Integrand size = 25, antiderivative size = 219 \[ \int \frac {\sqrt {2+3 x+5 x^2}}{\left (4+x-2 x^2\right )^3} \, dx=-\frac {(1-4 x) \sqrt {2+3 x+5 x^2}}{66 \left (4+x-2 x^2\right )^2}-\frac {(835-10358 x) \sqrt {2+3 x+5 x^2}}{676632 \left (4+x-2 x^2\right )}-\frac {\sqrt {\frac {38080240681-6621689063 \sqrt {33}}{2563}} \text {arctanh}\left (\frac {19-3 \sqrt {33}+2 \left (11-5 \sqrt {33}\right ) x}{2 \sqrt {2 \left (107-11 \sqrt {33}\right )} \sqrt {2+3 x+5 x^2}}\right )}{902176}+\frac {\sqrt {\frac {38080240681+6621689063 \sqrt {33}}{2563}} \text {arctanh}\left (\frac {19+3 \sqrt {33}+2 \left (11+5 \sqrt {33}\right ) x}{2 \sqrt {2 \left (107+11 \sqrt {33}\right )} \sqrt {2+3 x+5 x^2}}\right )}{902176} \] Output:
-1/66*(1-4*x)*(5*x^2+3*x+2)^(1/2)/(-2*x^2+x+4)^2-(835-10358*x)*(5*x^2+3*x+ 2)^(1/2)/(-1353264*x^2+676632*x+2706528)-1/2312277088*(97599656865403-1697 1389068469*33^(1/2))^(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))+1/2312277088*(97599656865403 +16971389068469*33^(1/2))^(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))
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 1.19 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.97 \[ \int \frac {\sqrt {2+3 x+5 x^2}}{\left (4+x-2 x^2\right )^3} \, dx=\frac {-\frac {1922 \sqrt {2+3 x+5 x^2} \left (13592-81605 x-12028 x^2+20716 x^3\right )}{\left (4+x-2 x^2\right )^2}-118995825 \text {RootSum}\left [-22+44 \sqrt {5} \text {$\#$1}-91 \text {$\#$1}^2+2 \sqrt {5} \text {$\#$1}^3+2 \text {$\#$1}^4\&,\frac {\log \left (-\sqrt {5} x+\sqrt {2+3 x+5 x^2}-\text {$\#$1}\right )}{22 \sqrt {5}-91 \text {$\#$1}+3 \sqrt {5} \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ]+220 \text {RootSum}\left [-22+44 \sqrt {5} \text {$\#$1}-91 \text {$\#$1}^2+2 \sqrt {5} \text {$\#$1}^3+2 \text {$\#$1}^4\&,\frac {2349842713 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {2+3 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}+1544419745 \log \left (-\sqrt {5} x+\sqrt {2+3 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{22 \sqrt {5}-91 \text {$\#$1}+3 \sqrt {5} \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ]-2 \text {RootSum}\left [-22+44 \sqrt {5} \text {$\#$1}-91 \text {$\#$1}^2+2 \sqrt {5} \text {$\#$1}^3+2 \text {$\#$1}^4\&,\frac {258395294519 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {2+3 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}+169921975927 \log \left (-\sqrt {5} x+\sqrt {2+3 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{22 \sqrt {5}-91 \text {$\#$1}+3 \sqrt {5} \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ]}{1300486704} \] Input:
Integrate[Sqrt[2 + 3*x + 5*x^2]/(4 + x - 2*x^2)^3,x]
Output:
((-1922*Sqrt[2 + 3*x + 5*x^2]*(13592 - 81605*x - 12028*x^2 + 20716*x^3))/( 4 + x - 2*x^2)^2 - 118995825*RootSum[-22 + 44*Sqrt[5]*#1 - 91*#1^2 + 2*Sqr t[5]*#1^3 + 2*#1^4 & , Log[-(Sqrt[5]*x) + Sqrt[2 + 3*x + 5*x^2] - #1]/(22* Sqrt[5] - 91*#1 + 3*Sqrt[5]*#1^2 + 4*#1^3) & ] + 220*RootSum[-22 + 44*Sqrt [5]*#1 - 91*#1^2 + 2*Sqrt[5]*#1^3 + 2*#1^4 & , (2349842713*Sqrt[5]*Log[-(S qrt[5]*x) + Sqrt[2 + 3*x + 5*x^2] - #1]*#1 + 1544419745*Log[-(Sqrt[5]*x) + Sqrt[2 + 3*x + 5*x^2] - #1]*#1^2)/(22*Sqrt[5] - 91*#1 + 3*Sqrt[5]*#1^2 + 4*#1^3) & ] - 2*RootSum[-22 + 44*Sqrt[5]*#1 - 91*#1^2 + 2*Sqrt[5]*#1^3 + 2 *#1^4 & , (258395294519*Sqrt[5]*Log[-(Sqrt[5]*x) + Sqrt[2 + 3*x + 5*x^2] - #1]*#1 + 169921975927*Log[-(Sqrt[5]*x) + Sqrt[2 + 3*x + 5*x^2] - #1]*#1^2 )/(22*Sqrt[5] - 91*#1 + 3*Sqrt[5]*#1^2 + 4*#1^3) & ])/1300486704
Time = 0.56 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.13, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {1302, 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.
