Integrand size = 25, antiderivative size = 223 \[ \int \frac {1}{\left (4+x-2 x^2\right )^2 \left (2+3 x+5 x^2\right )^{3/2}} \, dx=\frac {4752569+5254415 x}{222150588 \sqrt {2+3 x+5 x^2}}-\frac {235-214 x}{15378 \left (4+x-2 x^2\right ) \sqrt {2+3 x+5 x^2}}-\frac {\left (1668967+204919 \sqrt {33}\right ) \text {arctanh}\left (\frac {19-3 \sqrt {33}+2 \left (11-5 \sqrt {33}\right ) x}{2 \sqrt {2 \left (107-11 \sqrt {33}\right )} \sqrt {2+3 x+5 x^2}}\right )}{7166148 \sqrt {66 \left (107-11 \sqrt {33}\right )}}+\frac {\left (1668967-204919 \sqrt {33}\right ) \text {arctanh}\left (\frac {19+3 \sqrt {33}+2 \left (11+5 \sqrt {33}\right ) x}{2 \sqrt {2 \left (107+11 \sqrt {33}\right )} \sqrt {2+3 x+5 x^2}}\right )}{7166148 \sqrt {66 \left (107+11 \sqrt {33}\right )}} \] Output:
1/222150588*(4752569+5254415*x)/(5*x^2+3*x+2)^(1/2)-1/15378*(235-214*x)/(- 2*x^2+x+4)/(5*x^2+3*x+2)^(1/2)-1/7166148*(1668967+204919*33^(1/2))*arctanh (1/2*(19-3*33^(1/2)+2*(11-5*33^(1/2))*x)/(214-22*33^(1/2))^(1/2)/(5*x^2+3* x+2)^(1/2))/(7062-726*33^(1/2))^(1/2)+1/7166148*(1668967-204919*33^(1/2))* arctanh(1/2*(19+3*33^(1/2)+2*(11+5*33^(1/2))*x)/(214+22*33^(1/2))^(1/2)/(5 *x^2+3*x+2)^(1/2))/(7062+726*33^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 1.04 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.87 \[ \int \frac {1}{\left (4+x-2 x^2\right )^2 \left (2+3 x+5 x^2\right )^{3/2}} \, dx=\frac {-15615466-28861673 x+4250723 x^2+10508830 x^3}{222150588 \left (-4-x+2 x^2\right ) \sqrt {2+3 x+5 x^2}}+\frac {\text {RootSum}\left [-22+44 \sqrt {5} \text {$\#$1}-91 \text {$\#$1}^2+2 \sqrt {5} \text {$\#$1}^3+2 \text {$\#$1}^4\&,\frac {-134287 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {2+3 x+5 x^2}-\text {$\#$1}\right )+75530 \log \left (-\sqrt {5} x+\sqrt {2+3 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}+4554 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {2+3 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{22 \sqrt {5}-91 \text {$\#$1}+3 \sqrt {5} \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ]}{217156 \sqrt {5}}+\frac {\text {RootSum}\left [-22+44 \sqrt {5} \text {$\#$1}-91 \text {$\#$1}^2+2 \sqrt {5} \text {$\#$1}^3+2 \text {$\#$1}^4\&,\frac {5232437 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {2+3 x+5 x^2}-\text {$\#$1}\right )+4384450 \log \left (-\sqrt {5} x+\sqrt {2+3 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}+109274 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {2+3 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{22 \sqrt {5}-91 \text {$\#$1}+3 \sqrt {5} \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ]}{14332296 \sqrt {5}} \] Input:
Integrate[1/((4 + x - 2*x^2)^2*(2 + 3*x + 5*x^2)^(3/2)),x]
Output:
(-15615466 - 28861673*x + 4250723*x^2 + 10508830*x^3)/(222150588*(-4 - x + 2*x^2)*Sqrt[2 + 3*x + 5*x^2]) + RootSum[-22 + 44*Sqrt[5]*#1 - 91*#1^2 + 2 *Sqrt[5]*#1^3 + 2*#1^4 & , (-134287*Sqrt[5]*Log[-(Sqrt[5]*x) + Sqrt[2 + 3* x + 5*x^2] - #1] + 75530*Log[-(Sqrt[5]*x) + Sqrt[2 + 3*x + 5*x^2] - #1]*#1 + 4554*Sqrt[5]*Log[-(Sqrt[5]*x) + Sqrt[2 + 3*x + 5*x^2] - #1]*#1^2)/(22*S qrt[5] - 91*#1 + 3*Sqrt[5]*#1^2 + 4*#1^3) & ]/(217156*Sqrt[5]) + RootSum[- 22 + 44*Sqrt[5]*#1 - 91*#1^2 + 2*Sqrt[5]*#1^3 + 2*#1^4 & , (5232437*Sqrt[5 ]*Log[-(Sqrt[5]*x) + Sqrt[2 + 3*x + 5*x^2] - #1] + 4384450*Log[-(Sqrt[5]*x ) + Sqrt[2 + 3*x + 5*x^2] - #1]*#1 + 109274*Sqrt[5]*Log[-(Sqrt[5]*x) + Sqr t[2 + 3*x + 5*x^2] - #1]*#1^2)/(22*Sqrt[5] - 91*#1 + 3*Sqrt[5]*#1^2 + 4*#1 ^3) & ]/(14332296*Sqrt[5])
Time = 0.50 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {1305, 27, 2135, 27, 1365, 1154, 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.
\(\displaystyle \int \frac {1}{\left (-2 x^2+x+4\right )^2 \left (5 x^2+3 x+2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 1305 |
\(\displaystyle \frac {\int \frac {4280 x^2-4054 x+5893}{2 \left (-2 x^2+x+4\right ) \left (5 x^2+3 x+2\right )^{3/2}}dx}{15378}-\frac {235-214 x}{15378 \left (-2 x^2+x+4\right ) \sqrt {5 x^2+3 x+2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {4280 x^2-4054 x+5893}{\left (-2 x^2+x+4\right ) \left (5 x^2+3 x+2\right )^{3/2}}dx}{30756}-\frac {235-214 x}{15378 \left (-2 x^2+x+4\right ) \sqrt {5 x^2+3 x+2}}\) |
\(\Big \downarrow \) 2135 |
\(\displaystyle \frac {\frac {\int \frac {31 (936943-409838 x)}{2 \left (-2 x^2+x+4\right ) \sqrt {5 x^2+3 x+2}}dx}{7223}+\frac {5254415 x+4752569}{7223 \sqrt {5 x^2+3 x+2}}}{30756}-\frac {235-214 x}{15378 \left (-2 x^2+x+4\right ) \sqrt {5 x^2+3 x+2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {1}{466} \int \frac {936943-409838 x}{\left (-2 x^2+x+4\right ) \sqrt {5 x^2+3 x+2}}dx+\frac {5254415 x+4752569}{7223 \sqrt {5 x^2+3 x+2}}}{30756}-\frac {235-214 x}{15378 \left (-2 x^2+x+4\right ) \sqrt {5 x^2+3 x+2}}\) |
\(\Big \downarrow \) 1365 |
\(\displaystyle \frac {\frac {1}{466} \left (-\frac {2}{33} \left (6762327+1668967 \sqrt {33}\right ) \int \frac {1}{\left (-4 x-\sqrt {33}+1\right ) \sqrt {5 x^2+3 x+2}}dx-\frac {2}{33} \left (6762327-1668967 \sqrt {33}\right ) \int \frac {1}{\left (-4 x+\sqrt {33}+1\right ) \sqrt {5 x^2+3 x+2}}dx\right )+\frac {5254415 x+4752569}{7223 \sqrt {5 x^2+3 x+2}}}{30756}-\frac {235-214 x}{15378 \left (-2 x^2+x+4\right ) \sqrt {5 x^2+3 x+2}}\) |
\(\Big \downarrow \) 1154 |
