Integrand size = 22, antiderivative size = 210 \[ \int \frac {\sqrt {1+2 x}}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {\sqrt {1+2 x} (3+10 x)}{31 \left (2+3 x+5 x^2\right )}-\frac {1}{31} \sqrt {\frac {2}{217} \left (218+47 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )+\frac {1}{31} \sqrt {\frac {2}{217} \left (218+47 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )-\frac {1}{31} \sqrt {\frac {2}{217} \left (-218+47 \sqrt {35}\right )} \text {arctanh}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}}{5+\sqrt {35}+10 x}\right ) \] Output:
(1+2*x)^(1/2)*(3+10*x)/(155*x^2+93*x+62)-1/6727*(94612+20398*35^(1/2))^(1/ 2)*arctan(((20+10*35^(1/2))^(1/2)-10*(1+2*x)^(1/2))/(-20+10*35^(1/2))^(1/2 ))+1/6727*(94612+20398*35^(1/2))^(1/2)*arctan(((20+10*35^(1/2))^(1/2)+10*( 1+2*x)^(1/2))/(-20+10*35^(1/2))^(1/2))-1/6727*(-94612+20398*35^(1/2))^(1/2 )*arctanh((20+10*35^(1/2))^(1/2)*(1+2*x)^(1/2)/(5+35^(1/2)+10*x))
Result contains complex when optimal does not.
Time = 0.68 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt {1+2 x}}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {2 \left (\frac {217 \sqrt {1+2 x} (3+10 x)}{4+6 x+10 x^2}+\sqrt {217 \left (218-31 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )+\sqrt {217 \left (218+31 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )\right )}{6727} \] Input:
Integrate[Sqrt[1 + 2*x]/(2 + 3*x + 5*x^2)^2,x]
Output:
(2*((217*Sqrt[1 + 2*x]*(3 + 10*x))/(4 + 6*x + 10*x^2) + Sqrt[217*(218 - (3 1*I)*Sqrt[31])]*ArcTan[Sqrt[(-2 - I*Sqrt[31])/7]*Sqrt[1 + 2*x]] + Sqrt[217 *(218 + (31*I)*Sqrt[31])]*ArcTan[Sqrt[(I/7)*(2*I + Sqrt[31])]*Sqrt[1 + 2*x ]]))/6727
Time = 0.57 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.46, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1163, 25, 1197, 27, 1483, 27, 1142, 25, 1083, 217, 1103}
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.
| \(\displaystyle \int \frac {\sqrt {2 x+1}}{\left (5 x^2+3 x+2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1163 |
| \(\displaystyle \frac {\sqrt {2 x+1} (10 x+3)}{31 \left (5 x^2+3 x+2\right )}-\frac {1}{31} \int -\frac {10 x+7}{\sqrt {2 x+1} \left (5 x^2+3 x+2\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{31} \int \frac {10 x+7}{\sqrt {2 x+1} \left (5 x^2+3 x+2\right )}dx+\frac {\sqrt {2 x+1} (10 x+3)}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle \frac {2}{31} \int \frac {2 (5 (2 x+1)+2)}{5 (2 x+1)^2-4 (2 x+1)+7}d\sqrt {2 x+1}+\frac {\sqrt {2 x+1} (10 x+3)}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4}{31} \int \frac {5 (2 x+1)+2}{5 (2 x+1)^2-4 (2 x+1)+7}d\sqrt {2 x+1}+\frac {\sqrt {2 x+1} (10 x+3)}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {4}{31} \left (\frac {\int \frac {5 \left (2 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (2-\sqrt {35}\right ) \sqrt {2 x+1}\right )}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\int \frac {5 \left (\left (2-\sqrt {35}\right ) \sqrt {2 x+1}+2 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}\right )}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )+\frac {\sqrt {2 x+1} (10 x+3)}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4}{31} \left (\frac {5 \int \frac {2 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (2-\sqrt {35}\right ) \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \int \frac {\left (2-\sqrt {35}\right ) \sqrt {2 x+1}+2 