| \(\displaystyle \int \frac {\sqrt {5 x^2+3 x+2}}{\left (-2 x^2+x+4\right )^3} \, dx\) |
| \(\Big \downarrow \) 1302 |
| \(\displaystyle \frac {1}{66} \int \frac {80 x^2+70 x+51}{2 \left (-2 x^2+x+4\right )^2 \sqrt {5 x^2+3 x+2}}dx-\frac {(1-4 x) \sqrt {5 x^2+3 x+2}}{66 \left (-2 x^2+x+4\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{132} \int \frac {80 x^2+70 x+51}{\left (-2 x^2+x+4\right )^2 \sqrt {5 x^2+3 x+2}}dx-\frac {(1-4 x) \sqrt {5 x^2+3 x+2}}{66 \left (-2 x^2+x+4\right )^2}\) |
| \(\Big \downarrow \) 2135 |
| \(\displaystyle \frac {1}{132} \left (-\frac {\int -\frac {9 (24838 x+30317)}{2 \left (-2 x^2+x+4\right ) \sqrt {5 x^2+3 x+2}}dx}{15378}-\frac {\sqrt {5 x^2+3 x+2} (835-10358 x)}{5126 \left (-2 x^2+x+4\right )}\right )-\frac {(1-4 x) \sqrt {5 x^2+3 x+2}}{66 \left (-2 x^2+x+4\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{132} \left (\frac {3 \int \frac {24838 x+30317}{\left (-2 x^2+x+4\right ) \sqrt {5 x^2+3 x+2}}dx}{10252}-\frac {(835-10358 x) \sqrt {5 x^2+3 x+2}}{5126 \left (-2 x^2+x+4\right )}\right )-\frac {(1-4 x) \sqrt {5 x^2+3 x+2}}{66 \left (-2 x^2+x+4\right )^2}\) |
| \(\Big \downarrow \) 1365 |
| \(\displaystyle \frac {1}{132} \left (\frac {3 \left (\frac {2}{11} \left (136609-24351 \sqrt {33}\right ) \int \frac {1}{\left (-4 x-\sqrt {33}+1\right ) \sqrt {5 x^2+3 x+2}}dx+\frac {2}{11} \left (136609+24351 \sqrt {33}\right ) \int \frac {1}{\left (-4 x+\sqrt {33}+1\right ) \sqrt {5 x^2+3 x+2}}dx\right )}{10252}-\frac {(835-10358 x) \sqrt {5 x^2+3 x+2}}{5126 \left (-2 x^2+x+4\right )}\right )-\frac {(1-4 x) \sqrt {5 x^2+3 x+2}}{66 \left (-2 x^2+x+4\right )^2}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {1}{132} \left (\frac {3 \left (-\frac {4}{11} \left (136609-24351 \sqrt {33}\right ) \int \frac {1}{8 \left (107-11 \sqrt {33}\right )-\frac {\left (2 \left (11-5 \sqrt {33}\right ) x-3 \sqrt {33}+19\right )^2}{5 x^2+3 x+2}}d\left (-\frac {2 \left (11-5 \sqrt {33}\right ) x-3 \sqrt {33}+19}{\sqrt {5 x^2+3 x+2}}\right )-\frac {4}{11} \left (136609+24351 \sqrt {33}\right ) \int \frac {1}{8 \left (107+11 \sqrt {33}\right )-\frac {\left (2 \left (11+5 \sqrt {33}\right ) x+3 \sqrt {33}+19\right )^2}{5 x^2+3 x+2}}d\left (-\frac {2 \left (11+5 \sqrt {33}\right ) x+3 \sqrt {33}+19}{\sqrt {5 x^2+3 x+2}}\right )\right )}{10252}-\frac {(835-10358 x) \sqrt {5 x^2+3 x+2}}{5126 \left (-2 x^2+x+4\right )}\right )-\frac {(1-4 x) \sqrt {5 x^2+3 x+2}}{66 \left (-2 x^2+x+4\right )^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{132} \left (\frac {3 \left (\frac {1}{11} \left (136609-24351 \sqrt {33}\right ) \sqrt {\frac {2}{107-11 \sqrt {33}}} \text {arctanh}\left (\frac {2 \left (11-5 \sqrt {33}\right ) x-3 \sqrt {33}+19}{2 \sqrt {2 \left (107-11 \sqrt {33}\right )} \sqrt {5 x^2+3 x+2}}\right )+\frac {1}{11} \sqrt {\frac {2}{107+11 \sqrt {33}}} \left (136609+24351 \sqrt {33}\right ) \text {arctanh}\left (\frac {2 \left (11+5 \sqrt {33}\right ) x+3 \sqrt {33}+19}{2 \sqrt {2 \left (107+11 \sqrt {33}\right )} \sqrt {5 x^2+3 x+2}}\right )\right )}{10252}-\frac {(835-10358 x) \sqrt {5 x^2+3 x+2}}{5126 \left (-2 x^2+x+4\right )}\right )-\frac {(1-4 x) \sqrt {5 x^2+3 x+2}}{66 \left (-2 x^2+x+4\right )^2}\) |
Input:
Int[Sqrt[2 + 3*x + 5*x^2]/(4 + x - 2*x^2)^3,x]
Output:
-1/66*((1 - 4*x)*Sqrt[2 + 3*x + 5*x^2])/(4 + x - 2*x^2)^2 + (-1/5126*((835 - 10358*x)*Sqrt[2 + 3*x + 5*x^2])/(4 + x - 2*x^2) + (3*(((136609 - 24351* 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])])/11 + (Sqrt[2/(107 + 11*Sqrt[33])]*(136609 + 24351*Sqrt[33])*ArcTanh[(19 + 3*Sq rt[33] + 2*(11 + 5*Sqrt[33])*x)/(2*Sqrt[2*(107 + 11*Sqrt[33])]*Sqrt[2 + 3* x + 5*x^2])])/11))/10252)/132
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[(b + 2*c*x)*(a + b*x + c*x^2)^(p + 1)*((d + e *x + f*x^2)^q/((b^2 - 4*a*c)*(p + 1))), x] - Simp[1/((b^2 - 4*a*c)*(p + 1)) Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)*Simp[2*c*d*(2*p + 3) + b*e*q + (2*b*f*q + 2*c*e*(2*p + q + 3))*x + 2*c*f*(2*p + 2*q + 3)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ [e^2 - 4*d*f, 0] && LtQ[p, -1] && GtQ[q, 0] && !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 = 4.50 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.05
| method | result | size |
| risch | \(-\frac {\left (20716 x^{3}-12028 x^{2}-81605 x +13592\right ) \sqrt {5 x^{2}+3 x +2}}{676632 \left (2 x^{2}-x -4\right )^{2}}+\frac {\left (-73053+12419 \sqrt {33}\right ) \sqrt {33}\, \operatorname {arctanh}\left (\frac {214-22 \sqrt {33}+8 \left (\frac {11}{2}-\frac {5 \sqrt {33}}{2}\right ) \left (x -\frac {1}{4}+\frac {\sqrt {33}}{4}\right )}{\sqrt {214-22 \sqrt {33}}\, \sqrt {80 \left (x -\frac {1}{4}+\frac {\sqrt {33}}{4}\right )^{2}+16 \left (\frac {11}{2}-\frac {5 \sqrt {33}}{2}\right ) \left (x -\frac {1}{4}+\frac {\sqrt {33}}{4}\right )+214-22 \sqrt {33}}}\right )}{7442952 \sqrt {214-22 \sqrt {33}}}+\frac {\left (73053+12419 \sqrt {33}\right ) \sqrt {33}\, \operatorname {arctanh}\left (\frac {214+22 \sqrt {33}+8 \left (\frac {11}{2}+\frac {5 \sqrt {33}}{2}\right ) \left (x -\frac {\sqrt {33}}{4}-\frac {1}{4}\right )}{\sqrt {214+22 \sqrt {33}}\, \sqrt {80 \left (x -\frac {\sqrt {33}}{4}-\frac {1}{4}\right )^{2}+16 \left (\frac {11}{2}+\frac {5 \sqrt {33}}{2}\right ) \left (x -\frac {\sqrt {33}}{4}-\frac {1}{4}\right )+214+22 \sqrt {33}}}\right )}{7442952 \sqrt {214+22 \sqrt {33}}}\) | \(231\) |