\(\displaystyle \frac {\frac {1}{466} \left (\frac {4}{33} \left (6762327+1668967 \sqrt {33}\right ) \int \frac {1}{8 \left (107-11 \sqrt {33}\right )-\frac {\left (2 \left (11-5 \sqrt {33}\right ) x-3 \sqrt {33}+19\right )^2}{5 x^2+3 x+2}}d\left (-\frac {2 \left (11-5 \sqrt {33}\right ) x-3 \sqrt {33}+19}{\sqrt {5 x^2+3 x+2}}\right )+\frac {4}{33} \left (6762327-1668967 \sqrt {33}\right ) \int \frac {1}{8 \left (107+11 \sqrt {33}\right )-\frac {\left (2 \left (11+5 \sqrt {33}\right ) x+3 \sqrt {33}+19\right )^2}{5 x^2+3 x+2}}d\left (-\frac {2 \left (11+5 \sqrt {33}\right ) x+3 \sqrt {33}+19}{\sqrt {5 x^2+3 x+2}}\right )\right )+\frac {5254415 x+4752569}{7223 \sqrt {5 x^2+3 x+2}}}{30756}-\frac {235-214 x}{15378 \left (-2 x^2+x+4\right ) \sqrt {5 x^2+3 x+2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\frac {1}{466} \left (-\frac {1}{33} \sqrt {\frac {2}{107-11 \sqrt {33}}} \left (6762327+1668967 \sqrt {33}\right ) \text {arctanh}\left (\frac {2 \left (11-5 \sqrt {33}\right ) x-3 \sqrt {33}+19}{2 \sqrt {2 \left (107-11 \sqrt {33}\right )} \sqrt {5 x^2+3 x+2}}\right )-\frac {1}{33} \left (6762327-1668967 \sqrt {33}\right ) \sqrt {\frac {2}{107+11 \sqrt {33}}} \text {arctanh}\left (\frac {2 \left (11+5 \sqrt {33}\right ) x+3 \sqrt {33}+19}{2 \sqrt {2 \left (107+11 \sqrt {33}\right )} \sqrt {5 x^2+3 x+2}}\right )\right )+\frac {5254415 x+4752569}{7223 \sqrt {5 x^2+3 x+2}}}{30756}-\frac {235-214 x}{15378 \left (-2 x^2+x+4\right ) \sqrt {5 x^2+3 x+2}}\) |
Input:
Int[1/((4 + x - 2*x^2)^2*(2 + 3*x + 5*x^2)^(3/2)),x]
Output:
-1/15378*(235 - 214*x)/((4 + x - 2*x^2)*Sqrt[2 + 3*x + 5*x^2]) + ((4752569 + 5254415*x)/(7223*Sqrt[2 + 3*x + 5*x^2]) + (-1/33*(Sqrt[2/(107 - 11*Sqrt [33])]*(6762327 + 1668967*Sqrt[33])*ArcTanh[(19 - 3*Sqrt[33] + 2*(11 - 5*S qrt[33])*x)/(2*Sqrt[2*(107 - 11*Sqrt[33])]*Sqrt[2 + 3*x + 5*x^2])]) - ((67 62327 - 1668967*Sqrt[33])*Sqrt[2/(107 + 11*Sqrt[33])]*ArcTanh[(19 + 3*Sqrt [33] + 2*(11 + 5*Sqrt[33])*x)/(2*Sqrt[2*(107 + 11*Sqrt[33])]*Sqrt[2 + 3*x + 5*x^2])])/33)/466)/30756
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[(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[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], 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[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + ( e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Sim p[(2*c*g - h*(b - q))/q Int[1/((b - q + 2*c*x)*Sqrt[d + e*x + f*x^2]), x] , x] - Simp[(2*c*g - h*(b + q))/q Int[1/((b + q + 2*c*x)*Sqrt[d + e*x + f *x^2]), x], 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] && PosQ[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]
Time = 4.82 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.04
method | result | size |
risch | \(\frac {10508830 x^{3}+4250723 x^{2}-28861673 x -15615466}{222150588 \left (2 x^{2}-x -4\right ) \sqrt {5 x^{2}+3 x +2}}-\frac {\left (-1668967+204919 \sqrt {33}\right ) \sqrt {33}\, \operatorname {arctanh}\left (\frac {214+22 \sqrt {33}+8 \left (\frac {11}{2}+\frac {5 \sqrt {33}}{2}\right ) \left (x -\frac {\sqrt {33}}{4}-\frac {1}{4}\right )}{\sqrt {214+22 \sqrt {33}}\, \sqrt {80 \left (x -\frac {\sqrt {33}}{4}-\frac {1}{4}\right )^{2}+16 \left (\frac {11}{2}+\frac {5 \sqrt {33}}{2}\right ) \left (x -\frac {\sqrt {33}}{4}-\frac {1}{4}\right )+214+22 \sqrt {33}}}\right )}{236482884 \sqrt {214+22 \sqrt {33}}}-\frac {\left (1668967+204919 \sqrt {33}\right ) \sqrt {33}\, \operatorname {arctanh}\left (\frac {214-22 \sqrt {33}+8 \left (\frac {11}{2}-\frac {5 \sqrt {33}}{2}\right ) \left (x -\frac {1}{4}+\frac {\sqrt {33}}{4}\right )}{\sqrt {214-22 \sqrt {33}}\, \sqrt {80 \left (x -\frac {1}{4}+\frac {\sqrt {33}}{4}\right )^{2}+16 \left (\frac {11}{2}-\frac {5 \sqrt {33}}{2}\right ) \left (x -\frac {1}{4}+\frac {\sqrt {33}}{4}\right )+214-22 \sqrt {33}}}\right )}{236482884 \sqrt {214-22 \sqrt {33}}}\) | \(231\) |
trager | \(\text {Expression too large to display}\) | \(491\) |
default | \(\text {Expression too large to display}\) | \(1195\) |
Input:
int(1/(-2*x^2+x+4)^2/(5*x^2+3*x+2)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/222150588*(10508830*x^3+4250723*x^2-28861673*x-15615466)/(2*x^2-x-4)/(5* x^2+3*x+2)^(1/2)-1/236482884*(-1668967+204919*33^(1/2))*33^(1/2)/(214+22*3 3^(1/2))^(1/2)*arctanh(8*(107/4+11/4*33^(1/2)+(11/2+5/2*33^(1/2))*(x-1/4*3 3^(1/2)-1/4))/(214+22*33^(1/2))^(1/2)/(80*(x-1/4*33^(1/2)-1/4)^2+16*(11/2+ 5/2*33^(1/2))*(x-1/4*33^(1/2)-1/4)+214+22*33^(1/2))^(1/2))-1/236482884*(16 68967+204919*33^(1/2))*33^(1/2)/(214-22*33^(1/2))^(1/2)*arctanh(8*(107/4-1 1/4*33^(1/2)+(11/2-5/2*33^(1/2))*(x-1/4+1/4*33^(1/2)))/(214-22*33^(1/2))^( 1/2)/(80*(x-1/4+1/4*33^(1/2))^2+16*(11/2-5/2*33^(1/2))*(x-1/4+1/4*33^(1/2) )+214-22*33^(1/2))^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 366 vs. \(2 (169) = 338\).
Time = 0.11 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.64 \[ \int \frac {1}{\left (4+x-2 x^2\right )^2 \left (2+3 x+5 x^2\right )^{3/2}} \, dx=\frac {31 \, {\left (10 \, x^{4} + x^{3} - 19 \, x^{2} - 14 \, x - 8\right )} \sqrt {\frac {902057832467}{233} \, \sqrt {33} + \frac {173652621908003}{7689}} \log \left (-\frac {\sqrt {5 \, x^{2} + 3 \, x + 2} \sqrt {\frac {902057832467}{233} \, \sqrt {33} + \frac {173652621908003}{7689}} {\left (33638717 \, \sqrt {33} - 219326349\right )} + 174965195072 \, \sqrt {33} {\left (3 \, x + 4\right )} - 13472320020544 \, x - 3499303901440}{x}\right ) - 31 \, {\left (10 \, x^{4} + x^{3} - 19 \, x^{2} - 14 \, x - 8\right )} \sqrt {\frac {902057832467}{233} \, \sqrt {33} + \frac {173652621908003}{7689}} \log \left (\frac {\sqrt {5 \, x^{2} + 3 \, x + 2} \sqrt {\frac {902057832467}{233} \, \sqrt {33} + \frac {173652621908003}{7689}} {\left (33638717 \, \sqrt {33} - 219326349\right )} - 174965195072 \, \sqrt {33} {\left (3 \, x + 4\right )} + 13472320020544 \, x + 3499303901440}{x}\right ) + 31 \, {\left (10 \, x^{4} + x^{3} - 19 \, x^{2} - 14 \, x - 8\right )} \sqrt {-\frac {902057832467}{233} \, \sqrt {33} + \frac {173652621908003}{7689}} \log \left (\frac {\sqrt {5 \, x^{2} + 3 \, x + 2} {\left (33638717 \, \sqrt {33} + 219326349\right )} \sqrt {-\frac {902057832467}{233} \, \sqrt {33} + \frac {173652621908003}{7689}} + 174965195072 \, \sqrt {33} {\left (3 \, x + 4\right )} + 13472320020544 \, x + 3499303901440}{x}\right ) - 31 \, {\left (10 \, x^{4} + x^{3} - 19 \, x^{2} - 14 \, x - 8\right )} \sqrt {-\frac {902057832467}{233} \, \sqrt {33} + \frac {173652621908003}{7689}} \log \left (-\frac {\sqrt {5 \, x^{2} + 3 \, x + 2} {\left (33638717 \, \sqrt {33} + 219326349\right )} \sqrt {-\frac {902057832467}{233} \, \sqrt {33} + \frac {173652621908003}{7689}} - 174965195072 \, \sqrt {33} {\left (3 \, x + 4\right )} - 13472320020544 \, x - 3499303901440}{x}\right ) + 8 \, {\left (10508830 \, x^{3} + 4250723 \, x^{2} - 28861673 \, x - 15615466\right )} \sqrt {5 \, x^{2} + 3 \, x + 2}}{1777204704 \, {\left (10 \, x^{4} + x^{3} - 19 \, x^{2} - 14 \, x - 8\right )}} \] Input:
integrate(1/(-2*x^2+x+4)^2/(5*x^2+3*x+2)^(3/2),x, algorithm="fricas")
Output:
1/1777204704*(31*(10*x^4 + x^3 - 19*x^2 - 14*x - 8)*sqrt(902057832467/233* sqrt(33) + 173652621908003/7689)*log(-(sqrt(5*x^2 + 3*x + 2)*sqrt(90205783 2467/233*sqrt(33) + 173652621908003/7689)*(33638717*sqrt(33) - 219326349) + 174965195072*sqrt(33)*(3*x + 4) - 13472320020544*x - 3499303901440)/x) - 31*(10*x^4 + x^3 - 19*x^2 - 14*x - 8)*sqrt(902057832467/233*sqrt(33) + 17 3652621908003/7689)*log((sqrt(5*x^2 + 3*x + 2)*sqrt(902057832467/233*sqrt( 33) + 173652621908003/7689)*(33638717*sqrt(33) - 219326349) - 174965195072 *sqrt(33)*(3*x + 4) + 13472320020544*x + 3499303901440)/x) + 31*(10*x^4 + x^3 - 19*x^2 - 14*x - 8)*sqrt(-902057832467/233*sqrt(33) + 173652621908003 /7689)*log((sqrt(5*x^2 + 3*x + 2)*(33638717*sqrt(33) + 219326349)*sqrt(-90 2057832467/233*sqrt(33) + 173652621908003/7689) + 174965195072*sqrt(33)*(3 *x + 4) + 13472320020544*x + 3499303901440)/x) - 31*(10*x^4 + x^3 - 19*x^2 - 14*x - 8)*sqrt(-902057832467/233*sqrt(33) + 173652621908003/7689)*log(- (sqrt(5*x^2 + 3*x + 2)*(33638717*sqrt(33) + 219326349)*sqrt(-902057832467/ 233*sqrt(33) + 173652621908003/7689) - 174965195072*sqrt(33)*(3*x + 4) - 1 3472320020544*x - 3499303901440)/x) + 8*(10508830*x^3 + 4250723*x^2 - 2886 1673*x - 15615466)*sqrt(5*x^2 + 3*x + 2))/(10*x^4 + x^3 - 19*x^2 - 14*x - 8)
\[ \int \frac {1}{\left (4+x-2 x^2\right )^2 \left (2+3 x+5 x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (2 x^{2} - x - 4\right )^{2} \left (5 x^{2} + 3 x + 2\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/(-2*x**2+x+4)**2/(5*x**2+3*x+2)**(3/2),x)
Output:
Integral(1/((2*x**2 - x - 4)**2*(5*x**2 + 3*x + 2)**(3/2)), x)
\[ \int \frac {1}{\left (4+x-2 x^2\right )^2 \left (2+3 x+5 x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{\frac {3}{2}} {\left (2 \, x^{2} - x - 4\right )}^{2}} \,d x } \] Input:
integrate(1/(-2*x^2+x+4)^2/(5*x^2+3*x+2)^(3/2),x, algorithm="maxima")
