Integrand size = 22, antiderivative size = 193 \[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )} \, dx=-\frac {4}{7 \sqrt {1+2 x}}+\frac {1}{7} \sqrt {\frac {2}{217} \left (-178+35 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )-\frac {1}{7} \sqrt {\frac {2}{217} \left (-178+35 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )+\frac {1}{7} \sqrt {\frac {2}{217} \left (178+35 \sqrt {35}\right )} \text {arctanh}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}}{5+\sqrt {35}+10 x}\right ) \] Output:
-4/7/(1+2*x)^(1/2)+1/1519*(-77252+15190*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))-1/1519*(-77252+151 90*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))+1/1519*(77252+15190*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 = 0.63 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.59 \[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )} \, dx=\frac {2 \left (-\frac {434}{\sqrt {1+2 x}}-\sqrt {217 \left (-178-19 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )-\sqrt {217 \left (-178+19 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )\right )}{1519} \] Input:
Integrate[1/((1 + 2*x)^(3/2)*(2 + 3*x + 5*x^2)),x]
Output:
(2*(-434/Sqrt[1 + 2*x] - Sqrt[217*(-178 - (19*I)*Sqrt[31])]*ArcTan[Sqrt[(- 2 - I*Sqrt[31])/7]*Sqrt[1 + 2*x]] - Sqrt[217*(-178 + (19*I)*Sqrt[31])]*Arc Tan[Sqrt[(I/7)*(2*I + Sqrt[31])]*Sqrt[1 + 2*x]]))/1519
Time = 0.51 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.52, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1147, 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.
| \(\displaystyle \int \frac {1}{(2 x+1)^{3/2} \left (5 x^2+3 x+2\right )} \, dx\) |
| \(\Big \downarrow \) 1147 |
| \(\displaystyle \frac {1}{7} \int -\frac {10 x+1}{\sqrt {2 x+1} \left (5 x^2+3 x+2\right )}dx-\frac {4}{7 \sqrt {2 x+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{7} \int \frac {10 x+1}{\sqrt {2 x+1} \left (5 x^2+3 x+2\right )}dx-\frac {4}{7 \sqrt {2 x+1}}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle -\frac {2}{7} \int -\frac {2 (4-5 (2 x+1))}{5 (2 x+1)^2-4 (2 x+1)+7}d\sqrt {2 x+1}-\frac {4}{7 \sqrt {2 x+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4}{7} \int \frac {4-5 (2 x+1)}{5 (2 x+1)^2-4 (2 x+1)+7}d\sqrt {2 x+1}-\frac {4}{7 \sqrt {2 x+1}}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {4}{7} \left (\frac {\int \frac {5 \left (4 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (4+\sqrt {35}\right ) \sqrt {2 x+1}\right )}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\int \frac {5 \left (\left (4+\sqrt {35}\right ) \sqrt {2 x+1}+4 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}\right )}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {4}{7 \sqrt {2 x+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4}{7} \left (\frac {5 \int \frac {4 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (4+\sqrt {35}\right ) \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \int \frac {\left (4+\sqrt {35}\right ) \sqrt {2 x+1}+4 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {4}{7 \sqrt {2 x+1}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {4}{7} \left (\frac {5 \left (-\frac {1}{10} \sqrt {350 \sqrt {35}-1780} \int \frac {1}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\frac {1}{10} \left (4+\sqrt {35}\right ) \int -\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {1}{10} \left (4+\sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\frac {1}{10} \sqrt {350 \sqrt {35}-1780} \int \frac {1}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {4}{7 \sqrt {2 x+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {4}{7} \left (\frac {5 \left (\frac {1}{10} \left (4+\sqrt {35}\right ) \int \frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\frac {1}{10} \sqrt {350 \sqrt {35}-1780} \int \frac {1}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {1}{10} \left (4+\sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\frac {1}{10} \sqrt {350 \sqrt {35}-1780} \int \frac {1}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {4}{7 \sqrt {2 x+1}}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {4}{7} \left (\frac {5 \left (\frac {1}{5} \sqrt {350 \sqrt {35}-1780} \int \frac {1}{-2 x+10 \left (2-\sqrt {35}\right )-1}d\left (10 \sqrt {2 x+1}-\sqrt {10 \left (2+\sqrt {35}\right )}\right )+\frac {1}{10} \left (4+\sqrt {35}\right ) \int \frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {1}{5} \sqrt {350 \sqrt {35}-1780} \int \frac {1}{-2 x+10 \left (2-\sqrt {35}\right )-1}d\left (10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}\right )+\frac {1}{10} \left (4+\sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {4}{7 \sqrt {2 x+1}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {4}{7} \left (\frac {5 \left (\frac {1}{10} \left (4+\sqrt {35}\right ) \int \frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\frac {1}{5} \sqrt {\frac {350 \sqrt {35}-1780}{10 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}-\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {1}{10} \left (4+\sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\frac {1}{5} \sqrt {\frac {350 \sqrt {35}-1780}{10 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {4}{7 \sqrt {2 x+1}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {4}{7} \left (\frac {5 \left (-\frac {1}{5} \sqrt {\frac {350 \sqrt {35}-1780}{10 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}-\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )-\frac {1}{10} \left (4+\sqrt {35}\right ) \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {5 \left (\frac {1}{10} \left (4+\sqrt {35}\right ) \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )-\frac {1}{5} \sqrt {\frac {350 \sqrt {35}-1780}{10 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {4}{7 \sqrt {2 x+1}}\) |
Input:
Int[1/((1 + 2*x)^(3/2)*(2 + 3*x + 5*x^2)),x]
Output:
-4/(7*Sqrt[1 + 2*x]) + (4*((5*(-1/5*(Sqrt[(-1780 + 350*Sqrt[35])/(10*(-2 + Sqrt[35]))]*ArcTan[(-Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10* (-2 + Sqrt[35])]]) - ((4 + 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])]) + (5*(-1/ 5*(Sqrt[(-1780 + 350*Sqrt[35])/(10*(-2 + Sqrt[35]))]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]]) + ((4 + 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
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_)^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 = 2.52 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.31
| method | result | size |
| pseudoelliptic | \(-\frac {\left (\frac {\sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (14 \sqrt {5}+27 \sqrt {7}\right ) \left (\ln \left (-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+\sqrt {5}\, \sqrt {7}+5+10 x \right )-\ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{14}+\frac {248 \left (\sqrt {5}\, \sqrt {7}-\frac {35}{4}\right ) \left (\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )-\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )\right )}{7}\right ) \sqrt {1+2 x}+124 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}{217 \sqrt {1+2 x}\, \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(253\) |
| derivativedivides | \(-\frac {4}{7 \sqrt {1+2 x}}+\frac {\left (-70 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-135 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+\sqrt {5}\, \sqrt {7}+5+10 x \right )}{15190}+\frac {2 \left (248 \sqrt {5}\, \sqrt {7}+\frac {\left (-70 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-135 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{1519 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {\left (70 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+135 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{15190}+\frac {2 \left (248 \sqrt {5}\, \sqrt {7}-\frac {\left (70 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+135 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{1519 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(389\) |