| trager | \(\text {Expression too large to display}\) | \(484\) |
| default | \(\text {Expression too large to display}\) | \(2307\) |
Input:
int((5*x^2+3*x+2)^(1/2)/(-2*x^2+x+4)^3,x,method=_RETURNVERBOSE)
Output:
-1/676632*(20716*x^3-12028*x^2-81605*x+13592)/(2*x^2-x-4)^2*(5*x^2+3*x+2)^ (1/2)+1/7442952*(-73053+12419*33^(1/2))*33^(1/2)/(214-22*33^(1/2))^(1/2)*a rctanh(8*(107/4-11/4*33^(1/2)+(11/2-5/2*33^(1/2))*(x-1/4+1/4*33^(1/2)))/(2 14-22*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))+1/7442952*(73053+12419*33^(1/2) )*33^(1/2)/(214+22*33^(1/2))^(1/2)*arctanh(8*(107/4+11/4*33^(1/2)+(11/2+5/ 2*33^(1/2))*(x-1/4*33^(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))
Leaf count of result is larger than twice the leaf count of optimal. 380 vs. \(2 (167) = 334\).
Time = 0.10 (sec) , antiderivative size = 380, normalized size of antiderivative = 1.74 \[ \int \frac {\sqrt {2+3 x+5 x^2}}{\left (4+x-2 x^2\right )^3} \, dx=-\frac {3 \, {\left (4 \, x^{4} - 4 \, x^{3} - 15 \, x^{2} + 8 \, x + 16\right )} \sqrt {\frac {6621689063}{233} \, \sqrt {\frac {3}{11}} + \frac {38080240681}{2563}} \log \left (\frac {11 \, \sqrt {5 \, x^{2} + 3 \, x + 2} \sqrt {\frac {6621689063}{233} \, \sqrt {\frac {3}{11}} + \frac {38080240681}{2563}} {\left (215019 \, \sqrt {\frac {3}{11}} - 119807\right )} + 113253844 \, \sqrt {\frac {3}{11}} {\left (3 \, x + 4\right )} + 792776908 \, x + 205916080}{x}\right ) - 3 \, {\left (4 \, x^{4} - 4 \, x^{3} - 15 \, x^{2} + 8 \, x + 16\right )} \sqrt {\frac {6621689063}{233} \, \sqrt {\frac {3}{11}} + \frac {38080240681}{2563}} \log \left (-\frac {11 \, \sqrt {5 \, x^{2} + 3 \, x + 2} \sqrt {\frac {6621689063}{233} \, \sqrt {\frac {3}{11}} + \frac {38080240681}{2563}} {\left (215019 \, \sqrt {\frac {3}{11}} - 119807\right )} - 113253844 \, \sqrt {\frac {3}{11}} {\left (3 \, x + 4\right )} - 792776908 \, x - 205916080}{x}\right ) + 3 \, {\left (4 \, x^{4} - 4 \, x^{3} - 15 \, x^{2} + 8 \, x + 16\right )} \sqrt {-\frac {6621689063}{233} \, \sqrt {\frac {3}{11}} + \frac {38080240681}{2563}} \log \left (-\frac {11 \, \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (215019 \, \sqrt {\frac {3}{11}} + 119807\right )} \sqrt {-\frac {6621689063}{233} \, \sqrt {\frac {3}{11}} + \frac {38080240681}{2563}} + 113253844 \, \sqrt {\frac {3}{11}} {\left (3 \, x + 4\right )} - 792776908 \, x - 205916080}{x}\right ) - 3 \, {\left (4 \, x^{4} - 4 \, x^{3} - 15 \, x^{2} + 8 \, x + 16\right )} \sqrt {-\frac {6621689063}{233} \, \sqrt {\frac {3}{11}} + \frac {38080240681}{2563}} \log \left (\frac {11 \, \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (215019 \, \sqrt {\frac {3}{11}} + 119807\right )} \sqrt {-\frac {6621689063}{233} \, \sqrt {\frac {3}{11}} + \frac {38080240681}{2563}} - 113253844 \, \sqrt {\frac {3}{11}} {\left (3 \, x + 4\right )} + 792776908 \, x + 205916080}{x}\right ) + 8 \, {\left (20716 \, x^{3} - 12028 \, x^{2} - 81605 \, x + 13592\right )} \sqrt {5 \, x^{2} + 3 \, x + 2}}{5413056 \, {\left (4 \, x^{4} - 4 \, x^{3} - 15 \, x^{2} + 8 \, x + 16\right )}} \] Input:
integrate((5*x^2+3*x+2)^(1/2)/(-2*x^2+x+4)^3,x, algorithm="fricas")
Output:
-1/5413056*(3*(4*x^4 - 4*x^3 - 15*x^2 + 8*x + 16)*sqrt(6621689063/233*sqrt (3/11) + 38080240681/2563)*log((11*sqrt(5*x^2 + 3*x + 2)*sqrt(6621689063/2 33*sqrt(3/11) + 38080240681/2563)*(215019*sqrt(3/11) - 119807) + 113253844 *sqrt(3/11)*(3*x + 4) + 792776908*x + 205916080)/x) - 3*(4*x^4 - 4*x^3 - 1 5*x^2 + 8*x + 16)*sqrt(6621689063/233*sqrt(3/11) + 38080240681/2563)*log(- (11*sqrt(5*x^2 + 3*x + 2)*sqrt(6621689063/233*sqrt(3/11) + 38080240681/256 3)*(215019*sqrt(3/11) - 119807) - 113253844*sqrt(3/11)*(3*x + 4) - 7927769 08*x - 205916080)/x) + 3*(4*x^4 - 4*x^3 - 15*x^2 + 8*x + 16)*sqrt(-6621689 063/233*sqrt(3/11) + 38080240681/2563)*log(-(11*sqrt(5*x^2 + 3*x + 2)*(215 019*sqrt(3/11) + 119807)*sqrt(-6621689063/233*sqrt(3/11) + 38080240681/256 3) + 113253844*sqrt(3/11)*(3*x + 4) - 792776908*x - 205916080)/x) - 3*(4*x ^4 - 4*x^3 - 15*x^2 + 8*x + 16)*sqrt(-6621689063/233*sqrt(3/11) + 38080240 681/2563)*log((11*sqrt(5*x^2 + 3*x + 2)*(215019*sqrt(3/11) + 119807)*sqrt( -6621689063/233*sqrt(3/11) + 38080240681/2563) - 113253844*sqrt(3/11)*(3*x + 4) + 792776908*x + 205916080)/x) + 8*(20716*x^3 - 12028*x^2 - 81605*x + 13592)*sqrt(5*x^2 + 3*x + 2))/(4*x^4 - 4*x^3 - 15*x^2 + 8*x + 16)
\[ \int \frac {\sqrt {2+3 x+5 x^2}}{\left (4+x-2 x^2\right )^3} \, dx=- \int \frac {\sqrt {5 x^{2} + 3 x + 2}}{8 x^{6} - 12 x^{5} - 42 x^{4} + 47 x^{3} + 84 x^{2} - 48 x - 64}\, dx \] Input:
integrate((5*x**2+3*x+2)**(1/2)/(-2*x**2+x+4)**3,x)
Output:
-Integral(sqrt(5*x**2 + 3*x + 2)/(8*x**6 - 12*x**5 - 42*x**4 + 47*x**3 + 8 4*x**2 - 48*x - 64), x)
\[ \int \frac {\sqrt {2+3 x+5 x^2}}{\left (4+x-2 x^2\right )^3} \, dx=\int { -\frac {\sqrt {5 \, x^{2} + 3 \, x + 2}}{{\left (2 \, x^{2} - x - 4\right )}^{3}} \,d x } \] Input:
integrate((5*x^2+3*x+2)^(1/2)/(-2*x^2+x+4)^3,x, algorithm="maxima")
Output:
-integrate(sqrt(5*x^2 + 3*x + 2)/(2*x^2 - x - 4)^3, x)
Leaf count of result is larger than twice the leaf count of optimal. 378 vs. \(2 (167) = 334\).
