1) \left (5 x^2+3 x+2\right )^2}dx+\frac {1805 x+1983}{434 \left (5 x^2+3 x+2\right )^2}\right )+\frac {20 x+37}{651 \left (5 x^2+3 x+2\right )^3}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {8}{651} \left (\frac {3}{434} \left (\frac {1}{217} \int \frac {2 (203230 x+339943)}{(2 x+1) \left (5 x^2+3 x+2\right )}dx+\frac {203230 x+180133}{217 \left (5 x^2+3 x+2\right )}\right )+\frac {1805 x+1983}{434 \left (5 x^2+3 x+2\right )^2}\right )+\frac {20 x+37}{651 \left (5 x^2+3 x+2\right )^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {8}{651} \left (\frac {3}{434} \left (\frac {2}{217} \int \frac {203230 x+339943}{(2 x+1) \left (5 x^2+3 x+2\right )}dx+\frac {203230 x+180133}{217 \left (5 x^2+3 x+2\right )}\right )+\frac {1805 x+1983}{434 \left (5 x^2+3 x+2\right )^2}\right )+\frac {20 x+37}{651 \left (5 x^2+3 x+2\right )^3}\) |
| \(\Big \downarrow \) 1200 |
| \(\displaystyle \frac {8}{651} \left (\frac {3}{434} \left (\frac {2}{217} \int \left (\frac {472977-2383280 x}{7 \left (5 x^2+3 x+2\right )}+\frac {953312}{7 (2 x+1)}\right )dx+\frac {203230 x+180133}{217 \left (5 x^2+3 x+2\right )}\right )+\frac {1805 x+1983}{434 \left (5 x^2+3 x+2\right )^2}\right )+\frac {20 x+37}{651 \left (5 x^2+3 x+2\right )^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {8}{651} \left (\frac {3}{434} \left (\frac {2}{217} \left (\frac {2375922 \arctan \left (\frac {10 x+3}{\sqrt {31}}\right )}{7 \sqrt {31}}-\frac {238328}{7} \log \left (5 x^2+3 x+2\right )+\frac {476656}{7} \log (2 x+1)\right )+\frac {203230 x+180133}{217 \left (5 x^2+3 x+2\right )}\right )+\frac {1805 x+1983}{434 \left (5 x^2+3 x+2\right )^2}\right )+\frac {20 x+37}{651 \left (5 x^2+3 x+2\right )^3}\) |
(37 + 20*x)/(651*(2 + 3*x + 5*x^2)^3) + (8*((1983 + 1805*x)/(434*(2 + 3*x + 5*x^2)^2) + (3*((180133 + 203230*x)/(217*(2 + 3*x + 5*x^2)) + (2*((23759 22*ArcTan[(3 + 10*x)/Sqrt[31]])/(7*Sqrt[31]) + (476656*Log[1 + 2*x])/7 - ( 238328*Log[2 + 3*x + 5*x^2])/7))/217))/434))/651
3.23.31.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
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] )
Time = 17.25 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.71
| method | result | size |
| default | \(\frac {128 \ln \left (1+2 x \right )}{2401}-\frac {125 \left (-\frac {1138088}{29791} x^{5}-\frac {11872252}{148955} x^{4}-\frac {200767868}{2234325} x^{3}-\frac {245601636}{3723875} x^{2}-\frac {104844096}{3723875} x -\frac {19363631}{2234325}\right )}{2401 \left (5 x^{2}+3 x +2\right )^{3}}-\frac {64 \ln \left (5 x^{2}+3 x +2\right )}{2401}+\frac {19007376 \arctan \left (\frac {\left (3+10 x \right ) \sqrt {31}}{31}\right ) \sqrt {31}}{2217373921}\) | \(78\) |
