Integrand size = 25, antiderivative size = 181 \[ \int \frac {1}{\left (3-x+2 x^2\right )^3 \left (2+3 x+5 x^2\right )^3} \, dx=-\frac {5 (223707+77020 x)}{87308276 \left (2+3 x+5 x^2\right )^2}+\frac {13-6 x}{1012 \left (3-x+2 x^2\right )^2 \left (2+3 x+5 x^2\right )^2}+\frac {5 (302-35 x)}{64009 \left (3-x+2 x^2\right ) \left (2+3 x+5 x^2\right )^2}+\frac {15 (2618306+7140435 x)}{14886061058 \left (2+3 x+5 x^2\right )}-\frac {880575 \arctan \left (\frac {1-4 x}{\sqrt {23}}\right )}{340783916 \sqrt {23}}+\frac {2768835 \arctan \left (\frac {3+10 x}{\sqrt {31}}\right )}{619080044 \sqrt {31}}+\frac {405 \log \left (3-x+2 x^2\right )}{1288408}-\frac {405 \log \left (2+3 x+5 x^2\right )}{1288408} \]
-5/87308276*(223707+77020*x)/(5*x^2+3*x+2)^2+1/1012*(13-6*x)/(2*x^2-x+3)^2 /(5*x^2+3*x+2)^2+5/64009*(302-35*x)/(2*x^2-x+3)/(5*x^2+3*x+2)^2+15/1488606 1058*(2618306+7140435*x)/(5*x^2+3*x+2)+405/1288408*ln(2*x^2-x+3)-405/12884 08*ln(5*x^2+3*x+2)-880575/7838030068*arctan(1/23*(1-4*x)*23^(1/2))*23^(1/2 )+2768835/19191481364*arctan(1/31*(3+10*x)*31^(1/2))*31^(1/2)
Time = 0.06 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\left (3-x+2 x^2\right )^3 \left (2+3 x+5 x^2\right )^3} \, dx=\frac {-4342+11154 x-9275 x^2+6850 x^3}{345092 \left (6+7 x+16 x^2+x^3+10 x^4\right )^2}+\frac {5 \left (14085977+51156233 x-5711469 x^2+42842610 x^3\right )}{14886061058 \left (6+7 x+16 x^2+x^3+10 x^4\right )}+\frac {880575 \arctan \left (\frac {-1+4 x}{\sqrt {23}}\right )}{340783916 \sqrt {23}}+\frac {2768835 \arctan \left (\frac {3+10 x}{\sqrt {31}}\right )}{619080044 \sqrt {31}}+\frac {405 \log \left (3-x+2 x^2\right )}{1288408}-\frac {405 \log \left (2+3 x+5 x^2\right )}{1288408} \]
(-4342 + 11154*x - 9275*x^2 + 6850*x^3)/(345092*(6 + 7*x + 16*x^2 + x^3 + 10*x^4)^2) + (5*(14085977 + 51156233*x - 5711469*x^2 + 42842610*x^3))/(148 86061058*(6 + 7*x + 16*x^2 + x^3 + 10*x^4)) + (880575*ArcTan[(-1 + 4*x)/Sq rt[23]])/(340783916*Sqrt[23]) + (2768835*ArcTan[(3 + 10*x)/Sqrt[31]])/(619 080044*Sqrt[31]) + (405*Log[3 - x + 2*x^2])/1288408 - (405*Log[2 + 3*x + 5 *x^2])/1288408
Time = 0.66 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.14, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {1305, 27, 2135, 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.
