Integrand size = 22, antiderivative size = 219 \[ \int \frac {(1+2 x)^{7/2}}{2+3 x+5 x^2} \, dx=-\frac {76}{125} \sqrt {1+2 x}+\frac {16}{75} (1+2 x)^{3/2}+\frac {4}{25} (1+2 x)^{5/2}+\frac {1}{125} \sqrt {\frac {2}{155} \left (-168698+42875 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )-\frac {1}{125} \sqrt {\frac {2}{155} \left (-168698+42875 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )+\frac {1}{125} \sqrt {\frac {2}{155} \left (168698+42875 \sqrt {35}\right )} \text {arctanh}\left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}}{5+\sqrt {35}+10 x}\right ) \] Output:
-76/125*(1+2*x)^(1/2)+16/75*(1+2*x)^(3/2)+4/25*(1+2*x)^(5/2)+1/19375*(-522 96380+13291250*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))-1/19375*(-52296380+13291250*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)) +1/19375*(52296380+13291250*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 = 0.71 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.57 \[ \int \frac {(1+2 x)^{7/2}}{2+3 x+5 x^2} \, dx=\frac {1240 \sqrt {1+2 x} \left (-11+50 x+30 x^2\right )-6 \sqrt {155 \left (-168698-34021 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )-6 \sqrt {155 \left (-168698+34021 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )}{58125} \] Input:
Integrate[(1 + 2*x)^(7/2)/(2 + 3*x + 5*x^2),x]
Output:
(1240*Sqrt[1 + 2*x]*(-11 + 50*x + 30*x^2) - 6*Sqrt[155*(-168698 - (34021*I )*Sqrt[31])]*ArcTan[Sqrt[(-2 - I*Sqrt[31])/7]*Sqrt[1 + 2*x]] - 6*Sqrt[155* (-168698 + (34021*I)*Sqrt[31])]*ArcTan[Sqrt[(I/7)*(2*I + Sqrt[31])]*Sqrt[1 + 2*x]])/58125
Time = 0.67 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.51, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {1146, 25, 1196, 1196, 1197, 27, 1483, 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.
| \(\displaystyle \int \frac {(2 x+1)^{7/2}}{5 x^2+3 x+2} \, dx\) |
| \(\Big \downarrow \) 1146 |
| \(\displaystyle \frac {1}{5} \int -\frac {(3-8 x) (2 x+1)^{3/2}}{5 x^2+3 x+2}dx+\frac {4}{25} (2 x+1)^{5/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {4}{25} (2 x+1)^{5/2}-\frac {1}{5} \int \frac {(3-8 x) (2 x+1)^{3/2}}{5 x^2+3 x+2}dx\) |
| \(\Big \downarrow \) 1196 |
| \(\displaystyle \frac {1}{5} \left (\frac {16}{15} (2 x+1)^{3/2}-\frac {1}{5} \int \frac {\sqrt {2 x+1} (38 x+47)}{5 x^2+3 x+2}dx\right )+\frac {4}{25} (2 x+1)^{5/2}\) |
| \(\Big \downarrow \) 1196 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{5} \left (-\frac {1}{5} \int \frac {432 x+83}{\sqrt {2 x+1} \left (5 x^2+3 x+2\right )}dx-\frac {76}{5} \sqrt {2 x+1}\right )+\frac {16}{15} (2 x+1)^{3/2}\right )+\frac {4}{25} (2 x+1)^{5/2}\) |
| \(\Big \downarrow \) 1197 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{5} \left (-\frac {2}{5} \int -\frac {2 (133-216 (2 x+1))}{5 (2 x+1)^2-4 (2 x+1)+7}d\sqrt {2 x+1}-\frac {76}{5} \sqrt {2 x+1}\right )+\frac {16}{15} (2 x+1)^{3/2}\right )+\frac {4}{25} (2 x+1)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{5} \left (\frac {4}{5} \int \frac {133-216 (2 x+1)}{5 (2 x+1)^2-4 (2 x+1)+7}d\sqrt {2 x+1}-\frac {76}{5} \sqrt {2 x+1}\right )+\frac {16}{15} (2 x+1)^{3/2}\right )+\frac {4}{25} (2 x+1)^{5/2}\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{5} \left (\frac {4}{5} \left (\frac {\int \frac {133 \sqrt {10 \left (2+\sqrt {35}\right )}-\left (665+216 \sqrt {35}\right ) \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\int \frac {\left (665+216 \sqrt {35}\right ) \sqrt {2 x+1}+133 \sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {76}{5} \sqrt {2 x+1}\right )+\frac {16}{15} (2 x+1)^{3/2}\right )+\frac {4}{25} (2 x+1)^{5/2}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{5} \left (\frac {4}{5} \left (\frac {-\frac {1}{2} \sqrt {600250 \sqrt {35}-2361772} \int \frac {1}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\frac {1}{10} \left (665+216 \sqrt {35}\right ) \int -\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\frac {1}{10} \left (665+216 \sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\frac {1}{2} \sqrt {600250 \sqrt {35}-2361772} \int \frac {1}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {76}{5} \sqrt {2 x+1}\right )+\frac {16}{15} (2 x+1)^{3/2}\right )+\frac {4}{25} (2 x+1)^{5/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{5} \left (\frac {4}{5} \left (\frac {\frac {1}{10} \left (665+216 \sqrt {35}\right ) \int \frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\frac {1}{2} \sqrt {600250 \sqrt {35}-2361772} \int \frac {1}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\frac {1}{10} \left (665+216 \sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\frac {1}{2} \sqrt {600250 \sqrt {35}-2361772} \int \frac {1}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {76}{5} \sqrt {2 x+1}\right )+\frac {16}{15} (2 x+1)^{3/2}\right )+\frac {4}{25} (2 x+1)^{5/2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{5} \left (\frac {4}{5} \left (\frac {\sqrt {600250 \sqrt {35}-2361772} \int \frac {1}{-2 x+10 \left (2-\sqrt {35}\right )-1}d\left (10 \sqrt {2 x+1}-\sqrt {10 \left (2+\sqrt {35}\right )}\right )+\frac {1}{10} \left (665+216 \sqrt {35}\right ) \int \frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\sqrt {600250 \sqrt {35}-2361772} \int \frac {1}{-2 x+10 \left (2-\sqrt {35}\right )-1}d\left (10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}\right )+\frac {1}{10} \left (665+216 \sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {76}{5} \sqrt {2 x+1}\right )+\frac {16}{15} (2 x+1)^{3/2}\right )+\frac {4}{25} (2 x+1)^{5/2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{5} \left (\frac {4}{5} \left (\frac {\frac {1}{10} \left (665+216 \sqrt {35}\right ) \int \frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\sqrt {\frac {600250 \sqrt {35}-2361772}{10 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}-\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\frac {1}{10} \left (665+216 \sqrt {35}\right ) \int \frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}}d\sqrt {2 x+1}-\sqrt {\frac {600250 \sqrt {35}-2361772}{10 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {76}{5} \sqrt {2 x+1}\right )+\frac {16}{15} (2 x+1)^{3/2}\right )+\frac {4}{25} (2 x+1)^{5/2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{5} \left (\frac {4}{5} \left (\frac {-\sqrt {\frac {600250 \sqrt {35}-2361772}{10 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}-\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )-\frac {1}{10} \left (665+216 \sqrt {35}\right ) \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\frac {1}{10} \left (665+216 \sqrt {35}\right ) \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )-\sqrt {\frac {600250 \sqrt {35}-2361772}{10 \left (\sqrt {35}-2\right )}} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{2 \sqrt {14 \left (2+\sqrt {35}\right )}}\right )-\frac {76}{5} \sqrt {2 x+1}\right )+\frac {16}{15} (2 x+1)^{3/2}\right )+\frac {4}{25} (2 x+1)^{5/2}\) |
Input:
Int[(1 + 2*x)^(7/2)/(2 + 3*x + 5*x^2),x]
Output:
(4*(1 + 2*x)^(5/2))/25 + ((16*(1 + 2*x)^(3/2))/15 + ((-76*Sqrt[1 + 2*x])/5 + (4*((-(Sqrt[(-2361772 + 600250*Sqrt[35])/(10*(-2 + Sqrt[35]))]*ArcTan[( -Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]]) - ((665 + 216*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])]) + (-(Sqrt[(-2361772 + 600 250*Sqrt[35])/(10*(-2 + Sqrt[35]))]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*S qrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]]) + ((665 + 216*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)/5)/5
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)/(c*(m - 1))), x] + Simp[1/c Int[(d + e*x)^ (m - 2)*(Simp[c*d^2 - a*e^2 + e*(2*c*d - b*e)*x, x]/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[m, 1]
