Integrand size = 22, antiderivative size = 206 \[ \int \frac {1}{(1+2 x)^{5/2} \left (2+3 x+5 x^2\right )} \, dx=-\frac {4}{21 (1+2 x)^{3/2}}-\frac {16}{49 \sqrt {1+2 x}}+\frac {1}{49} \sqrt {\frac {2}{217} \left (7162+1225 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )-\frac {1}{49} \sqrt {\frac {2}{217} \left (7162+1225 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )+\frac {1}{49} \sqrt {\frac {2}{217} \left (-7162+1225 \sqrt {35}\right )} \text {arctanh}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}}{5+\sqrt {35}+10 x}\right ) \] Output:
-4/21/(1+2*x)^(3/2)-16/49/(1+2*x)^(1/2)+1/10633*(3108308+531650*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))-1/10633*(3108308+531650*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))+1/10633*(-3108308+531650*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 = 0.75 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.58 \[ \int \frac {1}{(1+2 x)^{5/2} \left (2+3 x+5 x^2\right )} \, dx=\frac {2 \left (-\frac {434 (19+24 x)}{(1+2 x)^{3/2}}-3 \sqrt {217 \left (7162-199 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )-3 \sqrt {217 \left (7162+199 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )\right )}{31899} \] Input:
Integrate[1/((1 + 2*x)^(5/2)*(2 + 3*x + 5*x^2)),x]
Output:
(2*((-434*(19 + 24*x))/(1 + 2*x)^(3/2) - 3*Sqrt[217*(7162 - (199*I)*Sqrt[3 1])]*ArcTan[Sqrt[(-2 - I*Sqrt[31])/7]*Sqrt[1 + 2*x]] - 3*Sqrt[217*(7162 + (199*I)*Sqrt[31])]*ArcTan[Sqrt[(I/7)*(2*I + Sqrt[31])]*Sqrt[1 + 2*x]]))/31 899
Time = 0.61 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.53, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {1147, 25, 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.
| \(\displaystyle \int \frac {1}{(2 x+1)^{5/2} \left (5 x^2+3 x+2\right )} \, dx\) |
| \(\Big \downarrow \) 1147 |
| \(\displaystyle \frac {1}{7} \int -\frac {10 x+1}{(2 x+1)^{3/2} \left (5 x^2+3 x+2\right )}dx-\frac {4}{21 (2 x+1)^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{7} \int \frac {10 x+1}{(2 x+1)^{3/2} \left (5 x^2+3 x+2\right )}dx-\frac {4}{21 (2 x+1)^{3/2}}\) |
| \(\Big \downarrow \) 1198 |
| \(\displaystyle \frac {1}{7} \left (-\frac {1}{7} \int \frac {40 x+39}{\sqrt {2 x+1} \left (5 x^2+3 x+2\right )}dx-\frac {16}{7 \sqrt {2 x+1}}\right )-\frac {4}{21 (2 x+1)^{3/2}}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle \frac {1}{7} \left (-\frac {2}{7} \int \frac {2 (20 (2 x+1)+19)}{5 (2 x+1)^2-4 (2 x+1)+7}d\sqrt {2 x+1}-\frac {16}{7 \sqrt {2 x+1}}\right )-\frac {4}{21 (2 x+1)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (-\frac {4}{7} \int \frac {20 (2 x+1)+19}{5 (2 x+1)^2-4 (2 x+1)+7}d\sqrt {2 x+1}-\frac {16}{7 \sqrt {2 x+1}}\right )-\frac {4}{21 (2 x+1)^{3/2}}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {1}{7} \left (-\frac {4}{7} \left (\frac {\int \frac {5 \left (19 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (19-4 \sqrt {35}\right ) \sqrt {2 x+1}\right )}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\int \frac {5 \left (\left (19-4 \sqrt {35}\right ) \sqrt {2 x+1}+19 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}\right )}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {16}{7 \sqrt {2 x+1}}\right )-\frac {4}{21 (2 x+1)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (-\frac {4}{7} \left (\frac {5 \int \frac {19 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (19-4 \sqrt {35}\right ) \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \int \frac {\left (19-4 \sqrt {35}\right ) \sqrt {2 x+1}+19 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {16}{7 \sqrt {2 x+1}}\right )-\frac {4}{21 (2 x+1)^{3/2}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{7} \left (-\frac {4}{7} \left (\frac {5 \left (\frac {1}{10} \sqrt {71620+12250 \sqrt {35}} \int \frac {1}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\frac {1}{10} \left (19-4 \sqrt {35}\right ) \int -\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {1}{10} \sqrt {71620+12250 \sqrt {35}} \int \frac {1}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}+\frac {1}{10} \left (19-4 \sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {16}{7 \sqrt {2 x+1}}\right )-\frac {4}{21 (2 x+1)^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{7} \left (-\frac {4}{7} \left (\frac {5 \left (\frac {1}{10} \sqrt {71620+12250 \sqrt {35}} \int \frac {1}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}+\frac {1}{10} \left (19-4 \sqrt {35}\right ) \int \frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {1}{10} \sqrt {71620+12250 \sqrt {35}} \int \frac {1}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}+\frac {1}{10} \left (19-4 \sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {16}{7 \sqrt {2 x+1}}\right )-\frac {4}{21 (2 x+1)^{3/2}}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{7} \left (-\frac {4}{7} \left (\frac {5 \left (\frac {1}{10} \left (19-4 \sqrt {35}\right ) \int \frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\frac {1}{5} \sqrt {71620+12250 \sqrt {35}} \int \frac {1}{-2 x+10 \left (2-\sqrt {35}\right )-1}d\left (10 \sqrt {2 x+1}-\sqrt {10 \left (2+\sqrt {35}\right )}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {1}{10} \left (19-4 \sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\frac {1}{5} \sqrt {71620+12250 \sqrt {35}} \int \frac {1}{-2 x+10 \left (2-\sqrt {35}\right )-1}d\left (10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {16}{7 \sqrt {2 x+1}}\right )-\frac {4}{21 (2 x+1)^{3/2}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{7} \left (-\frac {4}{7} \left (\frac {5 \left (\frac {1}{10} \left (19-4 \sqrt {35}\right ) \int \frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}+\frac {1}{5} \sqrt {\frac {71620+12250 \sqrt {35}}{10 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}-\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {1}{10} \left (19-4 \sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}+\frac {1}{5} \sqrt {\frac {71620+12250 \sqrt {35}}{10 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {16}{7 \sqrt {2 x+1}}\right )-\frac {4}{21 (2 x+1)^{3/2}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{7} \left (-\frac {4}{7} \left (\frac {5 \left (\frac {1}{5} \sqrt {\frac {71620+12250 \sqrt {35}}{10 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}-\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )-\frac {1}{10} \left (19-4 \sqrt {35}\right ) \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {1}{5} \sqrt {\frac {71620+12250 \sqrt {35}}{10 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )+\frac {1}{10} \left (19-4 \sqrt {35}\right ) \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {16}{7 \sqrt {2 x+1}}\right )-\frac {4}{21 (2 x+1)^{3/2}}\) |
