\(\int \frac {1}{(1+2 x)^{3/2} (2+3 x+5 x^2)^2} \, dx\) [2321]

Optimal result

Integrand size = 22, antiderivative size = 283 \[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^2} \, dx=-\frac {604}{1519 \sqrt {1+2 x}}+\frac {37+20 x}{217 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}+\frac {\sqrt {\frac {2}{217} \left (-5682718+968975 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )}{1519}-\frac {\sqrt {\frac {2}{217} \left (-5682718+968975 \sqrt {35}\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}+10 \sqrt {1+2 x}}{\sqrt {10 \left (-2+\sqrt {35}\right )}}\right )}{1519}-\frac {\sqrt {\frac {1}{434} \left (5682718+968975 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1519}+\frac {\sqrt {\frac {1}{434} \left (5682718+968975 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1519} \]

[Out]

-604/1519/(1+2*x)^(1/2)+1/217*(37+20*x)/(5*x^2+3*x+2)/(1+2*x)^(1/2)+1/329623*arctan((-10*(1+2*x)^(1/2)+(20+10*
35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(-2466299612+420535150*35^(1/2))^(1/2)-1/329623*arctan((10*(1+2*x)^(
1/2)+(20+10*35^(1/2))^(1/2))/(-20+10*35^(1/2))^(1/2))*(-2466299612+420535150*35^(1/2))^(1/2)-1/659246*ln(5+10*
x+35^(1/2)-(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(2466299612+420535150*35^(1/2))^(1/2)+1/659246*ln(5+10*x+35^(
1/2)+(1+2*x)^(1/2)*(20+10*35^(1/2))^(1/2))*(2466299612+420535150*35^(1/2))^(1/2)

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {754, 842, 840, 1183, 648, 632, 210, 642} \[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^2} \, dx=\frac {\sqrt {\frac {2}{217} \left (968975 \sqrt {35}-5682718\right )} \arctan \left (\frac {\sqrt {10 \left (2+\sqrt {35}\right )}-10 \sqrt {2 x+1}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{1519}-\frac {\sqrt {\frac {2}{217} \left (968975 \sqrt {35}-5682718\right )} \arctan \left (\frac {10 \sqrt {2 x+1}+\sqrt {10 \left (2+\sqrt {35}\right )}}{\sqrt {10 \left (\sqrt {35}-2\right )}}\right )}{1519}+\frac {20 x+37}{217 \sqrt {2 x+1} \left (5 x^2+3 x+2\right )}-\frac {604}{1519 \sqrt {2 x+1}}-\frac {\sqrt {\frac {1}{434} \left (5682718+968975 \sqrt {35}\right )} \log \left (5 (2 x+1)-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{1519}+\frac {\sqrt {\frac {1}{434} \left (5682718+968975 \sqrt {35}\right )} \log \left (5 (2 x+1)+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {2 x+1}+\sqrt {35}\right )}{1519} \]

[In]

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

[Out]

-604/(1519*Sqrt[1 + 2*x]) + (37 + 20*x)/(217*Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)) + (Sqrt[(2*(-5682718 + 968975*Sq
rt[35]))/217]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] - 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]])/1519 - (Sqrt[(2*(
-5682718 + 968975*Sqrt[35]))/217]*ArcTan[(Sqrt[10*(2 + Sqrt[35])] + 10*Sqrt[1 + 2*x])/Sqrt[10*(-2 + Sqrt[35])]
])/1519 - (Sqrt[(5682718 + 968975*Sqrt[35])/434]*Log[Sqrt[35] - Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5*(1 +
2*x)])/1519 + (Sqrt[(5682718 + 968975*Sqrt[35])/434]*Log[Sqrt[35] + Sqrt[10*(2 + Sqrt[35])]*Sqrt[1 + 2*x] + 5
*(1 + 2*x)])/1519

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 754

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

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 842

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] + Dist[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, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && FractionQ[m] && LtQ[m, -1]

