Integrand size = 35, antiderivative size = 187 \[ \int \frac {\left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2}}{1+4 x-7 x^2} \, dx=-\frac {1}{490} (397+35 x) \sqrt {3+2 x+5 x^2}-\frac {8233 \text {arcsinh}\left (\frac {1+5 x}{\sqrt {14}}\right )}{1715 \sqrt {5}}-\frac {3}{343} \sqrt {\frac {1}{11} \left (497041-146555 \sqrt {11}\right )} \text {arctanh}\left (\frac {23-\sqrt {11}+\left (17-5 \sqrt {11}\right ) x}{\sqrt {2 \left (125-17 \sqrt {11}\right )} \sqrt {3+2 x+5 x^2}}\right )+\frac {3}{343} \sqrt {\frac {1}{11} \left (497041+146555 \sqrt {11}\right )} \text {arctanh}\left (\frac {23+\sqrt {11}+\left (17+5 \sqrt {11}\right ) x}{\sqrt {2 \left (125+17 \sqrt {11}\right )} \sqrt {3+2 x+5 x^2}}\right ) \]
-8233/8575*arcsinh(1/14*(1+5*x)*14^(1/2))*5^(1/2)-1/490*(397+35*x)*(5*x^2+ 2*x+3)^(1/2)-3/3773*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))*(5467451-1612105*11^(1/2))^(1/2)+3/3773*arc tanh((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))*(5467451+1612105*11^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.35 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.25 \[ \int \frac {\left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2}}{1+4 x-7 x^2} \, dx=\frac {1}{490} (-397-35 x) \sqrt {3+2 x+5 x^2}+\frac {8233 \log \left (-1-5 x+\sqrt {5} \sqrt {3+2 x+5 x^2}\right )}{1715 \sqrt {5}}+\frac {6}{343} \text {RootSum}\left [83-16 \sqrt {5} \text {$\#$1}-70 \text {$\#$1}^2+8 \sqrt {5} \text {$\#$1}^3+7 \text {$\#$1}^4\&,\frac {3317 \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right )+676 \sqrt {5} \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}-1331 \log \left (-\sqrt {5} x+\sqrt {3+2 x+5 x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-4 \sqrt {5}-35 \text {$\#$1}+6 \sqrt {5} \text {$\#$1}^2+7 \text {$\#$1}^3}\&\right ] \]
((-397 - 35*x)*Sqrt[3 + 2*x + 5*x^2])/490 + (8233*Log[-1 - 5*x + Sqrt[5]*S qrt[3 + 2*x + 5*x^2]])/(1715*Sqrt[5]) + (6*RootSum[83 - 16*Sqrt[5]*#1 - 70 *#1^2 + 8*Sqrt[5]*#1^3 + 7*#1^4 & , (3317*Log[-(Sqrt[5]*x) + Sqrt[3 + 2*x + 5*x^2] - #1] + 676*Sqrt[5]*Log[-(Sqrt[5]*x) + Sqrt[3 + 2*x + 5*x^2] - #1 ]*#1 - 1331*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) & ])/343
Time = 0.58 (sec) , antiderivative size = 214, 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 = {2138, 27, 2143, 27, 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.
