\(\int \frac {(1+2 x)^{7/2}}{(2+3 x+5 x^2)^2} \, dx\) [2316]

Optimal result

Integrand size = 22, antiderivative size = 296 \[ \int \frac {(1+2 x)^{7/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {604}{775} \sqrt {1+2 x}-\frac {8}{155} (1+2 x)^{3/2}-\frac {(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {1}{775} \sqrt {\frac {2}{155} \left (-5682718+968975 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )-\frac {1}{775} \sqrt {\frac {2}{155} \left (-5682718+968975 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )+\frac {1}{775} \sqrt {\frac {1}{310} \left (5682718+968975 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )-\frac {1}{775} \sqrt {\frac {1}{310} \left (5682718+968975 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right ) \]

[Out]

-8/155*(1+2*x)^(3/2)-1/31*(5-4*x)*(1+2*x)^(5/2)/(5*x^2+3*x+2)+604/775*(1+2*x)^(1/2)+1/120125*arctan((-10*(1+2*
x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(-1761642580+300382250*35^(1/2))^(1/2)-1/120125*arct
an((10*(1+2*x)^(1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(-1761642580+300382250*35^(1/2))^(1/2)+1
/240250*ln(5+10*x+35^(1/2)-(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(1761642580+300382250*35^(1/2))^(1/2)-1/24025
0*ln(5+10*x+35^(1/2)+(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(1761642580+300382250*35^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {752, 838, 840, 1183, 648, 632, 210, 642} \[ \int \frac {(1+2 x)^{7/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {1}{775} \sqrt {\frac {2}{155} \left (968975 \sqrt {35}-5682718\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )-\frac {1}{775} \sqrt {\frac {2}{155} \left (968975 \sqrt {35}-5682718\right )} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )-\frac {(5-4 x) (2 x+1)^{5/2}}{31 \left (5 x^2+3 x+2\right )}-\frac {8}{155} (2 x+1)^{3/2}+\frac {604}{775} \sqrt {2 x+1}+\frac {1}{775} \sqrt {\frac {1}{310} \left (5682718+968975 \sqrt {35}\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )-\frac {1}{775} \sqrt {\frac {1}{310} \left (5682718+968975 \sqrt {35}\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right ) \]

[In]

Int[(1 + 2*x)^(7/2)/(2 + 3*x + 5*x^2)^2,x]

[Out]

(604*Sqrt[1 + 2*x])/775 - (8*(1 + 2*x)^(3/2))/155 - ((5 - 4*x)*(1 + 2*x)^(5/2))/(31*(2 + 3*x + 5*x^2)) + (Sqrt
[(2*(-5682718 + 968975*Sqrt[35]))/155]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[
35])]])/775 - (Sqrt[(2*(-5682718 + 968975*Sqrt[35]))/155]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/
Sqrt[10*(-2 + Sqrt[35])]])/775 + (Sqrt[(5682718 + 968975*Sqrt[35])/310]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]
*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/775 - (Sqrt[(5682718 + 968975*Sqrt[35])/310]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[3
5])]*Sqrt[1 + 2*x] + 5*(1 + 2*x)])/775

Rule 210

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])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[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]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 752

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m - 1)*(d
*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Dist[1/((p + 1)*(b^2 -
4*a*c)), Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*
c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] &
& NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, b, c, d,
e, m, p, x]

Rule 838

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] + Dist[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] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*
e^2, 0] && FractionQ[m] && GtQ[m, 0]

Rule 840

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[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] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 1183

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]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[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] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]

