Integrand size = 27, antiderivative size = 199 \[ \int \frac {1}{\left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )} \, dx=\frac {13-6 x}{759 \left (3-x+2 x^2\right )^{3/2}}+\frac {3603-658 x}{128018 \sqrt {3-x+2 x^2}}+\frac {1}{484} \sqrt {\frac {1}{682} \left (-15457+25000 \sqrt {2}\right )} \arctan \left (\frac {\sqrt {\frac {11}{31 \left (-15457+25000 \sqrt {2}\right )}} \left (443-98 \sqrt {2}+\left (247+345 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right )-\frac {1}{484} \sqrt {\frac {1}{682} \left (15457+25000 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {11}{31 \left (15457+25000 \sqrt {2}\right )}} \left (443+98 \sqrt {2}+\left (247-345 \sqrt {2}\right ) x\right )}{\sqrt {3-x+2 x^2}}\right ) \] Output:
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*(-10541674+17050000*2^(1/2))^(1/2)*arctan(11^(1/2)/(-479167+775000 *2^(1/2))^(1/2)*(443-98*2^(1/2)+(247+345*2^(1/2))*x)/(2*x^2-x+3)^(1/2))-1/ 330088*(10541674+17050000*2^(1/2))^(1/2)*arctanh(11^(1/2)/(479167+775000*2 ^(1/2))^(1/2)*(443+98*2^(1/2)+(247-345*2^(1/2))*x)/(2*x^2-x+3)^(1/2))
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.88 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.05 \[ \int \frac {1}{\left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )} \, dx=\frac {39005-19767 x+23592 x^2-3948 x^3}{384054 \left (3-x+2 x^2\right )^{3/2}}+\frac {1}{484} \text {RootSum}\left [-56-26 \sqrt {2} \text {$\#$1}+17 \text {$\#$1}^2+6 \sqrt {2} \text {$\#$1}^3-5 \text {$\#$1}^4\&,\frac {249 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right )+108 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}-65 \log \left (-\sqrt {2} x+\sqrt {3-x+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-13 \sqrt {2}+17 \text {$\#$1}+9 \sqrt {2} \text {$\#$1}^2-10 \text {$\#$1}^3}\&\right ] \] Input:
Integrate[1/((3 - x + 2*x^2)^(5/2)*(2 + 3*x + 5*x^2)),x]
Output:
(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 = 0.62 (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 |
| \(\displaystyle \frac {1}{506} \left (\frac {23}{22} \left (-\frac {\left (15457-25000 \sqrt {2}\right ) \arctan \left (\frac {\sqrt {\frac {11}{31 \left (25000 \sqrt {2}-15457\right )}} \left (\left (247+345 \sqrt {2}\right ) x-98 \sqrt {2}+443\right )}{\sqrt {2 x^2-x+3}}\right )}{\sqrt {682 \left (25000 \sqrt {2}-15457\right )}}-\sqrt {\frac {1}{682} \left (15457+25000 \sqrt {2}\right )} \text {arctanh}\left (\frac {\sqrt {\frac {11}{31 \left (15457+25000 \sqrt {2}\right )}} \left (\left (247-345 \sqrt {2}\right ) x+98 \sqrt {2}+443\right )}{\sqrt {2 x^2-x+3}}\right )\right )+\frac {3603-658 x}{253 \sqrt {2 x^2-x+3}}\right )+\frac {13-6 x}{759 \left (2 x^2-x+3\right )^{3/2}}\) |
Input:
Int[1/((3 - x + 2*x^2)^(5/2)*(2 + 3*x + 5*x^2)),x]
Output:
(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
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 = 4.58 (sec) , antiderivative size = 471, 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 {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}-\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 +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 )+34735243833848250 \operatorname {RootOf}\left (494371927368 \textit {\_Z}^{4}-3842440173 \textit {\_Z}^{2}+39062500\right )^{2} \sqrt {2 x^{2}-x +3}+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 -140419212500 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1356301584 \operatorname {RootOf}\left (494371927368 \textit {\_Z}^{4}-3842440173 \textit {\_Z}^{2}+39062500\right )^{2}-10541674\right )-120911390237000 \sqrt {2 x^{2}-x +3}}{994356 x \operatorname {RootOf}\left (494371927368 \textit {\_Z}^{4}-3842440173 \textit {\_Z}^{2}+39062500\right )^{2}+419 x +5711}\right )}{330088}\) | \(471\) |