| default | \(-\frac {4}{7 \sqrt {1+2 x}}+\frac {\left (-70 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-135 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+\sqrt {5}\, \sqrt {7}+5+10 x \right )}{15190}+\frac {2 \left (248 \sqrt {5}\, \sqrt {7}+\frac {\left (-70 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-135 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{1519 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {\left (70 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+135 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{15190}+\frac {2 \left (248 \sqrt {5}\, \sqrt {7}-\frac {\left (70 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+135 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{1519 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(389\) |
| trager | \(-\frac {4}{7 \sqrt {1+2 x}}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}-11036 \textit {\_Z}^{2}+6125\right )^{2}-77252\right ) \ln \left (-\frac {53816 \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}-11036 \textit {\_Z}^{2}+6125\right )^{2}-77252\right ) \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}-11036 \textit {\_Z}^{2}+6125\right )^{4} x -66495 \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}-11036 \textit {\_Z}^{2}+6125\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}-11036 \textit {\_Z}^{2}+6125\right )^{2}-77252\right ) x +2684073 \sqrt {1+2 x}\, \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}-11036 \textit {\_Z}^{2}+6125\right )^{2}-37696 \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}-11036 \textit {\_Z}^{2}+6125\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}-11036 \textit {\_Z}^{2}+6125\right )^{2}-77252\right )-15239 \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}-11036 \textit {\_Z}^{2}+6125\right )^{2}-77252\right ) x +8687161 \sqrt {1+2 x}-7448 \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}-11036 \textit {\_Z}^{2}+6125\right )^{2}-77252\right )}{217 \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}-11036 \textit {\_Z}^{2}+6125\right )^{2} x -121 x +76}\right )}{1519}+\frac {\operatorname {RootOf}\left (6727 \textit {\_Z}^{4}-11036 \textit {\_Z}^{2}+6125\right ) \ln \left (\frac {376712 x \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}-11036 \textit {\_Z}^{2}+6125\right )^{5}-770567 \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}-11036 \textit {\_Z}^{2}+6125\right )^{3} x +263872 \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}-11036 \textit {\_Z}^{2}+6125\right )^{3}+86583 \sqrt {1+2 x}\, \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}-11036 \textit {\_Z}^{2}+6125\right )^{2}+143595 x \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}-11036 \textit {\_Z}^{2}+6125\right )-485032 \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}-11036 \textit {\_Z}^{2}+6125\right )-422275 \sqrt {1+2 x}}{217 \operatorname {RootOf}\left (6727 \textit {\_Z}^{4}-11036 \textit {\_Z}^{2}+6125\right )^{2} x -235 x -76}\right )}{7}\) | \(431\) |
| risch | \(-\frac {4}{7 \sqrt {1+2 x}}-\frac {\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}}{217}-\frac {27 \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}}{3038}-\frac {10 \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 )}{217 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {27 \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}}{1519 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {16 \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}}{49 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {\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}}{217}+\frac {27 \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}}{3038}-\frac {10 \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 )}{217 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {27 \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}}{1519 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {16 \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}}{49 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(616\) |