Rubi steps \begin{align*} \text {integral}& = -\frac {(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {1}{31} \int \frac {(29-12 x) (1+2 x)^{3/2}}{2+3 x+5 x^2} \, dx \\ & = -\frac {8}{155} (1+2 x)^{3/2}-\frac {(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {1}{155} \int \frac {\sqrt {1+2 x} (193+302 x)}{2+3 x+5 x^2} \, dx \\ & = \frac {604}{775} \sqrt {1+2 x}-\frac {8}{155} (1+2 x)^{3/2}-\frac {(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {1}{775} \int \frac {-243+1628 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx \\ & = \frac {604}{775} \sqrt {1+2 x}-\frac {8}{155} (1+2 x)^{3/2}-\frac {(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {2}{775} \text {Subst}\left (\int \frac {-2114+1628 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right ) \\ & = \frac {604}{775} \sqrt {1+2 x}-\frac {8}{155} (1+2 x)^{3/2}-\frac {(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {\text {Subst}\left (\int \frac {-2114 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (-2114-1628 \sqrt {\frac {7}{5}}\right ) x}{\sqrt {\frac {7}{5}}-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{775 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\text {Subst}\left (\int \frac {-2114 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (-2114-1628 \sqrt {\frac {7}{5}}\right ) x}{\sqrt {\frac {7}{5}}+\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{775 \sqrt {14 \left (2+\sqrt {35}\right )}} \\ & = \frac {604}{775} \sqrt {1+2 x}-\frac {8}{155} (1+2 x)^{3/2}-\frac {(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}-\frac {\sqrt {1460631-245828 \sqrt {35}} \text {Subst}\left (\int \frac {1}{\sqrt {\frac {7}{5}}-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{3875}-\frac {\sqrt {1460631-245828 \sqrt {35}} \text {Subst}\left (\int \frac {1}{\sqrt {\frac {7}{5}}+\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )}{3875}+\frac {1}{775} \sqrt {\frac {1}{310} \left (5682718+968975 \sqrt {35}\right )} \text {Subst}\left (\int \frac {-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 x}{\sqrt {\frac {7}{5}}-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right )-\frac {1}{775} \sqrt {\frac {1}{310} \left (5682718+968975 \sqrt {35}\right )} \text {Subst}\left (\int \frac {\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 x}{\sqrt {\frac {7}{5}}+\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )} x+x^2} \, dx,x,\sqrt {1+2 x}\right ) \\ & = \frac {604}{775} \sqrt {1+2 x}-\frac {8}{155} (1+2 x)^{3/2}-\frac {(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {1}{775} \sqrt {\frac {1}{310} \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 )-\frac {1}{775} \sqrt {\frac {1}{310} \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 )+\frac {\left (2 \sqrt {1460631-245828 \sqrt {35}}\right ) \text {Subst}\left (\int \frac {1}{\frac {2}{5} \left (2-\sqrt {35}\right )-x^2} \, dx,x,-\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 \sqrt {1+2 x}\right )}{3875}+\frac {\left (2 \sqrt {1460631-245828 \sqrt {35}}\right ) \text {Subst}\left (\int \frac {1}{\frac {2}{5} \left (2-\sqrt {35}\right )-x^2} \, dx,x,\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 \sqrt {1+2 x}\right )}{3875} \\ & = \frac {604}{775} \sqrt {1+2 x}-\frac {8}{155} (1+2 x)^{3/2}-\frac {(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac {1}{775} \sqrt {\frac {2}{155} \left (-5682718+968975 \sqrt {35}\right )} \tan ^{-1}\left (\sqrt {\frac {5}{2 \left (-2+\sqrt {35}\right )}} \left (\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-2 \sqrt {1+2 x}\right )\right )-\frac {1}{775} \sqrt {\frac {2}{155} \left (-5682718+968975 \sqrt {35}\right )} \tan ^{-1}\left (\sqrt {\frac {5}{2 \left (-2+\sqrt {35}\right )}} \left (\sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+2 \sqrt {1+2 x}\right )\right )+\frac {1}{775} \sqrt {\frac {1}{310} \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 )-\frac {1}{775} \sqrt {\frac {1}{310} \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 ) \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.18 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.46 \[ \int \frac {(1+2 x)^{7/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {2 \left (\frac {155 \sqrt {1+2 x} \left (1003+1132 x+2480 x^2\right )}{4+6 x+10 x^2}-\sqrt {155 \left (-5682718+135439 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )-\sqrt {155 \left (-5682718-135439 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )\right )}{120125} \]

