Integrand size = 35, antiderivative size = 199 \[ \int \frac {\left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2}}{\left (1+4 x-7 x^2\right )^2} \, dx=\frac {3 (3+61 x) \sqrt {3+2 x+5 x^2}}{154 \left (1+4 x-7 x^2\right )}+\frac {1}{49} \sqrt {5} \text {arcsinh}\left (\frac {1+5 x}{\sqrt {14}}\right )-\frac {\sqrt {\frac {325022311+39132731 \sqrt {11}}{1397}} \text {arctanh}\left (\frac {23-\sqrt {11}+\left (17-5 \sqrt {11}\right ) x}{\sqrt {2 \left (125-17 \sqrt {11}\right )} \sqrt {3+2 x+5 x^2}}\right )}{2156}+\frac {\sqrt {\frac {325022311-39132731 \sqrt {11}}{1397}} \text {arctanh}\left (\frac {23+\sqrt {11}+\left (17+5 \sqrt {11}\right ) x}{\sqrt {2 \left (125+17 \sqrt {11}\right )} \sqrt {3+2 x+5 x^2}}\right )}{2156} \]
1/49*arcsinh(1/14*(1+5*x)*14^(1/2))*5^(1/2)+3/154*(3+61*x)*(5*x^2+2*x+3)^( 1/2)/(-7*x^2+4*x+1)+1/3011932*arctanh((23+11^(1/2)+x*(17+5*11^(1/2)))/(5*x ^2+2*x+3)^(1/2)/(250+34*11^(1/2))^(1/2))*(454056168467-54668425207*11^(1/2 ))^(1/2)-1/3011932*arctanh((23+x*(17-5*11^(1/2))-11^(1/2))/(5*x^2+2*x+3)^( 1/2)/(250-34*11^(1/2))^(1/2))*(454056168467+54668425207*11^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.53 (sec) , antiderivative size = 427, normalized size of antiderivative = 2.15 \[ \int \frac {\left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2}}{\left (1+4 x-7 x^2\right )^2} \, dx=\frac {-\frac {5145 (3+61 x) \sqrt {3+2 x+5 x^2}}{-1-4 x+7 x^2}-5390 \sqrt {5} \log \left (-1-5 x+\sqrt {5} \sqrt {3+2 x+5 x^2}\right )-55 \text {RootSum}\left [83-16 \sqrt {5} \text {$\#$1}-70 \text {$\#$1}^2+8 \sqrt {5} \text {$\#$1}^3+7 \text {$\#$1}^4\&,\frac {-314239 \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right )+28462 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}-11221 \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-4 \sqrt {5}-35 \text {$\#$1}+6 \sqrt {5} \text {$\#$1}^2+7 \text {$\#$1}^3}\&\right ]-6 \sqrt {5} \text {RootSum}\left [83-16 \sqrt {5} \text {$\#$1}-70 \text {$\#$1}^2+8 \sqrt {5} \text {$\#$1}^3+7 \text {$\#$1}^4\&,\frac {599633 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right )-391895 \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}+21462 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-4 \sqrt {5}-35 \text {$\#$1}+6 \sqrt {5} \text {$\#$1}^2+7 \text {$\#$1}^3}\&\right ]}{264110} \]
((-5145*(3 + 61*x)*Sqrt[3 + 2*x + 5*x^2])/(-1 - 4*x + 7*x^2) - 5390*Sqrt[5 ]*Log[-1 - 5*x + Sqrt[5]*Sqrt[3 + 2*x + 5*x^2]] - 55*RootSum[83 - 16*Sqrt[ 5]*#1 - 70*#1^2 + 8*Sqrt[5]*#1^3 + 7*#1^4 & , (-314239*Log[-(Sqrt[5]*x) + Sqrt[3 + 2*x + 5*x^2] - #1] + 28462*Sqrt[5]*Log[-(Sqrt[5]*x) + Sqrt[3 + 2* x + 5*x^2] - #1]*#1 - 11221*Log[-(Sqrt[5]*x) + Sqrt[3 + 2*x + 5*x^2] - #1] *#1^2)/(-4*Sqrt[5] - 35*#1 + 6*Sqrt[5]*#1^2 + 7*#1^3) & ] - 6*Sqrt[5]*Root Sum[83 - 16*Sqrt[5]*#1 - 70*#1^2 + 8*Sqrt[5]*#1^3 + 7*#1^4 & , (599633*Sqr t[5]*Log[-(Sqrt[5]*x) + Sqrt[3 + 2*x + 5*x^2] - #1] - 391895*Log[-(Sqrt[5] *x) + Sqrt[3 + 2*x + 5*x^2] - #1]*#1 + 21462*Sqrt[5]*Log[-(Sqrt[5]*x) + Sq rt[3 + 2*x + 5*x^2] - #1]*#1^2)/(-4*Sqrt[5] - 35*#1 + 6*Sqrt[5]*#1^2 + 7*# 1^3) & ])/264110
Time = 0.53 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.14, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2132, 27, 2143, 25, 1090, 222, 1365, 27, 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.
