Integrand size = 25, antiderivative size = 148 \[ \int \frac {1}{\left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )^3} \, dx=\frac {-9446+5765 x}{690184 \left (2+3 x+5 x^2\right )^2}+\frac {13-6 x}{506 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^2}+\frac {1765599+3996965 x}{235352744 \left (2+3 x+5 x^2\right )}-\frac {25557 \arctan \left (\frac {1-4 x}{\sqrt {23}}\right )}{5387888 \sqrt {23}}+\frac {4464079 \arctan \left (\frac {3+10 x}{\sqrt {31}}\right )}{225120016 \sqrt {31}}+\frac {97 \log \left (3-x+2 x^2\right )}{468512}-\frac {97 \log \left (2+3 x+5 x^2\right )}{468512} \] Output:
1/690184*(-9446+5765*x)/(5*x^2+3*x+2)^2+1/506*(13-6*x)/(2*x^2-x+3)/(5*x^2+ 3*x+2)^2+(1765599+3996965*x)/(1176763720*x^2+706058232*x+470705488)-25557/ 123921424*arctan(1/23*(1-4*x)*23^(1/2))*23^(1/2)+4464079/6978720496*arctan (1/31*(3+10*x)*31^(1/2))*31^(1/2)+97/468512*ln(2*x^2-x+3)-97/468512*ln(5*x ^2+3*x+2)
Time = 0.07 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )^3} \, dx=\frac {-11+90 x}{244904 \left (3-x+2 x^2\right )}+\frac {-98+345 x}{30008 \left (2+3 x+5 x^2\right )^2}+\frac {67573+164380 x}{10232728 \left (2+3 x+5 x^2\right )}+\frac {25557 \arctan \left (\frac {-1+4 x}{\sqrt {23}}\right )}{5387888 \sqrt {23}}+\frac {4464079 \arctan \left (\frac {3+10 x}{\sqrt {31}}\right )}{225120016 \sqrt {31}}+\frac {97 \log \left (3-x+2 x^2\right )}{468512}-\frac {97 \log \left (2+3 x+5 x^2\right )}{468512} \] Input:
Integrate[1/((3 - x + 2*x^2)^2*(2 + 3*x + 5*x^2)^3),x]
Output:
(-11 + 90*x)/(244904*(3 - x + 2*x^2)) + (-98 + 345*x)/(30008*(2 + 3*x + 5* x^2)^2) + (67573 + 164380*x)/(10232728*(2 + 3*x + 5*x^2)) + (25557*ArcTan[ (-1 + 4*x)/Sqrt[23]])/(5387888*Sqrt[23]) + (4464079*ArcTan[(3 + 10*x)/Sqrt [31]])/(225120016*Sqrt[31]) + (97*Log[3 - x + 2*x^2])/468512 - (97*Log[2 + 3*x + 5*x^2])/468512
Time = 0.55 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.14, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {1305, 27, 2135, 27, 2135, 27, 2141, 27, 1142, 25, 1083, 217, 1103}
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.