[In]

Integrate[(1 + 2*x)^(7/2)/(2 + 3*x + 5*x^2)^2,x]

[Out]

(2*((155*Sqrt[1 + 2*x]*(1003 + 1132*x + 2480*x^2))/(4 + 6*x + 10*x^2) - Sqrt[155*(-5682718 + (135439*I)*Sqrt[3
1])]*ArcTan[Sqrt[(-2 - I*Sqrt[31])/7]*Sqrt[1 + 2*x]] - Sqrt[155*(-5682718 - (135439*I)*Sqrt[31])]*ArcTan[Sqrt[
(I/7)*(2*I + Sqrt[31])]*Sqrt[1 + 2*x]]))/120125

Maple [A] (verified)

Time = 0.60 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.07

method	result	size
pseudoelliptic	\(\frac {18285 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right ) \left (\sqrt {5}+\frac {2560 \sqrt {7}}{3657}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )-18285 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right ) \left (\sqrt {5}+\frac {2560 \sqrt {7}}{3657}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )+768800 \left (x^{2}+\frac {283}{620} x +\frac {1003}{2480}\right ) \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \sqrt {1+2 x}+468100 \left (\sqrt {5}\, \sqrt {7}-\frac {814}{151}\right ) \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right ) \left (\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )-\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )\right )}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \left (1201250 x^{2}+720750 x +480500\right )}\)	\(317\)
derivativedivides	\(\frac {16 \sqrt {1+2 x}}{25}+\frac {-\frac {712 \left (1+2 x \right )^{\frac {3}{2}}}{3875}+\frac {756 \sqrt {1+2 x}}{3875}}{\left (1+2 x \right )^{2}+\frac {3}{5}-\frac {8 x}{5}}+\frac {\left (3657 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+2560 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{240250}+\frac {2 \left (-9362 \sqrt {5}\, \sqrt {7}+\frac {\left (3657 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+2560 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{24025 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {\left (-3657 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-2560 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{240250}+\frac {2 \left (-9362 \sqrt {5}\, \sqrt {7}-\frac {\left (-3657 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-2560 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{24025 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(424\)
default	\(\frac {16 \sqrt {1+2 x}}{25}+\frac {-\frac {712 \left (1+2 x \right )^{\frac {3}{2}}}{3875}+\frac {756 \sqrt {1+2 x}}{3875}}{\left (1+2 x \right )^{2}+\frac {3}{5}-\frac {8 x}{5}}+\frac {\left (3657 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+2560 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{240250}+\frac {2 \left (-9362 \sqrt {5}\, \sqrt {7}+\frac {\left (3657 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+2560 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{24025 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {\left (-3657 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-2560 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{240250}+\frac {2 \left (-9362 \sqrt {5}\, \sqrt {7}-\frac {\left (-3657 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-2560 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{24025 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(424\)
trager	\(\frac {\left (2480 x^{2}+1132 x +1003\right ) \sqrt {1+2 x}}{3875 x^{2}+2325 x +1550}+\frac {2 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}[phone]\right ) \ln \left (-\frac {160679200 x \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}[phone]\right )^{5}-1462685639320 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}[phone]\right )^{3} x -280802969920 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}[phone]\right )^{3}[phone] \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}[phone]\right )^{2} \sqrt {1+2 x}-2735202958459332 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}[phone]\right ) x -446948973045024 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}[phone]\right )+33331705773254525 \sqrt {1+2 x}}{620 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}[phone]\right )^{2} x -5276401 x +541756}\right )}{775}+\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}[phone]\right )^{2}-440410645\right ) \ln \left (-\frac {32135840 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}[phone]\right )^{2}-440410645\right ) \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}[phone]\right )^{4} x -885649429640 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}[phone]\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}[phone]\right )^{2}-440410645\right ) x [phone] \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}[phone]\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}[phone]\right )^{2}-440410645\right )-1081017958534500 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}[phone]\right )^{2} \sqrt {1+2 x}+4889233526572500 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}[phone]\right )^{2}-440410645\right ) x -1118889208556000 \operatorname {RootOf}\left (\textit {\_Z}^{2}+24025 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}[phone]\right )^{2}-440410645\right )+20849799689574944375 \sqrt {1+2 x}}{620 \operatorname {RootOf}\left (76880 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}[phone]\right )^{2} x -6089035 x -541756}\right )}{120125}\)	\(454\)
risch	\(\frac {\left (2480 x^{2}+1132 x +1003\right ) \sqrt {1+2 x}}{3875 x^{2}+2325 x +1550}+\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 {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{240250}+\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 {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{24025}+\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 ) \left (2 \sqrt {5}\, \sqrt {7}+4\right )}{24025 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {512 \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}}{24025 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {604 \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}}{775 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\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 {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{240250}-\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 {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{24025}+\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 ) \left (2 \sqrt {5}\, \sqrt {7}+4\right )}{24025 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {512 \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}}{24025 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {604 \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}}{775 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(638\)