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 {37+20 x}{217 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}+\frac {1}{217} \int \frac {181+60 x}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )} \, dx \\ & = -\frac {604}{1519 \sqrt {1+2 x}}+\frac {37+20 x}{217 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}+\frac {\int \frac {59-1510 x}{\sqrt {1+2 x} \left (2+3 x+5 x^2\right )} \, dx}{1519} \\ & = -\frac {604}{1519 \sqrt {1+2 x}}+\frac {37+20 x}{217 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}+\frac {2 \text {Subst}\left (\int \frac {1628-1510 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt {1+2 x}\right )}{1519} \\ & = -\frac {604}{1519 \sqrt {1+2 x}}+\frac {37+20 x}{217 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}+\frac {\text {Subst}\left (\int \frac {1628 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}-\left (1628+302 \sqrt {35}\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 )}{1519 \sqrt {14 \left (2+\sqrt {35}\right )}}+\frac {\text {Subst}\left (\int \frac {1628 \sqrt {\frac {2}{5} \left (2+\sqrt {35}\right )}+\left (1628+302 \sqrt {35}\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 )}{1519 \sqrt {14 \left (2+\sqrt {35}\right )}} \\ & = -\frac {604}{1519 \sqrt {1+2 x}}+\frac {37+20 x}{217 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}+\frac {\left (-5285+814 \sqrt {35}\right ) \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 )}{53165}+\frac {\left (-5285+814 \sqrt {35}\right ) \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 )}{53165}-\frac {\sqrt {\frac {1}{434} \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 )}{1519}+\frac {\sqrt {\frac {1}{434} \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 )}{1519} \\ & = -\frac {604}{1519 \sqrt {1+2 x}}+\frac {37+20 x}{217 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}-\frac {\sqrt {\frac {1}{434} \left (5682718+968975 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1519}+\frac {\sqrt {\frac {1}{434} \left (5682718+968975 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1519}+\frac {\left (2 \left (5285-814 \sqrt {35}\right )\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 )}{53165}+\frac {\left (2 \left (5285-814 \sqrt {35}\right )\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 )}{53165} \\ & = -\frac {604}{1519 \sqrt {1+2 x}}+\frac {37+20 x}{217 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}+\frac {\sqrt {\frac {2}{217} \left (-5682718+968975 \sqrt {35}\right )} \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 )}{1519}-\frac {\sqrt {\frac {2}{217} \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 )}{1519}-\frac {\sqrt {\frac {1}{434} \left (5682718+968975 \sqrt {35}\right )} \log \left (\sqrt {35}-\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1519}+\frac {\sqrt {\frac {1}{434} \left (5682718+968975 \sqrt {35}\right )} \log \left (\sqrt {35}+\sqrt {10 \left (2+\sqrt {35}\right )} \sqrt {1+2 x}+5 (1+2 x)\right )}{1519} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.20 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.49 \[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^2} \, dx=\frac {2 \left (-\frac {217 \left (949+1672 x+3020 x^2\right )}{2 \sqrt {1+2 x} \left (2+3 x+5 x^2\right )}-\sqrt {217 \left (-5682718-135439 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} \left (-2-i \sqrt {31}\right )} \sqrt {1+2 x}\right )-\sqrt {217 \left (-5682718+135439 i \sqrt {31}\right )} \arctan \left (\sqrt {\frac {1}{7} i \left (2 i+\sqrt {31}\right )} \sqrt {1+2 x}\right )\right )}{329623} \]

[In]

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

[Out]

(2*((-217*(949 + 1672*x + 3020*x^2))/(2*Sqrt[1 + 2*x]*(2 + 3*x + 5*x^2)) - Sqrt[217*(-5682718 - (135439*I)*Sqr
t[31])]*ArcTan[Sqrt[(-2 - I*Sqrt[31])/7]*Sqrt[1 + 2*x]] - Sqrt[217*(-5682718 + (135439*I)*Sqrt[31])]*ArcTan[Sq
rt[(I/7)*(2*I + Sqrt[31])]*Sqrt[1 + 2*x]]))/329623

Maple [A] (verified)