Input:
int(1/(1+2*x)^(3/2)/(5*x^2+3*x+2),x,method=_RETURNVERBOSE)
Output:
-1/217/(1+2*x)^(1/2)*((1/14*(10*5^(1/2)*7^(1/2)-20)^(1/2)*(14*5^(1/2)+27*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)+248/7*(5^(1/2)*7^(1/2)-35/4)* (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))))*(1+2*x)^(1/2)+124*(10*5^(1/2)*7^( 1/2)-20)^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2)
Time = 0.09 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.36 \[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )} \, dx=-\frac {2 \, {\left (2 \, x + 1\right )} \sqrt {\frac {35}{62} \, \sqrt {\frac {5}{7}} - \frac {89}{217}} \arctan \left (\frac {14}{19} \, \sqrt {2 \, x + 1} {\left (4 \, \sqrt {\frac {5}{7}} + 5\right )} \sqrt {\frac {35}{62} \, \sqrt {\frac {5}{7}} - \frac {89}{217}} + \frac {14}{19} \, {\left (7 \, \sqrt {\frac {5}{7}} + 2\right )} \sqrt {\frac {35}{62} \, \sqrt {\frac {5}{7}} + \frac {89}{217}} \sqrt {\frac {35}{62} \, \sqrt {\frac {5}{7}} - \frac {89}{217}}\right ) - 2 \, {\left (2 \, x + 1\right )} \sqrt {\frac {35}{62} \, \sqrt {\frac {5}{7}} - \frac {89}{217}} \arctan \left (-\frac {14}{19} \, \sqrt {2 \, x + 1} {\left (4 \, \sqrt {\frac {5}{7}} + 5\right )} \sqrt {\frac {35}{62} \, \sqrt {\frac {5}{7}} - \frac {89}{217}} + \frac {14}{19} \, {\left (7 \, \sqrt {\frac {5}{7}} + 2\right )} \sqrt {\frac {35}{62} \, \sqrt {\frac {5}{7}} + \frac {89}{217}} \sqrt {\frac {35}{62} \, \sqrt {\frac {5}{7}} - \frac {89}{217}}\right ) - {\left (2 \, x + 1\right )} \sqrt {\frac {35}{62} \, \sqrt {\frac {5}{7}} + \frac {89}{217}} \log \left (14 \, \sqrt {2 \, x + 1} {\left (27 \, \sqrt {\frac {5}{7}} - 10\right )} \sqrt {\frac {35}{62} \, \sqrt {\frac {5}{7}} + \frac {89}{217}} + 190 \, x + 133 \, \sqrt {\frac {5}{7}} + 95\right ) + {\left (2 \, x + 1\right )} \sqrt {\frac {35}{62} \, \sqrt {\frac {5}{7}} + \frac {89}{217}} \log \left (-14 \, \sqrt {2 \, x + 1} {\left (27 \, \sqrt {\frac {5}{7}} - 10\right )} \sqrt {\frac {35}{62} \, \sqrt {\frac {5}{7}} + \frac {89}{217}} + 190 \, x + 133 \, \sqrt {\frac {5}{7}} + 95\right ) + 4 \, \sqrt {2 \, x + 1}}{7 \, {\left (2 \, x + 1\right )}} \] Input:
integrate(1/(1+2*x)^(3/2)/(5*x^2+3*x+2),x, algorithm="fricas")
Output:
-1/7*(2*(2*x + 1)*sqrt(35/62*sqrt(5/7) - 89/217)*arctan(14/19*sqrt(2*x + 1 )*(4*sqrt(5/7) + 5)*sqrt(35/62*sqrt(5/7) - 89/217) + 14/19*(7*sqrt(5/7) + 2)*sqrt(35/62*sqrt(5/7) + 89/217)*sqrt(35/62*sqrt(5/7) - 89/217)) - 2*(2*x + 1)*sqrt(35/62*sqrt(5/7) - 89/217)*arctan(-14/19*sqrt(2*x + 1)*(4*sqrt(5 /7) + 5)*sqrt(35/62*sqrt(5/7) - 89/217) + 14/19*(7*sqrt(5/7) + 2)*sqrt(35/ 62*sqrt(5/7) + 89/217)*sqrt(35/62*sqrt(5/7) - 89/217)) - (2*x + 1)*sqrt(35 /62*sqrt(5/7) + 89/217)*log(14*sqrt(2*x + 1)*(27*sqrt(5/7) - 10)*sqrt(35/6 2*sqrt(5/7) + 89/217) + 190*x + 133*sqrt(5/7) + 95) + (2*x + 1)*sqrt(35/62 *sqrt(5/7) + 89/217)*log(-14*sqrt(2*x + 1)*(27*sqrt(5/7) - 10)*sqrt(35/62* sqrt(5/7) + 89/217) + 190*x + 133*sqrt(5/7) + 95) + 4*sqrt(2*x + 1))/(2*x + 1)
\[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )} \, dx=\int \frac {1}{\left (2 x + 1\right )^{\frac {3}{2}} \cdot \left (5 x^{2} + 3 x + 2\right )}\, dx \] Input:
integrate(1/(1+2*x)**(3/2)/(5*x**2+3*x+2),x)
Output:
Integral(1/((2*x + 1)**(3/2)*(5*x**2 + 3*x + 2)), x)
\[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )} \, dx=\int { \frac {1}{{\left (5 \, x^{2} + 3 \, x + 2\right )} {\left (2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(1+2*x)^(3/2)/(5*x^2+3*x+2),x, algorithm="maxima")
Output:
integrate(1/((5*x^2 + 3*x + 2)*(2*x + 1)^(3/2)), x)
Leaf count of result is larger than twice the leaf count of optimal. 594 vs. \(2 (134) = 268\).