[In]

int((1+2*x)^(7/2)/(5*x^2+3*x+2)^2,x,method=_RETURNVERBOSE)

[Out]

468100/(10*5^(1/2)*7^(1/2)-20)^(1/2)*(3657/93620*(10*5^(1/2)*7^(1/2)-20)^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)*(x^
2+3/5*x+2/5)*(5^(1/2)+2560/3657*7^(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)-3657/93620*(10*5^(1/2)*7^(1/2)-20)^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)*(x^2+3/5*x+2/5)*(5^(1/2)+2560/365
7*7^(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)+248/151*(x^2+283/620*x
+1003/2480)*(10*5^(1/2)*7^(1/2)-20)^(1/2)*(1+2*x)^(1/2)+(5^(1/2)*7^(1/2)-814/151)*(x^2+3/5*x+2/5)*(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))))/(1201250*x^2+720750*x+480500)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.40 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.85 \[ \int \frac {(1+2 x)^{7/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=-\frac {\sqrt {155} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {541756 i \, \sqrt {31} + 22730872} \log \left (\sqrt {155} \sqrt {541756 i \, \sqrt {31} + 22730872} {\left (512 i \, \sqrt {31} + 4681\right )} + 300382250 \, \sqrt {2 \, x + 1}\right ) - \sqrt {155} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {541756 i \, \sqrt {31} + 22730872} \log \left (\sqrt {155} \sqrt {541756 i \, \sqrt {31} + 22730872} {\left (-512 i \, \sqrt {31} - 4681\right )} + 300382250 \, \sqrt {2 \, x + 1}\right ) - \sqrt {155} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {-541756 i \, \sqrt {31} + 22730872} \log \left (\sqrt {155} {\left (512 i \, \sqrt {31} - 4681\right )} \sqrt {-541756 i \, \sqrt {31} + 22730872} + 300382250 \, \sqrt {2 \, x + 1}\right ) + \sqrt {155} {\left (5 \, x^{2} + 3 \, x + 2\right )} \sqrt {-541756 i \, \sqrt {31} + 22730872} \log \left (\sqrt {155} {\left (-512 i \, \sqrt {31} + 4681\right )} \sqrt {-541756 i \, \sqrt {31} + 22730872} + 300382250 \, \sqrt {2 \, x + 1}\right ) - 310 \, {\left (2480 \, x^{2} + 1132 \, x + 1003\right )} \sqrt {2 \, x + 1}}{240250 \, {\left (5 \, x^{2} + 3 \, x + 2\right )}} \]

[In]

integrate((1+2*x)^(7/2)/(5*x^2+3*x+2)^2,x, algorithm="fricas")