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )+\frac {\sqrt {2 x+1} (10 x+3)}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {4}{31} \left (\frac {5 \left (\frac {\left (2+\sqrt {35}\right )^{3/2} \int \frac {1}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{\sqrt {10}}-\frac {1}{10} \left (2-\sqrt {35}\right ) \int -\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {\left (2+\sqrt {35}\right )^{3/2} \int \frac {1}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{\sqrt {10}}+\frac {1}{10} \left (2-\sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )+\frac {\sqrt {2 x+1} (10 x+3)}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {4}{31} \left (\frac {5 \left (\frac {\left (2+\sqrt {35}\right )^{3/2} \int \frac {1}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{\sqrt {10}}+\frac {1}{10} \left (2-\sqrt {35}\right ) \int \frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {\left (2+\sqrt {35}\right )^{3/2} \int \frac {1}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{\sqrt {10}}+\frac {1}{10} \left (2-\sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )+\frac {\sqrt {2 x+1} (10 x+3)}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {4}{31} \left (\frac {5 \left (\frac {1}{10} \left (2-\sqrt {35}\right ) \int \frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\sqrt {\frac {2}{5}} \left (2+\sqrt {35}\right )^{3/2} \int \frac {1}{-2 x+10 \left (2-\sqrt {35}\right )-1}d\left (10 \sqrt {2 x+1}-\sqrt {10 \left (2+\sqrt {35}\right )}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {1}{10} \left (2-\sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\sqrt {\frac {2}{5}} \left (2+\sqrt {35}\right )^{3/2} \int \frac {1}{-2 x+10 \left (2-\sqrt {35}\right )-1}d\left (10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )+\frac {\sqrt {2 x+1} (10 x+3)}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {4}{31} \left (\frac {5 \left (\frac {1}{10} \left (2-\sqrt {35}\right ) \int \frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}+\frac {\left (2+\sqrt {35}\right )^{3/2} \arctan \left (\frac {10 \sqrt {2 x+1}-\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{5 \sqrt {\sqrt {35}-2}}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {1}{10} \left (2-\sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}+\frac {\left (2+\sqrt {35}\right )^{3/2} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{5 \sqrt {\sqrt {35}-2}}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )+\frac {\sqrt {2 x+1} (10 x+3)}{31 \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {4}{31} \left (\frac {5 \left (\frac {\left (2+\sqrt {35}\right )^{3/2} \arctan \left (\frac {10 \sqrt {2 x+1}-\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{5 \sqrt {\sqrt {35}-2}}-\frac {1}{10} \left (2-\sqrt {35}\right ) \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {\left (2+\sqrt {35}\right )^{3/2} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{5 \sqrt {\sqrt {35}-2}}+\frac {1}{10} \left (2-\sqrt {35}\right ) \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )+\frac {\sqrt {2 x+1} (10 x+3)}{31 \left (5 x^2+3 x+2\right )}\) |
Input:
Int[Sqrt[1 + 2*x]/(2 + 3*x + 5*x^2)^2,x]
Output:
(Sqrt[1 + 2*x]*(3 + 10*x))/(31*(2 + 3*x + 5*x^2)) + (4*((5*(((2 + Sqrt[35] )^(3/2)*ArcTan[(-Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/(5*Sqrt[-2 + Sqrt[35]]) - ((2 - Sqrt[35])*Log[Sqrt[35] - Sqr t[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/10))/(2*Sqrt[14*(2 + Sq rt[35])]) + (5*(((2 + Sqrt[35])^(3/2)*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10 *Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/(5*Sqrt[-2 + Sqrt[35]]) + ((2 - Sqrt[35])*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2 *x)])/10))/(2*Sqrt[14*(2 + Sqrt[35])])))/31
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_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^m*(b + 2*c*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)* (b^2 - 4*a*c))), x] - Simp[1/((p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^(m - 1 )*(b*e*m + 2*c*d*(2*p + 3) + 2*c*e*(m + 2*p + 3)*x)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 0] && (LtQ[ m, 1] || (ILtQ[m + 2*p + 3, 0] && NeQ[m, 2])) && IntQuadraticQ[a, b, c, d, e, m, p, x]
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; Fr eeQ[{a, b, c, d, e, f, g}, x]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) In t[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && N eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]
Time = 3.74 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.16
| method | result | size |
| pseudoelliptic | \(\frac {\frac {20 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (\sqrt {5}-\frac {5 \sqrt {7}}{2}\right ) \left (x +\frac {3}{10}\right ) \sqrt {1+2 x}}{31}+\frac {5 \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right ) \left (\frac {\sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (218 \sqrt {5}\, \sqrt {7}-1645\right ) \left (\ln \left (\sqrt {35}+5+10 x -\sqrt {1+2 x}\, \sqrt {20+10 \sqrt {35}}\right )-\ln \left (\sqrt {35}+5+10 x +\sqrt {1+2 x}\, \sqrt {20+10 \sqrt {35}}\right )\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{2}+9610 \left (\arctan \left (\frac {\sqrt {20+10 \sqrt {35}}-10 \sqrt {1+2 x}}{\sqrt {-20+10 \sqrt {35}}}\right )-\arctan \left (\frac {\sqrt {20+10 \sqrt {35}}+10 \sqrt {1+2 x}}{\sqrt {-20+10 \sqrt {35}}}\right )\right ) \sqrt {7}\right )}{6727}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (2 \sqrt {5}-5 \sqrt {7}\right ) \left (5 x^{2}+3 x +2\right )}\) | \(244\) |
| trager | \(\frac {\sqrt {1+2 x}\, \left (10 x +3\right )}{155 x^{2}+93 x +62}-\frac {2 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+54064 \textit {\_Z}^{2}+11045\right ) \ln \left (-\frac {159725888 x \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+54064 \textit {\_Z}^{2}+11045\right )^{5}+115157560 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+54064 \textit {\_Z}^{2}+11045\right )^{3} x +45635968 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+54064 \textit {\_Z}^{2}+11045\right )^{3}+3794028 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+54064 \textit {\_Z}^{2}+11045\right )^{2} \sqrt {1+2 x}+17595750 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+54064 \textit {\_Z}^{2}+11045\right ) x +10031600 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+54064 \textit {\_Z}^{2}+11045\right )-5325335 \sqrt {1+2 x}}{868 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+54064 \textit {\_Z}^{2}+11045\right )^{2} x +125 x -124}\right )}{31}-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+54064 \textit {\_Z}^{2}+11045\right )^{2}+23653\right ) \ln \left (-\frac {22817984 \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+54064 \textit {\_Z}^{2}+11045\right )^{2}+23653\right ) \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+54064 \textit {\_Z}^{2}+11045\right )^{4} x +6472056 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+54064 \textit {\_Z}^{2}+11045\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+54064 \textit {\_Z}^{2}+11045\right )^{2}+23653\right ) x -117614868 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+54064 \textit {\_Z}^{2}+11045\right )^{2} \sqrt {1+2 x}-6519424 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+54064 \textit {\_Z}^{2}+11045\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+54064 \textit {\_Z}^{2}+11045\right )^{2}+23653\right )+7426 \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+54064 \textit {\_Z}^{2}+11045\right )^{2}+23653\right ) x -224163821 \sqrt {1+2 x}-1841648 \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+54064 \textit {\_Z}^{2}+11045\right )^{2}+23653\right )}{868 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}+54064 \textit {\_Z}^{2}+11045\right )^{2} x +311 x +124}\right )}{6727}\) | \(449\) |
| derivativedivides | \(\frac {\frac {2 \left (-5425 \sqrt {7}+2170 \sqrt {5}\right ) \sqrt {1+2 x}}{33635 \left (2 \sqrt {5}-5 \sqrt {7}\right )}-\frac {\sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (-1085 \sqrt {5}+310 \sqrt {7}\right )}{33635 \left (2 \sqrt {5}-5 \sqrt {7}\right )}}{\frac {\sqrt {5}\, \sqrt {7}}{5}+1+2 x -\frac {\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}}{5}}+\frac {-\frac {\left (-218 \sqrt {2 \sqrt {35}+4}\, \sqrt {35}+1645 \sqrt {2 \sqrt {35}+4}\right ) \ln \left (\sqrt {35}+5+10 x -\sqrt {1+2 x}\, \sqrt {20+10 \sqrt {35}}\right )}{13454}-\frac {10 \left (868 \sqrt {5}-248 \sqrt {7}+\frac {\left (-218 \sqrt {2 \sqrt {35}+4}\, \sqrt {35}+1645 \sqrt {2 \sqrt {35}+4}\right ) \sqrt {20+10 \sqrt {35}}}{10}\right ) \arctan \left (\frac {10 \sqrt {1+2 x}-\sqrt {20+10 \sqrt {35}}}{\sqrt {-20+10 \sqrt {35}}}\right )}{6727 \sqrt {-20+10 \sqrt {35}}}}{2 \sqrt {5}-5 \sqrt {7}}+\frac {\frac {2 \left (-5425 \sqrt {7}+2170 \sqrt {5}\right ) \sqrt {1+2 x}}{33635 \left (2 \sqrt {5}-5 \sqrt {7}\right )}+\frac {5 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (-1085 \sqrt {5}+310 \sqrt {7}\right )}{6727 \left (50 \sqrt {5}-125 \sqrt {7}\right )}}{\frac {\sqrt {5}\, \sqrt {7}}{5}+1+2 x +\frac {\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}}{5}}+\frac {\frac {\left (-218 \sqrt {2 \sqrt {35}+4}\, \sqrt {35}+1645 \sqrt {2 \sqrt {35}+4}\right ) \ln \left (\sqrt {35}+5+10 x +\sqrt {1+2 x}\, \sqrt {20+10 \sqrt {35}}\right )}{13454}+\frac {10 \left (-868 \sqrt {5}+248 \sqrt {7}-\frac {\left (-218 \sqrt {2 \sqrt {35}+4}\, \sqrt {35}+1645 \sqrt {2 \sqrt {35}+4}\right ) \sqrt {20+10 \sqrt {35}}}{10}\right ) \arctan \left (\frac {\sqrt {20+10 \sqrt {35}}+10 \sqrt {1+2 x}}{\sqrt {-20+10 \sqrt {35}}}\right )}{6727 \sqrt {-20+10 \sqrt {35}}}}{2 \sqrt {5}-5 \sqrt {7}}\) | \(555\) |
| default | \(\frac {\frac {2 \left (-5425 \sqrt {7}+2170 \sqrt {5}\right ) \sqrt {1+2 x}}{33635 \left (2 \sqrt {5}-5 \sqrt {7}\right )}-\frac {\sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (-1085 \sqrt {5}+310 \sqrt {7}\right )}{33635 \left (2 \sqrt {5}-5 \sqrt {7}\right )}}{\frac {\sqrt {5}\, \sqrt {7}}{5}+1+2 x -\frac {\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}}{5}}+\frac {-\frac {\left (-218 \sqrt {2 \sqrt {35}+4}\, \sqrt {35}+1645 \sqrt {2 \sqrt {35}+4}\right ) \ln \left (\sqrt {35}+5+10 