Integrand size = 27, antiderivative size = 199 \[ \int \frac {1}{\left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )} \, dx=\frac {13-6 x}{759 \left (3-x+2 x^2\right )^{3/2}}+\frac {3603-658 x}{128018 \sqrt {3-x+2 x^2}}+\frac {1}{484} \sqrt {\frac {1}{682} \left (-15457+25000 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (-15457+25000 \sqrt {2}\right )}} \left (443-98 \sqrt {2}+\left (247+345 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )-\frac {1}{484} \sqrt {\frac {1}{682} \left (15457+25000 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {11}{31 \left (15457+25000 \sqrt {2}\right )}} \left (443+98 \sqrt {2}+\left (247-345 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right ) \]
1/759*(13-6*x)/(2*x^2-x+3)^(3/2)+1/128018*(3603-658*x)/(2*x^2-x+3)^(1/2)+1 /330088*arctan(1/31*(443-98*2^(1/2)+x*(247+345*2^(1/2)))*341^(1/2)/(-15457 +25000*2^(1/2))^(1/2)/(2*x^2-x+3)^(1/2))*(-10541674+17050000*2^(1/2))^(1/2 )-1/330088*arctanh(1/31*(443+x*(247-345*2^(1/2))+98*2^(1/2))*341^(1/2)/(15 457+25000*2^(1/2))^(1/2)/(2*x^2-x+3)^(1/2))*(10541674+17050000*2^(1/2))^(1 /2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.64 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.05 \[ \int \frac {1}{\left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )} \, dx=\frac {39005-19767 x+23592 x^2-3948 x^3}{384054 \left (3-x+2 x^2\right )^{3/2}}+\frac {1}{484} \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {249 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+108 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}-65 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ] \]
(39005 - 19767*x + 23592*x^2 - 3948*x^3)/(384054*(3 - x + 2*x^2)^(3/2)) + RootSum[-56 - 26*Sqrt[2]*#1 + 17*#1^2 + 6*Sqrt[2]*#1^3 - 5*#1^4 & , (249*L og[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1] + 108*Sqrt[2]*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2] - #1]*#1 - 65*Log[-(Sqrt[2]*x) + Sqrt[3 - x + 2*x^2 ] - #1]*#1^2)/(-13*Sqrt[2] + 17*#1 + 9*Sqrt[2]*#1^2 - 10*#1^3) & ]/484
Time = 0.60 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1305, 27, 2135, 27, 1368, 27, 1362, 217, 219}
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.
\(\displaystyle \int \frac {1}{\left (2 x^2-x+3\right )^{5/2} \left (5 x^2+3 x+2\right )} \, dx\) |
\(\Big \downarrow \) 1305 |
\(\displaystyle \frac {13-6 x}{759 \left (2 x^2-x+3\right )^{3/2}}-\frac {\int -\frac {33 \left (-40 x^2+91 x+168\right )}{2 \left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )}dx}{8349}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{506} \int \frac {-40 x^2+91 x+168}{\left (2 x^2-x+3\right )^{3/2} \left (5 x^2+3 x+2\right )}dx+\frac {13-6 x}{759 \left (2 x^2-x+3\right )^{3/2}}\) |
\(\Big \downarrow \) 2135 |
\(\displaystyle \frac {1}{506} \left (\frac {\int \frac {5819 (65 x+54)}{2 \sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2783}+\frac {3603-658 x}{253 \sqrt {2 x^2-x+3}}\right )+\frac {13-6 x}{759 \left (2 x^2-x+3\right )^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{506} \left (\frac {23}{22} \int \frac {65 x+54}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx+\frac {3603-658 x}{253 \sqrt {2 x^2-x+3}}\right )+\frac {13-6 x}{759 \left (2 x^2-x+3\right )^{3/2}}\) |