| \(\displaystyle \int \frac {\left (x^2+5 x+2\right ) \sqrt {5 x^2+2 x+3}}{-7 x^2+4 x+1} \, dx\) |
| \(\Big \downarrow \) 2138 |
| \(\displaystyle -\frac {1}{490} \int -\frac {2 \left (8233 x^2+6704 x+1721\right )}{\left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}dx-\frac {1}{490} \sqrt {5 x^2+2 x+3} (35 x+397)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{245} \int \frac {8233 x^2+6704 x+1721}{\left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}dx-\frac {1}{490} (35 x+397) \sqrt {5 x^2+2 x+3}\) |
| \(\Big \downarrow \) 2143 |
| \(\displaystyle \frac {1}{245} \left (-\frac {8233}{7} \int \frac {1}{\sqrt {5 x^2+2 x+3}}dx-\frac {1}{7} \int -\frac {60 (1331 x+338)}{\left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}dx\right )-\frac {1}{490} (35 x+397) \sqrt {5 x^2+2 x+3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{245} \left (\frac {60}{7} \int \frac {1331 x+338}{\left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}dx-\frac {8233}{7} \int \frac {1}{\sqrt {5 x^2+2 x+3}}dx\right )-\frac {1}{490} (35 x+397) \sqrt {5 x^2+2 x+3}\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {1}{245} \left (\frac {60}{7} \int \frac {1331 x+338}{\left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}dx-\frac {8233 \int \frac {1}{\sqrt {\frac {1}{56} (10 x+2)^2+1}}d(10 x+2)}{14 \sqrt {70}}\right )-\frac {1}{490} (35 x+397) \sqrt {5 x^2+2 x+3}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {1}{245} \left (\frac {60}{7} \int \frac {1331 x+338}{\left (-7 x^2+4 x+1\right ) \sqrt {5 x^2+2 x+3}}dx-\frac {8233 \text {arcsinh}\left (\frac {10 x+2}{2 \sqrt {14}}\right )}{7 \sqrt {5}}\right )-\frac {1}{490} (35 x+397) \sqrt {5 x^2+2 x+3}\) |
| \(\Big \downarrow \) 1365 |
| \(\displaystyle \frac {1}{245} \left (\frac {60}{7} \left (\frac {1}{11} \left (14641-5028 \sqrt {11}\right ) \int \frac {1}{2 \left (-7 x-\sqrt {11}+2\right ) \sqrt {5 x^2+2 x+3}}dx+\frac {1}{11} \left (14641+5028 \sqrt {11}\right ) \int \frac {1}{2 \left (-7 x+\sqrt {11}+2\right ) \sqrt {5 x^2+2 x+3}}dx\right )-\frac {8233 \text {arcsinh}\left (\frac {10 x+2}{2 \sqrt {14}}\right )}{7 \sqrt {5}}\right )-\frac {1}{490} (35 x+397) \sqrt {5 x^2+2 x+3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{245} \left (\frac {60}{7} \left (\frac {1}{22} \left (14641-5028 \sqrt {11}\right ) \int \frac {1}{\left (-7 x-\sqrt {11}+2\right ) \sqrt {5 x^2+2 x+3}}dx+\frac {1}{22} \left (14641+5028 \sqrt {11}\right ) \int \frac {1}{\left (-7 x+\sqrt {11}+2\right ) \sqrt {5 x^2+2 x+3}}dx\right )-\frac {8233 \text {arcsinh}\left (\frac {10 x+2}{2 \sqrt {14}}\right )}{7 \sqrt {5}}\right )-\frac {1}{490} (35 x+397) \sqrt {5 x^2+2 x+3}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {1}{245} \left (\frac {60}{7} \left (-\frac {1}{11} \left (14641-5028 \sqrt {11}\right ) \int \frac {1}{8 \left (125-17 \sqrt {11}\right )-\frac {4 \left (\left (17-5 \sqrt {11}\right ) x-\sqrt {11}+23\right )^2}{5 x^2+2 x+3}}d\left (-\frac {2 \left (\left (17-5 \sqrt {11}\right ) x-\sqrt {11}+23\right )}{\sqrt {5 x^2+2 x+3}}\right )-\frac {1}{11} \left (14641+5028 \sqrt {11}\right ) \int \frac {1}{8 \left (125+17 \sqrt {11}\right )-\frac {4 \left (\left (17+5 \sqrt {11}\right ) x+\sqrt {11}+23\right )^2}{5 x^2+2 x+3}}d\left (-\frac {2 \left (\left (17+5 \sqrt {11}\right ) x+\sqrt {11}+23\right )}{\sqrt {5 x^2+2 x+3}}\right )\right )-\frac {8233 \text {arcsinh}\left (\frac {10 x+2}{2 \sqrt {14}}\right )}{7 \sqrt {5}}\right )-\frac {1}{490} (35 x+397) \sqrt {5 x^2+2 x+3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{245} \left (\frac {60}{7} \left (\frac {\left (14641-5028 \sqrt {11}\right ) \text {arctanh}\left (\frac {\left (17-5 \sqrt {11}\right ) x-\sqrt {11}+23}{\sqrt {2 \left (125-17 \sqrt {11}\right )} \sqrt {5 x^2+2 x+3}}\right )}{22 \sqrt {2 \left (125-17 \sqrt {11}\right )}}+\frac {\left (14641+5028 \sqrt {11}\right ) \text {arctanh}\left (\frac {\left (17+5 \sqrt {11}\right ) x+\sqrt {11}+23}{\sqrt {2 \left (125+17 \sqrt {11}\right )} \sqrt {5 x^2+2 x+3}}\right )}{22 \sqrt {2 \left (125+17 \sqrt {11}\right )}}\right )-\frac {8233 \text {arcsinh}\left (\frac {10 x+2}{2 \sqrt {14}}\right )}{7 \sqrt {5}}\right )-\frac {1}{490} (35 x+397) \sqrt {5 x^2+2 x+3}\) |
-1/490*((397 + 35*x)*Sqrt[3 + 2*x + 5*x^2]) + ((-8233*ArcSinh[(2 + 10*x)/( 2*Sqrt[14])])/(7*Sqrt[5]) + (60*(((14641 - 5028*Sqrt[11])*ArcTanh[(23 - Sq rt[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])]) + ((14641 + 5028*Sqrt[11])*ArcT anh[(23 + Sqrt[11] + (17 + 5*Sqrt[11])*x)/(Sqrt[2*(125 + 17*Sqrt[11])]*Sqr t[3 + 2*x + 5*x^2])])/(22*Sqrt[2*(125 + 17*Sqrt[11])])))/7)/245
3.4.77.3.1 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[(B*c*f*(2*p + 2*q + 3) + C*(b*f*p - c*e*(2*p + q + 2)) + 2*c*C*f*(p + q + 1)*x)*(a + b*x + c*x^2)^p*((d + e*x + f*x^2)^(q + 1)/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3))), x] - Simp[1/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3)) Int[(a + b*x + c*x^2)^(p - 1)*(d + e*x + f*x^2)^q*Si mp[p*(b*d - a*e)*(C*(c*e - b*f)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(b^2*C*d*f*p + a*c*(C*(2*d*f - e^2*(2*p + q + 2)) + f*(B*e - 2* A*f)*(2*p + 2*q + 3))) + (2*p*(c*d - a*f)*(C*(c*e - b*f)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(C*e*f*p*(b^2 - 4*a*c) - b*c*(C*(e^2 - 4*d*f)*(2*p + q + 2) + f*(2*C*d - B*e + 2*A*f)*(2*p + 2*q + 3))))*x + (p*( c*e - b*f)*(C*(c*e - b*f)*(q + 1) - c*(C*e - B*f)*(2*p + 2*q + 3)) + (p + q + 1)*(C*f^2*p*(b^2 - 4*a*c) - c^2*(C*(e^2 - 4*d*f)*(2*p + q + 2) + f*(2*C* d - B*e + 2*A*f)*(2*p + 2*q + 3))))*x^2, x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && PolyQ[Px, x, 2] && GtQ[p, 0] && NeQ[p + q + 1, 0] && NeQ[2* p + 2*q + 3, 0] && !IGtQ[p, 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 = 0.62 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.19
| method | result | size |
