Integrand size = 22, antiderivative size = 270 \[ \int \frac {\sqrt {1+2 x}}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {\sqrt {1+2 x} (3+10 x)}{31 \left (2+3 x+5 x^2\right )}-\frac {1}{31} \sqrt {\frac {2}{217} \left (218+47 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )+\frac {1}{31} \sqrt {\frac {2}{217} \left (218+47 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )+\frac {1}{31} \sqrt {\frac {1}{434} \left (-218+47 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )-\frac {1}{31} \sqrt {\frac {1}{434} \left (-218+47 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right ) \]
1/31*(3+10*x)*(1+2*x)^(1/2)/(5*x^2+3*x+2)+1/13454*ln(5+10*x+35^(1/2)-(1+2* x)^(1/2)*(20+10*35^(1/2))^(1/2))*(-94612+20398*35^(1/2))^(1/2)-1/13454*ln( 5+10*x+35^(1/2)+(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(-94612+20398*35^(1/ 2))^(1/2)-1/6727*arctan((-10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10 *35^(1/2))^(1/2))*(94612+20398*35^(1/2))^(1/2)+1/6727*arctan((10*(1+2*x)^( 1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(94612+20398*35^(1/2 ))^(1/2)
Result contains complex when optimal does not.
Time = 0.58 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.48 \[ \int \frac {\sqrt {1+2 x}}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {2 \left (\frac {217 \sqrt {1+2 x} (3+10 x)}{4+6 x+10 x^2}+\sqrt {217 \left (218-31 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )+\sqrt {217 \left (218+31 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )\right )}{6727} \]
(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 = 0.59 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.14, 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 |
| \(\displaystyle \frac {4}{31} \left (\frac {5 \left (\frac {\left (2+\sqrt {35}\right )^{3/2} \arctan \left (\frac {10 \sqrt {2 x+1}-\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{5 \sqrt {\sqrt {35}-2}}-\frac {1}{10} \left (2-\sqrt {35}\right ) \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {\left (2+\sqrt {35}\right )^{3/2} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{5 \sqrt {\sqrt {35}-2}}+\frac {1}{10} \left (2-\sqrt {35}\right ) \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )+\frac {\sqrt {2 x+1} (10 x+3)}{31 \left (5 x^2+3 x+2\right )}\) |
(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
3.24.19.3.1 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[(-(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 = 1.09 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.90
| method | result | size |