Time = 0.55 (sec) , antiderivative size = 594, normalized size of antiderivative = 3.08 \[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )} \, dx=\text {Too large to display} \] Input:
integrate(1/(1+2*x)^(3/2)/(5*x^2+3*x+2),x, algorithm="giac")
Output:
-1/52101700*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) - 3920*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) - 7840*(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)) - 1/52101700*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) - 3920*sqrt(31)*(7/5)^(1/4)*sqr t(-140*sqrt(35) + 2450) - 7840*(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)) - 1/104203400*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) - 3920*sqrt(31 )*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) + 7840*(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) + 1/104203400*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 = 5.59 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.94 \[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )} \, dx=-\frac {4}{7\,\sqrt {2\,x+1}}-\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {178-\sqrt {31}\,19{}\mathrm {i}}\,\sqrt {2\,x+1}\,2432{}\mathrm {i}}{2100875\,\left (\frac {65664}{300125}+\frac {\sqrt {31}\,9728{}\mathrm {i}}{300125}\right )}-\frac {4864\,\sqrt {31}\,\sqrt {217}\,\sqrt {178-\sqrt {31}\,19{}\mathrm {i}}\,\sqrt {2\,x+1}}{65127125\,\left (\frac {65664}{300125}+\frac {\sqrt {31}\,9728{}\mathrm {i}}{300125}\right )}\right )\,\sqrt {178-\sqrt {31}\,19{}\mathrm {i}}\,2{}\mathrm {i}}{1519}+\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {178+\sqrt {31}\,19{}\mathrm {i}}\,\sqrt {2\,x+1}\,2432{}\mathrm {i}}{2100875\,\left (-\frac {65664}{300125}+\frac {\sqrt {31}\,9728{}\mathrm {i}}{300125}\right )}+\frac {4864\,\sqrt {31}\,\sqrt {217}\,\sqrt {178+\sqrt {31}\,19{}\mathrm {i}}\,\sqrt {2\,x+1}}{65127125\,\left (-\frac {65664}{300125}+\frac {\sqrt {31}\,9728{}\mathrm {i}}{300125}\right )}\right )\,\sqrt {178+\sqrt {31}\,19{}\mathrm {i}}\,2{}\mathrm {i}}{1519} \] Input:
int(1/((2*x + 1)^(3/2)*(3*x + 5*x^2 + 2)),x)
Output:
(217^(1/2)*atan((217^(1/2)*(31^(1/2)*19i + 178)^(1/2)*(2*x + 1)^(1/2)*2432 i)/(2100875*((31^(1/2)*9728i)/300125 - 65664/300125)) + (4864*31^(1/2)*217 ^(1/2)*(31^(1/2)*19i + 178)^(1/2)*(2*x + 1)^(1/2))/(65127125*((31^(1/2)*97 28i)/300125 - 65664/300125)))*(31^(1/2)*19i + 178)^(1/2)*2i)/1519 - (217^( 1/2)*atan((217^(1/2)*(178 - 31^(1/2)*19i)^(1/2)*(2*x + 1)^(1/2)*2432i)/(21 00875*((31^(1/2)*9728i)/300125 + 65664/300125)) - (4864*31^(1/2)*217^(1/2) *(178 - 31^(1/2)*19i)^(1/2)*(2*x + 1)^(1/2))/(65127125*((31^(1/2)*9728i)/3 00125 + 65664/300125)))*(178 - 31^(1/2)*19i)^(1/2)*2i)/1519 - 4/(7*(2*x + 1)^(1/2))