| risch | \(-\frac {\left (397+35 x \right ) \sqrt {5 x^{2}+2 x +3}}{490}-\frac {8233 \sqrt {5}\, \operatorname {arcsinh}\left (\frac {5 \sqrt {14}\, \left (x +\frac {1}{5}\right )}{14}\right )}{8575}+\frac {6 \left (-5028+1331 \sqrt {11}\right ) \sqrt {11}\, \operatorname {arctanh}\left (\frac {250-34 \sqrt {11}+\frac {49 \left (\frac {34}{7}-\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}+\frac {\sqrt {11}}{7}\right )}{2}}{\sqrt {250-34 \sqrt {11}}\, \sqrt {245 \left (x -\frac {2}{7}+\frac {\sqrt {11}}{7}\right )^{2}+49 \left (\frac {34}{7}-\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}+\frac {\sqrt {11}}{7}\right )+250-34 \sqrt {11}}}\right )}{3773 \sqrt {250-34 \sqrt {11}}}+\frac {6 \left (5028+1331 \sqrt {11}\right ) \sqrt {11}\, \operatorname {arctanh}\left (\frac {250+34 \sqrt {11}+\frac {49 \left (\frac {34}{7}+\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}-\frac {\sqrt {11}}{7}\right )}{2}}{\sqrt {250+34 \sqrt {11}}\, \sqrt {245 \left (x -\frac {2}{7}-\frac {\sqrt {11}}{7}\right )^{2}+49 \left (\frac {34}{7}+\frac {10 \sqrt {11}}{7}\right ) \left (x -\frac {2}{7}-\frac {\sqrt {11}}{7}\right )+250+34 \sqrt {11}}}\right )}{3773 \sqrt {250+34 \sqrt {11}}}\) | \(223\) |
| default | \(-\frac {\left (10 x +2\right ) \sqrt {5 x^{2}+2 x +3}}{140}-\frac {\sqrt {5}\, \operatorname {arcsinh}\left (\frac {5 \sqrt {14}\, \left (x +\frac {1}{5}\right )}{14}\right )}{25}-\frac {3 \left (-61+13 \sqrt {11}\right ) \sqrt {11}\, \left (\frac {\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}}}{49}+\frac {\left (\frac {34}{7}-\frac {10 \sqrt {11}}{7}\right ) \sqrt {5}\, \operatorname {arcsinh}\left (\frac {\sqrt {5}\, \left (x +\frac {1}{5}\right )}{\sqrt {\frac {250}{49}-\frac {34 \sqrt {11}}{49}-\frac {\left (\frac {34}{7}-\frac {10 \sqrt {11}}{7}\right )^{2}}{20}}}\right )}{70}-\frac {\left (\frac {250}{49}-\frac {34 \sqrt {11}}{49}\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 )}{\sqrt {250-34 \sqrt {11}}}\right )}{154}-\frac {3 \left (61+13 \sqrt {11}\right ) \sqrt {11}\, \left (\frac {\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}}}{49}+\frac {\left (\frac {34}{7}+\frac {10 \sqrt {11}}{7}\right ) \sqrt {5}\, \operatorname {arcsinh}\left (\frac {\sqrt {5}\, \left (x +\frac {1}{5}\right )}{\sqrt {\frac {250}{49}+\frac {34 \sqrt {11}}{49}-\frac {\left (\frac {34}{7}+\frac {10 \sqrt {11}}{7}\right )^{2}}{20}}}\right )}{70}-\frac {\left (\frac {250}{49}+\frac {34 \sqrt {11}}{49}\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 )}{\sqrt {250+34 \sqrt {11}}}\right )}{154}\) | \(403\) |
| trager | \(\left (-\frac {397}{490}-\frac {x}{14}\right ) \sqrt {5 x^{2}+2 x +3}+\frac {8233 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +5 \sqrt {5 x^{2}+2 x +3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )\right )}{8575}+\frac {\operatorname {RootOf}\left (5929 \textit {\_Z}^{4}-2460352950 \textit {\_Z}^{2}+11145463008750\right ) \ln \left (\frac {56289315313 x \operatorname {RootOf}\left (5929 \textit {\_Z}^{4}-2460352950 \textit {\_Z}^{2}+11145463008750\right )^{5}-22506793362674775 \operatorname {RootOf}\left (5929 \textit {\_Z}^{4}-2460352950 \textit {\_Z}^{2}+11145463008750\right )^{3} x +18202206743287875 \operatorname {RootOf}\left (5929 \textit {\_Z}^{4}-2460352950 \textit {\_Z}^{2}+11145463008750\right )^{3}-546053938612875000 \operatorname {RootOf}\left (5929 \textit {\_Z}^{4}-2460352950 \textit {\_Z}^{2}+11145463008750\right )^{2} \sqrt {5 x^{2}+2 x +3}-51336524264074357500 \operatorname {RootOf}\left (5929 \textit {\_Z}^{4}-2460352950 \textit {\_Z}^{2}+11145463008750\right ) x -7319277984915532462500 \operatorname {RootOf}\left (5929 \textit {\_Z}^{4}-2460352950 \textit {\_Z}^{2}+11145463008750\right )+293965802110893533906250 \sqrt {5 x^{2}+2 x +3}}{539 x \operatorname {RootOf}\left (5929 \textit {\_Z}^{4}-2460352950 \textit {\_Z}^{2}+11145463008750\right )^{2}-177783975 x -32974875}\right )}{245}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+5929 \operatorname {RootOf}\left (5929 \textit {\_Z}^{4}-2460352950 \textit {\_Z}^{2}+11145463008750\right )^{2}-2460352950\right ) \ln \left (\frac {1148761537 \operatorname {RootOf}\left (\textit {\_Z}^{2}+5929 \operatorname {RootOf}\left (5929 \textit {\_Z}^{4}-2460352950 \textit {\_Z}^{2}+11145463008750\right )^{2}-2460352950\right ) \operatorname {RootOf}\left (5929 \textit {\_Z}^{4}-2460352950 \textit {\_Z}^{2}+11145463008750\right )^{4} x -494079216718725 \operatorname {RootOf}\left (5929 \textit {\_Z}^{4}-2460352950 \textit {\_Z}^{2}+11145463008750\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+5929 \operatorname {RootOf}\left (5929 \textit {\_Z}^{4}-2460352950 \textit {\_Z}^{2}+11145463008750\right )^{2}-2460352950\right ) x +858084760677375000 \operatorname {RootOf}\left (5929 \textit {\_Z}^{4}-2460352950 \textit {\_Z}^{2}+11145463008750\right )^{2} \sqrt {5 x^{2}+2 x +3}-371473607005875 \operatorname {RootOf}\left (5929 \textit {\_Z}^{4}-2460352950 \textit {\_Z}^{2}+11145463008750\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+5929 \operatorname {RootOf}\left (5929 \textit {\_Z}^{4}-2460352950 \textit {\_Z}^{2}+11145463008750\right )^{2}-2460352950\right )+6163839638738538750 \operatorname {RootOf}\left (\textit {\_Z}^{2}+5929 \operatorname {RootOf}\left (5929 \textit {\_Z}^{4}-2460352950 \textit {\_Z}^{2}+11145463008750\right )^{2}-2460352950\right ) x +105867432279401497031250 \sqrt {5 x^{2}+2 x +3}+4777120706572068750 \operatorname {RootOf}\left (\textit {\_Z}^{2}+5929 \operatorname {RootOf}\left (5929 \textit {\_Z}^{4}-2460352950 \textit {\_Z}^{2}+11145463008750\right )^{2}-2460352950\right )}{539 x \operatorname {RootOf}\left (5929 \textit {\_Z}^{4}-2460352950 \textit {\_Z}^{2}+11145463008750\right )^{2}-45884475 x +32974875}\right )}{18865}\) | \(500\) |
-1/490*(397+35*x)*(5*x^2+2*x+3)^(1/2)-8233/8575*5^(1/2)*arcsinh(5/14*14^(1 /2)*(x+1/5))+6/3773*(-5028+1331*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))+6/3773*(5028+1331*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))
Leaf count of result is larger than twice the leaf count of optimal. 304 vs. \(2 (133) = 266\).
