Integrand size = 22, antiderivative size = 283 \[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^2} \, dx=-\frac {604}{1519 \sqrt {1+2 x}}+\frac {37+20 x}{217 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}+\frac {\sqrt {\frac {2}{217} \left (-5682718+968975 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )}{1519}-\frac {\sqrt {\frac {2}{217} \left (-5682718+968975 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )}{1519}-\frac {\sqrt {\frac {1}{434} \left (5682718+968975 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1519}+\frac {\sqrt {\frac {1}{434} \left (5682718+968975 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1519} \]
-604/1519/(1+2*x)^(1/2)+1/217*(37+20*x)/(5*x^2+3*x+2)/(1+2*x)^(1/2)+1/3296 23*arctan((-10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/ 2))*(-2466299612+420535150*35^(1/2))^(1/2)-1/329623*arctan((10*(1+2*x)^(1/ 2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(-2466299612+420535150 *35^(1/2))^(1/2)-1/659246*ln(5+10*x+35^(1/2)-(1+2*x)^(1/2)*(20+10*35^(1/2) )^(1/2))*(2466299612+420535150*35^(1/2))^(1/2)+1/659246*ln(5+10*x+35^(1/2) +(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(2466299612+420535150*35^(1/2))^(1/ 2)
Result contains complex when optimal does not.
Time = 1.20 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.49 \[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^2} \, dx=\frac {2 \left (-\frac {217 \left (949+1672 x+3020 x^2\right )}{2 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}-\sqrt {217 \left (-5682718-135439 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )-\sqrt {217 \left (-5682718+135439 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )\right )}{329623} \]
(2*((-217*(949 + 1672*x + 3020*x^2))/(2*Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)) - Sqrt[217*(-5682718 - (135439*I)*Sqrt[31])]*ArcTan[Sqrt[(-2 - I*Sqrt[31])/ 7]*Sqrt[1 + 2*x]] - Sqrt[217*(-5682718 + (135439*I)*Sqrt[31])]*ArcTan[Sqrt [(I/7)*(2*I + Sqrt[31])]*Sqrt[1 + 2*x]]))/329623
Time = 0.63 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.18, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1165, 1198, 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.
| \(\displaystyle \int \frac {1}{(2 x+1)^{3/2} \left (5 x^2+3 x+2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1165 |
| \(\displaystyle \frac {1}{217} \int \frac {60 x+181}{(2 x+1)^{3/2} \left (5 x^2+3 x+2\right )}dx+\frac {20 x+37}{217 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 1198 |
| \(\displaystyle \frac {1}{217} \left (\frac {1}{7} \int \frac {59-1510 x}{\sqrt {2 x+1} \left (5 x^2+3 x+2\right )}dx-\frac {604}{7 \sqrt {2 x+1}}\right )+\frac {20 x+37}{217 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle \frac {1}{217} \left (\frac {2}{7} \int \frac {2 (814-755 (2 x+1))}{5 (2 x+1)^2-4 (2 x+1)+7}d\sqrt {2 x+1}-\frac {604}{7 \sqrt {2 x+1}}\right )+\frac {20 x+37}{217 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{217} \left (\frac {4}{7} \int \frac {814-755 (2 x+1)}{5 (2 x+1)^2-4 (2 x+1)+7}d\sqrt {2 x+1}-\frac {604}{7 \sqrt {2 x+1}}\right )+\frac {20 x+37}{217 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {1}{217} \left (\frac {4}{7} \left (\frac {\int \frac {5 \left (814 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (814+151 \sqrt {35}\right ) \sqrt {2 x+1}\right )}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\int \frac {5 \left (\left (814+151 \sqrt {35}\right ) \sqrt {2 x+1}+814 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}\right )}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {604}{7 \sqrt {2 x+1}}\right )+\frac {20 x+37}{217 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{217} \left (\frac {4}{7} \left (\frac {5 \int \frac {814 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (814+151 \sqrt {35}\right ) \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \int \frac {\left (814+151 \sqrt {35}\right ) \sqrt {2 x+1}+814 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {604}{7 \sqrt {2 x+1}}\right )+\frac {20 x+37}{217 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{217} \left (\frac {4}{7} \left (\frac {5 \left (-\frac {1}{10} \sqrt {9689750 \sqrt {35}-56827180} \int \frac {1}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\frac {1}{10} \left (814+151 \sqrt {35}\right ) \int -\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {1}{10} \left (814+151 \sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\frac {1}{10} \sqrt {9689750 \sqrt {35}-56827180} \int \frac {1}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {604}{7 \sqrt {2 x+1}}\right )+\frac {20 x+37}{217 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{217} \left (\frac {4}{7} \left (\frac {5 \left (\frac {1}{10} \left (814+151 \sqrt {35}\right ) \int \frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\frac {1}{10} \sqrt {9689750 \sqrt {35}-56827180} \int \frac {1}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {1}{10} \left (814+151 \sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\frac {1}{10} \sqrt {9689750 \sqrt {35}-56827180} \int \frac {1}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {604}{7 \sqrt {2 x+1}}\right )+\frac {20 x+37}{217 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{217} \left (\frac {4}{7} \left (\frac {5 \left (\frac {1}{5} \sqrt {9689750 \sqrt {35}-56827180} \int \frac {1}{-2 x+10 \left (2-\sqrt {35}\right )-1}d\left (10 \sqrt {2 x+1}-\sqrt {10 \left (2+\sqrt {35}\right )}\right )+\frac {1}{10} \left (814+151 \sqrt {35}\right ) \int \frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {1}{5} \sqrt {9689750 \sqrt {35}-56827180} \int \frac {1}{-2 x+10 \left (2-\sqrt {35}\right )-1}d\left (10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}\right )+\frac {1}{10} \left (814+151 \sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {604}{7 \sqrt {2 x+1}}\right )+\frac {20 x+37}{217 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{217} \left (\frac {4}{7} \left (\frac {5 \left (\frac {1}{10} \left (814+151 \sqrt {35}\right ) \int \frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\frac {1}{5} \sqrt {\frac {9689750 \sqrt {35}-56827180}{10 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}-\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {1}{10} \left (814+151 \sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\frac {1}{5} \sqrt {\frac {9689750 \sqrt {35}-56827180}{10 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {604}{7 \sqrt {2 x+1}}\right )+\frac {20 x+37}{217 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{217} \left (\frac {4}{7} \left (\frac {5 \left (-\frac {1}{5} \sqrt {\frac {9689750 \sqrt {35}-56827180}{10 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}-\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )-\frac {1}{10} \left (814+151 \sqrt {35}\right ) \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {1}{10} \left (814+151 \sqrt {35}\right ) \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )-\frac {1}{5} \sqrt {\frac {9689750 \sqrt {35}-56827180}{10 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {604}{7 \sqrt {2 x+1}}\right )+\frac {20 x+37}{217 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )}\) |
(37 + 20*x)/(217*Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)) + (-604/(7*Sqrt[1 + 2*x] ) + (4*((5*(-1/5*(Sqrt[(-56827180 + 9689750*Sqrt[35])/(10*(-2 + Sqrt[35])) ]*ArcTan[(-Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[ 35])]]) - ((814 + 151*Sqrt[35])*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqr t[1 + 2*x] + 5*(1 + 2*x)])/10))/(2*Sqrt[14*(2 + Sqrt[35])]) + (5*(-1/5*(Sq rt[(-56827180 + 9689750*Sqrt[35])/(10*(-2 + Sqrt[35]))]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]]) + ((814 + 151 *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])])))/7)/217
3.24.21.3.1 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 + 1)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e) *x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^ 2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m + 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p, -1] && 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_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*f - d*g)*((d + e*x)^(m + 1)/((m + 1)*(c *d^2 - b*d*e + a*e^2))), x] + Simp[1/(c*d^2 - b*d*e + a*e^2) Int[(d + e*x )^(m + 1)*(Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x]/(a + b*x + c*x^ 2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && FractionQ[m] && LtQ[m, -1 ]
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 = 0.70 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.19
| method | result | size |
| pseudoelliptic | \(\frac {-17920 \left (\sqrt {5}+\frac {3657 \sqrt {7}}{3584}\right ) \sqrt {1+2 x}\, \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )+17920 \left (\sqrt {5}+\frac {3657 \sqrt {7}}{3584}\right ) \sqrt {1+2 x}\, \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )+504680 \left (\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )-\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )\right ) \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right ) \left (\sqrt {5}\, \sqrt {7}-\frac {5285}{814}\right ) \sqrt {1+2 x}-1310680 \left (x^{2}+\frac {418}{755} x +\frac {949}{3020}\right ) \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \sqrt {1+2 x}\, \left (3296230 x^{2}+1977738 x +1318492\right )}\) | \(338\) |