| \(\displaystyle \int \frac {1}{\left (2 x^2-x+3\right )^3 \left (5 x^2+3 x+2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1305 |
| \(\displaystyle \frac {13-6 x}{1012 \left (2 x^2-x+3\right )^2 \left (5 x^2+3 x+2\right )^2}-\frac {\int -\frac {110 \left (-21 x^2+40 x+41\right )}{\left (2 x^2-x+3\right )^2 \left (5 x^2+3 x+2\right )^3}dx}{11132}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{506} \int \frac {-21 x^2+40 x+41}{\left (2 x^2-x+3\right )^2 \left (5 x^2+3 x+2\right )^3}dx+\frac {13-6 x}{1012 \left (2 x^2-x+3\right )^2 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 2135 |
| \(\displaystyle \frac {5}{506} \left (\frac {\int \frac {22 \left (-1750 x^2+17315 x+6819\right )}{\left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^3}dx}{5566}+\frac {2 (302-35 x)}{253 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}\right )+\frac {13-6 x}{1012 \left (2 x^2-x+3\right )^2 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{506} \left (\frac {1}{253} \int \frac {-1750 x^2+17315 x+6819}{\left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^3}dx+\frac {2 (302-35 x)}{253 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}\right )+\frac {13-6 x}{1012 \left (2 x^2-x+3\right )^2 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 2135 |
| \(\displaystyle \frac {5}{506} \left (\frac {1}{253} \left (\frac {\int \frac {264 \left (-38510 x^2-114479 x+45248\right )}{\left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}dx}{15004}-\frac {77020 x+223707}{682 \left (5 x^2+3 x+2\right )^2}\right )+\frac {2 (302-35 x)}{253 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}\right )+\frac {13-6 x}{1012 \left (2 x^2-x+3\right )^2 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{506} \left (\frac {1}{253} \left (\frac {6}{341} \int \frac {-38510 x^2-114479 x+45248}{\left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}dx-\frac {77020 x+223707}{682 \left (5 x^2+3 x+2\right )^2}\right )+\frac {2 (302-35 x)}{253 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}\right )+\frac {13-6 x}{1012 \left (2 x^2-x+3\right )^2 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 2135 |
| \(\displaystyle \frac {5}{506} \left (\frac {1}{253} \left (\frac {6}{341} \left (\frac {\int \frac {11 \left (14280870 x^2-5235733 x+5790640\right )}{\left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )}dx}{7502}+\frac {7140435 x+2618306}{682 \left (5 x^2+3 x+2\right )}\right )-\frac {77020 x+223707}{682 \left (5 x^2+3 x+2\right )^2}\right )+\frac {2 (302-35 x)}{253 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}\right )+\frac {13-6 x}{1012 \left (2 x^2-x+3\right )^2 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{506} \left (\frac {1}{253} \left (\frac {6}{341} \left (\frac {1}{682} \int \frac {14280870 x^2-5235733 x+5790640}{\left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )}dx+\frac {7140435 x+2618306}{682 \left (5 x^2+3 x+2\right )}\right )-\frac {77020 x+223707}{682 \left (5 x^2+3 x+2\right )^2}\right )+\frac {2 (302-35 x)}{253 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}\right )+\frac {13-6 x}{1012 \left (2 x^2-x+3\right )^2 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 2141 |