| \(\displaystyle \int \frac {1}{\left (2 x^2-x+3\right )^2 \left (5 x^2+3 x+2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1305 |
| \(\displaystyle \frac {13-6 x}{506 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}-\frac {\int -\frac {11 \left (-150 x^2+321 x+250\right )}{\left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^3}dx}{5566}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{506} \int \frac {-150 x^2+321 x+250}{\left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^3}dx+\frac {13-6 x}{506 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 2135 |
| \(\displaystyle \frac {1}{506} \left (\frac {\int \frac {11 \left (34590 x^2-106699 x+68191\right )}{\left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}dx}{15004}-\frac {9446-5765 x}{1364 \left (5 x^2+3 x+2\right )^2}\right )+\frac {13-6 x}{506 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{506} \left (\frac {\int \frac {34590 x^2-106699 x+68191}{\left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}dx}{1364}-\frac {9446-5765 x}{1364 \left (5 x^2+3 x+2\right )^2}\right )+\frac {13-6 x}{506 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 2135 |
| \(\displaystyle \frac {1}{506} \left (\frac {\frac {\int \frac {22 \left (7993930 x^2-1730927 x+7580866\right )}{\left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )}dx}{7502}+\frac {3996965 x+1765599}{341 \left (5 x^2+3 x+2\right )}}{1364}-\frac {9446-5765 x}{1364 \left (5 x^2+3 x+2\right )^2}\right )+\frac {13-6 x}{506 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{506} \left (\frac {\frac {1}{341} \int \frac {7993930 x^2-1730927 x+7580866}{\left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )}dx+\frac {3996965 x+1765599}{341 \left (5 x^2+3 x+2\right )}}{1364}-\frac {9446-5765 x}{1364 \left (5 x^2+3 x+2\right )^2}\right )+\frac {13-6 x}{506 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 2141 |
| \(\displaystyle \frac {1}{506} \left (\frac {\frac {1}{341} \left (\frac {1}{242} \int \frac {10571 (4462 x+11663)}{2 x^2-x+3}dx+\frac {1}{242} \int \frac {253 (2092214-466085 x)}{5 x^2+3 x+2}dx\right )+\frac {3996965 x+1765599}{341 \left (5 x^2+3 x+2\right )}}{1364}-\frac {9446-5765 x}{1364 \left (5 x^2+3 x+2\right )^2}\right )+\frac {13-6 x}{506 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{506} \left (\frac {\frac {1}{341} \left (\frac {961}{22} \int \frac {4462 x+11663}{2 x^2-x+3}dx+\frac {23}{22} \int \frac {2092214-466085 x}{5 x^2+3 x+2}dx\right )+\frac {3996965 x+1765599}{341 \left (5 x^2+3 x+2\right )}}{1364}-\frac {9446-5765 x}{1364 \left (5 x^2+3 x+2\right )^2}\right )+\frac {13-6 x}{506 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{506} \left (\frac {\frac {1}{341} \left (\frac {961}{22} \left (\frac {25557}{2} \int \frac {1}{2 x^2-x+3}dx+\frac {2231}{2} \int -\frac {1-4 x}{2 x^2-x+3}dx\right )+\frac {23}{22} \left (\frac {4464079}{2} \int \frac {1}{5 x^2+3 x+2}dx-\frac {93217}{2} \int \frac {10 x+3}{5 x^2+3 x+2}dx\right )\right )+\frac {3996965 x+1765599}{341 \left (5 x^2+3 x+2\right )}}{1364}-\frac {9446-5765 x}{1364 \left (5 x^2+3 x+2\right )^2}\right )+\frac {13-6 x}{506 