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[g*((d + e*x)^m/(c*m)), x] + Simp[1/c Int [(d + e*x)^(m - 1)*(Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x]/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && FractionQ[m] & & GtQ[m, 0]
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 = 4.82 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.15
| method | result | size |
| pseudoelliptic | \(-\frac {-24800 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (x^{2}+\frac {5}{3} x -\frac {11}{30}\right ) \sqrt {1+2 x}+\sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (233 \sqrt {5}+890 \sqrt {7}\right ) \left (\ln \left (-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+\sqrt {5}\, \sqrt {7}+5+10 x \right )-\ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+11780 \left (\sqrt {5}\, \sqrt {7}-\frac {216}{19}\right ) \left (\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )-\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )\right )}{38750 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(251\) |
| derivativedivides | \(\frac {4 \left (1+2 x \right )^{\frac {5}{2}}}{25}+\frac {16 \left (1+2 x \right )^{\frac {3}{2}}}{75}-\frac {76 \sqrt {1+2 x}}{125}-\frac {\left (890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+233 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+\sqrt {5}\, \sqrt {7}+5+10 x \right )}{38750}-\frac {2 \left (-1178 \sqrt {5}\, \sqrt {7}+\frac {\left (890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+233 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{3875 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {\left (890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+233 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{38750}+\frac {2 \left (1178 \sqrt {5}\, \sqrt {7}-\frac {\left (890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+233 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{3875 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(407\) |
| default | \(\frac {4 \left (1+2 x \right )^{\frac {5}{2}}}{25}+\frac {16 \left (1+2 x \right )^{\frac {3}{2}}}{75}-\frac {76 \sqrt {1+2 x}}{125}-\frac {\left (890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+233 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+\sqrt {5}\, \sqrt {7}+5+10 x \right )}{38750}-\frac {2 \left (-1178 \sqrt {5}\, \sqrt {7}+\frac {\left (890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+233 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{3875 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {\left (890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+233 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{38750}+\frac {2 \left (1178 \sqrt {5}\, \sqrt {7}-\frac {\left (890 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+233 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{3875 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(407\) |
| trager | \(\left (\frac {16}{25} x^{2}+\frac {16}{15} x -\frac {88}{375}\right ) \sqrt {1+2 x}-\frac {\operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}+12867859375\right ) \ln \left (\frac {-408425 x \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}+12867859375\right )^{5}+990852845 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}+12867859375\right )^{3} x +7453543125 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}+12867859375\right )^{2} \sqrt {1+2 x}-717162680 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}+12867859375\right )^{3}+525953389168 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}+12867859375\right ) x -83912677753625 \sqrt {1+2 x}+2061191406976 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}+12867859375\right )}{155 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}+12867859375\right )^{2} x -270761 x -136084}\right )}{125}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}+12867859375\right )^{2}-52296380\right ) \ln \left (-\frac {81685 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}+12867859375\right )^{2}-52296380\right ) \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}+12867859375\right )^{4} x -157444815 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}+12867859375\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}+12867859375\right )^{2}-52296380\right ) x +231059836875 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}+12867859375\right )^{2} \sqrt {1+2 x}-143432536 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}+12867859375\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}+12867859375\right )^{2}-52296380\right )-149515537500 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}+12867859375\right )^{2}-52296380\right ) x +2098333883121875 \sqrt {1+2 x}-100021740000 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}+12867859375\right )^{2}-52296380\right )}{155 \operatorname {RootOf}\left (4805 \textit {\_Z}^{4}-10459276 \textit {\_Z}^{2}+12867859375\right )^{2} x -66635 x +136084}\right )}{19375}\) | \(440\) |
| risch | \(\frac {8 \left (30 x^{2}+50 x -11\right ) \sqrt {1+2 x}}{375}-\frac {89 \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}}{3875}-\frac {233 \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}}{38750}-\frac {178 \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}}{3875 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {233 \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 )}{3875 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {76 \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}}{125 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {89 \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}}{3875}+\frac {233 \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}}{38750}-\frac {178 \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}}{3875 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {233 \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 )}{3875 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {76 \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}}{125 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\) | \(626\) |
Input:
int((1+2*x)^(7/2)/(5*x^2+3*x+2),x,method=_RETURNVERBOSE)
Output:
-1/38750/(10*5^(1/2)*7^(1/2)-20)^(1/2)*(-24800*(10*5^(1/2)*7^(1/2)-20)^(1/ 2)*(x^2+5/3*x-11/30)*(1+2*x)^(1/2)+(10*5^(1/2)*7^(1/2)-20)^(1/2)*(233*5^(1 /2)+890*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)+11780*(5^(1/2)*7^(1/ 2)-216/19)*(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))))
Time = 0.08 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.11 \[ \int \frac {(1+2 x)^{7/2}}{2+3 x+5 x^2} \, dx=\frac {8}{375} \, {\left (30 \, x^{2} + 50 \, x - 11\right )} \sqrt {2 \, x + 1} - \frac {2}{125} \, \sqrt {\frac {42875}{62} \, \sqrt {\frac {7}{5}} - \frac {84349}{155}} \arctan \left (\frac {10}{34021} \, \sqrt {2 \, x + 1} \sqrt {\frac {42875}{62} \, \sqrt {\frac {7}{5}} - \frac {84349}{155}} {\left (95 \, \sqrt {\frac {7}{5}} + 216\right )} + \frac {10}{34021} \, \sqrt {\frac {42875}{62} \, \sqrt {\frac {7}{5}} + \frac {84349}{155}} \sqrt {\frac {42875}{62} \, \sqrt {\frac {7}{5}} - \frac {84349}{155}} {\left (5 \, \sqrt {\frac {7}{5}} + 2\right )}\right ) + \frac {2}{125} \, \sqrt {\frac {42875}{62} \, \sqrt {\frac {7}{5}} - \frac {84349}{155}} \arctan \left (-\frac {10}{34021} \, \sqrt {2 \, x + 1} \sqrt {\frac {42875}{62} \, \sqrt {\frac {7}{5}} - \frac {84349}{155}} {\left (95 \, \sqrt {\frac {7}{5}} + 216\right )} + \frac {10}{34021} \, \sqrt {\frac {42875}{62} \, \sqrt {\frac {7}{5}} + \frac {84349}{155}} \sqrt {\frac {42875}{62} \, \sqrt {\frac {7}{5}} - \frac {84349}{155}} {\left (5 \, \sqrt {\frac {7}{5}} + 2\right )}\right ) + \frac {1}{125} \, \sqrt {\frac {42875}{62} \, \sqrt {\frac {7}{5}} + \frac {84349}{155}} \log \left (2 \, \sqrt {2 \, x + 1} {\left (890 \, \sqrt {\frac {7}{5}} - 233\right )} \sqrt {\frac {42875}{62} \, \sqrt {\frac {7}{5}} + \frac {84349}{155}} + 68042 \, x + 34021 \, \sqrt {\frac {7}{5}} + 34021\right ) - \frac {1}{125} \, \sqrt {\frac {42875}{62} \, \sqrt {\frac {7}{5}} + \frac {84349}{155}} \log \left (-2 \, \sqrt {2 \, x + 1} {\left (890 \, \sqrt {\frac {7}{5}} - 233\right )} \sqrt {\frac {42875}{62} \, \sqrt {\frac {7}{5}} + \frac {84349}{155}} + 68042 \, x + 34021 \, \sqrt {\frac {7}{5}} + 34021\right ) \] Input:
integrate((1+2*x)^(7/2)/(5*x^2+3*x+2),x, algorithm="fricas")
Output:
8/375*(30*x^2 + 50*x - 11)*sqrt(2*x + 1) - 2/125*sqrt(42875/62*sqrt(7/5) - 84349/155)*arctan(10/34021*sqrt(2*x + 1)*sqrt(42875/62*sqrt(7/5) - 84349/ 155)*(95*sqrt(7/5) + 216) + 10/34021*sqrt(42875/62*sqrt(7/5) + 84349/155)* sqrt(42875/62*sqrt(7/5) - 84349/155)*(5*sqrt(7/5) + 2)) + 2/125*sqrt(42875 /62*sqrt(7/5) - 84349/155)*arctan(-10/34021*sqrt(2*x + 1)*sqrt(42875/62*sq rt(7/5) - 84349/155)*(95*sqrt(7/5) + 216) + 10/34021*sqrt(42875/62*sqrt(7/ 5) + 84349/155)*sqrt(42875/62*sqrt(7/5) - 84349/155)*(5*sqrt(7/5) + 2)) + 1/125*sqrt(42875/62*sqrt(7/5) + 84349/155)*log(2*sqrt(2*x + 1)*(890*sqrt(7 /5) - 233)*sqrt(42875/62*sqrt(7/5) + 84349/155) + 68042*x + 34021*sqrt(7/5 ) + 34021) - 1/125*sqrt(42875/62*sqrt(7/5) + 84349/155)*log(-2*sqrt(2*x + 1)*(890*sqrt(7/5) - 233)*sqrt(42875/62*sqrt(7/5) + 84349/155) + 68042*x + 34021*sqrt(7/5) + 34021)
\[ \int \frac {(1+2 x)^{7/2}}{2+3 x+5 x^2} \, dx=\int \frac {\left (2 x + 1\right )^{\frac {7}{2}}}{5 x^{2} + 3 x + 2}\, dx \] Input:
integrate((1+2*x)**(7/2)/(5*x**2+3*x+2),x)
Output:
Integral((2*x + 1)**(7/2)/(5*x**2 + 3*x + 2), x)
\[ \int \frac {(1+2 x)^{7/2}}{2+3 x+5 x^2} \, dx=\int { \frac {{\left (2 \, x + 1\right )}^{\frac {7}{2}}}{5 \, x^{2} + 3 \, x + 2} \,d x } \] Input:
integrate((1+2*x)^(7/2)/(5*x^2+3*x+2),x, algorithm="maxima")
Output:
integrate((2*x + 1)^(7/2)/(5*x^2 + 3*x + 2), x)
Leaf count of result is larger than twice the leaf count of optimal. 614 vs. \(2 (152) = 304\).
Time = 0.91 (sec) , antiderivative size = 614, normalized size of antiderivative = 2.80 \[ \int \frac {(1+2 x)^{7/2}}{2+3 x+5 x^2} \, dx=\text {Too large to display} \] Input:
integrate((1+2*x)^(7/2)/(5*x^2+3*x+2),x, algorithm="giac")
Output:
4/25*(2*x + 1)^(5/2) - 1/1162984375*sqrt(31)*(11340*sqrt(31)*(7/5)^(3/4)*( 2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - 54*sqrt(31)*(7/5)^(3/4)*(-14 0*sqrt(35) + 2450)^(3/2) + 108*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 2 2680*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 162925*sqrt (31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) - 325850*(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)) - 1/1162984375*sqrt(31) *(11340*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - 54*sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 108*(7/5)^(3/4)*( 140*sqrt(35) + 2450)^(3/2) + 22680*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*( 2*sqrt(35) - 35) - 162925*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) - 325850*(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)) - 1/2325968750*sqrt(31)*(54*sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 2 450)^(3/2) + 11340*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt( 35) - 35) - 22680*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 108*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) - 162925*sqrt(31)*(7/5)^(1 /4)*sqrt(140*sqrt(35) + 2450) + 325850*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 24 50))*log(2*(7/5)^(1/4)*sqrt(2*x + 1)*sqrt(1/35*sqrt(35) + 1/2) + 2*x + sqr t(7/5) + 1) + 1/2325968750*sqrt(31)*(54*sqrt(31)*(7/5)^(3/4)*(140*sqrt(...