| risch | \(\frac {\frac {20323000}{10218313} x^{5}+\frac {42400900}{10218313} x^{4}+\frac {143405620}{30654939} x^{3}+\frac {35085948}{10218313} x^{2}+\frac {14977728}{10218313} x +\frac {13831165}{30654939}}{\left (5 x^{2}+3 x +2\right )^{3}}-\frac {64 \ln \left (100 x^{2}+60 x +40\right )}{2401}+\frac {19007376 \arctan \left (\frac {\left (3+10 x \right ) \sqrt {31}}{31}\right ) \sqrt {31}}{2217373921}+\frac {128 \ln \left (1+2 x \right )}{2401}\) | \(78\) |
128/2401*ln(1+2*x)-125/2401*(-1138088/29791*x^5-11872252/148955*x^4-200767 868/2234325*x^3-245601636/3723875*x^2-104844096/3723875*x-19363631/2234325 )/(5*x^2+3*x+2)^3-64/2401*ln(5*x^2+3*x+2)+19007376/2217373921*arctan(1/31* (3+10*x)*31^(1/2))*31^(1/2)
Time = 0.39 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.69 \[ \int \frac {1}{(1+2 x) \left (2+3 x+5 x^2\right )^4} \, dx=\frac {13230273000 \, x^{5} + 27602985900 \, x^{4} + 31119019540 \, x^{3} + 57022128 \, \sqrt {31} {\left (125 \, x^{6} + 225 \, x^{5} + 285 \, x^{4} + 207 \, x^{3} + 114 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + 22840952148 \, x^{2} - 177316032 \, {\left (125 \, x^{6} + 225 \, x^{5} + 285 \, x^{4} + 207 \, x^{3} + 114 \, x^{2} + 36 \, x + 8\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) + 354632064 \, {\left (125 \, x^{6} + 225 \, x^{5} + 285 \, x^{4} + 207 \, x^{3} + 114 \, x^{2} + 36 \, x + 8\right )} \log \left (2 \, x + 1\right ) + 9750500928 \, x + 3001362805}{6652121763 \, {\left (125 \, x^{6} + 225 \, x^{5} + 285 \, x^{4} + 207 \, x^{3} + 114 \, x^{2} + 36 \, x + 8\right )}} \]
1/6652121763*(13230273000*x^5 + 27602985900*x^4 + 31119019540*x^3 + 570221 28*sqrt(31)*(125*x^6 + 225*x^5 + 285*x^4 + 207*x^3 + 114*x^2 + 36*x + 8)*a rctan(1/31*sqrt(31)*(10*x + 3)) + 22840952148*x^2 - 177316032*(125*x^6 + 2 25*x^5 + 285*x^4 + 207*x^3 + 114*x^2 + 36*x + 8)*log(5*x^2 + 3*x + 2) + 35 4632064*(125*x^6 + 225*x^5 + 285*x^4 + 207*x^3 + 114*x^2 + 36*x + 8)*log(2 *x + 1) + 9750500928*x + 3001362805)/(125*x^6 + 225*x^5 + 285*x^4 + 207*x^ 3 + 114*x^2 + 36*x + 8)
Time = 0.14 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(1+2 x) \left (2+3 x+5 x^2\right )^4} \, dx=\frac {60969000 x^{5} + 127202700 x^{4} + 143405620 x^{3} + 105257844 x^{2} + 44933184 x + 13831165}{3831867375 x^{6} + 6897361275 x^{5} + 8736657615 x^{4} + 6345572373 x^{3} + 3494663046 x^{2} + 1103577804 x + 245239512} + \frac {128 \log {\left (x + \frac {1}{2} \right )}}{2401} - \frac {64 \log {\left (x^{2} + \frac {3 x}{5} + \frac {2}{5} \right )}}{2401} + \frac {19007376 \sqrt {31} \operatorname {atan}{\left (\frac {10 \sqrt {31} x}{31} + \frac {3 \sqrt {31}}{31} \right )}}{2217373921} \]
(60969000*x**5 + 127202700*x**4 + 143405620*x**3 + 105257844*x**2 + 449331 84*x + 13831165)/(3831867375*x**6 + 6897361275*x**5 + 8736657615*x**4 + 63 45572373*x**3 + 3494663046*x**2 + 1103577804*x + 245239512) + 128*log(x + 1/2)/2401 - 64*log(x**2 + 3*x/5 + 2/5)/2401 + 19007376*sqrt(31)*atan(10*sq rt(31)*x/31 + 3*sqrt(31)/31)/2217373921