Time = 0.70 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.19

method	result	size
pseudoelliptic	\(\frac {-17920 \left (\sqrt {5}+\frac {3657 \sqrt {7}}{3584}\right ) \sqrt {1+2 x}\, \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )+17920 \left (\sqrt {5}+\frac {3657 \sqrt {7}}{3584}\right ) \sqrt {1+2 x}\, \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right ) \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )+504680 \left (\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )-\arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )\right ) \left (x^{2}+\frac {3}{5} x +\frac {2}{5}\right ) \left (\sqrt {5}\, \sqrt {7}-\frac {5285}{814}\right ) \sqrt {1+2 x}-1310680 \left (x^{2}+\frac {418}{755} x +\frac {949}{3020}\right ) \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}\, \sqrt {1+2 x}\, \left (3296230 x^{2}+1977738 x +1318492\right )}\)	\(338\)
derivativedivides	\(-\frac {16}{49 \sqrt {1+2 x}}-\frac {16 \left (\frac {27 \left (1+2 x \right )^{\frac {3}{2}}}{124}-\frac {89 \sqrt {1+2 x}}{310}\right )}{49 \left (\left (1+2 x \right )^{2}+\frac {3}{5}-\frac {8 x}{5}\right )}-\frac {\left (17920 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+18285 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{3296230}-\frac {2 \left (-50468 \sqrt {5}\, \sqrt {7}+\frac {\left (17920 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+18285 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{329623 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {\left (-17920 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-18285 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{3296230}-\frac {2 \left (-50468 \sqrt {5}\, \sqrt {7}-\frac {\left (-17920 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-18285 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{329623 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(424\)
default	\(-\frac {16}{49 \sqrt {1+2 x}}-\frac {16 \left (\frac {27 \left (1+2 x \right )^{\frac {3}{2}}}{124}-\frac {89 \sqrt {1+2 x}}{310}\right )}{49 \left (\left (1+2 x \right )^{2}+\frac {3}{5}-\frac {8 x}{5}\right )}-\frac {\left (17920 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+18285 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{3296230}-\frac {2 \left (-50468 \sqrt {5}\, \sqrt {7}+\frac {\left (17920 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+18285 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{329623 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {\left (-17920 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-18285 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right )}{3296230}-\frac {2 \left (-50468 \sqrt {5}\, \sqrt {7}-\frac {\left (-17920 \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}-18285 \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}\right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{10}\right ) \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right )}{329623 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(424\)
trager	\(-\frac {\left (3020 x^{2}+1672 x +949\right ) \sqrt {1+2 x}}{1519 \left (10 x^{3}+11 x^{2}+7 x +2\right )}-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}[phone]\right )^{2}-616574903\right ) \ln \left (-\frac {160586944 \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}[phone]\right )^{2}-616574903\right ) \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}[phone]\right )^{4} x -1718943720888 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}[phone]\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}[phone]\right )^{2}-616574903\right ) x -200458388096 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}[phone]\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}[phone]\right )^{2}-616574903\right )+2118795198727620 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}[phone]\right )^{2} \sqrt {1+2 x}+3759951379038470 \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}[phone]\right )^{2}-616574903\right ) x [phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}+47089 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}[phone]\right )^{2}-616574903\right )+1446596030559246385 \sqrt {1+2 x}}{868 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}[phone]\right )^{2} x -5276401 x +541756}\right )}{329623}-\frac {2 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}[phone]\right ) \ln \left (-\frac {1124108608 x \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}[phone]\right )^{5}-17405146122616 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}[phone]\right )^{3} x [phone] \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}[phone]\right )^{3}-68348232217020 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}[phone]\right )^{2} \sqrt {1+2 x}+61493196747600690 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}[phone]\right ) x -14072568607442864 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}[phone]\right )+941603856948545875 \sqrt {1+2 x}}{868 \operatorname {RootOf}\left (107632 \textit {\_Z}^{4}-1409314064 \textit {\_Z}^{2}[phone]\right )^{2} x -6089035 x -541756}\right )}{1519}\)	\(459\)
risch	\(-\frac {3020 x^{2}+1672 x +949}{1519 \left (5 x^{2}+3 x +2\right ) \sqrt {1+2 x}}-\frac {256 \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{47089}-\frac {3657 \ln \left (\sqrt {5}\, \sqrt {7}-\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right ) \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{659246}-\frac {2560 \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \left (2 \sqrt {5}\, \sqrt {7}+4\right )}{47089 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {3657 \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \sqrt {5}\, \left (2 \sqrt {5}\, \sqrt {7}+4\right ) \sqrt {7}}{329623 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {3256 \arctan \left (\frac {-\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \sqrt {5}\, \sqrt {7}}{10633 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {256 \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right ) \sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{47089}+\frac {3657 \ln \left (\sqrt {5}\, \sqrt {7}+\sqrt {2 \sqrt {5}\, \sqrt {7}+4}\, \sqrt {5}\, \sqrt {1+2 x}+5+10 x \right ) \sqrt {7}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}}{659246}-\frac {2560 \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \left (2 \sqrt {5}\, \sqrt {7}+4\right )}{47089 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}-\frac {3657 \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \sqrt {5}\, \left (2 \sqrt {5}\, \sqrt {7}+4\right ) \sqrt {7}}{329623 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}+\frac {3256 \arctan \left (\frac {\sqrt {5}\, \sqrt {2 \sqrt {5}\, \sqrt {7}+4}+10 \sqrt {1+2 x}}{\sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\right ) \sqrt {5}\, \sqrt {7}}{10633 \sqrt {10 \sqrt {5}\, \sqrt {7}-20}}\)	\(638\)