| \(\displaystyle \frac {5}{506} \left (\frac {1}{253} \left (\frac {6}{341} \left (\frac {1}{682} \left (\frac {1}{242} \int \frac {10571 (28566 x+22211)}{2 x^2-x+3}dx+\frac {1}{242} \int \frac {5819 (53374-129735 x)}{5 x^2+3 x+2}dx\right )+\frac {7140435 x+2618306}{682 \left (5 x^2+3 x+2\right )}\right )-\frac {77020 x+223707}{682 \left (5 x^2+3 x+2\right )^2}\right )+\frac {2 (302-35 x)}{253 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}\right )+\frac {13-6 x}{1012 \left (2 x^2-x+3\right )^2 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{506} \left (\frac {1}{253} \left (\frac {6}{341} \left (\frac {1}{682} \left (\frac {961}{22} \int \frac {28566 x+22211}{2 x^2-x+3}dx+\frac {529}{22} \int \frac {53374-129735 x}{5 x^2+3 x+2}dx\right )+\frac {7140435 x+2618306}{682 \left (5 x^2+3 x+2\right )}\right )-\frac {77020 x+223707}{682 \left (5 x^2+3 x+2\right )^2}\right )+\frac {2 (302-35 x)}{253 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}\right )+\frac {13-6 x}{1012 \left (2 x^2-x+3\right )^2 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {5}{506} \left (\frac {1}{253} \left (\frac {6}{341} \left (\frac {1}{682} \left (\frac {961}{22} \left (\frac {58705}{2} \int \frac {1}{2 x^2-x+3}dx+\frac {14283}{2} \int -\frac {1-4 x}{2 x^2-x+3}dx\right )+\frac {529}{22} \left (\frac {184589}{2} \int \frac {1}{5 x^2+3 x+2}dx-\frac {25947}{2} \int \frac {10 x+3}{5 x^2+3 x+2}dx\right )\right )+\frac {7140435 x+2618306}{682 \left (5 x^2+3 x+2\right )}\right )-\frac {77020 x+223707}{682 \left (5 x^2+3 x+2\right )^2}\right )+\frac {2 (302-35 x)}{253 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}\right )+\frac {13-6 x}{1012 \left (2 x^2-x+3\right )^2 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5}{506} \left (\frac {1}{253} \left (\frac {6}{341} \left (\frac {1}{682} \left (\frac {961}{22} \left (\frac {58705}{2} \int \frac {1}{2 x^2-x+3}dx-\frac {14283}{2} \int \frac {1-4 x}{2 x^2-x+3}dx\right )+\frac {529}{22} \left (\frac {184589}{2} \int \frac {1}{5 x^2+3 x+2}dx-\frac {25947}{2} \int \frac {10 x+3}{5 x^2+3 x+2}dx\right )\right )+\frac {7140435 x+2618306}{682 \left (5 x^2+3 x+2\right )}\right )-\frac {77020 x+223707}{682 \left (5 x^2+3 x+2\right )^2}\right )+\frac {2 (302-35 x)}{253 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}\right )+\frac {13-6 x}{1012 \left (2 x^2-x+3\right )^2 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {5}{506} \left (\frac {1}{253} \left (\frac {6}{341} \left (\frac {1}{682} \left (\frac {961}{22} \left (-\frac {14283}{2} \int \frac {1-4 x}{2 x^2-x+3}dx-58705 \int \frac {1}{-(4 x-1)^2-23}d(4 x-1)\right )+\frac {529}{22} \left (-\frac {25947}{2} \int \frac {10 x+3}{5 x^2+3 x+2}dx-184589 \int \frac {1}{-(10 x+3)^2-31}d(10 x+3)\right )\right )+\frac {7140435 x+2618306}{682 \left (5 x^2+3 x+2\right )}\right )-\frac {77020 x+223707}{682 \left (5 x^2+3 x+2\right )^2}\right )+\frac {2 (302-35 x)}{253 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}\right )+\frac {13-6 x}{1012 \left (2 x^2-x+3\right )^2 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {5}{506} \left (\frac {1}{253} \left (\frac {6}{341} \left (\frac {1}{682} \left (\frac {961}{22} \left (\frac {58705 \arctan \left (\frac {4 x-1}{\sqrt {23}}\right )}{\sqrt {23}}-\frac {14283}{2} \int \frac {1-4 x}{2 x^2-x+3}dx\right )+\frac {529}{22} \left (\frac {184589 \arctan \left (\frac {10 x+3}{\sqrt {31}}\right )}{\sqrt {31}}-\frac {25947}{2} \int \frac {10 x+3}{5 x^2+3 x+2}dx\right )\right )+\frac {7140435 x+2618306}{682 \left (5 x^2+3 x+2\right )}\right )-\frac {77020 x+223707}{682 \left (5 x^2+3 x+2\right )^2}\right )+\frac {2 (302-35 x)}{253 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}\right )+\frac {13-6 x}{1012 \left (2 x^2-x+3\right )^2 \left (5 x^2+3 x+2\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {5}{506} \left (\frac {1}{253} \left (\frac {6}{341} \left (\frac {1}{682} \left (\frac {961}{22} \left (\frac {58705 \arctan \left (\frac {4 x-1}{\sqrt {23}}\right )}{\sqrt {23}}+\frac {14283}{2} \log \left (2 x^2-x+3\right )\right )+\frac {529}{22} \left (\frac {184589 \arctan \left (\frac {10 x+3}{\sqrt {31}}\right )}{\sqrt {31}}-\frac {25947}{2} \log \left (5 x^2+3 x+2\right )\right )\right )+\frac {7140435 x+2618306}{682 \left (5 x^2+3 x+2\right )}\right )-\frac {77020 x+223707}{682 \left (5 x^2+3 x+2\right )^2}\right )+\frac {2 (302-35 x)}{253 \left (2 x^2-x+3\right ) \left (5 x^2+3 x+2\right )^2}\right )+\frac {13-6 x}{1012 \left (2 x^2-x+3\right )^2 \left (5 x^2+3 x+2\right )^2}\) |
(13 - 6*x)/(1012*(3 - x + 2*x^2)^2*(2 + 3*x + 5*x^2)^2) + (5*((2*(302 - 35 *x))/(253*(3 - x + 2*x^2)*(2 + 3*x + 5*x^2)^2) + (-1/682*(223707 + 77020*x )/(2 + 3*x + 5*x^2)^2 + (6*((2618306 + 7140435*x)/(682*(2 + 3*x + 5*x^2)) + ((961*((58705*ArcTan[(-1 + 4*x)/Sqrt[23]])/Sqrt[23] + (14283*Log[3 - x + 2*x^2])/2))/22 + (529*((184589*ArcTan[(3 + 10*x)/Sqrt[31]])/Sqrt[31] - (2 5947*Log[2 + 3*x + 5*x^2])/2))/22)/682))/341)/253))/506
3.1.57.3.1 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[((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 = 0.86 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.65
| method | result | size |
| default | \(-\frac {25 \left (-\frac {3013197}{961} x^{3}-\frac {14516062}{4805} x^{2}-\frac {51193868}{24025} x -\frac {5423968}{24025}\right )}{2576816 \left (5 x^{2}+3 x +2\right )^{2}}-\frac {405 \ln \left (5 x^{2}+3 x +2\right )}{1288408}+\frac {2768835 \arctan \left (\frac {\left (10 x +3\right ) \sqrt {31}}{31}\right ) \sqrt {31}}{19191481364}+\frac {\frac {302907}{529} x^{3}-\frac {368291}{529} x^{2}+\frac {2501587}{2116} x -\frac {665819}{1058}}{644204 \left (2 x^{2}-x +3\right )^{2}}+\frac {405 \ln \left (2 x^{2}-x +3\right )}{1288408}+\frac {880575 \sqrt {23}\, \arctan \left (\frac {\left (-1+4 x \right ) \sqrt {23}}{23}\right )}{7838030068}\) | \(118\) |