Time = 0.24 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.73 \[ \int \frac {\sqrt {2+3 x+5 x^2}}{\left (4+x-2 x^2\right )^3} \, dx=\frac {149028 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{7} - 41640 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{6} - 12183040 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{5} - 46222852 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{4} - 183950431 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{3} - 50816348 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{2} - 19592650 \, \sqrt {5} x + 544984 \, \sqrt {5} + 19592650 \, \sqrt {5 \, x^{2} + 3 \, x + 2}}{676632 \, {\left (2 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{4} - 2 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{3} - 91 \, {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{2} - 44 \, \sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )} - 22\right )}^{2}} + 0.00604061027347214 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 3 \, x + 2} + 8.38267526007000\right ) - 0.000141101376098677 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 3 \, x + 2} - 0.312157316296000\right ) - 0.00604061027347214 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 3 \, x + 2} - 0.842024981991000\right ) + 0.000141101376098677 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 3 \, x + 2} - 4.99242498429000\right ) \] Input:
integrate((5*x^2+3*x+2)^(1/2)/(-2*x^2+x+4)^3,x, algorithm="giac")
Output:
1/676632*(149028*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^7 - 41640*sqrt(5)*(sq rt(5)*x - sqrt(5*x^2 + 3*x + 2))^6 - 12183040*(sqrt(5)*x - sqrt(5*x^2 + 3* x + 2))^5 - 46222852*sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^4 - 18395 0431*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^3 - 50816348*sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^2 - 19592650*sqrt(5)*x + 544984*sqrt(5) + 19592650 *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*sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 3*x + 2)) - 22)^2 + 0.006 04061027347214*log(-sqrt(5)*x + sqrt(5*x^2 + 3*x + 2) + 8.38267526007000) - 0.000141101376098677*log(-sqrt(5)*x + sqrt(5*x^2 + 3*x + 2) - 0.31215731 6296000) - 0.00604061027347214*log(-sqrt(5)*x + sqrt(5*x^2 + 3*x + 2) - 0. 842024981991000) + 0.000141101376098677*log(-sqrt(5)*x + sqrt(5*x^2 + 3*x + 2) - 4.99242498429000)
Timed out. \[ \int \frac {\sqrt {2+3 x+5 x^2}}{\left (4+x-2 x^2\right )^3} \, dx=\int \frac {\sqrt {5\,x^2+3\,x+2}}{{\left (-2\,x^2+x+4\right )}^3} \,d x \] Input:
int((3*x + 5*x^2 + 2)^(1/2)/(x - 2*x^2 + 4)^3,x)
Output:
int((3*x + 5*x^2 + 2)^(1/2)/(x - 2*x^2 + 4)^3, x)
\[ \int \frac {\sqrt {2+3 x+5 x^2}}{\left (4+x-2 x^2\right )^3} \, dx=\int \frac {\sqrt {5 x^{2}+3 x +2}}{\left (-2 x^{2}+x +4\right )^{3}}d x \] Input:
int((5*x^2+3*x+2)^(1/2)/(-2*x^2+x+4)^3,x)
Output:
int((5*x^2+3*x+2)^(1/2)/(-2*x^2+x+4)^3,x)