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{506} \left (\frac {\frac {1}{341} \left (\frac {961}{22} \left (\frac {25557}{2} \int \frac {1}{2 x^2-x+3}dx-\frac {2231}{2} \int \frac {1-4 x}{2 x^2-x+3}dx\right )+\frac {23}{22} \left (\frac {4464079}{2} \int \frac {1}{5 x^2+3 x+2}dx-\frac {93217}{2} \int \frac {10 x+3}{5 x^2+3 x+2}dx\right )\right )+\frac {3996965 x+1765599}{341 \left (5 x^2+3 x+2\right )}}{1364}-\frac {9446-5765 x}{1364 \left (5 x^2+3 x+2\right )^2}\right )+\frac {13-6 x}{506 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{506} \left (\frac {\frac {1}{341} \left (\frac {961}{22} \left (-\frac {2231}{2} \int \frac {1-4 x}{2 x^2-x+3}dx-25557 \int \frac {1}{-(4 x-1)^2-23}d(4 x-1)\right )+\frac {23}{22} \left (-\frac {93217}{2} \int \frac {10 x+3}{5 x^2+3 x+2}dx-4464079 \int \frac {1}{-(10 x+3)^2-31}d(10 x+3)\right )\right )+\frac {3996965 x+1765599}{341 \left (5 x^2+3 x+2\right )}}{1364}-\frac {9446-5765 x}{1364 \left (5 x^2+3 x+2\right )^2}\right )+\frac {13-6 x}{506 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{506} \left (\frac {\frac {1}{341} \left (\frac {961}{22} \left (\frac {25557 \arctan \left (\frac {4 x-1}{\sqrt {23}}\right )}{\sqrt {23}}-\frac {2231}{2} \int \frac {1-4 x}{2 x^2-x+3}dx\right )+\frac {23}{22} \left (\frac {4464079 \arctan \left (\frac {10 x+3}{\sqrt {31}}\right )}{\sqrt {31}}-\frac {93217}{2} \int \frac {10 x+3}{5 x^2+3 x+2}dx\right )\right )+\frac {3996965 x+1765599}{341 \left (5 x^2+3 x+2\right )}}{1364}-\frac {9446-5765 x}{1364 \left (5 x^2+3 x+2\right )^2}\right )+\frac {13-6 x}{506 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{506} \left (\frac {\frac {1}{341} \left (\frac {961}{22} \left (\frac {25557 \arctan \left (\frac {4 x-1}{\sqrt {23}}\right )}{\sqrt {23}}+\frac {2231}{2} \log \left (2 x^2-x+3\right )\right )+\frac {23}{22} \left (\frac {4464079 \arctan \left (\frac {10 x+3}{\sqrt {31}}\right )}{\sqrt {31}}-\frac {93217}{2} \log \left (5 x^2+3 x+2\right )\right )\right )+\frac {3996965 x+1765599}{341 \left (5 x^2+3 x+2\right )}}{1364}-\frac {9446-5765 x}{1364 \left (5 x^2+3 x+2\right )^2}\right )+\frac {13-6 x}{506 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}\) |
Input:
Int[1/((3 - x + 2*x^2)^2*(2 + 3*x + 5*x^2)^3),x]
Output:
(13 - 6*x)/(506*(3 - x + 2*x^2)*(2 + 3*x + 5*x^2)^2) + (-1/1364*(9446 - 57 65*x)/(2 + 3*x + 5*x^2)^2 + ((1765599 + 3996965*x)/(341*(2 + 3*x + 5*x^2)) + ((961*((25557*ArcTan[(-1 + 4*x)/Sqrt[23]])/Sqrt[23] + (2231*Log[3 - x + 2*x^2])/2))/22 + (23*((4464079*ArcTan[(3 + 10*x)/Sqrt[31]])/Sqrt[31] - (9 3217*Log[2 + 3*x + 5*x^2])/2))/22)/341)/1364)/506
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + (f_.)*(x _)^2)^(q_), x_Symbol] :> Simp[(2*a*c^2*e - b^2*c*e + b^3*f + b*c*(c*d - 3*a *f) + c*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))*x)*(a + b*x + c*x^2)^(p + 1)*(( d + e*x + f*x^2)^(q + 1)/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1))), x] - Simp[1/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*( c*e - b*f))*(p + 1)) Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Si mp[2*c*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(a*f*(p + 1) - c*d*(p + 2)) - e*(b^2*c*e - 2*a*c^2*e - b^3*f - b*c*(c*d - 3*a*f))*(p + q + 2) + (2*f*(2*a*c^2*e - b^2*c*e + b^3*f + b*c*(c*d - 3*a*f))*(p + q + 2) - (2*c^2*d + b^2*f - c*(b*e + 2*a*f))*(b*f *(p + 1) - c*e*(2*p + q + 4)))*x + c*f*(2*c^2*d + b^2*f - c*(b*e + 2*a*f))* (2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[b ^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 0] && !( !IntegerQ[p] && ILtQ[q, -1]) && !IGtQ[q , 0]