x -\sqrt {1+2 x}\, \sqrt {20+10 \sqrt {35}}\right )}{13454}-\frac {10 \left (868 \sqrt {5}-248 \sqrt {7}+\frac {\left (-218 \sqrt {2 \sqrt {35}+4}\, \sqrt {35}+1645 \sqrt {2 \sqrt {35}+4}\right ) \sqrt {20+10 \sqrt {35}}}{10}\right ) \arctan \left (\frac {10 \sqrt {1+2 x}-\sqrt {20+10 \sqrt {35}}}{\sqrt {-20+10 \sqrt {35}}}\right )}{6727 \sqrt {-20+10 \sqrt {35}}}}{2 \sqrt {5}-5 \sqrt {7}}+\frac {\frac {2 \left (-5425 \sqrt {7}+2170 \sqrt {5}\right ) \sqrt {1+2 x}}{33635 \left (2 \sqrt {5}-5 \sqrt {7}\right )}+\frac {5 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (-1085 \sqrt {5}+310 \sqrt {7}\right )}{6727 \left (50 \sqrt {5}-125 \sqrt {7}\right )}}{\frac {\sqrt {5}\, \sqrt {7}}{5}+1+2 x +\frac {\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}}{5}}+\frac {\frac {\left (-218 \sqrt {2 \sqrt {35}+4}\, \sqrt {35}+1645 \sqrt {2 \sqrt {35}+4}\right ) \ln \left (\sqrt {35}+5+10 x +\sqrt {1+2 x}\, \sqrt {20+10 \sqrt {35}}\right )}{13454}+\frac {10 \left (-868 \sqrt {5}+248 \sqrt {7}-\frac {\left (-218 \sqrt {2 \sqrt {35}+4}\, \sqrt {35}+1645 \sqrt {2 \sqrt {35}+4}\right ) \sqrt {20+10 \sqrt {35}}}{10}\right ) \arctan \left (\frac {\sqrt {20+10 \sqrt {35}}+10 \sqrt {1+2 x}}{\sqrt {-20+10 \sqrt {35}}}\right )}{6727 \sqrt {-20+10 \sqrt {35}}}}{2 \sqrt {5}-5 \sqrt {7}}\) | \(555\) |
| risch | \(\frac {\sqrt {1+2 x}\, \left (10 x +3\right )}{155 x^{2}+93 x +62}+\frac {39 \ln \left (-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+\sqrt {5}\, \sqrt {7}+5+10 x \right ) \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{13454}-\frac {2 \ln \left (-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+\sqrt {5}\, \sqrt {7}+5+10 x \right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{961}+\frac {39 \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \sqrt {5}\, \left (2 \sqrt {5}\, \sqrt {7}+4\right ) \sqrt {7}}{6727 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {20 \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \left (2 \sqrt {5}\, \sqrt {7}+4\right )}{961 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {8 \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \sqrt {5}\, \sqrt {7}}{217 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {39 \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right ) \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{13454}+\frac {2 \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{961}+\frac {39 \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \sqrt {5}\, \left (2 \sqrt {5}\, \sqrt {7}+4\right ) \sqrt {7}}{6727 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {20 \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \left (2 \sqrt {5}\, \sqrt {7}+4\right )}{961 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {8 \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \sqrt {5}\, \sqrt {7}}{217 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(633\) |
Input:
int((1+2*x)^(1/2)/(5*x^2+3*x+2)^2,x,method=_RETURNVERBOSE)
Output:
5/6727*(868*(10*5^(1/2)*7^(1/2)-20)^(1/2)*(5^(1/2)-5/2*7^(1/2))*(x+3/10)*( 1+2*x)^(1/2)+(x^2+3/5*x+2/5)*(1/2*(10*5^(1/2)*7^(1/2)-20)^(1/2)*(218*5^(1/ 2)*7^(1/2)-1645)*(ln(35^(1/2)+5+10*x-(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2)) -ln(35^(1/2)+5+10*x+(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2)))*(2*5^(1/2)*7^(1 /2)+4)^(1/2)+9610*(arctan(((20+10*35^(1/2))^(1/2)-10*(1+2*x)^(1/2))/(-20+1 0*35^(1/2))^(1/2))-arctan(((20+10*35^(1/2))^(1/2)+10*(1+2*x)^(1/2))/(-20+1 0*35^(1/2))^(1/2)))*7^(1/2)))/(10*5^(1/2)*7^(1/2)-20)^(1/2)/(2*5^(1/2)-5*7 ^(1/2))/(5*x^2+3*x+2)