\(\Big \downarrow \) 1368 |
\(\displaystyle \frac {1}{506} \left (\frac {23}{22} \left (\frac {\int \frac {11 \left (\left (119+65 \sqrt {2}\right ) x+54 \sqrt {2}+11\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}-\frac {\int \frac {11 \left (\left (119-65 \sqrt {2}\right ) x-54 \sqrt {2}+11\right )}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{22 \sqrt {2}}\right )+\frac {3603-658 x}{253 \sqrt {2 x^2-x+3}}\right )+\frac {13-6 x}{759 \left (2 x^2-x+3\right )^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{506} \left (\frac {23}{22} \left (\frac {\int \frac {\left (119+65 \sqrt {2}\right ) x+54 \sqrt {2}+11}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}-\frac {\int \frac {\left (119-65 \sqrt {2}\right ) x-54 \sqrt {2}+11}{\sqrt {2 x^2-x+3} \left (5 x^2+3 x+2\right )}dx}{2 \sqrt {2}}\right )+\frac {3603-658 x}{253 \sqrt {2 x^2-x+3}}\right )+\frac {13-6 x}{759 \left (2 x^2-x+3\right )^{3/2}}\) |
\(\Big \downarrow \) 1362 |
\(\displaystyle \frac {1}{506} \left (\frac {23}{22} \left (\frac {\left (15457+25000 \sqrt {2}\right ) \int \frac {1}{31 \left (15457+25000 \sqrt {2}\right )-\frac {11 \left (\left (247-345 \sqrt {2}\right ) x+98 \sqrt {2}+443\right )^2}{2 x^2-x+3}}d\left (-\frac {\left (247-345 \sqrt {2}\right ) x+98 \sqrt {2}+443}{\sqrt {2 x^2-x+3}}\right )}{\sqrt {2}}-\frac {\left (15457-25000 \sqrt {2}\right ) \int \frac {1}{31 \left (15457-25000 \sqrt {2}\right )-\frac {11 \left (\left (247+345 \sqrt {2}\right ) x-98 \sqrt {2}+443\right )^2}{2 x^2-x+3}}d\left (-\frac {\left (247+345 \sqrt {2}\right ) x-98 \sqrt {2}+443}{\sqrt {2 x^2-x+3}}\right )}{\sqrt {2}}\right )+\frac {3603-658 x}{253 \sqrt {2 x^2-x+3}}\right )+\frac {13-6 x}{759 \left (2 x^2-x+3\right )^{3/2}}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{506} \left (\frac {23}{22} \left (\frac {\left (15457+25000 \sqrt {2}\right ) \int \frac {1}{31 \left (15457+25000 \sqrt {2}\right )-\frac {11 \left (\left (247-345 \sqrt {2}\right ) x+98 \sqrt {2}+443\right )^2}{2 x^2-x+3}}d\left (-\frac {\left (247-345 \sqrt {2}\right ) x+98 \sqrt {2}+443}{\sqrt {2 x^2-x+3}}\right )}{\sqrt {2}}-\frac {\left (15457-25000 \sqrt {2}\right ) \arctan \left (\frac {\sqrt {\frac {11}{31 \left (25000 \sqrt {2}-15457\right )}} \left (\left (247+345 \sqrt {2}\right ) x-98 \sqrt {2}+443\right )}{\sqrt {2 x^2-x+3}}\right )}{\sqrt {682 \left (25000 \sqrt {2}-15457\right )}}\right )+\frac {3603-658 x}{253 \sqrt {2 x^2-x+3}}\right )+\frac {13-6 x}{759 \left (2 x^2-x+3\right )^{3/2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{506} \left (\frac {23}{22} \left (-\frac {\left (15457-25000 \sqrt {2}\right ) \arctan \left (\frac {\sqrt {\frac {11}{31 \left (25000 \sqrt {2}-15457\right )}} \left (\left (247+345 \sqrt {2}\right ) x-98 \sqrt {2}+443\right )}{\sqrt {2 x^2-x+3}}\right )}{\sqrt {682 \left (25000 \sqrt {2}-15457\right )}}-\sqrt {\frac {1}{682} \left (15457+25000 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {11}{31 \left (15457+25000 \sqrt {2}\right )}} \left (\left (247-345 \sqrt {2}\right ) x+98 \sqrt {2}+443\right )}{\sqrt {2 x^2-x+3}}\right )\right )+\frac {3603-658 x}{253 \sqrt {2 x^2-x+3}}\right )+\frac {13-6 x}{759 \left (2 x^2-x+3\right )^{3/2}}\) |