| \(\displaystyle \int \frac {\left (x^2+5 x+2\right ) \sqrt {5 x^2+2 x+3}}{\left (-7 x^2+4 x+1\right )^2} \, dx\) |
| \(\Big \downarrow \) 2132 |
| \(\displaystyle \frac {3 (61 x+3) \sqrt {5 x^2+2 x+3}}{154 \left (-7 x^2+4 x+1\right )}-\frac {1}{308} \int -\frac {4 \left (-55 x^2+47 x+237\right )}{\left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{77} \int \frac {-55 x^2+47 x+237}{\left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}dx+\frac {3 \sqrt {5 x^2+2 x+3} (61 x+3)}{154 \left (-7 x^2+4 x+1\right )}\) |
| \(\Big \downarrow \) 2143 |
| \(\displaystyle \frac {1}{77} \left (\frac {55}{7} \int \frac {1}{\sqrt {5 x^2+2 x+3}}dx-\frac {1}{7} \int -\frac {109 x+1604}{\left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}dx\right )+\frac {3 \sqrt {5 x^2+2 x+3} (61 x+3)}{154 \left (-7 x^2+4 x+1\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{77} \left (\frac {55}{7} \int \frac {1}{\sqrt {5 x^2+2 x+3}}dx+\frac {1}{7} \int \frac {109 x+1604}{\left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}dx\right )+\frac {3 \sqrt {5 x^2+2 x+3} (61 x+3)}{154 \left (-7 x^2+4 x+1\right )}\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {1}{77} \left (\frac {1}{7} \int \frac {109 x+1604}{\left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}dx+\frac {11}{14} \sqrt {\frac {5}{14}} \int \frac {1}{\sqrt {\frac {1}{56} (10 x+2)^2+1}}d(10 x+2)\right )+\frac {3 \sqrt {5 x^2+2 x+3} (61 x+3)}{154 \left (-7 x^2+4 x+1\right )}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {1}{77} \left (\frac {1}{7} \int \frac {109 x+1604}{\left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}dx+\frac {11}{7} \sqrt {5} \text {arcsinh}\left (\frac {10 x+2}{2 \sqrt {14}}\right )\right )+\frac {3 \sqrt {5 x^2+2 x+3} (61 x+3)}{154 \left (-7 x^2+4 x+1\right )}\) |
| \(\Big \downarrow \) 1365 |
| \(\displaystyle \frac {1}{77} \left (\frac {1}{7} \left (\frac {1}{11} \left (1199-11446 \sqrt {11}\right ) \int \frac {1}{2 \left (-7 x-\sqrt {11}+2\right ) \sqrt {5 x^2+2 x+3}}dx+\frac {1}{11} \left (1199+11446 \sqrt {11}\right ) \int \frac {1}{2 \left (-7 x+\sqrt {11}+2\right ) \sqrt {5 x^2+2 x+3}}dx\right )+\frac {11}{7} \sqrt {5} \text {arcsinh}\left (\frac {10 x+2}{2 \sqrt {14}}\right )\right )+\frac {3 \sqrt {5 x^2+2 x+3} (61 x+3)}{154 \left (-7 x^2+4 x+1\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{77} \left (\frac {1}{7} \left (\frac {1}{22} \left (1199-11446 \sqrt {11}\right ) \int \frac {1}{\left (-7 x-\sqrt {11}+2\right ) \sqrt {5 x^2+2 x+3}}dx+\frac {1}{22} \left (1199+11446 \sqrt {11}\right ) \int \frac {1}{\left (-7 x+\sqrt {11}+2\right ) \sqrt {5 x^2+2 x+3}}dx\right )+\frac {11}{7} \sqrt {5} \text {arcsinh}\left (\frac {10 x+2}{2 \sqrt {14}}\right )\right )+\frac {3 \sqrt {5 x^2+2 x+3} (61 x+3)}{154 \left (-7 x^2+4 x+1\right )}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {1}{77} \left (\frac {1}{7} \left (-\frac {1}{11} \left (1199-11446 \sqrt {11}\right ) \int \frac {1}{8 \left (125-17 \sqrt {11}\right )-\frac {4 \left (\left (17-5 \sqrt {11}\right ) x-\sqrt {11}+23\right )^2}{5 x^2+2 x+3}}d\left (-\frac {2 \left (\left (17-5 \sqrt {11}\right ) x-\sqrt {11}+23\right )}{\sqrt {5 x^2+2 x+3}}\right )-\frac {1}{11} \left (1199+11446 \sqrt {11}\right ) \int \frac {1}{8 \left (125+17 \sqrt {11}\right )-\frac {4 \left (\left (17+5 \sqrt {11}\right ) x+\sqrt {11}+23\right )^2}{5 x^2+2 x+3}}d\left (-\frac {2 \left (\left (17+5 \sqrt {11}\right ) x+\sqrt {11}+23\right )}{\sqrt {5 x^2+2 x+3}}\right )\right )+\frac {11}{7} \sqrt {5} \text {arcsinh}\left (\frac {10 x+2}{2 \sqrt {14}}\right )\right )+\frac {3 \sqrt {5 x^2+2 x+3} (61 x+3)}{154 \left (-7 x^2+4 x+1\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{77} \left (\frac {11}{7} \sqrt {5} \text {arcsinh}\left (\frac {10 x+2}{2 \sqrt {14}}\right )+\frac {1}{7} \left (\frac {\left (1199-11446 \sqrt {11}\right ) \text {arctanh}\left (\frac {\left (17-5 \sqrt {11}\right ) x-\sqrt {11}+23}{\sqrt {2 \left (125-17 \sqrt {11}\right )} \sqrt {5 x^2+2 x+3}}\right )}{22 \sqrt {2 \left (125-17 \sqrt {11}\right )}}+\frac {\left (1199+11446 \sqrt {11}\right ) \text {arctanh}\left (\frac {\left (17+5 \sqrt {11}\right ) x+\sqrt {11}+23}{\sqrt {2 \left (125+17 \sqrt {11}\right )} \sqrt {5 x^2+2 x+3}}\right )}{22 \sqrt {2 \left (125+17 \sqrt {11}\right )}}\right )\right )+\frac {3 \sqrt {5 x^2+2 x+3} (61 x+3)}{154 \left (-7 x^2+4 x+1\right )}\) |
(3*(3 + 61*x)*Sqrt[3 + 2*x + 5*x^2])/(154*(1 + 4*x - 7*x^2)) + ((11*Sqrt[5 ]*ArcSinh[(2 + 10*x)/(2*Sqrt[14])])/7 + (((1199 - 11446*Sqrt[11])*ArcTanh[ (23 - Sqrt[11] + (17 - 5*Sqrt[11])*x)/(Sqrt[2*(125 - 17*Sqrt[11])]*Sqrt[3 + 2*x + 5*x^2])])/(22*Sqrt[2*(125 - 17*Sqrt[11])]) + ((1199 + 11446*Sqrt[1 1])*ArcTanh[(23 + Sqrt[11] + (17 + 5*Sqrt[11])*x)/(Sqrt[2*(125 + 17*Sqrt[1 1])]*Sqrt[3 + 2*x + 5*x^2])])/(22*Sqrt[2*(125 + 17*Sqrt[11])]))/7)/77
3.4.78.3.1 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[(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/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/(2*c*(-4* (c/(b^2 - 4*a*c)))^p) Subst[Int[Simp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 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[((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*c - 2*a*B*c + a*b*C - (c*(b*B - 2*A*c) - C*(b^2 - 2*a*c))*x)*(a + b*x + c*x^2)^(p + 1)*((d + e*x + f*x^2)^q/(c*(b^2 - 4*a*c)*(p + 1))), x] - Simp[1/(c*(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[e*q*(A*b*c - 2*a*B*c + a*b*C) - d*(c*(b*B - 2*A*c)*(2*p + 3) + C*(2*a*c - b^2*(p + 2))) + (2*f*q*(A*b*c - 2*a*B*c + a*b*C) - e*(c*(b*B - 2*A*c)*(2*p + q + 3) + C*(2*a*c*(q + 1) - b^2*(p + q + 2))))*x - f*(c*(b*B - 2*A*c)*(2*p + 2*q + 3) + C*(2*a*c*(2*q + 1) - b^2*(p + 2*q + 2)))*x^2, x], x], x]] /; FreeQ[{a, b, c, d, e, f}, x] && PolyQ[Px, x, 2] && LtQ[p, -1] && GtQ[q, 0] && !IGtQ[q, 0]