| risch | \(\frac {\frac {1071065250}{7443030529} x^{7}-\frac {35680200}{7443030529} x^{6}+\frac {5956663105}{14886061058} x^{5}+\frac {2002653845}{14886061058} x^{4}+\frac {5543790435}{14886061058} x^{3}+\frac {4691822415}{29772122116} x^{2}+\frac {1254420353}{7443030529} x +\frac {235280627}{14886061058}}{\left (2 x^{2}-x +3\right )^{2} \left (5 x^{2}+3 x +2\right )^{2}}+\frac {405 \ln \left (16 x^{2}-8 x +24\right )}{1288408}+\frac {880575 \sqrt {23}\, \arctan \left (\frac {\left (-1+4 x \right ) \sqrt {23}}{23}\right )}{7838030068}-\frac {405 \ln \left (100 x^{2}+60 x +40\right )}{1288408}+\frac {2768835 \arctan \left (\frac {\left (10 x +3\right ) \sqrt {31}}{31}\right ) \sqrt {31}}{19191481364}\) | \(121\) |
-25/2576816*(-3013197/961*x^3-14516062/4805*x^2-51193868/24025*x-5423968/2 4025)/(5*x^2+3*x+2)^2-405/1288408*ln(5*x^2+3*x+2)+2768835/19191481364*arct an(1/31*(10*x+3)*31^(1/2))*31^(1/2)+1/644204*(302907/529*x^3-368291/529*x^ 2+2501587/2116*x-665819/1058)/(2*x^2-x+3)^2+405/1288408*ln(2*x^2-x+3)+8805 75/7838030068*23^(1/2)*arctan(1/23*(-1+4*x)*23^(1/2))
Time = 0.29 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.64 \[ \int \frac {1}{\left (3-x+2 x^2\right )^3 \left (2+3 x+5 x^2\right )^3} \, dx=\frac {67202918046000 \, x^{7} - 2238718468800 \, x^{6} + 186872434930060 \, x^{5} + 62827256425340 \, x^{4} + 173919793526820 \, x^{3} + 67376830890 \, \sqrt {31} {\left (100 \, x^{8} + 20 \, x^{7} + 321 \, x^{6} + 172 \, x^{5} + 390 \, x^{4} + 236 \, x^{3} + 241 \, x^{2} + 84 \, x + 36\right )} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + 52466419650 \, \sqrt {23} {\left (100 \, x^{8} + 20 \, x^{7} + 321 \, x^{6} + 172 \, x^{5} + 390 \, x^{4} + 236 \, x^{3} + 241 \, x^{2} + 84 \, x + 36\right )} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + 73595926401690 \, x^{2} - 146799174285 \, {\left (100 \, x^{8} + 20 \, x^{7} + 321 \, x^{6} + 172 \, x^{5} + 390 \, x^{4} + 236 \, x^{3} + 241 \, x^{2} + 84 \, x + 36\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) + 146799174285 \, {\left (100 \, x^{8} + 20 \, x^{7} + 321 \, x^{6} + 172 \, x^{5} + 390 \, x^{4} + 236 \, x^{3} + 241 \, x^{2} + 84 \, x + 36\right )} \log \left (2 \, x^{2} - x + 3\right ) + 78707350628632 \, x + 7381223830244}{467005507511576 \, {\left (100 \, x^{8} + 20 \, x^{7} + 321 \, x^{6} + 172 \, x^{5} + 390 \, x^{4} + 236 \, x^{3} + 241 \, x^{2} + 84 \, x + 36\right )}} \]
1/467005507511576*(67202918046000*x^7 - 2238718468800*x^6 + 18687243493006 0*x^5 + 62827256425340*x^4 + 173919793526820*x^3 + 67376830890*sqrt(31)*(1 00*x^8 + 20*x^7 + 321*x^6 + 172*x^5 + 390*x^4 + 236*x^3 + 241*x^2 + 84*x + 36)*arctan(1/31*sqrt(31)*(10*x + 3)) + 52466419650*sqrt(23)*(100*x^8 + 20 *x^7 + 321*x^6 + 172*x^5 + 390*x^4 + 236*x^3 + 241*x^2 + 84*x + 36)*arctan (1/23*sqrt(23)*(4*x - 1)) + 73595926401690*x^2 - 146799174285*(100*x^8 + 2 0*x^7 + 321*x^6 + 172*x^5 + 390*x^4 + 236*x^3 + 241*x^2 + 84*x + 36)*log(5 *x^2 + 3*x + 2) + 146799174285*(100*x^8 + 20*x^7 + 321*x^6 + 172*x^5 + 390 *x^4 + 236*x^3 + 241*x^2 + 84*x + 36)*log(2*x^2 - x + 3) + 78707350628632* x + 7381223830244)/(100*x^8 + 20*x^7 + 321*x^6 + 172*x^5 + 390*x^4 + 236*x ^3 + 241*x^2 + 84*x + 36)