[In]

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

[Out]

504680/(10*5^(1/2)*7^(1/2)-20)^(1/2)*(-448/12617*(5^(1/2)+3657/3584*7^(1/2))*(1+2*x)^(1/2)*(x^2+3/5*x+2/5)*(2*
5^(1/2)*7^(1/2)+4)^(1/2)*(10*5^(1/2)*7^(1/2)-20)^(1/2)*ln(5^(1/2)*7^(1/2)-(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*
(1+2*x)^(1/2)+5+10*x)+448/12617*(5^(1/2)+3657/3584*7^(1/2))*(1+2*x)^(1/2)*(x^2+3/5*x+2/5)*(2*5^(1/2)*7^(1/2)+4
)^(1/2)*(10*5^(1/2)*7^(1/2)-20)^(1/2)*ln(5^(1/2)*7^(1/2)+(2*5^(1/2)*7^(1/2)+4)^(1/2)*5^(1/2)*(1+2*x)^(1/2)+5+1
0*x)+(arctan((5^(1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)+10*(1+2*x)^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2))-arctan((5^(
1/2)*(2*5^(1/2)*7^(1/2)+4)^(1/2)-10*(1+2*x)^(1/2))/(10*5^(1/2)*7^(1/2)-20)^(1/2)))*(x^2+3/5*x+2/5)*(5^(1/2)*7^
(1/2)-5285/814)*(1+2*x)^(1/2)-1057/407*(x^2+418/755*x+949/3020)*(10*5^(1/2)*7^(1/2)-20)^(1/2))/(1+2*x)^(1/2)/(
3296230*x^2+1977738*x+1318492)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.98 \[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^2} \, dx=-\frac {\sqrt {217} {\left (10 \, x^{3} + 11 \, x^{2} + 7 \, x + 2\right )} \sqrt {541756 i \, \sqrt {31} + 22730872} \log \left (\sqrt {217} \sqrt {541756 i \, \sqrt {31} + 22730872} {\left (3657 i \, \sqrt {31} - 25234\right )} + 2102675750 \, \sqrt {2 \, x + 1}\right ) - \sqrt {217} {\left (10 \, x^{3} + 11 \, x^{2} + 7 \, x + 2\right )} \sqrt {541756 i \, \sqrt {31} + 22730872} \log \left (\sqrt {217} \sqrt {541756 i \, \sqrt {31} + 22730872} {\left (-3657 i \, \sqrt {31} + 25234\right )} + 2102675750 \, \sqrt {2 \, x + 1}\right ) - \sqrt {217} {\left (10 \, x^{3} + 11 \, x^{2} + 7 \, x + 2\right )} \sqrt {-541756 i \, \sqrt {31} + 22730872} \log \left (\sqrt {217} {\left (3657 i \, \sqrt {31} + 25234\right )} \sqrt {-541756 i \, \sqrt {31} + 22730872} + 2102675750 \, \sqrt {2 \, x + 1}\right ) + \sqrt {217} {\left (10 \, x^{3} + 11 \, x^{2} + 7 \, x + 2\right )} \sqrt {-541756 i \, \sqrt {31} + 22730872} \log \left (\sqrt {217} {\left (-3657 i \, \sqrt {31} - 25234\right )} \sqrt {-541756 i \, \sqrt {31} + 22730872} + 2102675750 \, \sqrt {2 \, x + 1}\right ) + 434 \, {\left (3020 \, x^{2} + 1672 \, x + 949\right )} \sqrt {2 \, x + 1}}{659246 \, {\left (10 \, x^{3} + 11 \, x^{2} + 7 \, x + 2\right )}} \]