[Out]

-1/240250*(sqrt(155)*(5*x^2 + 3*x + 2)*sqrt(541756*I*sqrt(31) + 22730872)*log(sqrt(155)*sqrt(541756*I*sqrt(31)
+ 22730872)*(512*I*sqrt(31) + 4681) + 300382250*sqrt(2*x + 1)) - sqrt(155)*(5*x^2 + 3*x + 2)*sqrt(541756*I*sq
rt(31) + 22730872)*log(sqrt(155)*sqrt(541756*I*sqrt(31) + 22730872)*(-512*I*sqrt(31) - 4681) + 300382250*sqrt(
2*x + 1)) - sqrt(155)*(5*x^2 + 3*x + 2)*sqrt(-541756*I*sqrt(31) + 22730872)*log(sqrt(155)*(512*I*sqrt(31) - 46
81)*sqrt(-541756*I*sqrt(31) + 22730872) + 300382250*sqrt(2*x + 1)) + sqrt(155)*(5*x^2 + 3*x + 2)*sqrt(-541756*
I*sqrt(31) + 22730872)*log(sqrt(155)*(-512*I*sqrt(31) + 4681)*sqrt(-541756*I*sqrt(31) + 22730872) + 300382250*
sqrt(2*x + 1)) - 310*(2480*x^2 + 1132*x + 1003)*sqrt(2*x + 1))/(5*x^2 + 3*x + 2)

Sympy [F]

\[ \int \frac {(1+2 x)^{7/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\int \frac {\left (2 x + 1\right )^{\frac {7}{2}}}{\left (5 x^{2} + 3 x + 2\right )^{2}}\, dx \]

[In]

integrate((1+2*x)**(7/2)/(5*x**2+3*x+2)**2,x)

[Out]

Integral((2*x + 1)**(7/2)/(5*x**2 + 3*x + 2)**2, x)

Maxima [F]

\[ \int \frac {(1+2 x)^{7/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\int { \frac {{\left (2 \, x + 1\right )}^{\frac {7}{2}}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{2}} \,d x } \]

[In]

integrate((1+2*x)^(7/2)/(5*x^2+3*x+2)^2,x, algorithm="maxima")

[Out]

integrate((2*x + 1)^(7/2)/(5*x^2 + 3*x + 2)^2, x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 633 vs. \(2 (205) = 410\).

Time = 0.68 (sec) , antiderivative size = 633, normalized size of antiderivative = 2.14 \[ \int \frac {(1+2 x)^{7/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

integrate((1+2*x)^(7/2)/(5*x^2+3*x+2)^2,x, algorithm="giac")

[Out]

1/14421006250*sqrt(31)*(85470*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - 407*sqrt(31)
*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 814*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 170940*(7/5)^(3/4)*s
qrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 2589650*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) - 5179300
*(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)) + 1/14421006250*sqrt(31)*(85470*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sq
rt(-140*sqrt(35) + 2450) - 407*sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 814*(7/5)^(3/4)*(140*sqrt(3
5) + 2450)^(3/2) + 170940*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 2589650*sqrt(31)*(7/5)^(1/
4)*sqrt(-140*sqrt(35) + 2450) - 5179300*(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)) + 1/28842012500*sqrt(31)*(407*sqr
t(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 85470*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(3
5) - 35) - 170940*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 814*(7/5)^(3/4)*(-140*sqrt(35) +
2450)^(3/2) - 2589650*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) + 5179300*(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) - 1/28842012500*sqrt
(31)*(407*sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 85470*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 24
50)*(2*sqrt(35) - 35) - 170940*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 814*(7/5)^(3/4)*(-14
0*sqrt(35) + 2450)^(3/2) - 2589650*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) + 5179300*(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) + 16/2
5*sqrt(2*x + 1) - 4/775*(178*(2*x + 1)^(3/2) - 189*sqrt(2*x + 1))/(5*(2*x + 1)^2 - 8*x + 3)