Int[(Px_)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_ .)*(x_)^2]), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2]}, Simp[C/c Int[1/Sqrt[d + e*x + f*x^2], x], x] + Simp[ 1/c Int[(A*c - a*C + (B*c - b*C)*x)/((a + b*x + c*x^2)*Sqrt[d + e*x + f*x ^2]), x], x]] /; FreeQ[{a, b, c, d, e, f}, x] && PolyQ[Px, x, 2]
Time = 0.50 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.18
| method | result | size |
| risch | \(-\frac {3 \left (3+61 x \right ) \sqrt {5 x^{2}+2 x +3}}{154 \left (7 x^{2}-4 x -1\right )}+\frac {\sqrt {5}\, \operatorname {arcsinh}\left (\frac {5 \sqrt {14}\, \left (x +\frac {1}{5}\right )}{14}\right )}{49}+\frac {\left (-11446+109 \sqrt {11}\right ) \sqrt {11}\, \operatorname {arctanh}\left (\frac {250-34 \sqrt {11}+\frac {49 \left (\frac {34}{7}-\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}+\frac {\sqrt {11}}{7}\right )}{2}}{\sqrt {250-34 \sqrt {11}}\, \sqrt {245 \left (x -\frac {2}{7}+\frac {\sqrt {11}}{7}\right )^{2}+49 \left (\frac {34}{7}-\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}+\frac {\sqrt {11}}{7}\right )+250-34 \sqrt {11}}}\right )}{11858 \sqrt {250-34 \sqrt {11}}}+\frac {\sqrt {11}\, \left (11446+109 \sqrt {11}\right ) \operatorname {arctanh}\left (\frac {250+34 \sqrt {11}+\frac {49 \left (\frac {34}{7}+\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}-\frac {\sqrt {11}}{7}\right )}{2}}{\sqrt {250+34 \sqrt {11}}\, \sqrt {245 \left (x -\frac {2}{7}-\frac {\sqrt {11}}{7}\right )^{2}+49 \left (\frac {34}{7}+\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}-\frac {\sqrt {11}}{7}\right )+250+34 \sqrt {11}}}\right )}{11858 \sqrt {250+34 \sqrt {11}}}\) | \(235\) |
| trager | \(\text {Expression too large to display}\) | \(512\) |
| default | \(\text {Expression too large to display}\) | \(1084\) |
-3/154*(3+61*x)/(7*x^2-4*x-1)*(5*x^2+2*x+3)^(1/2)+1/49*5^(1/2)*arcsinh(5/1 4*14^(1/2)*(x+1/5))+1/11858*(-11446+109*11^(1/2))*11^(1/2)/(250-34*11^(1/2 ))^(1/2)*arctanh(49/2*(500/49-68/49*11^(1/2)+(34/7-10/7*11^(1/2))*(x-2/7+1 /7*11^(1/2)))/(250-34*11^(1/2))^(1/2)/(245*(x-2/7+1/7*11^(1/2))^2+49*(34/7 -10/7*11^(1/2))*(x-2/7+1/7*11^(1/2))+250-34*11^(1/2))^(1/2))+1/11858*11^(1 /2)*(11446+109*11^(1/2))/(250+34*11^(1/2))^(1/2)*arctanh(49/2*(500/49+68/4 9*11^(1/2)+(34/7+10/7*11^(1/2))*(x-2/7-1/7*11^(1/2)))/(250+34*11^(1/2))^(1 /2)/(245*(x-2/7-1/7*11^(1/2))^2+49*(34/7+10/7*11^(1/2))*(x-2/7-1/7*11^(1/2 ))+250+34*11^(1/2))^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 378 vs. \(2 (145) = 290\).