[In]

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

[Out]

-1/659246*(sqrt(217)*(10*x^3 + 11*x^2 + 7*x + 2)*sqrt(541756*I*sqrt(31) + 22730872)*log(sqrt(217)*sqrt(541756*
I*sqrt(31) + 22730872)*(3657*I*sqrt(31) - 25234) + 2102675750*sqrt(2*x + 1)) - sqrt(217)*(10*x^3 + 11*x^2 + 7*
x + 2)*sqrt(541756*I*sqrt(31) + 22730872)*log(sqrt(217)*sqrt(541756*I*sqrt(31) + 22730872)*(-3657*I*sqrt(31) +
25234) + 2102675750*sqrt(2*x + 1)) - sqrt(217)*(10*x^3 + 11*x^2 + 7*x + 2)*sqrt(-541756*I*sqrt(31) + 22730872
)*log(sqrt(217)*(3657*I*sqrt(31) + 25234)*sqrt(-541756*I*sqrt(31) + 22730872) + 2102675750*sqrt(2*x + 1)) + sq
rt(217)*(10*x^3 + 11*x^2 + 7*x + 2)*sqrt(-541756*I*sqrt(31) + 22730872)*log(sqrt(217)*(-3657*I*sqrt(31) - 2523
4)*sqrt(-541756*I*sqrt(31) + 22730872) + 2102675750*sqrt(2*x + 1)) + 434*(3020*x^2 + 1672*x + 949)*sqrt(2*x +
1))/(10*x^3 + 11*x^2 + 7*x + 2)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 633 vs. \(2 (196) = 392\).

Time = 0.54 (sec) , antiderivative size = 633, normalized size of antiderivative = 2.24 \[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/11306068900*sqrt(31)*(31710*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) - 151*sqrt(31
)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 302*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 63420*(7/5)^(3/4)*s
qrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 797720*sqrt(31)*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 2450) - 1595440*
(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450))*arctan(5/7*(7/5)^(3/4)*((7/5)^(1/4)*sqrt(1/35*sqrt(35) + 1/2) + sqrt(2*
x + 1))/sqrt(-1/35*sqrt(35) + 1/2)) - 1/11306068900*sqrt(31)*(31710*sqrt(31)*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqr
t(-140*sqrt(35) + 2450) - 151*sqrt(31)*(7/5)^(3/4)*(-140*sqrt(35) + 2450)^(3/2) + 302*(7/5)^(3/4)*(140*sqrt(35
) + 2450)^(3/2) + 63420*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35) - 35) - 797720*sqrt(31)*(7/5)^(1/4)*
sqrt(-140*sqrt(35) + 2450) - 1595440*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450))*arctan(-5/7*(7/5)^(3/4)*((7/5)^(1/
4)*sqrt(1/35*sqrt(35) + 1/2) - sqrt(2*x + 1))/sqrt(-1/35*sqrt(35) + 1/2)) - 1/22612137800*sqrt(31)*(151*sqrt(3
1)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 31710*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(2*sqrt(35)
- 35) - 63420*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 302*(7/5)^(3/4)*(-140*sqrt(35) + 2450
)^(3/2) - 797720*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) + 1595440*(7/5)^(1/4)*sqrt(-140*sqrt(35) + 245
0))*log(2*(7/5)^(1/4)*sqrt(2*x + 1)*sqrt(1/35*sqrt(35) + 1/2) + 2*x + sqrt(7/5) + 1) + 1/22612137800*sqrt(31)*
(151*sqrt(31)*(7/5)^(3/4)*(140*sqrt(35) + 2450)^(3/2) + 31710*sqrt(31)*(7/5)^(3/4)*sqrt(140*sqrt(35) + 2450)*(
2*sqrt(35) - 35) - 63420*(7/5)^(3/4)*(2*sqrt(35) + 35)*sqrt(-140*sqrt(35) + 2450) + 302*(7/5)^(3/4)*(-140*sqrt
(35) + 2450)^(3/2) - 797720*sqrt(31)*(7/5)^(1/4)*sqrt(140*sqrt(35) + 2450) + 1595440*(7/5)^(1/4)*sqrt(-140*sqr
t(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) - 4/1519*(755
*(2*x + 1)^2 - 1348*x + 194)/(5*(2*x + 1)^(5/2) - 4*(2*x + 1)^(3/2) + 7*sqrt(2*x + 1))