Time = 5.76 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.91 \[ \int \frac {(1+2 x)^{7/2}}{2+3 x+5 x^2} \, dx=\frac {16\,{\left (2\,x+1\right )}^{3/2}}{75}-\frac {76\,\sqrt {2\,x+1}}{125}+\frac {4\,{\left (2\,x+1\right )}^{5/2}}{25}+\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {168698+\sqrt {31}\,34021{}\mathrm {i}}\,\sqrt {2\,x+1}\,4354688{}\mathrm {i}}{6103515625\,\left (-\frac {5425941248}{1220703125}+\frac {\sqrt {31}\,579173504{}\mathrm {i}}{1220703125}\right )}+\frac {8709376\,\sqrt {31}\,\sqrt {155}\,\sqrt {168698+\sqrt {31}\,34021{}\mathrm {i}}\,\sqrt {2\,x+1}}{189208984375\,\left (-\frac {5425941248}{1220703125}+\frac {\sqrt {31}\,579173504{}\mathrm {i}}{1220703125}\right )}\right )\,\sqrt {168698+\sqrt {31}\,34021{}\mathrm {i}}\,2{}\mathrm {i}}{19375}-\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {168698-\sqrt {31}\,34021{}\mathrm {i}}\,\sqrt {2\,x+1}\,4354688{}\mathrm {i}}{6103515625\,\left (\frac {5425941248}{1220703125}+\frac {\sqrt {31}\,579173504{}\mathrm {i}}{1220703125}\right )}-\frac {8709376\,\sqrt {31}\,\sqrt {155}\,\sqrt {168698-\sqrt {31}\,34021{}\mathrm {i}}\,\sqrt {2\,x+1}}{189208984375\,\left (\frac {5425941248}{1220703125}+\frac {\sqrt {31}\,579173504{}\mathrm {i}}{1220703125}\right )}\right )\,\sqrt {168698-\sqrt {31}\,34021{}\mathrm {i}}\,2{}\mathrm {i}}{19375} \] Input:
int((2*x + 1)^(7/2)/(3*x + 5*x^2 + 2),x)
Output:
(16*(2*x + 1)^(3/2))/75 - (76*(2*x + 1)^(1/2))/125 + (4*(2*x + 1)^(5/2))/2 5 + (155^(1/2)*atan((155^(1/2)*(31^(1/2)*34021i + 168698)^(1/2)*(2*x + 1)^ (1/2)*4354688i)/(6103515625*((31^(1/2)*579173504i)/1220703125 - 5425941248 /1220703125)) + (8709376*31^(1/2)*155^(1/2)*(31^(1/2)*34021i + 168698)^(1/ 2)*(2*x + 1)^(1/2))/(189208984375*((31^(1/2)*579173504i)/1220703125 - 5425 941248/1220703125)))*(31^(1/2)*34021i + 168698)^(1/2)*2i)/19375 - (155^(1/ 2)*atan((155^(1/2)*(168698 - 31^(1/2)*34021i)^(1/2)*(2*x + 1)^(1/2)*435468 8i)/(6103515625*((31^(1/2)*579173504i)/1220703125 + 5425941248/1220703125) ) - (8709376*31^(1/2)*155^(1/2)*(168698 - 31^(1/2)*34021i)^(1/2)*(2*x + 1) ^(1/2))/(189208984375*((31^(1/2)*579173504i)/1220703125 + 5425941248/12207 03125)))*(168698 - 31^(1/2)*34021i)^(1/2)*2i)/19375