Time = 0.27 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.63 \[ \int \frac {\left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2}}{1+4 x-7 x^2} \, dx=\frac {3}{7546} \, \sqrt {11} \sqrt {146555 \, \sqrt {11} + 497041} \log \left (\frac {6 \, {\left (\sqrt {5 \, x^{2} + 2 \, x + 3} \sqrt {146555 \, \sqrt {11} + 497041} {\left (87 \, \sqrt {11} - 265\right )} + 6517 \, \sqrt {11} {\left (x + 3\right )} + 19551 \, x - 32585\right )}}{x}\right ) - \frac {3}{7546} \, \sqrt {11} \sqrt {146555 \, \sqrt {11} + 497041} \log \left (-\frac {6 \, {\left (\sqrt {5 \, x^{2} + 2 \, x + 3} \sqrt {146555 \, \sqrt {11} + 497041} {\left (87 \, \sqrt {11} - 265\right )} - 6517 \, \sqrt {11} {\left (x + 3\right )} - 19551 \, x + 32585\right )}}{x}\right ) - \frac {1}{15092} \, \sqrt {11} \sqrt {-5275980 \, \sqrt {11} + 17893476} \log \left (-\frac {\sqrt {5 \, x^{2} + 2 \, x + 3} {\left (87 \, \sqrt {11} + 265\right )} \sqrt {-5275980 \, \sqrt {11} + 17893476} + 39102 \, \sqrt {11} {\left (x + 3\right )} - 117306 \, x + 195510}{x}\right ) + \frac {1}{15092} \, \sqrt {11} \sqrt {-5275980 \, \sqrt {11} + 17893476} \log \left (\frac {\sqrt {5 \, x^{2} + 2 \, x + 3} {\left (87 \, \sqrt {11} + 265\right )} \sqrt {-5275980 \, \sqrt {11} + 17893476} - 39102 \, \sqrt {11} {\left (x + 3\right )} + 117306 \, x - 195510}{x}\right ) - \frac {1}{490} \, \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (35 \, x + 397\right )} + \frac {8233}{17150} \, \sqrt {5} \log \left (\sqrt {5} \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (5 \, x + 1\right )} - 25 \, x^{2} - 10 \, x - 8\right ) \]
3/7546*sqrt(11)*sqrt(146555*sqrt(11) + 497041)*log(6*(sqrt(5*x^2 + 2*x + 3 )*sqrt(146555*sqrt(11) + 497041)*(87*sqrt(11) - 265) + 6517*sqrt(11)*(x + 3) + 19551*x - 32585)/x) - 3/7546*sqrt(11)*sqrt(146555*sqrt(11) + 497041)* log(-6*(sqrt(5*x^2 + 2*x + 3)*sqrt(146555*sqrt(11) + 497041)*(87*sqrt(11) - 265) - 6517*sqrt(11)*(x + 3) - 19551*x + 32585)/x) - 1/15092*sqrt(11)*sq rt(-5275980*sqrt(11) + 17893476)*log(-(sqrt(5*x^2 + 2*x + 3)*(87*sqrt(11) + 265)*sqrt(-5275980*sqrt(11) + 17893476) + 39102*sqrt(11)*(x + 3) - 11730 6*x + 195510)/x) + 1/15092*sqrt(11)*sqrt(-5275980*sqrt(11) + 17893476)*log ((sqrt(5*x^2 + 2*x + 3)*(87*sqrt(11) + 265)*sqrt(-5275980*sqrt(11) + 17893 476) - 39102*sqrt(11)*(x + 3) + 117306*x - 195510)/x) - 1/490*sqrt(5*x^2 + 2*x + 3)*(35*x + 397) + 8233/17150*sqrt(5)*log(sqrt(5)*sqrt(5*x^2 + 2*x + 3)*(5*x + 1) - 25*x^2 - 10*x - 8)
\[ \int \frac {\left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2}}{1+4 x-7 x^2} \, dx=- \int \frac {2 \sqrt {5 x^{2} + 2 x + 3}}{7 x^{2} - 4 x - 1}\, dx - \int \frac {5 x \sqrt {5 x^{2} + 2 x + 3}}{7 x^{2} - 4 x - 1}\, dx - \int \frac {x^{2} \sqrt {5 x^{2} + 2 x + 3}}{7 x^{2} - 4 x - 1}\, dx \]
-Integral(2*sqrt(5*x**2 + 2*x + 3)/(7*x**2 - 4*x - 1), x) - Integral(5*x*s qrt(5*x**2 + 2*x + 3)/(7*x**2 - 4*x - 1), x) - Integral(x**2*sqrt(5*x**2 + 2*x + 3)/(7*x**2 - 4*x - 1), x)
Leaf count of result is larger than twice the leaf count of optimal. 500 vs. \(2 (133) = 266\).