Input:
Int[1/((1 + 2*x)^(5/2)*(2 + 3*x + 5*x^2)),x]
Output:
-4/(21*(1 + 2*x)^(3/2)) + (-16/(7*Sqrt[1 + 2*x]) - (4*((5*((Sqrt[(71620 + 12250*Sqrt[35])/(10*(-2 + Sqrt[35]))]*ArcTan[(-Sqrt[10*(2 + Sqrt[35])] + 1 0*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/5 - ((19 - 4*Sqrt[35])*Log[Sqr t[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/10))/(2*Sqrt [14*(2 + Sqrt[35])]) + (5*((Sqrt[(71620 + 12250*Sqrt[35])/(10*(-2 + Sqrt[3 5]))]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sq rt[35])]])/5 + ((19 - 4*Sqrt[35])*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*S qrt[1 + 2*x] + 5*(1 + 2*x)])/10))/(2*Sqrt[14*(2 + Sqrt[35])])))/7)/7
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), x_Symbol ] :> Simp[e*((d + e*x)^(m + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Sim p[1/(c*d^2 - b*d*e + a*e^2) Int[(d + e*x)^(m + 1)*(Simp[c*d - b*e - c*e*x , x]/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[m, - 1]
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 = 2.38 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.24
| method | result | size |
| pseudoelliptic | \(\frac {\frac {\left (1+2 x \right )^{\frac {3}{2}} \left (\sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (189 \sqrt {5}-178 \sqrt {7}\right ) \left (\ln \left (-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+\sqrt {5}\, \sqrt {7}+5+10 x \right )-\ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+2356 \left (\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )-\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )\right ) \left (\sqrt {5}\, \sqrt {7}+\frac {140}{19}\right )\right )}{14}-992 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (x +\frac {19}{24}\right )}{1519 \left (1+2 x \right )^{\frac {3}{2}} \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(256\) |
| derivativedivides | \(-\frac {4}{21 \left (1+2 x \right )^{\frac {3}{2}}}-\frac {16}{49 \sqrt {1+2 x}}+\frac {\left (-945 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{106330}+\frac {2 \left (-1178 \sqrt {5}\, \sqrt {7}-\frac {\left (-945 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{10633 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {\left (-945 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+\sqrt {5}\, \sqrt {7}+5+10 x \right )}{106330}-\frac {2 \left (1178 \sqrt {5}\, \sqrt {7}+\frac {\left (-945 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{10633 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(398\) |
| default | \(-\frac {4}{21 \left (1+2 x \right )^{\frac {3}{2}}}-\frac {16}{49 \sqrt {1+2 x}}+\frac {\left (-945 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{106330}+\frac {2 \left (-1178 \sqrt {5}\, \sqrt {7}-\frac {\left (-945 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{10633 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {\left (-945 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+\sqrt {5}\, \sqrt {7}+5+10 x \right )}{106330}-\frac {2 \left (1178 \sqrt {5}\, \sqrt {7}+\frac {\left (-945 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{10633 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(398\) |