| pseudoelliptic | \(-\frac {50 \left (-\frac {14 \left (\sqrt {5}-\frac {5 \sqrt {7}}{2}\right ) \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (x +\frac {3}{10}\right ) \sqrt {1+2 x}}{155}+\left (-\frac {\sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (218 \sqrt {5}\, \sqrt {7}-1645\right ) \left (\ln \left (5+10 x +\sqrt {35}-\sqrt {1+2 x}\, \sqrt {20+10 \sqrt {35}}\right )-\ln \left (5+10 x +\sqrt {35}+\sqrt {1+2 x}\, \sqrt {20+10 \sqrt {35}}\right )\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{19220}+\sqrt {7}\, \left (\arctan \left (\frac {10 \sqrt {1+2 x}+\sqrt {20+10 \sqrt {35}}}{\sqrt {-20+10 \sqrt {35}}}\right )+\arctan \left (\frac {-\sqrt {20+10 \sqrt {35}}+10 \sqrt {1+2 x}}{\sqrt {-20+10 \sqrt {35}}}\right )\right )\right ) \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right )\right )}{7 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (2 \sqrt {5}-5 \sqrt {7}\right ) \left (5 x^{2}+3 x +2\right )}\) | \(243\) |
| trager | \(\frac {\left (3+10 x \right ) \sqrt {1+2 x}}{155 x^{2}+93 x +62}+\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}-\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}\) | \(448\) |
| derivativedivides | \(\frac {\frac {2 \left (-5425 \sqrt {7}+2170 \sqrt {5}\right ) \sqrt {1+2 x}}{33635 \left (2 \sqrt {5}-5 \sqrt {7}\right )}-\frac {\sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (-1085 \sqrt {5}+310 \sqrt {7}\right )}{33635 \left (2 \sqrt {5}-5 \sqrt {7}\right )}}{\frac {\sqrt {5}\, \sqrt {7}}{5}-\frac {\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}}{5}+1+2 x}+\frac {-\frac {\left (-218 \sqrt {2 \sqrt {35}+4}\, \sqrt {35}+1645 \sqrt {2 \sqrt {35}+4}\right ) \ln \left (5+10 x +\sqrt {35}-\sqrt {1+2 x}\, \sqrt {20+10 \sqrt {35}}\right )}{13454}-\frac {10 \left (868 \sqrt {5}-248 \sqrt {7}+\frac {\left (-218 \sqrt {2 \sqrt {35}+4}\, \sqrt {35}+1645 \sqrt {2 \sqrt {35}+4}\right ) \sqrt {20+10 \sqrt {35}}}{10}\right ) \arctan \left (\frac {-\sqrt {20+10 \sqrt {35}}+10 \sqrt {1+2 x}}{\sqrt {-20+10 \sqrt {35}}}\right )}{6727 \sqrt {-20+10 \sqrt {35}}}}{2 \sqrt {5}-5 \sqrt {7}}+\frac {\frac {2 \left (-5425 \sqrt {7}+2170 \sqrt {5}\right ) \sqrt {1+2 x}}{33635 \left (2 \sqrt {5}-5 \sqrt {7}\right )}+\frac {5 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (-1085 \sqrt {5}+310 \sqrt {7}\right )}{6727 \left (50 \sqrt {5}-125 \sqrt {7}\right )}}{\frac {\sqrt {5}\, \sqrt {7}}{5}+\frac {\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}}{5}+1+2 x}+\frac {\frac {\left (-218 \sqrt {2 \sqrt {35}+4}\, \sqrt {35}+1645 \sqrt {2 \sqrt {35}+4}\right ) \ln \left (5+10 x +\sqrt {35}+\sqrt {1+2 x}\, \sqrt {20+10 \sqrt {35}}\right )}{13454}+\frac {10 \left (-868 \sqrt {5}+248 \sqrt {7}-\frac {\left (-218 \sqrt {2 \sqrt {35}+4}\, \sqrt {35}+1645 \sqrt {2 \sqrt {35}+4}\right ) \sqrt {20+10 \sqrt {35}}}{10}\right ) \arctan \left (\frac {10 \sqrt {1+2 x}+\sqrt {20+10 \sqrt {35}}}{\sqrt {-20+10 \sqrt {35}}}\right )}{6727 \sqrt {-20+10 \sqrt {35}}}}{2 \sqrt {5}-5 \sqrt {7}}\) | \(555\) |