(13 - 6*x)/(759*(3 - x + 2*x^2)^(3/2)) + ((3603 - 658*x)/(253*Sqrt[3 - x + 2*x^2]) + (23*(-(((15457 - 25000*Sqrt[2])*ArcTan[(Sqrt[11/(31*(-15457 + 2 5000*Sqrt[2]))]*(443 - 98*Sqrt[2] + (247 + 345*Sqrt[2])*x))/Sqrt[3 - x + 2 *x^2]])/Sqrt[682*(-15457 + 25000*Sqrt[2])]) - Sqrt[(15457 + 25000*Sqrt[2]) /682]*ArcTanh[(Sqrt[11/(31*(15457 + 25000*Sqrt[2]))]*(443 + 98*Sqrt[2] + ( 247 - 345*Sqrt[2])*x))/Sqrt[3 - x + 2*x^2]]))/22)/506
3.1.97.3.1 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_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
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[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[-2*g*(g*b - 2*a*h) Subst[I nt[1/Simp[g*(g*b - 2*a*h)*(b^2 - 4*a*c) - (b*d - a*e)*x^2, x], x], x, Simp[ g*b - 2*a*h - (b*h - 2*g*c)*x, x]/Sqrt[d + e*x + f*x^2]], x] /; FreeQ[{a, b , c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && Ne Q[b*d - a*e, 0] && EqQ[h^2*(b*d - a*e) - 2*g*h*(c*d - a*f) + g^2*(c*e - b*f ), 0]
Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 2]}, Simp[1/(2*q) Int[Simp[h*(b*d - a*e) - g*(c*d - a*f - q) - (g*(c*e - b*f) - h*(c*d - a*f + q))*x, x]/((a + b*x + c*x^2)*Sqr t[d + e*x + f*x^2]), x], x] - Simp[1/(2*q) Int[Simp[h*(b*d - a*e) - g*(c* d - a*f + q) - (g*(c*e - b*f) - h*(c*d - a*f - q))*x, x]/((a + b*x + c*x^2) *Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && N eQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && NeQ[b*d - a*e, 0] && NegQ[b^2 - 4*a*c]
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.64 (sec) , antiderivative size = 472, normalized size of antiderivative = 2.37
method | result | size |
trager | \(-\frac {3948 x^{3}-23592 x^{2}+19767 x -39005}{384054 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1356301584 \operatorname {RootOf}\left (494371927368 \textit {\_Z}^{4}-3842440173 \textit {\_Z}^{2}+39062500\right )^{2}-10541674\right ) \ln \left (-\frac {584162228676213 x \operatorname {RootOf}\left (494371927368 \textit {\_Z}^{4}-3842440173 \textit {\_Z}^{2}+39062500\right )^{4} \operatorname {RootOf}\left (\textit {\_Z}^{2}+1356301584 \operatorname {RootOf}\left (494371927368 \textit {\_Z}^{4}-3842440173 \textit {\_Z}^{2}+39062500\right )^{2}-10541674\right )-13400087377743 \operatorname {RootOf}\left (494371927368 \textit {\_Z}^{4}-3842440173 \textit {\_Z}^{2}+39062500\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+1356301584 \operatorname {RootOf}\left (494371927368 \textit {\_Z}^{4}-3842440173 \textit {\_Z}^{2}+39062500\right )^{2}-10541674\right ) x -34735243833848250 \operatorname {RootOf}\left (494371927368 \textit {\_Z}^{4}-3842440173 \textit {\_Z}^{2}+39062500\right )^{2} \sqrt {2 x^{2}-x +3}+14587333029225 \operatorname {RootOf}\left (494371927368 \textit {\_Z}^{4}-3842440173 \textit {\_Z}^{2}+39062500\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+1356301584 \operatorname {RootOf}\left (494371927368 \textit {\_Z}^{4}-3842440173 \textit {\_Z}^{2}+39062500\right )^{2}-10541674\right )+74861069500 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1356301584 \operatorname {RootOf}\left (494371927368 \textit {\_Z}^{4}-3842440173 \textit {\_Z}^{2}+39062500\right )^{2}-10541674\right ) x +120911390237000 \sqrt {2 x^{2}-x +3}-140419212500 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1356301584 \operatorname {RootOf}\left (494371927368 \textit {\_Z}^{4}-3842440173 \textit {\_Z}^{2}+39062500\right )^{2}-10541674\right )}{994356 x \operatorname {RootOf}\left (494371927368 \textit {\_Z}^{4}-3842440173 \textit {\_Z}^{2}+39062500\right )^{2}+419 x +5711}\right )}{330088}+\frac {27 \operatorname {RootOf}\left (494371927368 \textit {\_Z}^{4}-3842440173 \textit {\_Z}^{2}+39062500\right ) \ln \left (-\frac {1009432331152496064 x \operatorname {RootOf}\left (494371927368 \textit {\_Z}^{4}-3842440173 \textit {\_Z}^{2}+39062500\right )^{5}+7463993424020496 \operatorname {RootOf}\left (494371927368 \textit {\_Z}^{4}-3842440173 \textit {\_Z}^{2}+39062500\right )^{3} x -25206911474500800 \operatorname {RootOf}\left (494371927368 \textit {\_Z}^{4}-3842440173 \textit {\_Z}^{2}+39062500\right )^{3}+1629806162292000 \operatorname {RootOf}\left (494371927368 \textit {\_Z}^{4}-3842440173 \textit {\_Z}^{2}+39062500\right )^{2} \sqrt {2 x^{2}-x +3}+10367533373175 \operatorname {RootOf}\left (494371927368 \textit {\_Z}^{4}-3842440173 \textit {\_Z}^{2}+39062500\right ) x -46727026216200 \operatorname {RootOf}\left (494371927368 \textit {\_Z}^{4}-3842440173 \textit {\_Z}^{2}+39062500\right )-6994190312500 \sqrt {2 x^{2}-x +3}}{1988712 x \operatorname {RootOf}\left (494371927368 \textit {\_Z}^{4}-3842440173 \textit {\_Z}^{2}+39062500\right )^{2}-16295 x -11422}\right )}{242}\) | \(472\) |
default | \(\frac {\sqrt {\frac {8 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}\, \sqrt {2}\, \left (10111 \sqrt {-775687+549362 \sqrt {2}}\, \sqrt {2}\, \sqrt {-8866+6820 \sqrt {2}}\, \arctan \left (\frac {\sqrt {-775687+549362 \sqrt {2}}\, \sqrt {-23 \left (8+3 \sqrt {2}\right ) \left (-\frac {23 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+24 \sqrt {2}-41\right )}\, \left (\frac {6485 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {10368 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+22379 \sqrt {2}+32016\right ) \left (\sqrt {2}-1+x \right ) \left (8+3 \sqrt {2}\right )}{11692487 \left (\frac {23 \left (\sqrt {2}-1+x \right )^{4}}{\left (\sqrt {2}+1-x \right )^{4}}+\frac {82 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+23\right ) \left (\sqrt {2}+1-x \right )}\right )+13910 \sqrt {-775687+549362 \sqrt {2}}\, \sqrt {-8866+6820 \sqrt {2}}\, \arctan \left (\frac {\sqrt {-775687+549362 \sqrt {2}}\, \sqrt {-23 \left (8+3 \sqrt {2}\right ) \left (-\frac {23 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+24 \sqrt {2}-41\right )}\, \left (\frac {6485 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {10368 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+22379 \sqrt {2}+32016\right ) \left (\sqrt {2}-1+x \right ) \left (8+3 \sqrt {2}\right )}{11692487 \left (\frac {23 \left (\sqrt {2}-1+x \right )^{4}}{\left (\sqrt {2}+1-x \right )^{4}}+\frac {82 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+23\right ) \left (\sqrt {2}+1-x \right )}\right )-993674 \,\operatorname {arctanh}\left (\frac {31 \sqrt {\frac {8 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}}{2 \sqrt {-8866+6820 \sqrt {2}}}\right ) \sqrt {2}-42685698 \,\operatorname {arctanh}\left (\frac {31 \sqrt {\frac {8 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}}{2 \sqrt {-8866+6820 \sqrt {2}}}\right )\right )}{10232728 \sqrt {\frac {\frac {8 \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (\sqrt {2}-1+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}{\left (1+\frac {\sqrt {2}-1+x}{\sqrt {2}+1-x}\right )^{2}}}\, \left (1+\frac {\sqrt {2}-1+x}{\sqrt {2}+1-x}\right ) \left (8+3 \sqrt {2}\right ) \sqrt {-8866+6820 \sqrt {2}}}-\frac {329 \left (-1+4 x \right )}{256036 \sqrt {2 x^{2}-x +3}}+\frac {13}{484 \sqrt {2 x^{2}-x +3}}-\frac {-1+4 x}{506 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {1}{66 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}\) | \(751\) |
-1/384054*(3948*x^3-23592*x^2+19767*x-39005)/(2*x^2-x+3)^(3/2)+1/330088*Ro otOf(_Z^2+1356301584*RootOf(494371927368*_Z^4-3842440173*_Z^2+39062500)^2- 10541674)*ln(-(584162228676213*x*RootOf(494371927368*_Z^4-3842440173*_Z^2+ 39062500)^4*RootOf(_Z^2+1356301584*RootOf(494371927368*_Z^4-3842440173*_Z^ 2+39062500)^2-10541674)-13400087377743*RootOf(494371927368*_Z^4-3842440173 *_Z^2+39062500)^2*RootOf(_Z^2+1356301584*RootOf(494371927368*_Z^4-38424401 73*_Z^2+39062500)^2-10541674)*x-34735243833848250*RootOf(494371927368*_Z^4 -3842440173*_Z^2+39062500)^2*(2*x^2-x+3)^(1/2)+14587333029225*RootOf(49437 1927368*_Z^4-3842440173*_Z^2+39062500)^2*RootOf(_Z^2+1356301584*RootOf(494 371927368*_Z^4-3842440173*_Z^2+39062500)^2-10541674)+74861069500*RootOf(_Z ^2+1356301584*RootOf(494371927368*_Z^4-3842440173*_Z^2+39062500)^2-1054167 4)*x+120911390237000*(2*x^2-x+3)^(1/2)-140419212500*RootOf(_Z^2+1356301584 *RootOf(494371927368*_Z^4-3842440173*_Z^2+39062500)^2-10541674))/(994356*x *RootOf(494371927368*_Z^4-3842440173*_Z^2+39062500)^2+419*x+5711))+27/242* RootOf(494371927368*_Z^4-3842440173*_Z^2+39062500)*ln(-(100943233115249606 4*x*RootOf(494371927368*_Z^4-3842440173*_Z^2+39062500)^5+7463993424020496* RootOf(494371927368*_Z^4-3842440173*_Z^2+39062500)^3*x-25206911474500800*R ootOf(494371927368*_Z^4-3842440173*_Z^2+39062500)^3+1629806162292000*RootO f(494371927368*_Z^4-3842440173*_Z^2+39062500)^2*(2*x^2-x+3)^(1/2)+10367533 373175*RootOf(494371927368*_Z^4-3842440173*_Z^2+39062500)*x-46727026216...