| derivativedivides | \(-\frac {16}{49 \sqrt {1+2 x}}-\frac {16 \left (\frac {27 \left (1+2 x \right )^{\frac {3}{2}}}{124}-\frac {89 \sqrt {1+2 x}}{310}\right )}{49 \left (\left (1+2 x \right )^{2}+\frac {3}{5}-\frac {8 x}{5}\right )}-\frac {\left (17920 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+18285 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{3296230}-\frac {2 \left (-50468 \sqrt {5}\, \sqrt {7}+\frac {\left (17920 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+18285 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{329623 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {\left (-17920 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-18285 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{3296230}-\frac {2 \left (-50468 \sqrt {5}\, \sqrt {7}-\frac {\left (-17920 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-18285 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{329623 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(424\) |
| default | \(-\frac {16}{49 \sqrt {1+2 x}}-\frac {16 \left (\frac {27 \left (1+2 x \right )^{\frac {3}{2}}}{124}-\frac {89 \sqrt {1+2 x}}{310}\right )}{49 \left (\left (1+2 x \right )^{2}+\frac {3}{5}-\frac {8 x}{5}\right )}-\frac {\left (17920 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+18285 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{3296230}-\frac {2 \left (-50468 \sqrt {5}\, \sqrt {7}+\frac {\left (17920 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+18285 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{329623 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {\left (-17920 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-18285 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{3296230}-\frac {2 \left (-50468 \sqrt {5}\, \sqrt {7}-\frac {\left (-17920 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-18285 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{329623 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(424\) |
| trager | \(-\frac {\left (3020 x^{2}+1672 x +949\right ) \sqrt {1+2 x}}{1519 \left (10 x^{3}+11 x^{2}+7 x +2\right )}-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}+4694562753125\right )^{2}-616574903\right ) \ln \left (-\frac {160586944 \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}+4694562753125\right )^{2}-616574903\right ) \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}+4694562753125\right )^{4} x -1718943720888 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}+4694562753125\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}+4694562753125\right )^{2}-616574903\right ) x -200458388096 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}+4694562753125\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}+4694562753125\right )^{2}-616574903\right )+2118795198727620 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}+4694562753125\right )^{2} \sqrt {1+2 x}+3759951379038470 \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}+4694562753125\right )^{2}-616574903\right ) x +614399162725040 \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}+4694562753125\right )^{2}-616574903\right )+1446596030559246385 \sqrt {1+2 x}}{868 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}+4694562753125\right )^{2} x -5276401 x +541756}\right )}{329623}-\frac {2 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}+4694562753125\right ) \ln \left (-\frac {1124108608 x \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}+4694562753125\right )^{5}-17405146122616 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}+4694562753125\right )^{3} x +1403208716672 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}+4694562753125\right )^{3}-68348232217020 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}+4694562753125\right )^{2} \sqrt {1+2 x}+61493196747600690 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}+4694562753125\right ) x -14072568607442864 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}+4694562753125\right )+941603856948545875 \sqrt {1+2 x}}{868 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}+4694562753125\right )^{2} x -6089035 x -541756}\right )}{1519}\) | \(459\) |