Time = 0.33 (sec) , antiderivative size = 500, normalized size of antiderivative = 2.67 \[ \int \frac {\left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2}}{1+4 x-7 x^2} \, dx=\frac {1}{188650} \, \sqrt {11} {\left (975 \, \sqrt {11} \sqrt {2} \sqrt {17 \, \sqrt {11} + 125} \operatorname {arsinh}\left (\frac {5 \, \sqrt {11} \sqrt {7} \sqrt {2} x}{7 \, {\left | 14 \, x - 2 \, \sqrt {11} - 4 \right |}} + \frac {17 \, \sqrt {7} \sqrt {2} x}{7 \, {\left | 14 \, x - 2 \, \sqrt {11} - 4 \right |}} + \frac {\sqrt {11} \sqrt {7} \sqrt {2}}{7 \, {\left | 14 \, x - 2 \, \sqrt {11} - 4 \right |}} + \frac {23 \, \sqrt {7} \sqrt {2}}{7 \, {\left | 14 \, x - 2 \, \sqrt {11} - 4 \right |}}\right ) - 1225 \, \sqrt {11} \sqrt {5 \, x^{2} + 2 \, x + 3} x - 16466 \, \sqrt {11} \sqrt {5} \operatorname {arsinh}\left (\frac {5}{14} \, \sqrt {7} \sqrt {2} x + \frac {1}{14} \, \sqrt {7} \sqrt {2}\right ) - 6825 \, \sqrt {11} \sqrt {-\frac {34}{49} \, \sqrt {11} + \frac {250}{49}} \operatorname {arsinh}\left (\frac {5 \, \sqrt {11} \sqrt {7} \sqrt {2} x}{7 \, {\left | 14 \, x + 2 \, \sqrt {11} - 4 \right |}} - \frac {17 \, \sqrt {7} \sqrt {2} x}{7 \, {\left | 14 \, x + 2 \, \sqrt {11} - 4 \right |}} + \frac {\sqrt {11} \sqrt {7} \sqrt {2}}{7 \, {\left | 14 \, x + 2 \, \sqrt {11} - 4 \right |}} - \frac {23 \, \sqrt {7} \sqrt {2}}{7 \, {\left | 14 \, x + 2 \, \sqrt {11} - 4 \right |}}\right ) + 4575 \, \sqrt {2} \sqrt {17 \, \sqrt {11} + 125} \operatorname {arsinh}\left (\frac {5 \, \sqrt {11} \sqrt {7} \sqrt {2} x}{7 \, {\left | 14 \, x - 2 \, \sqrt {11} - 4 \right |}} + \frac {17 \, \sqrt {7} \sqrt {2} x}{7 \, {\left | 14 \, x - 2 \, \sqrt {11} - 4 \right |}} + \frac {\sqrt {11} \sqrt {7} \sqrt {2}}{7 \, {\left | 14 \, x - 2 \, \sqrt {11} - 4 \right |}} + \frac {23 \, \sqrt {7} \sqrt {2}}{7 \, {\left | 14 \, x - 2 \, \sqrt {11} - 4 \right |}}\right ) + 32025 \, \sqrt {-\frac {34}{49} \, \sqrt {11} + \frac {250}{49}} \operatorname {arsinh}\left (\frac {5 \, \sqrt {11} \sqrt {7} \sqrt {2} x}{7 \, {\left | 14 \, x + 2 \, \sqrt {11} - 4 \right |}} - \frac {17 \, \sqrt {7} \sqrt {2} x}{7 \, {\left | 14 \, x + 2 \, \sqrt {11} - 4 \right |}} + \frac {\sqrt {11} \sqrt {7} \sqrt {2}}{7 \, {\left | 14 \, x + 2 \, \sqrt {11} - 4 \right |}} - \frac {23 \, \sqrt {7} \sqrt {2}}{7 \, {\left | 14 \, x + 2 \, \sqrt {11} - 4 \right |}}\right ) - 13895 \, \sqrt {11} \sqrt {5 \, x^{2} + 2 \, x + 3}\right )} \]