| default | \(\frac {\frac {2 \left (-5425 \sqrt {7}+2170 \sqrt {5}\right ) \sqrt {1+2 x}}{33635 \left (2 \sqrt {5}-5 \sqrt {7}\right )}-\frac {\sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (-1085 \sqrt {5}+310 \sqrt {7}\right )}{33635 \left (2 \sqrt {5}-5 \sqrt {7}\right )}}{\frac {\sqrt {5}\, \sqrt {7}}{5}-\frac {\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}}{5}+1+2 x}+\frac {\frac {\left (218 \sqrt {2 \sqrt {35}+4}\, \sqrt {35}-1645 \sqrt {2 \sqrt {35}+4}\right ) \ln \left (5+10 x +\sqrt {35}-\sqrt {1+2 x}\, \sqrt {20+10 \sqrt {35}}\right )}{13454}+\frac {10 \left (-868 \sqrt {5}+248 \sqrt {7}+\frac {\left (218 \sqrt {2 \sqrt {35}+4}\, \sqrt {35}-1645 \sqrt {2 \sqrt {35}+4}\right ) \sqrt {20+10 \sqrt {35}}}{10}\right ) \arctan \left (\frac {-\sqrt {20+10 \sqrt {35}}+10 \sqrt {1+2 x}}{\sqrt {-20+10 \sqrt {35}}}\right )}{6727 \sqrt {-20+10 \sqrt {35}}}}{2 \sqrt {5}-5 \sqrt {7}}+\frac {\frac {2 \left (-5425 \sqrt {7}+2170 \sqrt {5}\right ) \sqrt {1+2 x}}{33635 \left (2 \sqrt {5}-5 \sqrt {7}\right )}+\frac {5 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (-1085 \sqrt {5}+310 \sqrt {7}\right )}{6727 \left (50 \sqrt {5}-125 \sqrt {7}\right )}}{\frac {\sqrt {5}\, \sqrt {7}}{5}+\frac {\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}}{5}+1+2 x}+\frac {\frac {\left (-218 \sqrt {2 \sqrt {35}+4}\, \sqrt {35}+1645 \sqrt {2 \sqrt {35}+4}\right ) \ln \left (5+10 x +\sqrt {35}+\sqrt {1+2 x}\, \sqrt {20+10 \sqrt {35}}\right )}{13454}+\frac {10 \left (-868 \sqrt {5}+248 \sqrt {7}-\frac {\left (-218 \sqrt {2 \sqrt {35}+4}\, \sqrt {35}+1645 \sqrt {2 \sqrt {35}+4}\right ) \sqrt {20+10 \sqrt {35}}}{10}\right ) \arctan \left (\frac {10 \sqrt {1+2 x}+\sqrt {20+10 \sqrt {35}}}{\sqrt {-20+10 \sqrt {35}}}\right )}{6727 \sqrt {-20+10 \sqrt {35}}}}{2 \sqrt {5}-5 \sqrt {7}}\) | \(555\) |
| risch | \(\frac {\left (3+10 x \right ) \sqrt {1+2 x}}{155 x^{2}+93 x +62}+\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}}-\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}}\) | \(633\) |
-50/7*(-14/155*(5^(1/2)-5/2*7^(1/2))*(10*5^(1/2)*7^(1/2)-20)^(1/2)*(x+3/10 )*(1+2*x)^(1/2)+(-1/19220*(10*5^(1/2)*7^(1/2)-20)^(1/2)*(218*5^(1/2)*7^(1/ 2)-1645)*(ln(5+10*x+35^(1/2)-(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))-ln(5+10 *x+35^(1/2)+(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2)))*(2*5^(1/2)*7^(1/2)+4)^( 1/2)+7^(1/2)*(arctan((10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*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))))*(x^2+3/5*x+2/5))/(10*5^(1/2)*7^(1/2)-20)^(1/2)/(2*5^(1/2) -5*7^(1/2))/(5*x^2+3*x+2)
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {1+2 x}}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {\sqrt {217} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {124 i \, \sqrt {31} - 872} \log \left (\sqrt {217} \sqrt {124 i \, \sqrt {31} - 872} {\left (39 i \, \sqrt {31} + 62\right )} + 101990 \, \sqrt {2 \, x + 1}\right ) - \sqrt {217} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {124 i \, \sqrt {31} - 872} \log \left (\sqrt {217} \sqrt {124 i \, \sqrt {31} - 872} {\left (-39 i \, \sqrt {31} - 62\right )} + 101990 \, \sqrt {2 \, x + 1}\right ) - \sqrt {217} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {-124 i \, \sqrt {31} - 872} \log \left (\sqrt {217} {\left (39 i \, \sqrt {31} - 62\right )} \sqrt {-124 i \, \sqrt {31} - 872} + 101990 \, \sqrt {2 \, x + 1}\right ) + \sqrt {217} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {-124 i \, \sqrt {31} - 872} \log \left (\sqrt {217} {\left (-39 i \, \sqrt {31} + 62\right )} \sqrt {-124 i \, \sqrt {31} - 872} + 101990 \, \sqrt {2 \, x + 1}\right ) + 434 \, {\left (10 \, x + 3\right )} \sqrt {2 \, x + 1}}{13454 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}} \]
1/13454*(sqrt(217)*(5*x^2 + 3*x + 2)*sqrt(124*I*sqrt(31) - 872)*log(sqrt(2 17)*sqrt(124*I*sqrt(31) - 872)*(39*I*sqrt(31) + 62) + 101990*sqrt(2*x + 1) ) - sqrt(217)*(5*x^2 + 3*x + 2)*sqrt(124*I*sqrt(31) - 872)*log(sqrt(217)*s qrt(124*I*sqrt(31) - 872)*(-39*I*sqrt(31) - 62) + 101990*sqrt(2*x + 1)) - sqrt(217)*(5*x^2 + 3*x + 2)*sqrt(-124*I*sqrt(31) - 872)*log(sqrt(217)*(39* I*sqrt(31) - 62)*sqrt(-124*I*sqrt(31) - 872) + 101990*sqrt(2*x + 1)) + sqr t(217)*(5*x^2 + 3*x + 2)*sqrt(-124*I*sqrt(31) - 872)*log(sqrt(217)*(-39*I* sqrt(31) + 62)*sqrt(-124*I*sqrt(31) - 872) + 101990*sqrt(2*x + 1)) + 434*( 10*x + 3)*sqrt(2*x + 1))/(5*x^2 + 3*x + 2)
\[ \int \frac {\sqrt {1+2 x}}{\left (2+3 x+5 x^2\right )^2} \, dx=\int \frac {\sqrt {2 x + 1}}{\left (5 x^{2} + 3 x + 2\right )^{2}}\, dx \]
\[ \int \frac {\sqrt {1+2 x}}{\left (2+3 x+5 x^2\right )^2} \, dx=\int { \frac {\sqrt {2 \, x + 1}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{2}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 622 vs. \(2 (187) = 374\).
Time = 0.67 (sec) , antiderivative size = 622, normalized size of antiderivative = 2.30 \[ \int \frac {\sqrt {1+2 x}}{\left (2+3 x+5 x^2\right )^2} \, dx=\text {Too large to display} \]
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 = 9.95 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {1+2 x}}{\left (2+3 x+5 x^2\right )^2} \, dx=-\frac {\frac {8\,\sqrt {2\,x+1}}{155}-\frac {4\,{\left (2\,x+1\right )}^{3/2}}{31}}{{\left (2\,x+1\right )}^2-\frac {8\,x}{5}+\frac {3}{5}}-\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-218-\sqrt {31}\,31{}\mathrm {i}}\,\sqrt {2\,x+1}\,128{}\mathrm {i}}{5886125\,\left (-\frac {4992}{840875}+\frac {\sqrt {31}\,256{}\mathrm {i}}{840875}\right )}-\frac {256\,\sqrt {31}\,\sqrt {217}\,\sqrt {-218-\sqrt {31}\,31{}\mathrm {i}}\,\sqrt {2\,x+1}}{182469875\,\left (-\frac {4992}{840875}+\frac {\sqrt {31}\,256{}\mathrm {i}}{840875}\right )}\right )\,\sqrt {-218-\sqrt {31}\,31{}\mathrm {i}}\,2{}\mathrm {i}}{6727}+\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-218+\sqrt {31}\,31{}\mathrm {i}}\,\sqrt {2\,x+1}\,128{}\mathrm {i}}{5886125\,\left (\frac {4992}{840875}+\frac {\sqrt {31}\,256{}\mathrm {i}}{840875}\right )}+\frac {256\,\sqrt {31}\,\sqrt {217}\,\sqrt {-218+\sqrt {31}\,31{}\mathrm {i}}\,\sqrt {2\,x+1}}{182469875\,\left (\frac {4992}{840875}+\frac {\sqrt {31}\,256{}\mathrm {i}}{840875}\right )}\right )\,\sqrt {-218+\sqrt {31}\,31{}\mathrm {i}}\,2{}\mathrm {i}}{6727} \]
(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)