Time = 0.26 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.90 \[ \int \frac {\left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2}}{\left (1+4 x-7 x^2\right )^2} \, dx=-\frac {\sqrt {1397} {\left (7 \, x^{2} - 4 \, x - 1\right )} \sqrt {39132731 \, \sqrt {11} + 325022311} \log \left (-\frac {\sqrt {1397} \sqrt {5 \, x^{2} + 2 \, x + 3} \sqrt {39132731 \, \sqrt {11} + 325022311} {\left (16943 \, \sqrt {11} + 235367\right )} + 26119953475 \, \sqrt {11} {\left (x + 3\right )} - 78359860425 \, x + 130599767375}{x}\right ) - \sqrt {1397} {\left (7 \, x^{2} - 4 \, x - 1\right )} \sqrt {39132731 \, \sqrt {11} + 325022311} \log \left (\frac {\sqrt {1397} \sqrt {5 \, x^{2} + 2 \, x + 3} \sqrt {39132731 \, \sqrt {11} + 325022311} {\left (16943 \, \sqrt {11} + 235367\right )} - 26119953475 \, \sqrt {11} {\left (x + 3\right )} + 78359860425 \, x - 130599767375}{x}\right ) + \sqrt {1397} {\left (7 \, x^{2} - 4 \, x - 1\right )} \sqrt {-39132731 \, \sqrt {11} + 325022311} \log \left (\frac {\sqrt {1397} \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (16943 \, \sqrt {11} - 235367\right )} \sqrt {-39132731 \, \sqrt {11} + 325022311} + 26119953475 \, \sqrt {11} {\left (x + 3\right )} + 78359860425 \, x - 130599767375}{x}\right ) - \sqrt {1397} {\left (7 \, x^{2} - 4 \, x - 1\right )} \sqrt {-39132731 \, \sqrt {11} + 325022311} \log \left (-\frac {\sqrt {1397} \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (16943 \, \sqrt {11} - 235367\right )} \sqrt {-39132731 \, \sqrt {11} + 325022311} - 26119953475 \, \sqrt {11} {\left (x + 3\right )} - 78359860425 \, x + 130599767375}{x}\right ) - 61468 \, \sqrt {5} {\left (7 \, x^{2} - 4 \, x - 1\right )} \log \left (-\sqrt {5} \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (5 \, x + 1\right )} - 25 \, x^{2} - 10 \, x - 8\right ) + 117348 \, \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (61 \, x + 3\right )}}{6023864 \, {\left (7 \, x^{2} - 4 \, x - 1\right )}} \]
-1/6023864*(sqrt(1397)*(7*x^2 - 4*x - 1)*sqrt(39132731*sqrt(11) + 32502231 1)*log(-(sqrt(1397)*sqrt(5*x^2 + 2*x + 3)*sqrt(39132731*sqrt(11) + 3250223 11)*(16943*sqrt(11) + 235367) + 26119953475*sqrt(11)*(x + 3) - 78359860425 *x + 130599767375)/x) - sqrt(1397)*(7*x^2 - 4*x - 1)*sqrt(39132731*sqrt(11 ) + 325022311)*log((sqrt(1397)*sqrt(5*x^2 + 2*x + 3)*sqrt(39132731*sqrt(11 ) + 325022311)*(16943*sqrt(11) + 235367) - 26119953475*sqrt(11)*(x + 3) + 78359860425*x - 130599767375)/x) + sqrt(1397)*(7*x^2 - 4*x - 1)*sqrt(-3913 2731*sqrt(11) + 325022311)*log((sqrt(1397)*sqrt(5*x^2 + 2*x + 3)*(16943*sq rt(11) - 235367)*sqrt(-39132731*sqrt(11) + 325022311) + 26119953475*sqrt(1 1)*(x + 3) + 78359860425*x - 130599767375)/x) - sqrt(1397)*(7*x^2 - 4*x - 1)*sqrt(-39132731*sqrt(11) + 325022311)*log(-(sqrt(1397)*sqrt(5*x^2 + 2*x + 3)*(16943*sqrt(11) - 235367)*sqrt(-39132731*sqrt(11) + 325022311) - 2611 9953475*sqrt(11)*(x + 3) - 78359860425*x + 130599767375)/x) - 61468*sqrt(5 )*(7*x^2 - 4*x - 1)*log(-sqrt(5)*sqrt(5*x^2 + 2*x + 3)*(5*x + 1) - 25*x^2 - 10*x - 8) + 117348*sqrt(5*x^2 + 2*x + 3)*(61*x + 3))/(7*x^2 - 4*x - 1)
\[ \int \frac {\left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2}}{\left (1+4 x-7 x^2\right )^2} \, dx=\int \frac {\left (x^{2} + 5 x + 2\right ) \sqrt {5 x^{2} + 2 x + 3}}{\left (7 x^{2} - 4 x - 1\right )^{2}}\, dx \]
\[ \int \frac {\left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2}}{\left (1+4 x-7 x^2\right )^2} \, dx=\int { \frac {\sqrt {5 \, x^{2} + 2 \, x + 3} {\left (x^{2} + 5 \, x + 2\right )}}{{\left (7 \, x^{2} - 4 \, x - 1\right )}^{2}} \,d x } \]
Exception generated. \[ \int \frac {\left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2}}{\left (1+4 x-7 x^2\right )^2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{184473632,[8]%%%}+%%%{%%{[421654016,0]:[1,0,-5]%%},[7]%%%} +%%%{-248
Timed out. \[ \int \frac {\left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2}}{\left (1+4 x-7 x^2\right )^2} \, dx=\int \frac {\left (x^2+5\,x+2\right )\,\sqrt {5\,x^2+2\,x+3}}{{\left (-7\,x^2+4\,x+1\right )}^2} \,d x \]