1/188650*sqrt(11)*(975*sqrt(11)*sqrt(2)*sqrt(17*sqrt(11) + 125)*arcsinh(5/ 7*sqrt(11)*sqrt(7)*sqrt(2)*x/abs(14*x - 2*sqrt(11) - 4) + 17/7*sqrt(7)*sqr t(2)*x/abs(14*x - 2*sqrt(11) - 4) + 1/7*sqrt(11)*sqrt(7)*sqrt(2)/abs(14*x - 2*sqrt(11) - 4) + 23/7*sqrt(7)*sqrt(2)/abs(14*x - 2*sqrt(11) - 4)) - 122 5*sqrt(11)*sqrt(5*x^2 + 2*x + 3)*x - 16466*sqrt(11)*sqrt(5)*arcsinh(5/14*s qrt(7)*sqrt(2)*x + 1/14*sqrt(7)*sqrt(2)) - 6825*sqrt(11)*sqrt(-34/49*sqrt( 11) + 250/49)*arcsinh(5/7*sqrt(11)*sqrt(7)*sqrt(2)*x/abs(14*x + 2*sqrt(11) - 4) - 17/7*sqrt(7)*sqrt(2)*x/abs(14*x + 2*sqrt(11) - 4) + 1/7*sqrt(11)*s qrt(7)*sqrt(2)/abs(14*x + 2*sqrt(11) - 4) - 23/7*sqrt(7)*sqrt(2)/abs(14*x + 2*sqrt(11) - 4)) + 4575*sqrt(2)*sqrt(17*sqrt(11) + 125)*arcsinh(5/7*sqrt (11)*sqrt(7)*sqrt(2)*x/abs(14*x - 2*sqrt(11) - 4) + 17/7*sqrt(7)*sqrt(2)*x /abs(14*x - 2*sqrt(11) - 4) + 1/7*sqrt(11)*sqrt(7)*sqrt(2)/abs(14*x - 2*sq rt(11) - 4) + 23/7*sqrt(7)*sqrt(2)/abs(14*x - 2*sqrt(11) - 4)) + 32025*sqr t(-34/49*sqrt(11) + 250/49)*arcsinh(5/7*sqrt(11)*sqrt(7)*sqrt(2)*x/abs(14* x + 2*sqrt(11) - 4) - 17/7*sqrt(7)*sqrt(2)*x/abs(14*x + 2*sqrt(11) - 4) + 1/7*sqrt(11)*sqrt(7)*sqrt(2)/abs(14*x + 2*sqrt(11) - 4) - 23/7*sqrt(7)*sqr t(2)/abs(14*x + 2*sqrt(11) - 4)) - 13895*sqrt(11)*sqrt(5*x^2 + 2*x + 3))
Time = 0.31 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.77 \[ \int \frac {\left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2}}{1+4 x-7 x^2} \, dx=-\frac {1}{490} \, \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (35 \, x + 397\right )} + \frac {8233}{8575} \, \sqrt {5} \log \left (-5 \, \sqrt {5} x - \sqrt {5} + 5 \, \sqrt {5 \, x^{2} + 2 \, x + 3}\right ) + 2.61475869687464 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 2 \, x + 3} + 4.41924736459000\right ) - 0.276245077121866 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 2 \, x + 3} + 1.25295163054000\right ) - 2.61475869687464 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 2 \, x + 3} - 1.02258038113000\right ) + 0.276245077121866 \, \log \left (-\sqrt {5} x + \sqrt {5 \, x^{2} + 2 \, x + 3} - 2.09411235400000\right ) \]
-1/490*sqrt(5*x^2 + 2*x + 3)*(35*x + 397) + 8233/8575*sqrt(5)*log(-5*sqrt( 5)*x - sqrt(5) + 5*sqrt(5*x^2 + 2*x + 3)) + 2.61475869687464*log(-sqrt(5)* x + sqrt(5*x^2 + 2*x + 3) + 4.41924736459000) - 0.276245077121866*log(-sqr t(5)*x + sqrt(5*x^2 + 2*x + 3) + 1.25295163054000) - 2.61475869687464*log( -sqrt(5)*x + sqrt(5*x^2 + 2*x + 3) - 1.02258038113000) + 0.276245077121866 *log(-sqrt(5)*x + sqrt(5*x^2 + 2*x + 3) - 2.09411235400000)
Timed out. \[ \int \frac {\left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2}}{1+4 x-7 x^2} \, dx=\int \frac {\left (x^2+5\,x+2\right )\,\sqrt {5\,x^2+2\,x+3}}{-7\,x^2+4\,x+1} \,d x \]