Mupad [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.73 \[ \int \frac {(1+2 x)^{7/2}}{\left (2+3 x+5 x^2\right )^2} \, dx=\frac {16\,\sqrt {2\,x+1}}{25}+\frac {\frac {756\,\sqrt {2\,x+1}}{3875}-\frac {712\,{\left (2\,x+1\right )}^{3/2}}{3875}}{{\left (2\,x+1\right )}^2-\frac {8\,x}{5}+\frac {3}{5}}-\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {5682718-\sqrt {31}\,135439{}\mathrm {i}}\,\sqrt {2\,x+1}\,559232{}\mathrm {i}}{46923828125\,\left (-\frac {2004287488}{9384765625}+\frac {\sqrt {31}\,591108224{}\mathrm {i}}{9384765625}\right )}+\frac {1118464\,\sqrt {31}\,\sqrt {155}\,\sqrt {5682718-\sqrt {31}\,135439{}\mathrm {i}}\,\sqrt {2\,x+1}}{1454638671875\,\left (-\frac {2004287488}{9384765625}+\frac {\sqrt {31}\,591108224{}\mathrm {i}}{9384765625}\right )}\right )\,\sqrt {5682718-\sqrt {31}\,135439{}\mathrm {i}}\,2{}\mathrm {i}}{120125}+\frac {\sqrt {155}\,\mathrm {atan}\left (\frac {\sqrt {155}\,\sqrt {5682718+\sqrt {31}\,135439{}\mathrm {i}}\,\sqrt {2\,x+1}\,559232{}\mathrm {i}}{46923828125\,\left (\frac {2004287488}{9384765625}+\frac {\sqrt {31}\,591108224{}\mathrm {i}}{9384765625}\right )}-\frac {1118464\,\sqrt {31}\,\sqrt {155}\,\sqrt {5682718+\sqrt {31}\,135439{}\mathrm {i}}\,\sqrt {2\,x+1}}{1454638671875\,\left (\frac {2004287488}{9384765625}+\frac {\sqrt {31}\,591108224{}\mathrm {i}}{9384765625}\right )}\right )\,\sqrt {5682718+\sqrt {31}\,135439{}\mathrm {i}}\,2{}\mathrm {i}}{120125} \]

[In]

int((2*x + 1)^(7/2)/(3*x + 5*x^2 + 2)^2,x)

[Out]

(16*(2*x + 1)^(1/2))/25 + ((756*(2*x + 1)^(1/2))/3875 - (712*(2*x + 1)^(3/2))/3875)/((2*x + 1)^2 - (8*x)/5 + 3
/5) - (155^(1/2)*atan((155^(1/2)*(5682718 - 31^(1/2)*135439i)^(1/2)*(2*x + 1)^(1/2)*559232i)/(46923828125*((31
^(1/2)*591108224i)/9384765625 - 2004287488/9384765625)) + (1118464*31^(1/2)*155^(1/2)*(5682718 - 31^(1/2)*1354
39i)^(1/2)*(2*x + 1)^(1/2))/(1454638671875*((31^(1/2)*591108224i)/9384765625 - 2004287488/9384765625)))*(56827
18 - 31^(1/2)*135439i)^(1/2)*2i)/120125 + (155^(1/2)*atan((155^(1/2)*(31^(1/2)*135439i + 5682718)^(1/2)*(2*x +
1)^(1/2)*559232i)/(46923828125*((31^(1/2)*591108224i)/9384765625 + 2004287488/9384765625)) - (1118464*31^(1/2
)*155^(1/2)*(31^(1/2)*135439i + 5682718)^(1/2)*(2*x + 1)^(1/2))/(1454638671875*((31^(1/2)*591108224i)/93847656
25 + 2004287488/9384765625)))*(31^(1/2)*135439i + 5682718)^(1/2)*2i)/120125