| risch | \(-\frac {3020 x^{2}+1672 x +949}{1519 \left (5 x^{2}+3 x +2\right ) \sqrt {1+2 x}}-\frac {256 \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}}{47089}-\frac {3657 \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}}{659246}-\frac {2560 \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 )}{47089 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {3657 \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}}{329623 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {3256 \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}}{10633 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {256 \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}}{47089}+\frac {3657 \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}}{659246}-\frac {2560 \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 )}{47089 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {3657 \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}}{329623 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {3256 \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}}{10633 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(638\) |
504680/(10*5^(1/2)*7^(1/2)-20)^(1/2)*(-448/12617*(5^(1/2)+3657/3584*7^(1/2 ))*(1+2*x)^(1/2)*(x^2+3/5*x+2/5)*(2*5^(1/2)*7^(1/2)+4)^(1/2)*(10*5^(1/2)*7 ^(1/2)-20)^(1/2)*ln(5^(1/2)*7^(1/2)-(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(1 +2*x)^(1/2)+5+10*x)+448/12617*(5^(1/2)+3657/3584*7^(1/2))*(1+2*x)^(1/2)*(x ^2+3/5*x+2/5)*(2*5^(1/2)*7^(1/2)+4)^(1/2)*(10*5^(1/2)*7^(1/2)-20)^(1/2)*ln (5^(1/2)*7^(1/2)+(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(1+2*x)^(1/2)+5+10*x) +(arctan((5^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)+10*(1+2*x)^(1/2))/(10*5^(1/2 )*7^(1/2)-20)^(1/2))-arctan((5^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)-10*(1+2*x )^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2)))*(x^2+3/5*x+2/5)*(5^(1/2)*7^(1/2)- 5285/814)*(1+2*x)^(1/2)-1057/407*(x^2+418/755*x+949/3020)*(10*5^(1/2)*7^(1 /2)-20)^(1/2))/(1+2*x)^(1/2)/(3296230*x^2+1977738*x+1318492)
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.98 \[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^2} \, dx=-\frac {\sqrt {217} {\left (10 \, x^{3} + 11 \, x^{2} + 7 \, x + 2\right )} \sqrt {541756 i \, \sqrt {31} + 22730872} \log \left (\sqrt {217} \sqrt {541756 i \, \sqrt {31} + 22730872} {\left (3657 i \, \sqrt {31} - 25234\right )} + 2102675750 \, \sqrt {2 \, x + 1}\right ) - \sqrt {217} {\left (10 \, x^{3} + 11 \, x^{2} + 7 \, x + 2\right )} \sqrt {541756 i \, \sqrt {31} + 22730872} \log \left (\sqrt {217} \sqrt {541756 i \, \sqrt {31} + 22730872} {\left (-3657 i \, \sqrt {31} + 25234\right )} + 2102675750 \, \sqrt {2 \, x + 1}\right ) - \sqrt {217} {\left (10 \, x^{3} + 11 \, x^{2} + 7 \, x + 2\right )} \sqrt {-541756 i \, \sqrt {31} + 22730872} \log \left (\sqrt {217} {\left (3657 i \, \sqrt {31} + 25234\right )} \sqrt {-541756 i \, \sqrt {31} + 22730872} + 2102675750 \, \sqrt {2 \, x + 1}\right ) + \sqrt {217} {\left (10 \, x^{3} + 11 \, x^{2} + 7 \, x + 2\right )} \sqrt {-541756 i \, \sqrt {31} + 22730872} \log \left (\sqrt {217} {\left (-3657 i \, \sqrt {31} - 25234\right )} \sqrt {-541756 i \, \sqrt {31} + 22730872} + 2102675750 \, \sqrt {2 \, x + 1}\right ) + 434 \, {\left (3020 \, x^{2} + 1672 \, x + 949\right )} \sqrt {2 \, x + 1}}{659246 \, {\left (10 \, x^{3} + 11 \, x^{2} + 7 \, x + 2\right )}} \]
-1/659246*(sqrt(217)*(10*x^3 + 11*x^2 + 7*x + 2)*sqrt(541756*I*sqrt(31) + 22730872)*log(sqrt(217)*sqrt(541756*I*sqrt(31) + 22730872)*(3657*I*sqrt(31 ) - 25234) + 2102675750*sqrt(2*x + 1)) - sqrt(217)*(10*x^3 + 11*x^2 + 7*x + 2)*sqrt(541756*I*sqrt(31) + 22730872)*log(sqrt(217)*sqrt(541756*I*sqrt(3 1) + 22730872)*(-3657*I*sqrt(31) + 25234) + 2102675750*sqrt(2*x + 1)) - sq rt(217)*(10*x^3 + 11*x^2 + 7*x + 2)*sqrt(-541756*I*sqrt(31) + 22730872)*lo g(sqrt(217)*(3657*I*sqrt(31) + 25234)*sqrt(-541756*I*sqrt(31) + 22730872) + 2102675750*sqrt(2*x + 1)) + sqrt(217)*(10*x^3 + 11*x^2 + 7*x + 2)*sqrt(- 541756*I*sqrt(31) + 22730872)*log(sqrt(217)*(-3657*I*sqrt(31) - 25234)*sqr t(-541756*I*sqrt(31) + 22730872) + 2102675750*sqrt(2*x + 1)) + 434*(3020*x ^2 + 1672*x + 949)*sqrt(2*x + 1))/(10*x^3 + 11*x^2 + 7*x + 2)
\[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^2} \, dx=\int \frac {1}{\left (2 x + 1\right )^{\frac {3}{2}} \left (5 x^{2} + 3 x + 2\right )^{2}}\, dx \]
\[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^2} \, dx=\int { \frac {1}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{2} {\left (2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 633 vs. \(2 (196) = 392\).