Time = 0.23 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.88 \[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )} \, dx=\frac {54 \sqrt {2 x +1}\, \sqrt {\sqrt {35}-2}\, \sqrt {14}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {35}+2}\, \sqrt {2}-2 \sqrt {2 x +1}\, \sqrt {5}}{\sqrt {\sqrt {35}-2}\, \sqrt {2}}\right )-28 \sqrt {2 x +1}\, \sqrt {\sqrt {35}-2}\, \sqrt {10}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {35}+2}\, \sqrt {2}-2 \sqrt {2 x +1}\, \sqrt {5}}{\sqrt {\sqrt {35}-2}\, \sqrt {2}}\right )-54 \sqrt {2 x +1}\, \sqrt {\sqrt {35}-2}\, \sqrt {14}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {35}+2}\, \sqrt {2}+2 \sqrt {2 x +1}\, \sqrt {5}}{\sqrt {\sqrt {35}-2}\, \sqrt {2}}\right )+28 \sqrt {2 x +1}\, \sqrt {\sqrt {35}-2}\, \sqrt {10}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {35}+2}\, \sqrt {2}+2 \sqrt {2 x +1}\, \sqrt {5}}{\sqrt {\sqrt {35}-2}\, \sqrt {2}}\right )-27 \sqrt {2 x +1}\, \sqrt {\sqrt {35}+2}\, \sqrt {14}\, \mathrm {log}\left (-\sqrt {2 x +1}\, \sqrt {\sqrt {35}+2}\, \sqrt {2}+\sqrt {7}+2 \sqrt {5}\, x +\sqrt {5}\right )+27 \sqrt {2 x +1}\, \sqrt {\sqrt {35}+2}\, \sqrt {14}\, \mathrm {log}\left (\sqrt {2 x +1}\, \sqrt {\sqrt {35}+2}\, \sqrt {2}+\sqrt {7}+2 \sqrt {5}\, x +\sqrt {5}\right )-14 \sqrt {2 x +1}\, \sqrt {\sqrt {35}+2}\, \sqrt {10}\, \mathrm {log}\left (-\sqrt {2 x +1}\, \sqrt {\sqrt {35}+2}\, \sqrt {2}+\sqrt {7}+2 \sqrt {5}\, x +\sqrt {5}\right )+14 \sqrt {2 x +1}\, \sqrt {\sqrt {35}+2}\, \sqrt {10}\, \mathrm {log}\left (\sqrt {2 x +1}\, \sqrt {\sqrt {35}+2}\, \sqrt {2}+\sqrt {7}+2 \sqrt {5}\, x +\sqrt {5}\right )-1736}{3038 \sqrt {2 x +1}} \] Input:
int(1/(1+2*x)^(3/2)/(5*x^2+3*x+2),x)
Output:
(54*sqrt(2*x + 1)*sqrt(sqrt(35) - 2)*sqrt(14)*atan((sqrt(sqrt(35) + 2)*sqr t(2) - 2*sqrt(2*x + 1)*sqrt(5))/(sqrt(sqrt(35) - 2)*sqrt(2))) - 28*sqrt(2* x + 1)*sqrt(sqrt(35) - 2)*sqrt(10)*atan((sqrt(sqrt(35) + 2)*sqrt(2) - 2*sq rt(2*x + 1)*sqrt(5))/(sqrt(sqrt(35) - 2)*sqrt(2))) - 54*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))) + 28*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))) - 27*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)) + 27*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)) - 14*sqrt( 2*x + 1)*sqrt(sqrt(35) + 2)*sqrt(10)*log( - sqrt(2*x + 1)*sqrt(sqrt(35) + 2)*sqrt(2) + sqrt(7) + 2*sqrt(5)*x + sqrt(5)) + 14*sqrt(2*x + 1)*sqrt(sqrt (35) + 2)*sqrt(10)*log(sqrt(2*x + 1)*sqrt(sqrt(35) + 2)*sqrt(2) + sqrt(7) + 2*sqrt(5)*x + sqrt(5)) - 1736)/(3038*sqrt(2*x + 1))