Mupad [B] (verification not implemented)

Time = 9.98 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.77 \[ \int \frac {1}{(1+2 x)^{3/2} \left (2+3 x+5 x^2\right )^2} \, dx=-\frac {\frac {604\,{\left (2\,x+1\right )}^2}{1519}-\frac {5392\,x}{7595}+\frac {776}{7595}}{\frac {7\,\sqrt {2\,x+1}}{5}-\frac {4\,{\left (2\,x+1\right )}^{3/2}}{5}+{\left (2\,x+1\right )}^{5/2}}-\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {5682718-\sqrt {31}\,135439{}\mathrm {i}}\,\sqrt {2\,x+1}\,559232{}\mathrm {i}}{692496720125\,\left (\frac {2045111424}{98928102875}+\frac {\sqrt {31}\,455214848{}\mathrm {i}}{98928102875}\right )}-\frac {1118464\,\sqrt {31}\,\sqrt {217}\,\sqrt {5682718-\sqrt {31}\,135439{}\mathrm {i}}\,\sqrt {2\,x+1}}{21467398323875\,\left (\frac {2045111424}{98928102875}+\frac {\sqrt {31}\,455214848{}\mathrm {i}}{98928102875}\right )}\right )\,\sqrt {5682718-\sqrt {31}\,135439{}\mathrm {i}}\,2{}\mathrm {i}}{329623}+\frac {\sqrt {217}\,\mathrm {atan}\left (\frac {\sqrt {217}\,\sqrt {5682718+\sqrt {31}\,135439{}\mathrm {i}}\,\sqrt {2\,x+1}\,559232{}\mathrm {i}}{692496720125\,\left (-\frac {2045111424}{98928102875}+\frac {\sqrt {31}\,455214848{}\mathrm {i}}{98928102875}\right )}+\frac {1118464\,\sqrt {31}\,\sqrt {217}\,\sqrt {5682718+\sqrt {31}\,135439{}\mathrm {i}}\,\sqrt {2\,x+1}}{21467398323875\,\left (-\frac {2045111424}{98928102875}+\frac {\sqrt {31}\,455214848{}\mathrm {i}}{98928102875}\right )}\right )\,\sqrt {5682718+\sqrt {31}\,135439{}\mathrm {i}}\,2{}\mathrm {i}}{329623} \]

[In]

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

[Out]

(217^(1/2)*atan((217^(1/2)*(31^(1/2)*135439i + 5682718)^(1/2)*(2*x + 1)^(1/2)*559232i)/(692496720125*((31^(1/2
)*455214848i)/98928102875 - 2045111424/98928102875)) + (1118464*31^(1/2)*217^(1/2)*(31^(1/2)*135439i + 5682718
)^(1/2)*(2*x + 1)^(1/2))/(21467398323875*((31^(1/2)*455214848i)/98928102875 - 2045111424/98928102875)))*(31^(1
/2)*135439i + 5682718)^(1/2)*2i)/329623 - (217^(1/2)*atan((217^(1/2)*(5682718 - 31^(1/2)*135439i)^(1/2)*(2*x +
1)^(1/2)*559232i)/(692496720125*((31^(1/2)*455214848i)/98928102875 + 2045111424/98928102875)) - (1118464*31^(
1/2)*217^(1/2)*(5682718 - 31^(1/2)*135439i)^(1/2)*(2*x + 1)^(1/2))/(21467398323875*((31^(1/2)*455214848i)/9892
8102875 + 2045111424/98928102875)))*(5682718 - 31^(1/2)*135439i)^(1/2)*2i)/329623 - ((604*(2*x + 1)^2)/1519 -
(5392*x)/7595 + 776/7595)/((7*(2*x + 1)^(1/2))/5 - (4*(2*x + 1)^(3/2))/5 + (2*x + 1)^(5/2))