Time = 0.54 (sec) , antiderivative size = 633, normalized size of antiderivative = 2.24 \[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^2} \, dx=\text {Too large to display} \]
-1/11306068900*sqrt(31)*(31710*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt (-140*sqrt(35) + 2450) - 151*sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^( 3/2) + 302*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 63420*(7/5)^(3/4)*sqr t(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 797720*sqrt(31)*(7/5)^(1/4)*sqr t(-140*sqrt(35) + 2450) - 1595440*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450))*a rctan(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)) - 1/11306068900*sqrt(31)*(31710*sqrt(31)*( 7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - 151*sqrt(31)*(7/ 5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 302*(7/5)^(3/4)*(140*sqrt(35) + 24 50)^(3/2) + 63420*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 797720*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) - 1595440*(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*sqrt(35) + 1/2)) - 1/2261 2137800*sqrt(31)*(151*sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 3 1710*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 63 420*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 302*(7/5)^( 3/4)*(-140*sqrt(35) + 2450)^(3/2) - 797720*sqrt(31)*(7/5)^(1/4)*sqrt(140*s qrt(35) + 2450) + 1595440*(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) + 1/22612137800*sqrt(31)*(151*sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)...
Time = 9.98 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.77 \[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^2} \, dx=-\frac {\frac {604\,{\left (2\,x+1\right )}^2}{1519}-\frac {5392\,x}{7595}+\frac {776}{7595}}{\frac {7\,\sqrt {2\,x+1}}{5}-\frac {4\,{\left (2\,x+1\right )}^{3/2}}{5}+{\left (2\,x+1\right )}^{5/2}}-\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {5682718-\sqrt {31}\,135439{}\mathrm {i}}\,\sqrt {2\,x+1}\,559232{}\mathrm {i}}{692496720125\,\left (\frac {2045111424}{98928102875}+\frac {\sqrt {31}\,455214848{}\mathrm {i}}{98928102875}\right )}-\frac {1118464\,\sqrt {31}\,\sqrt {217}\,\sqrt {5682718-\sqrt {31}\,135439{}\mathrm {i}}\,\sqrt {2\,x+1}}{21467398323875\,\left (\frac {2045111424}{98928102875}+\frac {\sqrt {31}\,455214848{}\mathrm {i}}{98928102875}\right )}\right )\,\sqrt {5682718-\sqrt {31}\,135439{}\mathrm {i}}\,2{}\mathrm {i}}{329623}+\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {5682718+\sqrt {31}\,135439{}\mathrm {i}}\,\sqrt {2\,x+1}\,559232{}\mathrm {i}}{692496720125\,\left (-\frac {2045111424}{98928102875}+\frac {\sqrt {31}\,455214848{}\mathrm {i}}{98928102875}\right )}+\frac {1118464\,\sqrt {31}\,\sqrt {217}\,\sqrt {5682718+\sqrt {31}\,135439{}\mathrm {i}}\,\sqrt {2\,x+1}}{21467398323875\,\left (-\frac {2045111424}{98928102875}+\frac {\sqrt {31}\,455214848{}\mathrm {i}}{98928102875}\right )}\right )\,\sqrt {5682718+\sqrt {31}\,135439{}\mathrm {i}}\,2{}\mathrm {i}}{329623} \]
(217^(1/2)*atan((217^(1/2)*(31^(1/2)*135439i + 5682718)^(1/2)*(2*x + 1)^(1 /2)*559232i)/(692496720125*((31^(1/2)*455214848i)/98928102875 - 2045111424 /98928102875)) + (1118464*31^(1/2)*217^(1/2)*(31^(1/2)*135439i + 5682718)^ (1/2)*(2*x + 1)^(1/2))/(21467398323875*((31^(1/2)*455214848i)/98928102875 - 2045111424/98928102875)))*(31^(1/2)*135439i + 5682718)^(1/2)*2i)/329623 - (217^(1/2)*atan((217^(1/2)*(5682718 - 31^(1/2)*135439i)^(1/2)*(2*x + 1)^ (1/2)*559232i)/(692496720125*((31^(1/2)*455214848i)/98928102875 + 20451114 24/98928102875)) - (1118464*31^(1/2)*217^(1/2)*(5682718 - 31^(1/2)*135439i )^(1/2)*(2*x + 1)^(1/2))/(21467398323875*((31^(1/2)*455214848i)/9892810287 5 + 2045111424/98928102875)))*(5682718 - 31^(1/2)*135439i)^(1/2)*2i)/32962 3 - ((604*(2*x + 1)^2)/1519 - (5392*x)/7595 + 776/7595)/((7*(2*x + 1)^(1/2 ))/5 - (4*(2*x + 1)^(3/2))/5 + (2*x + 1)^(5/2))