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 398, normalized size of antiderivative = 2.00 \[ \int \frac {1}{\left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )} \, dx=-\frac {1587 \, \sqrt {341} {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \sqrt {5711 i \, \sqrt {31} + 15457} \log \left (\frac {\sqrt {341} \sqrt {2 \, x^{2} - x + 3} \sqrt {5711 i \, \sqrt {31} + 15457} {\left (443 i \, \sqrt {31} - 341\right )} - 387500 \, \sqrt {31} {\left (i \, x - 6 i\right )} + 7362500 \, x - 8525000}{x}\right ) - 1587 \, \sqrt {341} {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \sqrt {5711 i \, \sqrt {31} + 15457} \log \left (\frac {\sqrt {341} \sqrt {2 \, x^{2} - x + 3} \sqrt {5711 i \, \sqrt {31} + 15457} {\left (-443 i \, \sqrt {31} + 341\right )} - 387500 \, \sqrt {31} {\left (i \, x - 6 i\right )} + 7362500 \, x - 8525000}{x}\right ) - 1587 \, \sqrt {341} {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \sqrt {-5711 i \, \sqrt {31} + 15457} \log \left (\frac {\sqrt {341} \sqrt {2 \, x^{2} - x + 3} {\left (443 i \, \sqrt {31} + 341\right )} \sqrt {-5711 i \, \sqrt {31} + 15457} - 387500 \, \sqrt {31} {\left (-i \, x + 6 i\right )} + 7362500 \, x - 8525000}{x}\right ) + 1587 \, \sqrt {341} {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \sqrt {-5711 i \, \sqrt {31} + 15457} \log \left (\frac {\sqrt {341} \sqrt {2 \, x^{2} - x + 3} {\left (-443 i \, \sqrt {31} - 341\right )} \sqrt {-5711 i \, \sqrt {31} + 15457} - 387500 \, \sqrt {31} {\left (-i \, x + 6 i\right )} + 7362500 \, x - 8525000}{x}\right ) + 2728 \, {\left (3948 \, x^{3} - 23592 \, x^{2} + 19767 \, x - 39005\right )} \sqrt {2 \, x^{2} - x + 3}}{1047699312 \, {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} \]
-1/1047699312*(1587*sqrt(341)*(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)*sqrt(5711 *I*sqrt(31) + 15457)*log((sqrt(341)*sqrt(2*x^2 - x + 3)*sqrt(5711*I*sqrt(3 1) + 15457)*(443*I*sqrt(31) - 341) - 387500*sqrt(31)*(I*x - 6*I) + 7362500 *x - 8525000)/x) - 1587*sqrt(341)*(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)*sqrt( 5711*I*sqrt(31) + 15457)*log((sqrt(341)*sqrt(2*x^2 - x + 3)*sqrt(5711*I*sq rt(31) + 15457)*(-443*I*sqrt(31) + 341) - 387500*sqrt(31)*(I*x - 6*I) + 73 62500*x - 8525000)/x) - 1587*sqrt(341)*(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)* sqrt(-5711*I*sqrt(31) + 15457)*log((sqrt(341)*sqrt(2*x^2 - x + 3)*(443*I*s qrt(31) + 341)*sqrt(-5711*I*sqrt(31) + 15457) - 387500*sqrt(31)*(-I*x + 6* I) + 7362500*x - 8525000)/x) + 1587*sqrt(341)*(4*x^4 - 4*x^3 + 13*x^2 - 6* x + 9)*sqrt(-5711*I*sqrt(31) + 15457)*log((sqrt(341)*sqrt(2*x^2 - x + 3)*( -443*I*sqrt(31) - 341)*sqrt(-5711*I*sqrt(31) + 15457) - 387500*sqrt(31)*(- I*x + 6*I) + 7362500*x - 8525000)/x) + 2728*(3948*x^3 - 23592*x^2 + 19767* x - 39005)*sqrt(2*x^2 - x + 3))/(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)
\[ \int \frac {1}{\left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )} \, dx=\int \frac {1}{\left (2 x^{2} - x + 3\right )^{\frac {5}{2}} \cdot \left (5 x^{2} + 3 x + 2\right )}\, dx \]
\[ \int \frac {1}{\left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )} \, dx=\int { \frac {1}{{\left (5 \, x^{2} + 3 \, x + 2\right )} {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}}} \,d x } \]
Exception generated. \[ \int \frac {1}{\left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Francis algorithm failure for[-1.0, infinity,infinity,infinity,infinity]proot error [1.0,infinity,infinity,inf inity,inf
Timed out. \[ \int \frac {1}{\left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )} \, dx=\int \frac {1}{{\left (2\,x^2-x+3\right )}^{5/2}\,\left (5\,x^2+3\,x+2\right )} \,d x \]