Int[(Px_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_. )*(x_)^2)^(q_), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1] , C = Coeff[Px, x, 2]}, Simp[(a + b*x + c*x^2)^(p + 1)*((d + e*x + f*x^2)^( q + 1)/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)))*( (A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b *e + 2*a*f)) + c*(A*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) - B*(b*c*d - 2*a*c* e + a*b*f) + C*(b^2*d - a*b*e - 2*a*(c*d - a*f)))*x), x] + Simp[1/((b^2 - 4 *a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)) Int[(a + b*x + c *x^2)^(p + 1)*(d + e*x + f*x^2)^q*Simp[(b*B - 2*A*c - 2*a*C)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) + (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B )*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (2*f*((A*c - a*C)*(2*a *c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*( c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(b*f*(p + 1) - c*e*(2*p + q + 4))) *x - c*f*(b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x]] /; F reeQ[{a, b, c, d, e, f, q}, x] && PolyQ[Px, x, 2] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 0] && !( !IntegerQ[p] && ILtQ[q, -1]) && !IGtQ[q, 0]
Int[(Px_)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_) + (e_.)*(x_) + (f_.)*(x _)^2)), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Co eff[Px, x, 2], q = c^2*d^2 - b*c*d*e + a*c*e^2 + b^2*d*f - 2*a*c*d*f - a*b* e*f + a^2*f^2}, Simp[1/q Int[(A*c^2*d - a*c*C*d - A*b*c*e + a*B*c*e + A*b ^2*f - a*b*B*f - a*A*c*f + a^2*C*f + c*(B*c*d - b*C*d - A*c*e + a*C*e + A*b *f - a*B*f)*x)/(a + b*x + c*x^2), x], x] + Simp[1/q Int[(c*C*d^2 - B*c*d* e + A*c*e^2 + b*B*d*f - A*c*d*f - a*C*d*f - A*b*e*f + a*A*f^2 - f*(B*c*d - b*C*d - A*c*e + a*C*e + A*b*f - a*B*f)*x)/(d + e*x + f*x^2), x], x] /; NeQ[ q, 0]] /; FreeQ[{a, b, c, d, e, f}, x] && PolyQ[Px, x, 2]
Time = 2.96 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.72
| method | result | size |
| default | \(\frac {\frac {990 x}{23}-\frac {121}{23}}{234256 x^{2}-117128 x +351384}+\frac {97 \ln \left (2 x^{2}-x +3\right )}{468512}+\frac {25557 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{123921424}-\frac {25 \left (-\frac {723272}{961} x^{3}-\frac {3656422}{4805} x^{2}-\frac {14280728}{24025} x -\frac {2238016}{24025}\right )}{234256 \left (5 x^{2}+3 x +2\right )^{2}}-\frac {97 \ln \left (5 x^{2}+3 x +2\right )}{468512}+\frac {4464079 \arctan \left (\frac {\left (10 x +3\right ) \sqrt {31}}{31}\right ) \sqrt {31}}{6978720496}\) | \(106\) |