| default | \(-\frac {329 \left (4 x -1\right )}{256036 \sqrt {2 x^{2}-x +3}}+\frac {13}{484 \sqrt {2 x^{2}-x +3}}-\frac {4 x -1}{506 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {1}{66 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {\sqrt {\frac {8 \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}\, \sqrt {2}\, \left (10111 \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 (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+24 \sqrt {2}-41\right )}\, \left (\frac {6485 \sqrt {2}\, \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {10368 \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+22379 \sqrt {2}+32016\right ) \left (-1+\sqrt {2}+x \right ) \left (8+3 \sqrt {2}\right )}{11692487 \left (\frac {23 \left (-1+\sqrt {2}+x \right )^{4}}{\left (\sqrt {2}+1-x \right )^{4}}+\frac {82 \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+23\right ) \left (\sqrt {2}+1-x \right )}\right ) \sqrt {-775687+549362 \sqrt {2}}+13910 \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 (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+24 \sqrt {2}-41\right )}\, \left (\frac {6485 \sqrt {2}\, \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {10368 \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+22379 \sqrt {2}+32016\right ) \left (-1+\sqrt {2}+x \right ) \left (8+3 \sqrt {2}\right )}{11692487 \left (\frac {23 \left (-1+\sqrt {2}+x \right )^{4}}{\left (\sqrt {2}+1-x \right )^{4}}+\frac {82 \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+23\right ) \left (\sqrt {2}+1-x \right )}\right ) \sqrt {-775687+549362 \sqrt {2}}-993674 \,\operatorname {arctanh}\left (\frac {31 \sqrt {\frac {8 \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (-1+\sqrt {2}+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 (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (-1+\sqrt {2}+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 (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+\frac {3 \sqrt {2}\, \left (-1+\sqrt {2}+x \right )^{2}}{\left (\sqrt {2}+1-x \right )^{2}}+8-3 \sqrt {2}}{\left (1+\frac {-1+\sqrt {2}+x}{\sqrt {2}+1-x}\right )^{2}}}\, \left (1+\frac {-1+\sqrt {2}+x}{\sqrt {2}+1-x}\right ) \left (8+3 \sqrt {2}\right ) \sqrt {-8866+6820 \sqrt {2}}}\) | \(751\) |
Input:
int(1/(2*x^2-x+3)^(5/2)/(5*x^2+3*x+2),x,method=_RETURNVERBOSE)
Output:
-1/384054*(3948*x^3-23592*x^2+19767*x-39005)/(2*x^2-x+3)^(3/2)+27/242*Root Of(494371927368*_Z^4-3842440173*_Z^2+39062500)*ln(-(1009432331152496064*x* RootOf(494371927368*_Z^4-3842440173*_Z^2+39062500)^5+7463993424020496*Root Of(494371927368*_Z^4-3842440173*_Z^2+39062500)^3*x-25206911474500800*RootO f(494371927368*_Z^4-3842440173*_Z^2+39062500)^3+1629806162292000*RootOf(49 4371927368*_Z^4-3842440173*_Z^2+39062500)^2*(2*x^2-x+3)^(1/2)+103675333731 75*RootOf(494371927368*_Z^4-3842440173*_Z^2+39062500)*x-46727026216200*Roo tOf(494371927368*_Z^4-3842440173*_Z^2+39062500)-6994190312500*(2*x^2-x+3)^ (1/2))/(1988712*x*RootOf(494371927368*_Z^4-3842440173*_Z^2+39062500)^2-162 95*x-11422))-1/330088*RootOf(_Z^2+1356301584*RootOf(494371927368*_Z^4-3842 440173*_Z^2+39062500)^2-10541674)*ln((584162228676213*x*RootOf(49437192736 8*_Z^4-3842440173*_Z^2+39062500)^4*RootOf(_Z^2+1356301584*RootOf(494371927 368*_Z^4-3842440173*_Z^2+39062500)^2-10541674)-13400087377743*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)*x+14587333029225*Root Of(494371927368*_Z^4-3842440173*_Z^2+39062500)^2*RootOf(_Z^2+1356301584*Ro otOf(494371927368*_Z^4-3842440173*_Z^2+39062500)^2-10541674)+3473524383384 8250*RootOf(494371927368*_Z^4-3842440173*_Z^2+39062500)^2*(2*x^2-x+3)^(1/2 )+74861069500*RootOf(_Z^2+1356301584*RootOf(494371927368*_Z^4-3842440173*_ Z^2+39062500)^2-10541674)*x-140419212500*RootOf(_Z^2+1356301584*RootOf(...
Leaf count of result is larger than twice the leaf count of optimal. 605 vs. \(2 (146) = 292\).