Time = 0.09 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.31 \[ \int \frac {\sqrt {1+2 x}}{\left (2+3 x+5 x^2\right )^2} \, dx=-\frac {2 \, {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {\frac {47}{62} \, \sqrt {\frac {5}{7}} + \frac {109}{217}} \arctan \left (\frac {14}{31} \, {\left (\sqrt {2 \, x + 1} {\left (2 \, \sqrt {\frac {5}{7}} - 5\right )} + {\left (7 \, \sqrt {\frac {5}{7}} + 2\right )} \sqrt {\frac {47}{62} \, \sqrt {\frac {5}{7}} - \frac {109}{217}}\right )} \sqrt {\frac {47}{62} \, \sqrt {\frac {5}{7}} + \frac {109}{217}}\right ) - 2 \, {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {\frac {47}{62} \, \sqrt {\frac {5}{7}} + \frac {109}{217}} \arctan \left (-\frac {14}{31} \, {\left (\sqrt {2 \, x + 1} {\left (2 \, \sqrt {\frac {5}{7}} - 5\right )} - {\left (7 \, \sqrt {\frac {5}{7}} + 2\right )} \sqrt {\frac {47}{62} \, \sqrt {\frac {5}{7}} - \frac {109}{217}}\right )} \sqrt {\frac {47}{62} \, \sqrt {\frac {5}{7}} + \frac {109}{217}}\right ) + {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {\frac {47}{62} \, \sqrt {\frac {5}{7}} - \frac {109}{217}} \log \left (14 \, \sqrt {2 \, x + 1} {\left (39 \, \sqrt {\frac {5}{7}} + 20\right )} \sqrt {\frac {47}{62} \, \sqrt {\frac {5}{7}} - \frac {109}{217}} + 310 \, x + 217 \, \sqrt {\frac {5}{7}} + 155\right ) - {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {\frac {47}{62} \, \sqrt {\frac {5}{7}} - \frac {109}{217}} \log \left (-14 \, \sqrt {2 \, x + 1} {\left (39 \, \sqrt {\frac {5}{7}} + 20\right )} \sqrt {\frac {47}{62} \, \sqrt {\frac {5}{7}} - \frac {109}{217}} + 310 \, x + 217 \, \sqrt {\frac {5}{7}} + 155\right ) - {\left (10 \, x + 3\right )} \sqrt {2 \, x + 1}}{31 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}} \] Input:
integrate((1+2*x)^(1/2)/(5*x^2+3*x+2)^2,x, algorithm="fricas")
Output:
-1/31*(2*(5*x^2 + 3*x + 2)*sqrt(47/62*sqrt(5/7) + 109/217)*arctan(14/31*(s qrt(2*x + 1)*(2*sqrt(5/7) - 5) + (7*sqrt(5/7) + 2)*sqrt(47/62*sqrt(5/7) - 109/217))*sqrt(47/62*sqrt(5/7) + 109/217)) - 2*(5*x^2 + 3*x + 2)*sqrt(47/6 2*sqrt(5/7) + 109/217)*arctan(-14/31*(sqrt(2*x + 1)*(2*sqrt(5/7) - 5) - (7 *sqrt(5/7) + 2)*sqrt(47/62*sqrt(5/7) - 109/217))*sqrt(47/62*sqrt(5/7) + 10 9/217)) + (5*x^2 + 3*x + 2)*sqrt(47/62*sqrt(5/7) - 109/217)*log(14*sqrt(2* x + 1)*(39*sqrt(5/7) + 20)*sqrt(47/62*sqrt(5/7) - 109/217) + 310*x + 217*s qrt(5/7) + 155) - (5*x^2 + 3*x + 2)*sqrt(47/62*sqrt(5/7) - 109/217)*log(-1 4*sqrt(2*x + 1)*(39*sqrt(5/7) + 20)*sqrt(47/62*sqrt(5/7) - 109/217) + 310* x + 217*sqrt(5/7) + 155) - (10*x + 3)*sqrt(2*x + 1))/(5*x^2 + 3*x + 2)
\[ \int \frac {\sqrt {1+2 x}}{\left (2+3 x+5 x^2\right )^2} \, dx=\int \frac {\sqrt {2 x + 1}}{\left (5 x^{2} + 3 x + 2\right )^{2}}\, dx \] Input:
integrate((1+2*x)**(1/2)/(5*x**2+3*x+2)**2,x)
Output:
Integral(sqrt(2*x + 1)/(5*x**2 + 3*x + 2)**2, x)
\[ \int \frac {\sqrt {1+2 x}}{\left (2+3 x+5 x^2\right )^2} \, dx=\int { \frac {\sqrt {2 \, x + 1}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{2}} \,d x } \] Input:
integrate((1+2*x)^(1/2)/(5*x^2+3*x+2)^2,x, algorithm="maxima")
Output:
integrate(sqrt(2*x + 1)/(5*x^2 + 3*x + 2)^2, x)
Leaf count of result is larger than twice the leaf count of optimal. 622 vs. \(2 (151) = 302\).