Output:
integrate(1/((5*x^2 + 3*x + 2)^(3/2)*(2*x^2 - x - 4)^2), x)
Time = 0.17 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.32 \[ \int \frac {1}{\left (4+x-2 x^2\right )^2 \left (2+3 x+5 x^2\right )^{3/2}} \, dx=\frac {97745 \, x + 64617}{3365918 \, \sqrt {5 \, x^{2} + 3 \, x + 2}} - \frac {109274 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{3} - 493082 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{2} - 930819 \, \sqrt {5} x + 31592 \, \sqrt {5} + 930819 \, \sqrt {5 \, x^{2} + 3 \, x + 2}}{7166148 \, {\left (2 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{4} - 2 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{3} - 91 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{2} - 44 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )} - 22\right )}} + 0.000647531417292805 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 3 \, x + 2} + 8.38267526007000\right ) - 0.00738605225631678 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 3 \, x + 2} - 0.312157316296000\right ) - 0.000647531417292805 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 3 \, x + 2} - 0.842024981991000\right ) + 0.00738605225631678 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 3 \, x + 2} - 4.99242498429000\right ) \] Input:
integrate(1/(-2*x^2+x+4)^2/(5*x^2+3*x+2)^(3/2),x, algorithm="giac")
Output:
1/3365918*(97745*x + 64617)/sqrt(5*x^2 + 3*x + 2) - 1/7166148*(109274*(sqr t(5)*x - sqrt(5*x^2 + 3*x + 2))^3 - 493082*sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^2 - 930819*sqrt(5)*x + 31592*sqrt(5) + 930819*sqrt(5*x^2 + 3* x + 2))/(2*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^4 - 2*sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^3 - 91*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^2 - 44*s qrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2)) - 22) + 0.000647531417292805*lo g(-sqrt(5)*x + sqrt(5*x^2 + 3*x + 2) + 8.38267526007000) - 0.0073860522563 1678*log(-sqrt(5)*x + sqrt(5*x^2 + 3*x + 2) - 0.312157316296000) - 0.00064 7531417292805*log(-sqrt(5)*x + sqrt(5*x^2 + 3*x + 2) - 0.842024981991000) + 0.00738605225631678*log(-sqrt(5)*x + sqrt(5*x^2 + 3*x + 2) - 4.992424984 29000)
Timed out. \[ \int \frac {1}{\left (4+x-2 x^2\right )^2 \left (2+3 x+5 x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (-2\,x^2+x+4\right )}^2\,{\left (5\,x^2+3\,x+2\right )}^{3/2}} \,d x \] Input:
int(1/((x - 2*x^2 + 4)^2*(3*x + 5*x^2 + 2)^(3/2)),x)
Output:
int(1/((x - 2*x^2 + 4)^2*(3*x + 5*x^2 + 2)^(3/2)), x)
\[ \int \frac {1}{\left (4+x-2 x^2\right )^2 \left (2+3 x+5 x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (-2 x^{2}+x +4\right )^{2} \left (5 x^{2}+3 x +2\right )^{\frac {3}{2}}}d x \] Input:
int(1/(-2*x^2+x+4)^2/(5*x^2+3*x+2)^(3/2),x)
Output:
int(1/(-2*x^2+x+4)^2/(5*x^2+3*x+2)^(3/2),x)