Time = 0.19 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\left (3-x+2 x^2\right )^3 \left (2+3 x+5 x^2\right )^3} \, dx=\frac {4284261000 x^{7} - 142720800 x^{6} + 11913326210 x^{5} + 4005307690 x^{4} + 11087580870 x^{3} + 4691822415 x^{2} + 5017681412 x + 470561254}{2977212211600 x^{8} + 595442442320 x^{7} + 9556851199236 x^{6} + 5120805003952 x^{5} + 11611127625240 x^{4} + 7026220819376 x^{3} + 7175081429956 x^{2} + 2500858257744 x + 1071796396176} + \frac {405 \log {\left (x^{2} - \frac {x}{2} + \frac {3}{2} \right )}}{1288408} - \frac {405 \log {\left (x^{2} + \frac {3 x}{5} + \frac {2}{5} \right )}}{1288408} + \frac {880575 \sqrt {23} \operatorname {atan}{\left (\frac {4 \sqrt {23} x}{23} - \frac {\sqrt {23}}{23} \right )}}{7838030068} + \frac {2768835 \sqrt {31} \operatorname {atan}{\left (\frac {10 \sqrt {31} x}{31} + \frac {3 \sqrt {31}}{31} \right )}}{19191481364} \]
(4284261000*x**7 - 142720800*x**6 + 11913326210*x**5 + 4005307690*x**4 + 1 1087580870*x**3 + 4691822415*x**2 + 5017681412*x + 470561254)/(29772122116 00*x**8 + 595442442320*x**7 + 9556851199236*x**6 + 5120805003952*x**5 + 11 611127625240*x**4 + 7026220819376*x**3 + 7175081429956*x**2 + 250085825774 4*x + 1071796396176) + 405*log(x**2 - x/2 + 3/2)/1288408 - 405*log(x**2 + 3*x/5 + 2/5)/1288408 + 880575*sqrt(23)*atan(4*sqrt(23)*x/23 - sqrt(23)/23) /7838030068 + 2768835*sqrt(31)*atan(10*sqrt(31)*x/31 + 3*sqrt(31)/31)/1919 1481364
Time = 0.26 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\left (3-x+2 x^2\right )^3 \left (2+3 x+5 x^2\right )^3} \, dx=\frac {2768835}{19191481364} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {880575}{7838030068} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {4284261000 \, x^{7} - 142720800 \, x^{6} + 11913326210 \, x^{5} + 4005307690 \, x^{4} + 11087580870 \, x^{3} + 4691822415 \, x^{2} + 5017681412 \, x + 470561254}{29772122116 \, {\left (100 \, x^{8} + 20 \, x^{7} + 321 \, x^{6} + 172 \, x^{5} + 390 \, x^{4} + 236 \, x^{3} + 241 \, x^{2} + 84 \, x + 36\right )}} - \frac {405}{1288408} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) + \frac {405}{1288408} \, \log \left (2 \, x^{2} - x + 3\right ) \]
2768835/19191481364*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 880575/783 8030068*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) + 1/29772122116*(42842610 00*x^7 - 142720800*x^6 + 11913326210*x^5 + 4005307690*x^4 + 11087580870*x^ 3 + 4691822415*x^2 + 5017681412*x + 470561254)/(100*x^8 + 20*x^7 + 321*x^6 + 172*x^5 + 390*x^4 + 236*x^3 + 241*x^2 + 84*x + 36) - 405/1288408*log(5* x^2 + 3*x + 2) + 405/1288408*log(2*x^2 - x + 3)