Time = 0.28 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(1+2 x) \left (2+3 x+5 x^2\right )^4} \, dx=\frac {19007376}{2217373921} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {60969000 \, x^{5} + 127202700 \, x^{4} + 143405620 \, x^{3} + 105257844 \, x^{2} + 44933184 \, x + 13831165}{30654939 \, {\left (125 \, x^{6} + 225 \, x^{5} + 285 \, x^{4} + 207 \, x^{3} + 114 \, x^{2} + 36 \, x + 8\right )}} - \frac {64}{2401} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) + \frac {128}{2401} \, \log \left (2 \, x + 1\right ) \]
19007376/2217373921*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 1/30654939 *(60969000*x^5 + 127202700*x^4 + 143405620*x^3 + 105257844*x^2 + 44933184* x + 13831165)/(125*x^6 + 225*x^5 + 285*x^4 + 207*x^3 + 114*x^2 + 36*x + 8) - 64/2401*log(5*x^2 + 3*x + 2) + 128/2401*log(2*x + 1)
Time = 0.26 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.71 \[ \int \frac {1}{(1+2 x) \left (2+3 x+5 x^2\right )^4} \, dx=\frac {19007376}{2217373921} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {60969000 \, x^{5} + 127202700 \, x^{4} + 143405620 \, x^{3} + 105257844 \, x^{2} + 44933184 \, x + 13831165}{30654939 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}^{3}} - \frac {64}{2401} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) + \frac {128}{2401} \, \log \left ({\left | 2 \, x + 1 \right |}\right ) \]
19007376/2217373921*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 1/30654939 *(60969000*x^5 + 127202700*x^4 + 143405620*x^3 + 105257844*x^2 + 44933184* x + 13831165)/(5*x^2 + 3*x + 2)^3 - 64/2401*log(5*x^2 + 3*x + 2) + 128/240 1*log(abs(2*x + 1))
Time = 0.11 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.93 \[ \int \frac {1}{(1+2 x) \left (2+3 x+5 x^2\right )^4} \, dx=\frac {128\,\ln \left (x+\frac {1}{2}\right )}{2401}-\ln \left (x+\frac {3}{10}-\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (\frac {64}{2401}+\frac {\sqrt {31}\,9503688{}\mathrm {i}}{2217373921}\right )+\ln \left (x+\frac {3}{10}+\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (-\frac {64}{2401}+\frac {\sqrt {31}\,9503688{}\mathrm {i}}{2217373921}\right )+\frac {\frac {162584\,x^5}{10218313}+\frac {1696036\,x^4}{51091565}+\frac {28681124\,x^3}{766373475}+\frac {35085948\,x^2}{1277289125}+\frac {14977728\,x}{1277289125}+\frac {2766233}{766373475}}{x^6+\frac {9\,x^5}{5}+\frac {57\,x^4}{25}+\frac {207\,x^3}{125}+\frac {114\,x^2}{125}+\frac {36\,x}{125}+\frac {8}{125}} \]
(128*log(x + 1/2))/2401 - log(x - (31^(1/2)*1i)/10 + 3/10)*((31^(1/2)*9503 688i)/2217373921 + 64/2401) + log(x + (31^(1/2)*1i)/10 + 3/10)*((31^(1/2)* 9503688i)/2217373921 - 64/2401) + ((14977728*x)/1277289125 + (35085948*x^2 )/1277289125 + (28681124*x^3)/766373475 + (1696036*x^4)/51091565 + (162584 *x^5)/10218313 + 2766233/766373475)/((36*x)/125 + (114*x^2)/125 + (207*x^3 )/125 + (57*x^4)/25 + (9*x^5)/5 + x^6 + 8/125)