Time = 0.74 (sec) , antiderivative size = 622, normalized size of antiderivative = 2.96 \[ \int \frac {\sqrt {1+2 x}}{\left (2+3 x+5 x^2\right )^2} \, dx=\text {Too large to display} \] Input:
integrate((1+2*x)^(1/2)/(5*x^2+3*x+2)^2,x, algorithm="giac")
Output:
1/230736100*sqrt(31)*(210*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140 *sqrt(35) + 2450) - sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 2* (7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 420*(7/5)^(3/4)*sqrt(140*sqrt(35 ) + 2450)*(2*sqrt(35) - 35) + 1960*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) + 3920*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450))*arctan(5/7*(7/5)^(3 /4)*((7/5)^(1/4)*sqrt(1/35*sqrt(35) + 1/2) + sqrt(2*x + 1))/sqrt(-1/35*sqr t(35) + 1/2)) + 1/230736100*sqrt(31)*(210*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 2*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 420*(7/5)^(3/4)* sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) + 1960*sqrt(31)*(7/5)^(1/4)*sq rt(-140*sqrt(35) + 2450) + 3920*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450))*arc tan(-5/7*(7/5)^(3/4)*((7/5)^(1/4)*sqrt(1/35*sqrt(35) + 1/2) - sqrt(2*x + 1 ))/sqrt(-1/35*sqrt(35) + 1/2)) + 1/461472200*sqrt(31)*(sqrt(31)*(7/5)^(3/4 )*(140*sqrt(35) + 2450)^(3/2) + 210*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 420*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*s qrt(35) + 2450) + 2*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 1960*sqrt(3 1)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) - 3920*(7/5)^(1/4)*sqrt(-140*sqrt (35) + 2450))*log(2*(7/5)^(1/4)*sqrt(2*x + 1)*sqrt(1/35*sqrt(35) + 1/2) + 2*x + sqrt(7/5) + 1) - 1/461472200*sqrt(31)*(sqrt(31)*(7/5)^(3/4)*(140*sqr t(35) + 2450)^(3/2) + 210*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450...
Time = 5.65 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt {1+2 x}}{\left (2+3 x+5 x^2\right )^2} \, dx=-\frac {\frac {8\,\sqrt {2\,x+1}}{155}-\frac {4\,{\left (2\,x+1\right )}^{3/2}}{31}}{{\left (2\,x+1\right )}^2-\frac {8\,x}{5}+\frac {3}{5}}-\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-218-\sqrt {31}\,31{}\mathrm {i}}\,\sqrt {2\,x+1}\,128{}\mathrm {i}}{5886125\,\left (-\frac {4992}{840875}+\frac {\sqrt {31}\,256{}\mathrm {i}}{840875}\right )}-\frac {256\,\sqrt {31}\,\sqrt {217}\,\sqrt {-218-\sqrt {31}\,31{}\mathrm {i}}\,\sqrt {2\,x+1}}{182469875\,\left (-\frac {4992}{840875}+\frac {\sqrt {31}\,256{}\mathrm {i}}{840875}\right )}\right )\,\sqrt {-218-\sqrt {31}\,31{}\mathrm {i}}\,2{}\mathrm {i}}{6727}+\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-218+\sqrt {31}\,31{}\mathrm {i}}\,\sqrt {2\,x+1}\,128{}\mathrm {i}}{5886125\,\left (\frac {4992}{840875}+\frac {\sqrt {31}\,256{}\mathrm {i}}{840875}\right )}+\frac {256\,\sqrt {31}\,\sqrt {217}\,\sqrt {-218+\sqrt {31}\,31{}\mathrm {i}}\,\sqrt {2\,x+1}}{182469875\,\left (\frac {4992}{840875}+\frac {\sqrt {31}\,256{}\mathrm {i}}{840875}\right )}\right )\,\sqrt {-218+\sqrt {31}\,31{}\mathrm {i}}\,2{}\mathrm {i}}{6727} \] Input:
int((2*x + 1)^(1/2)/(3*x + 5*x^2 + 2)^2,x)
Output:
(217^(1/2)*atan((217^(1/2)*(31^(1/2)*31i - 218)^(1/2)*(2*x + 1)^(1/2)*128i )/(5886125*((31^(1/2)*256i)/840875 + 4992/840875)) + (256*31^(1/2)*217^(1/ 2)*(31^(1/2)*31i - 218)^(1/2)*(2*x + 1)^(1/2))/(182469875*((31^(1/2)*256i) /840875 + 4992/840875)))*(31^(1/2)*31i - 218)^(1/2)*2i)/6727 - (217^(1/2)* atan((217^(1/2)*(- 31^(1/2)*31i - 218)^(1/2)*(2*x + 1)^(1/2)*128i)/(588612 5*((31^(1/2)*256i)/840875 - 4992/840875)) - (256*31^(1/2)*217^(1/2)*(- 31^ (1/2)*31i - 218)^(1/2)*(2*x + 1)^(1/2))/(182469875*((31^(1/2)*256i)/840875 - 4992/840875)))*(- 31^(1/2)*31i - 218)^(1/2)*2i)/6727 - ((8*(2*x + 1)^(1 /2))/155 - (4*(2*x + 1)^(3/2))/31)/((2*x + 1)^2 - (8*x)/5 + 3/5)
Time = 0.23 (sec) , antiderivative size = 969, normalized size of antiderivative = 4.61 \[ \int \frac {\sqrt {1+2 x}}{\left (2+3 x+5 x^2\right )^2} \, dx =\text {Too large to display} \] Input:
int((1+2*x)^(1/2)/(5*x^2+3*x+2)^2,x)
Output:
( - 390*sqrt(sqrt(35) - 2)*sqrt(14)*atan((sqrt(sqrt(35) + 2)*sqrt(2) - 2*s qrt(2*x + 1)*sqrt(5))/(sqrt(sqrt(35) - 2)*sqrt(2)))*x**2 - 234*sqrt(sqrt(3 5) - 2)*sqrt(14)*atan((sqrt(sqrt(35) + 2)*sqrt(2) - 2*sqrt(2*x + 1)*sqrt(5 ))/(sqrt(sqrt(35) - 2)*sqrt(2)))*x - 156*sqrt(sqrt(35) - 2)*sqrt(14)*atan( (sqrt(sqrt(35) + 2)*sqrt(2) - 2*sqrt(2*x + 1)*sqrt(5))/(sqrt(sqrt(35) - 2) *sqrt(2))) - 280*sqrt(sqrt(35) - 2)*sqrt(10)*atan((sqrt(sqrt(35) + 2)*sqrt (2) - 2*sqrt(2*x + 1)*sqrt(5))/(sqrt(sqrt(35) - 2)*sqrt(2)))*x**2 - 168*sq rt(sqrt(35) - 2)*sqrt(10)*atan((sqrt(sqrt(35) + 2)*sqrt(2) - 2*sqrt(2*x + 1)*sqrt(5))/(sqrt(sqrt(35) - 2)*sqrt(2)))*x - 112*sqrt(sqrt(35) - 2)*sqrt( 10)*atan((sqrt(sqrt(35) + 2)*sqrt(2) - 2*sqrt(2*x + 1)*sqrt(5))/(sqrt(sqrt (35) - 2)*sqrt(2))) + 390*sqrt(sqrt(35) - 2)*sqrt(14)*atan((sqrt(sqrt(35) + 2)*sqrt(2) + 2*sqrt(2*x + 1)*sqrt(5))/(sqrt(sqrt(35) - 2)*sqrt(2)))*x**2 + 234*sqrt(sqrt(35) - 2)*sqrt(14)*atan((sqrt(sqrt(35) + 2)*sqrt(2) + 2*sq rt(2*x + 1)*sqrt(5))/(sqrt(sqrt(35) - 2)*sqrt(2)))*x + 156*sqrt(sqrt(35) - 2)*sqrt(14)*atan((sqrt(sqrt(35) + 2)*sqrt(2) + 2*sqrt(2*x + 1)*sqrt(5))/( sqrt(sqrt(35) - 2)*sqrt(2))) + 280*sqrt(sqrt(35) - 2)*sqrt(10)*atan((sqrt( sqrt(35) + 2)*sqrt(2) + 2*sqrt(2*x + 1)*sqrt(5))/(sqrt(sqrt(35) - 2)*sqrt( 2)))*x**2 + 168*sqrt(sqrt(35) - 2)*sqrt(10)*atan((sqrt(sqrt(35) + 2)*sqrt( 2) + 2*sqrt(2*x + 1)*sqrt(5))/(sqrt(sqrt(35) - 2)*sqrt(2)))*x + 112*sqrt(s qrt(35) - 2)*sqrt(10)*atan((sqrt(sqrt(35) + 2)*sqrt(2) + 2*sqrt(2*x + 1...