Time = 0.09 (sec) , antiderivative size = 605, normalized size of antiderivative = 3.04 \[ \int \frac {1}{\left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )} \, dx =\text {Too large to display} \] Input:
integrate(1/(2*x^2-x+3)^(5/2)/(5*x^2+3*x+2),x, algorithm="fricas")
Output:
1/3072432*(3174*(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)*sqrt(12500/341*sqrt(2) - 15457/682)*arctan(-22/5711*(4*(1986*x^3 - 3256*x^2 - sqrt(2)*(115*x^3 - 1149*x^2 - 2768*x - 264) + 2256*x + 2592)*sqrt(2*x^2 - x + 3)*sqrt(12500/3 41*sqrt(2) - 15457/682) + (171*x^4 + 1212*x^3 - 1640*x^2 - 176*sqrt(2)*(6* x^4 + 5*x^3 + 5*x^2 + 12*x) - 3936*x)*sqrt(12500/341*sqrt(2) + 15457/682)* sqrt(12500/341*sqrt(2) - 15457/682))/(343*x^4 - 400*x^3 + 1136*x^2 + 384*x - 576)) - 3174*(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)*sqrt(12500/341*sqrt(2) - 15457/682)*arctan(22/5711*(4*(1986*x^3 - 3256*x^2 - sqrt(2)*(115*x^3 - 1 149*x^2 - 2768*x - 264) + 2256*x + 2592)*sqrt(2*x^2 - x + 3)*sqrt(12500/34 1*sqrt(2) - 15457/682) - (171*x^4 + 1212*x^3 - 1640*x^2 - 176*sqrt(2)*(6*x ^4 + 5*x^3 + 5*x^2 + 12*x) - 3936*x)*sqrt(12500/341*sqrt(2) + 15457/682)*s qrt(12500/341*sqrt(2) - 15457/682))/(343*x^4 - 400*x^3 + 1136*x^2 + 384*x - 576)) + 1587*(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)*sqrt(12500/341*sqrt(2) + 15457/682)*log((279839*x^2 + 22*sqrt(2*x^2 - x + 3)*(sqrt(2)*(2019*x - 19 25) - 94*x - 3944)*sqrt(12500/341*sqrt(2) + 15457/682) + 251284*sqrt(2)*(2 *x^2 - x + 3) - 862361*x + 1142200)/x^2) - 1587*(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)*sqrt(12500/341*sqrt(2) + 15457/682)*log((279839*x^2 - 22*sqrt(2*x ^2 - x + 3)*(sqrt(2)*(2019*x - 1925) - 94*x - 3944)*sqrt(12500/341*sqrt(2) + 15457/682) + 251284*sqrt(2)*(2*x^2 - x + 3) - 862361*x + 1142200)/x^2) - 8*(3948*x^3 - 23592*x^2 + 19767*x - 39005)*sqrt(2*x^2 - x + 3))/(4*x^...
\[ \int \frac {1}{\left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )} \, dx=\int \frac {1}{\left (2 x^{2} - x + 3\right )^{\frac {5}{2}} \cdot \left (5 x^{2} + 3 x + 2\right )}\, dx \] Input:
integrate(1/(2*x**2-x+3)**(5/2)/(5*x**2+3*x+2),x)
Output:
Integral(1/((2*x**2 - x + 3)**(5/2)*(5*x**2 + 3*x + 2)), x)
\[ \int \frac {1}{\left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )} \, dx=\int { \frac {1}{{\left (5 \, x^{2} + 3 \, x + 2\right )} {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(2*x^2-x+3)^(5/2)/(5*x^2+3*x+2),x, algorithm="maxima")
Output:
integrate(1/((5*x^2 + 3*x + 2)*(2*x^2 - x + 3)^(5/2)), x)
Exception generated. \[ \int \frac {1}{\left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/(2*x^2-x+3)^(5/2)/(5*x^2+3*x+2),x, algorithm="giac")
Output:
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. \[ \int \frac {1}{\left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )} \, dx=\int \frac {1}{{\left (2\,x^2-x+3\right )}^{5/2}\,\left (5\,x^2+3\,x+2\right )} \,d x \] Input:
int(1/((2*x^2 - x + 3)^(5/2)*(3*x + 5*x^2 + 2)),x)
Output:
int(1/((2*x^2 - x + 3)^(5/2)*(3*x + 5*x^2 + 2)), x)
\[ \int \frac {1}{\left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )} \, dx=\int \frac {\sqrt {2 x^{2}-x +3}}{40 x^{8}-36 x^{7}+190 x^{6}-83 x^{5}+288 x^{4}-20 x^{3}+180 x^{2}+27 x +54}d x \] Input:
int(1/(2*x^2-x+3)^(5/2)/(5*x^2+3*x+2),x)
Output:
int(sqrt(2*x**2 - x + 3)/(40*x**8 - 36*x**7 + 190*x**6 - 83*x**5 + 288*x** 4 - 20*x**3 + 180*x**2 + 27*x + 54),x)