| trager | \(-\frac {4 \left (24 x +19\right )}{147 \left (1+2 x \right )^{\frac {3}{2}}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{2}+3108308\right ) \ln \left (\frac {733243 \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{2}+3108308\right ) \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{4} x +43311371 \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{2}+3108308\right ) x -983924655 \sqrt {1+2 x}\, \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{2}-5379368 \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{2}+3108308\right )+633205440 \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{2}+3108308\right ) x -74281021535 \sqrt {1+2 x}-174737920 \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{2}+3108308\right )}{217 \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{2} x +7759 x +796}\right )}{10633}-\frac {\operatorname {RootOf}\left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right ) \ln \left (-\frac {5132701 x \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{5}+374431547 \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{3} x +37655576 \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{3}-31739505 \sqrt {1+2 x}\, \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{2}+6784080780 x \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )+1262449632 \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )+301062125 \sqrt {1+2 x}}{217 \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}+444044 \textit {\_Z}^{2}+7503125\right )^{2} x +6565 x -796}\right )}{49}\) | \(436\) |
Input:
int(1/(1+2*x)^(5/2)/(5*x^2+3*x+2),x,method=_RETURNVERBOSE)
Output:
1/1519*(1/14*(1+2*x)^(3/2)*((10*5^(1/2)*7^(1/2)-20)^(1/2)*(189*5^(1/2)-178 *7^(1/2))*(ln(-(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(1+2*x)^(1/2)+5^(1/2)*7 ^(1/2)+5+10*x)-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))*(2*5^(1/2)*7^(1/2)+4)^(1/2)+2356*(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))-ar ctan((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)))*(5^(1/2)*7^(1/2)+140/19))-992*(10*5^(1/2)*7^(1/2)-20)^(1 /2)*(x+19/24))/(1+2*x)^(3/2)/(10*5^(1/2)*7^(1/2)-20)^(1/2)
Time = 0.09 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.34 \[ \int \frac {1}{(1+2 x)^{5/2} \left (2+3 x+5 x^2\right )} \, dx=\frac {6 \, {\left (4 \, x^{2} + 4 \, x + 1\right )} \sqrt {\frac {1225}{62} \, \sqrt {\frac {5}{7}} + \frac {3581}{217}} \arctan \left (\frac {14}{199} \, {\left (\sqrt {2 \, x + 1} {\left (19 \, \sqrt {\frac {5}{7}} - 20\right )} + \sqrt {\frac {1225}{62} \, \sqrt {\frac {5}{7}} - \frac {3581}{217}} {\left (7 \, \sqrt {\frac {5}{7}} + 2\right )}\right )} \sqrt {\frac {1225}{62} \, \sqrt {\frac {5}{7}} + \frac {3581}{217}}\right ) - 6 \, {\left (4 \, x^{2} + 4 \, x + 1\right )} \sqrt {\frac {1225}{62} \, \sqrt {\frac {5}{7}} + \frac {3581}{217}} \arctan \left (-\frac {14}{199} \, {\left (\sqrt {2 \, x + 1} {\left (19 \, \sqrt {\frac {5}{7}} - 20\right )} - \sqrt {\frac {1225}{62} \, \sqrt {\frac {5}{7}} - \frac {3581}{217}} {\left (7 \, \sqrt {\frac {5}{7}} + 2\right )}\right )} \sqrt {\frac {1225}{62} \, \sqrt {\frac {5}{7}} + \frac {3581}{217}}\right ) + 3 \, {\left (4 \, x^{2} + 4 \, x + 1\right )} \sqrt {\frac {1225}{62} \, \sqrt {\frac {5}{7}} - \frac {3581}{217}} \log \left (14 \, \sqrt {2 \, x + 1} {\left (178 \, \sqrt {\frac {5}{7}} + 135\right )} \sqrt {\frac {1225}{62} \, \sqrt {\frac {5}{7}} - \frac {3581}{217}} + 1990 \, x + 1393 \, \sqrt {\frac {5}{7}} + 995\right ) - 3 \, {\left (4 \, x^{2} + 4 \, x + 1\right )} \sqrt {\frac {1225}{62} \, \sqrt {\frac {5}{7}} - \frac {3581}{217}} \log \left (-14 \, \sqrt {2 \, x + 1} {\left (178 \, \sqrt {\frac {5}{7}} + 135\right )} \sqrt {\frac {1225}{62} \, \sqrt {\frac {5}{7}} - \frac {3581}{217}} + 1990 \, x + 1393 \, \sqrt {\frac {5}{7}} + 995\right ) - 4 \, {\left (24 \, x + 19\right )} \sqrt {2 \, x + 1}}{147 \, {\left (4 \, x^{2} + 4 \, x + 1\right )}} \] Input:
integrate(1/(1+2*x)^(5/2)/(5*x^2+3*x+2),x, algorithm="fricas")
Output:
1/147*(6*(4*x^2 + 4*x + 1)*sqrt(1225/62*sqrt(5/7) + 3581/217)*arctan(14/19 9*(sqrt(2*x + 1)*(19*sqrt(5/7) - 20) + sqrt(1225/62*sqrt(5/7) - 3581/217)* (7*sqrt(5/7) + 2))*sqrt(1225/62*sqrt(5/7) + 3581/217)) - 6*(4*x^2 + 4*x + 1)*sqrt(1225/62*sqrt(5/7) + 3581/217)*arctan(-14/199*(sqrt(2*x + 1)*(19*sq rt(5/7) - 20) - sqrt(1225/62*sqrt(5/7) - 3581/217)*(7*sqrt(5/7) + 2))*sqrt (1225/62*sqrt(5/7) + 3581/217)) + 3*(4*x^2 + 4*x + 1)*sqrt(1225/62*sqrt(5/ 7) - 3581/217)*log(14*sqrt(2*x + 1)*(178*sqrt(5/7) + 135)*sqrt(1225/62*sqr t(5/7) - 3581/217) + 1990*x + 1393*sqrt(5/7) + 995) - 3*(4*x^2 + 4*x + 1)* sqrt(1225/62*sqrt(5/7) - 3581/217)*log(-14*sqrt(2*x + 1)*(178*sqrt(5/7) + 135)*sqrt(1225/62*sqrt(5/7) - 3581/217) + 1990*x + 1393*sqrt(5/7) + 995) - 4*(24*x + 19)*sqrt(2*x + 1))/(4*x^2 + 4*x + 1)
\[ \int \frac {1}{(1+2 x)^{5/2} \left (2+3 x+5 x^2\right )} \, dx=\int \frac {1}{\left (2 x + 1\right )^{\frac {5}{2}} \cdot \left (5 x^{2} + 3 x + 2\right )}\, dx \] Input:
integrate(1/(1+2*x)**(5/2)/(5*x**2+3*x+2),x)
Output:
Integral(1/((2*x + 1)**(5/2)*(5*x**2 + 3*x + 2)), x)
\[ \int \frac {1}{(1+2 x)^{5/2} \left (2+3 x+5 x^2\right )} \, dx=\int { \frac {1}{{\left (5 \, x^{2} + 3 \, x + 2\right )} {\left (2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(1+2*x)^(5/2)/(5*x^2+3*x+2),x, algorithm="maxima")
Output:
integrate(1/((5*x^2 + 3*x + 2)*(2*x + 1)^(5/2)), x)
Leaf count of result is larger than twice the leaf count of optimal. 599 vs. \(2 (143) = 286\).
Time = 0.60 (sec) , antiderivative size = 599, normalized size of antiderivative = 2.91 \[ \int \frac {1}{(1+2 x)^{5/2} \left (2+3 x+5 x^2\right )} \, dx=\text {Too large to display} \] Input:
integrate(1/(1+2*x)^(5/2)/(5*x^2+3*x+2),x, algorithm="giac")
Output:
-1/91177975*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) + 4655*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) - 9310*(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)) - 1/91177975*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) + 2 450)^(3/2) - 2*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) - 420*(7/5)^(3/4)*s qrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) + 4655*sqrt(31)*(7/5)^(1/4)*sqr t(-140*sqrt(35) + 2450) - 9310*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450))*arct an(-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)) - 1/182355950*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*sq rt(35) + 2450) - 2*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 4655*sqrt(31 )*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) + 9310*(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) + 1/182355950*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)...