Time = 0.19 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.51 \[ \int \frac {(1+2 x)^{7/2}}{2+3 x+5 x^2} \, dx=\frac {178 \sqrt {\sqrt {35}-2}\, \sqrt {14}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {35}+2}\, \sqrt {2}-2 \sqrt {2 x +1}\, \sqrt {5}}{\sqrt {\sqrt {35}-2}\, \sqrt {2}}\right )}{3875}-\frac {233 \sqrt {\sqrt {35}-2}\, \sqrt {10}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {35}+2}\, \sqrt {2}-2 \sqrt {2 x +1}\, \sqrt {5}}{\sqrt {\sqrt {35}-2}\, \sqrt {2}}\right )}{19375}-\frac {178 \sqrt {\sqrt {35}-2}\, \sqrt {14}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {35}+2}\, \sqrt {2}+2 \sqrt {2 x +1}\, \sqrt {5}}{\sqrt {\sqrt {35}-2}\, \sqrt {2}}\right )}{3875}+\frac {233 \sqrt {\sqrt {35}-2}\, \sqrt {10}\, \mathit {atan} \left (\frac {\sqrt {\sqrt {35}+2}\, \sqrt {2}+2 \sqrt {2 x +1}\, \sqrt {5}}{\sqrt {\sqrt {35}-2}\, \sqrt {2}}\right )}{19375}-\frac {89 \sqrt {\sqrt {35}+2}\, \sqrt {14}\, \mathrm {log}\left (-\sqrt {2 x +1}\, \sqrt {\sqrt {35}+2}\, \sqrt {2}+\sqrt {7}+2 \sqrt {5}\, x +\sqrt {5}\right )}{3875}+\frac {89 \sqrt {\sqrt {35}+2}\, \sqrt {14}\, \mathrm {log}\left (\sqrt {2 x +1}\, \sqrt {\sqrt {35}+2}\, \sqrt {2}+\sqrt {7}+2 \sqrt {5}\, x +\sqrt {5}\right )}{3875}-\frac {233 \sqrt {\sqrt {35}+2}\, \sqrt {10}\, \mathrm {log}\left (-\sqrt {2 x +1}\, \sqrt {\sqrt {35}+2}\, \sqrt {2}+\sqrt {7}+2 \sqrt {5}\, x +\sqrt {5}\right )}{38750}+\frac {233 \sqrt {\sqrt {35}+2}\, \sqrt {10}\, \mathrm {log}\left (\sqrt {2 x +1}\, \sqrt {\sqrt {35}+2}\, \sqrt {2}+\sqrt {7}+2 \sqrt {5}\, x +\sqrt {5}\right )}{38750}+\frac {16 \sqrt {2 x +1}\, x^{2}}{25}+\frac {16 \sqrt {2 x +1}\, x}{15}-\frac {88 \sqrt {2 x +1}}{375} \] Input:
int((1+2*x)^(7/2)/(5*x^2+3*x+2),x)
Output:
(5340*sqrt(sqrt(35) - 2)*sqrt(14)*atan((sqrt(sqrt(35) + 2)*sqrt(2) - 2*sqr t(2*x + 1)*sqrt(5))/(sqrt(sqrt(35) - 2)*sqrt(2))) - 1398*sqrt(sqrt(35) - 2 )*sqrt(10)*atan((sqrt(sqrt(35) + 2)*sqrt(2) - 2*sqrt(2*x + 1)*sqrt(5))/(sq rt(sqrt(35) - 2)*sqrt(2))) - 5340*sqrt(sqrt(35) - 2)*sqrt(14)*atan((sqrt(s qrt(35) + 2)*sqrt(2) + 2*sqrt(2*x + 1)*sqrt(5))/(sqrt(sqrt(35) - 2)*sqrt(2 ))) + 1398*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))) - 2670*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)) + 2670*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)) - 699*sq rt(sqrt(35) + 2)*sqrt(10)*log( - sqrt(2*x + 1)*sqrt(sqrt(35) + 2)*sqrt(2) + sqrt(7) + 2*sqrt(5)*x + sqrt(5)) + 699*sqrt(sqrt(35) + 2)*sqrt(10)*log(s qrt(2*x + 1)*sqrt(sqrt(35) + 2)*sqrt(2) + sqrt(7) + 2*sqrt(5)*x + sqrt(5)) + 74400*sqrt(2*x + 1)*x**2 + 124000*sqrt(2*x + 1)*x - 27280*sqrt(2*x + 1) )/116250