Time = 0.28 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.64 \[ \int \frac {1}{\left (3-x+2 x^2\right )^3 \left (2+3 x+5 x^2\right )^3} \, dx=\frac {2768835}{19191481364} \, \sqrt {31} \arctan \left (\frac {1}{31} \, \sqrt {31} {\left (10 \, x + 3\right )}\right ) + \frac {880575}{7838030068} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {4284261000 \, x^{7} - 142720800 \, x^{6} + 11913326210 \, x^{5} + 4005307690 \, x^{4} + 11087580870 \, x^{3} + 4691822415 \, x^{2} + 5017681412 \, x + 470561254}{29772122116 \, {\left (10 \, x^{4} + x^{3} + 16 \, x^{2} + 7 \, x + 6\right )}^{2}} - \frac {405}{1288408} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) + \frac {405}{1288408} \, \log \left (2 \, x^{2} - x + 3\right ) \]
2768835/19191481364*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)) + 880575/783 8030068*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) + 1/29772122116*(42842610 00*x^7 - 142720800*x^6 + 11913326210*x^5 + 4005307690*x^4 + 11087580870*x^ 3 + 4691822415*x^2 + 5017681412*x + 470561254)/(10*x^4 + x^3 + 16*x^2 + 7* x + 6)^2 - 405/1288408*log(5*x^2 + 3*x + 2) + 405/1288408*log(2*x^2 - x + 3)
Time = 0.20 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\left (3-x+2 x^2\right )^3 \left (2+3 x+5 x^2\right )^3} \, dx=\frac {\frac {21421305\,x^7}{14886061058}-\frac {356802\,x^6}{7443030529}+\frac {1191332621\,x^5}{297721221160}+\frac {400530769\,x^4}{297721221160}+\frac {1108758087\,x^3}{297721221160}+\frac {938364483\,x^2}{595442442320}+\frac {1254420353\,x}{744303052900}+\frac {235280627}{1488606105800}}{x^8+\frac {x^7}{5}+\frac {321\,x^6}{100}+\frac {43\,x^5}{25}+\frac {39\,x^4}{10}+\frac {59\,x^3}{25}+\frac {241\,x^2}{100}+\frac {21\,x}{25}+\frac {9}{25}}+\ln \left (x-\frac {1}{4}+\frac {\sqrt {23}\,1{}\mathrm {i}}{4}\right )\,\left (\frac {405}{1288408}+\frac {\sqrt {23}\,880575{}\mathrm {i}}{15676060136}\right )-\ln \left (x+\frac {3}{10}-\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (\frac {405}{1288408}+\frac {\sqrt {31}\,2768835{}\mathrm {i}}{38382962728}\right )+\ln \left (x+\frac {3}{10}+\frac {\sqrt {31}\,1{}\mathrm {i}}{10}\right )\,\left (-\frac {405}{1288408}+\frac {\sqrt {31}\,2768835{}\mathrm {i}}{38382962728}\right )-\ln \left (x-\frac {1}{4}-\frac {\sqrt {23}\,1{}\mathrm {i}}{4}\right )\,\left (-\frac {405}{1288408}+\frac {\sqrt {23}\,880575{}\mathrm {i}}{15676060136}\right ) \]
log(x + (23^(1/2)*1i)/4 - 1/4)*((23^(1/2)*880575i)/15676060136 + 405/12884 08) - log(x - (23^(1/2)*1i)/4 - 1/4)*((23^(1/2)*880575i)/15676060136 - 405 /1288408) - log(x - (31^(1/2)*1i)/10 + 3/10)*((31^(1/2)*2768835i)/38382962 728 + 405/1288408) + log(x + (31^(1/2)*1i)/10 + 3/10)*((31^(1/2)*2768835i) /38382962728 - 405/1288408) + ((1254420353*x)/744303052900 + (938364483*x^ 2)/595442442320 + (1108758087*x^3)/297721221160 + (400530769*x^4)/29772122 1160 + (1191332621*x^5)/297721221160 - (356802*x^6)/7443030529 + (21421305 *x^7)/14886061058 + 235280627/1488606105800)/((21*x)/25 + (241*x^2)/100 + (59*x^3)/25 + (39*x^4)/10 + (43*x^5)/25 + (321*x^6)/100 + x^7/5 + x^8 + 9/ 25)