| risch | \(\frac {\frac {19984825}{117676372} x^{5}+\frac {21652955}{235352744} x^{4}+\frac {69648769}{235352744} x^{3}+\frac {23910151}{117676372} x^{2}+\frac {5333615}{29419093} x +\frac {158567}{5348926}}{\left (2 x^{2}-x +3\right ) \left (5 x^{2}+3 x +2\right )^{2}}+\frac {97 \ln \left (16 x^{2}-8 x +24\right )}{468512}+\frac {25557 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{123921424}-\frac {97 \ln \left (100 x^{2}+60 x +40\right )}{468512}+\frac {4464079 \arctan \left (\frac {\left (10 x +3\right ) \sqrt {31}}{31}\right ) \sqrt {31}}{6978720496}\) | \(111\) |
Input:
int(1/(2*x^2-x+3)^2/(5*x^2+3*x+2)^3,x,method=_RETURNVERBOSE)
Output:
1/234256*(990/23*x-121/23)/(x^2-1/2*x+3/2)+97/468512*ln(2*x^2-x+3)+25557/1 23921424*23^(1/2)*arctan(1/23*(4*x-1)*23^(1/2))-25/234256*(-723272/961*x^3 -3656422/4805*x^2-14280728/24025*x-2238016/24025)/(5*x^2+3*x+2)^2-97/46851 2*ln(5*x^2+3*x+2)+4464079/6978720496*arctan(1/31*(10*x+3)*31^(1/2))*31^(1/ 2)
Time = 0.07 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.60 \[ \int \frac {1}{\left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )^3} \, dx=\frac {1253927859800 \, x^{5} + 679296504260 \, x^{4} + 2185021181068 \, x^{3} + 4722995582 \, \sqrt {31} {\left (50 \, x^{6} + 35 \, x^{5} + 103 \, x^{4} + 85 \, x^{3} + 83 \, x^{2} + 32 \, x + 12\right )} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + 1522737174 \, \sqrt {23} {\left (50 \, x^{6} + 35 \, x^{5} + 103 \, x^{4} + 85 \, x^{3} + 83 \, x^{2} + 32 \, x + 12\right )} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + 1500218514344 \, x^{2} - 1528665583 \, {\left (50 \, x^{6} + 35 \, x^{5} + 103 \, x^{4} + 85 \, x^{3} + 83 \, x^{2} + 32 \, x + 12\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) + 1528665583 \, {\left (50 \, x^{6} + 35 \, x^{5} + 103 \, x^{4} + 85 \, x^{3} + 83 \, x^{2} + 32 \, x + 12\right )} \log \left (2 \, x^{2} - x + 3\right ) + 1338609358240 \, x + 218880812656}{7383486284768 \, {\left (50 \, x^{6} + 35 \, x^{5} + 103 \, x^{4} + 85 \, x^{3} + 83 \, x^{2} + 32 \, x + 12\right )}} \] Input:
integrate(1/(2*x^2-x+3)^2/(5*x^2+3*x+2)^3,x, algorithm="fricas")
Output:
1/7383486284768*(1253927859800*x^5 + 679296504260*x^4 + 2185021181068*x^3 + 4722995582*sqrt(31)*(50*x^6 + 35*x^5 + 103*x^4 + 85*x^3 + 83*x^2 + 32*x + 12)*arctan(1/31*sqrt(31)*(10*x + 3)) + 1522737174*sqrt(23)*(50*x^6 + 35* x^5 + 103*x^4 + 85*x^3 + 83*x^2 + 32*x + 12)*arctan(1/23*sqrt(23)*(4*x - 1 )) + 1500218514344*x^2 - 1528665583*(50*x^6 + 35*x^5 + 103*x^4 + 85*x^3 + 83*x^2 + 32*x + 12)*log(5*x^2 + 3*x + 2) + 1528665583*(50*x^6 + 35*x^5 + 1 03*x^4 + 85*x^3 + 83*x^2 + 32*x + 12)*log(2*x^2 - x + 3) + 1338609358240*x + 218880812656)/(50*x^6 + 35*x^5 + 103*x^4 + 85*x^3 + 83*x^2 + 32*x + 12)
Time = 0.19 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.97 \[ \int \frac {1}{\left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )^3} \, dx=\frac {39969650 x^{5} + 21652955 x^{4} + 69648769 x^{3} + 47820302 x^{2} + 42668920 x + 6976948}{11767637200 x^{6} + 8237346040 x^{5} + 24241332632 x^{4} + 20004983240 x^{3} + 19534277752 x^{2} + 7531287808 x + 2824232928} + \frac {97 \log {\left (x^{2} - \frac {x}{2} + \frac {3}{2} \right )}}{468512} - \frac {97 \log {\left (x^{2} + \frac {3 x}{5} + \frac {2}{5} \right )}}{468512} + \frac {25557 \sqrt {23} \operatorname {atan}{\left (\frac {4 \sqrt {23} x}{23} - \frac {\sqrt {23}}{23} \right )}}{123921424} + \frac {4464079 \sqrt {31} \operatorname {atan}{\left (\frac {10 \sqrt {31} x}{31} + \frac {3 \sqrt {31}}{31} \right )}}{6978720496} \] Input:
integrate(1/(2*x**2-x+3)**2/(5*x**2+3*x+2)**3,x)
Output:
(39969650*x**5 + 21652955*x**4 + 69648769*x**3 + 47820302*x**2 + 42668920* x + 6976948)/(11767637200*x**6 + 8237346040*x**5 + 24241332632*x**4 + 2000 4983240*x**3 + 19534277752*x**2 + 7531287808*x + 2824232928) + 97*log(x**2 - x/2 + 3/2)/468512 - 97*log(x**2 + 3*x/5 + 2/5)/468512 + 25557*sqrt(23)* atan(4*sqrt(23)*x/23 - sqrt(23)/23)/123921424 + 4464079*sqrt(31)*atan(10*s qrt(31)*x/31 + 3*sqrt(31)/31)/6978720496
Time = 0.11 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )^3} \, dx=\frac {4464079}{6978720496} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {25557}{123921424} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {39969650 \, x^{5} + 21652955 \, x^{4} + 69648769 \, x^{3} + 47820302 \, x^{2} + 42668920 \, x + 6976948}{235352744 \, {\left (50 \, x^{6} + 35 \, x^{5} + 103 \, x^{4} + 85 \, x^{3} + 83 \, x^{2} + 32 \, x + 12\right )}} - \frac {97}{468512} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) + \frac {97}{468512} \, \log \left (2 \, x^{2} - x + 3\right ) \] Input:
integrate(1/(2*x^2-x+3)^2/(5*x^2+3*x+2)^3,x, algorithm="maxima")
Output:
4464079/6978720496*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 25557/12392 1424*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) + 1/235352744*(39969650*x^5 + 21652955*x^4 + 69648769*x^3 + 47820302*x^2 + 42668920*x + 6976948)/(50*x ^6 + 35*x^5 + 103*x^4 + 85*x^3 + 83*x^2 + 32*x + 12) - 97/468512*log(5*x^2 + 3*x + 2) + 97/468512*log(2*x^2 - x + 3)
Time = 0.11 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.74 \[ \int \frac {1}{\left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )^3} \, dx=\frac {4464079}{6978720496} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {25557}{123921424} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {39969650 \, x^{5} + 21652955 \, x^{4} + 69648769 \, x^{3} + 47820302 \, x^{2} + 42668920 \, x + 6976948}{235352744 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}^{2} {\left (2 \, x^{2} - x + 3\right )}} - \frac {97}{468512} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) + \frac {97}{468512} \, \log \left (2 \, x^{2} - x + 3\right ) \] Input:
integrate(1/(2*x^2-x+3)^2/(5*x^2+3*x+2)^3,x, algorithm="giac")
Output:
4464079/6978720496*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 25557/12392 1424*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) + 1/235352744*(39969650*x^5 + 21652955*x^4 + 69648769*x^3 + 47820302*x^2 + 42668920*x + 6976948)/((5*x ^2 + 3*x + 2)^2*(2*x^2 - x + 3)) - 97/468512*log(5*x^2 + 3*x + 2) + 97/468 512*log(2*x^2 - x + 3)