Time = 0.09 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.91 \[ \int \frac {1}{(1+2 x)^{5/2} \left (2+3 x+5 x^2\right )} \, dx=-\frac {\frac {32\,x}{49}+\frac {76}{147}}{{\left (2\,x+1\right )}^{3/2}}+\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-7162-\sqrt {31}\,199{}\mathrm {i}}\,\sqrt {2\,x+1}\,25472{}\mathrm {i}}{720600125\,\left (-\frac {4534016}{102942875}+\frac {\sqrt {31}\,483968{}\mathrm {i}}{102942875}\right )}-\frac {50944\,\sqrt {31}\,\sqrt {217}\,\sqrt {-7162-\sqrt {31}\,199{}\mathrm {i}}\,\sqrt {2\,x+1}}{22338603875\,\left (-\frac {4534016}{102942875}+\frac {\sqrt {31}\,483968{}\mathrm {i}}{102942875}\right )}\right )\,\sqrt {-7162-\sqrt {31}\,199{}\mathrm {i}}\,2{}\mathrm {i}}{10633}-\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {-7162+\sqrt {31}\,199{}\mathrm {i}}\,\sqrt {2\,x+1}\,25472{}\mathrm {i}}{720600125\,\left (\frac {4534016}{102942875}+\frac {\sqrt {31}\,483968{}\mathrm {i}}{102942875}\right )}+\frac {50944\,\sqrt {31}\,\sqrt {217}\,\sqrt {-7162+\sqrt {31}\,199{}\mathrm {i}}\,\sqrt {2\,x+1}}{22338603875\,\left (\frac {4534016}{102942875}+\frac {\sqrt {31}\,483968{}\mathrm {i}}{102942875}\right )}\right )\,\sqrt {-7162+\sqrt {31}\,199{}\mathrm {i}}\,2{}\mathrm {i}}{10633} \] Input:
int(1/((2*x + 1)^(5/2)*(3*x + 5*x^2 + 2)),x)
Output:
(217^(1/2)*atan((217^(1/2)*(- 31^(1/2)*199i - 7162)^(1/2)*(2*x + 1)^(1/2)* 25472i)/(720600125*((31^(1/2)*483968i)/102942875 - 4534016/102942875)) - ( 50944*31^(1/2)*217^(1/2)*(- 31^(1/2)*199i - 7162)^(1/2)*(2*x + 1)^(1/2))/( 22338603875*((31^(1/2)*483968i)/102942875 - 4534016/102942875)))*(- 31^(1/ 2)*199i - 7162)^(1/2)*2i)/10633 - ((32*x)/49 + 76/147)/(2*x + 1)^(3/2) - ( 217^(1/2)*atan((217^(1/2)*(31^(1/2)*199i - 7162)^(1/2)*(2*x + 1)^(1/2)*254 72i)/(720600125*((31^(1/2)*483968i)/102942875 + 4534016/102942875)) + (509 44*31^(1/2)*217^(1/2)*(31^(1/2)*199i - 7162)^(1/2)*(2*x + 1)^(1/2))/(22338 603875*((31^(1/2)*483968i)/102942875 + 4534016/102942875)))*(31^(1/2)*199i - 7162)^(1/2)*2i)/10633
Time = 0.23 (sec) , antiderivative size = 730, normalized size of antiderivative = 3.54 \[ \int \frac {1}{(1+2 x)^{5/2} \left (2+3 x+5 x^2\right )} \, dx =\text {Too large to display} \] Input:
int(1/(1+2*x)^(5/2)/(5*x^2+3*x+2),x)
Output:
(2136*sqrt(2*x + 1)*sqrt(sqrt(35) - 2)*sqrt(14)*atan((sqrt(sqrt(35) + 2)*s qrt(2) - 2*sqrt(2*x + 1)*sqrt(5))/(sqrt(sqrt(35) - 2)*sqrt(2)))*x + 1068*s qrt(2*x + 1)*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))) + 2268*sqrt(2*x + 1)*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 + 1134*sqrt(2*x + 1)*sqr t(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))) - 2136*sqrt(2*x + 1)*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 - 1068*sqrt(2*x + 1)*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))) - 2268*sqrt(2*x + 1)*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 - 1134*sqrt(2*x + 1)*sqrt(sqrt(35) - 2)*sqrt(10)*atan((s qrt(sqrt(35) + 2)*sqrt(2) + 2*sqrt(2*x + 1)*sqrt(5))/(sqrt(sqrt(35) - 2)*s qrt(2))) - 1068*sqrt(2*x + 1)*sqrt(sqrt(35) + 2)*sqrt(14)*log( - sqrt(2*x + 1)*sqrt(sqrt(35) + 2)*sqrt(2) + sqrt(7) + 2*sqrt(5)*x + sqrt(5))*x - 534 *sqrt(2*x + 1)*sqrt(sqrt(35) + 2)*sqrt(14)*log( - sqrt(2*x + 1)*sqrt(sqrt( 35) + 2)*sqrt(2) + sqrt(7) + 2*sqrt(5)*x + sqrt(5)) + 1068*sqrt(2*x + 1)*s qrt(sqrt(35) + 2)*sqrt(14)*log(sqrt(2*x + 1)*sqrt(sqrt(35) + 2)*sqrt(2)...