Time = 15.26 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )^3} \, dx=\frac {\frac {799393\,x^5}{235352744}+\frac {4330591\,x^4}{2353527440}+\frac {69648769\,x^3}{11767637200}+\frac {23910151\,x^2}{5883818600}+\frac {1066723\,x}{294190930}+\frac {158567}{267446300}}{x^6+\frac {7\,x^5}{10}+\frac {103\,x^4}{50}+\frac {17\,x^3}{10}+\frac {83\,x^2}{50}+\frac {16\,x}{25}+\frac {6}{25}}+\ln \left (x-\frac {1}{4}+\frac {\sqrt {23}\,1{}\mathrm {i}}{4}\right )\,\left (\frac {97}{468512}+\frac {\sqrt {23}\,25557{}\mathrm {i}}{247842848}\right )-\ln \left (x-\frac {1}{4}-\frac {\sqrt {23}\,1{}\mathrm {i}}{4}\right )\,\left (-\frac {97}{468512}+\frac {\sqrt {23}\,25557{}\mathrm {i}}{247842848}\right )-\ln \left (x+\frac {3}{10}-\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (\frac {97}{468512}+\frac {\sqrt {31}\,4464079{}\mathrm {i}}{13957440992}\right )+\ln \left (x+\frac {3}{10}+\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (-\frac {97}{468512}+\frac {\sqrt {31}\,4464079{}\mathrm {i}}{13957440992}\right ) \] Input:
int(1/((2*x^2 - x + 3)^2*(3*x + 5*x^2 + 2)^3),x)
Output:
log(x + (23^(1/2)*1i)/4 - 1/4)*((23^(1/2)*25557i)/247842848 + 97/468512) - log(x - (23^(1/2)*1i)/4 - 1/4)*((23^(1/2)*25557i)/247842848 - 97/468512) + ((1066723*x)/294190930 + (23910151*x^2)/5883818600 + (69648769*x^3)/1176 7637200 + (4330591*x^4)/2353527440 + (799393*x^5)/235352744 + 158567/26744 6300)/((16*x)/25 + (83*x^2)/50 + (17*x^3)/10 + (103*x^4)/50 + (7*x^5)/10 + x^6 + 6/25) - log(x - (31^(1/2)*1i)/10 + 3/10)*((31^(1/2)*4464079i)/13957 440992 + 97/468512) + log(x + (31^(1/2)*1i)/10 + 3/10)*((31^(1/2)*4464079i )/13957440992 - 97/468512)
Time = 0.19 (sec) , antiderivative size = 514, normalized size of antiderivative = 3.47 \[ \int \frac {1}{\left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )^3} \, dx =\text {Too large to display} \] Input:
int(1/(2*x^2-x+3)^2/(5*x^2+3*x+2)^3,x)
Output:
(236149779100*sqrt(31)*atan((10*x + 3)/sqrt(31))*x**6 + 165304845370*sqrt( 31)*atan((10*x + 3)/sqrt(31))*x**5 + 486468544946*sqrt(31)*atan((10*x + 3) /sqrt(31))*x**4 + 401454624470*sqrt(31)*atan((10*x + 3)/sqrt(31))*x**3 + 3 92008633306*sqrt(31)*atan((10*x + 3)/sqrt(31))*x**2 + 151135858624*sqrt(31 )*atan((10*x + 3)/sqrt(31))*x + 56675946984*sqrt(31)*atan((10*x + 3)/sqrt( 31)) + 76136858700*sqrt(23)*atan((4*x - 1)/sqrt(23))*x**6 + 53295801090*sq rt(23)*atan((4*x - 1)/sqrt(23))*x**5 + 156841928922*sqrt(23)*atan((4*x - 1 )/sqrt(23))*x**4 + 129432659790*sqrt(23)*atan((4*x - 1)/sqrt(23))*x**3 + 1 26387185442*sqrt(23)*atan((4*x - 1)/sqrt(23))*x**2 + 48727589568*sqrt(23)* atan((4*x - 1)/sqrt(23))*x + 18272846088*sqrt(23)*atan((4*x - 1)/sqrt(23)) - 76433279150*log(5*x**2 + 3*x + 2)*x**6 - 53503295405*log(5*x**2 + 3*x + 2)*x**5 - 157452555049*log(5*x**2 + 3*x + 2)*x**4 - 129936574555*log(5*x* *2 + 3*x + 2)*x**3 - 126879243389*log(5*x**2 + 3*x + 2)*x**2 - 48917298656 *log(5*x**2 + 3*x + 2)*x - 18343986996*log(5*x**2 + 3*x + 2) + 76433279150 *log(2*x**2 - x + 3)*x**6 + 53503295405*log(2*x**2 - x + 3)*x**5 + 1574525 55049*log(2*x**2 - x + 3)*x**4 + 129936574555*log(2*x**2 - x + 3)*x**3 + 1 26879243389*log(2*x**2 - x + 3)*x**2 + 48917298656*log(2*x**2 - x + 3)*x + 18343986996*log(2*x**2 - x + 3) - 1791325514000*x**6 - 3010834054580*x**4 - 860232192732*x**3 - 1473381838896*x**2 + 192161029280*x - 211037310704) /(7383486284768*(50*x**6 + 35*x**5 + 103*x**4 + 85*x**3 + 83*x**2 + 32*...