\(\int \frac {(1-2 x)^{3/2} (2+3 x)^4}{(3+5 x)^2} \, dx\) [1909]

Optimal result

Integrand size = 24, antiderivative size = 128 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{(3+5 x)^2} \, dx=\frac {258 \sqrt {1-2 x}}{15625}-\frac {2}{875} (1-2 x)^{3/2} (2+3 x)^2+\frac {11}{75} (1-2 x)^{3/2} (2+3 x)^3-\frac {(1-2 x)^{3/2} (2+3 x)^4}{5 (3+5 x)}-\frac {(1-2 x)^{3/2} (5678+3663 x)}{9375}-\frac {258 \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625} \]

[Out]

-2/875*(1-2*x)^(3/2)*(2+3*x)^2+11/75*(1-2*x)^(3/2)*(2+3*x)^3-1/5*(1-2*x)^(3/2)*(2+3*x)^4/(3+5*x)-1/9375*(1-2*x
)^(3/2)*(5678+3663*x)-258/78125*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+258/15625*(1-2*x)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {99, 158, 152, 52, 65, 212} \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{(3+5 x)^2} \, dx=-\frac {258 \sqrt {\frac {11}{5}} \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625}-\frac {(1-2 x)^{3/2} (3 x+2)^4}{5 (5 x+3)}+\frac {11}{75} (1-2 x)^{3/2} (3 x+2)^3-\frac {2}{875} (1-2 x)^{3/2} (3 x+2)^2-\frac {(1-2 x)^{3/2} (3663 x+5678)}{9375}+\frac {258 \sqrt {1-2 x}}{15625} \]

[In]

Int[((1 - 2*x)^(3/2)*(2 + 3*x)^4)/(3 + 5*x)^2,x]

[Out]

(258*Sqrt[1 - 2*x])/15625 - (2*(1 - 2*x)^(3/2)*(2 + 3*x)^2)/875 + (11*(1 - 2*x)^(3/2)*(2 + 3*x)^3)/75 - ((1 -
2*x)^(3/2)*(2 + 3*x)^4)/(5*(3 + 5*x)) - ((1 - 2*x)^(3/2)*(5678 + 3663*x))/9375 - (258*Sqrt[11/5]*ArcTanh[Sqrt[
5/11]*Sqrt[1 - 2*x]])/15625

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 99

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

Rule 152

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)
^(m + 1)*((c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d
*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1
)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)
^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]

Rule 158

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps \begin{align*} \text {integral}& = -\frac {(1-2 x)^{3/2} (2+3 x)^4}{5 (3+5 x)}+\frac {1}{5} \int \frac {(6-33 x) \sqrt {1-2 x} (2+3 x)^3}{3+5 x} \, dx \\ & = \frac {11}{75} (1-2 x)^{3/2} (2+3 x)^3-\frac {(1-2 x)^{3/2} (2+3 x)^4}{5 (3+5 x)}-\frac {1}{225} \int \frac {(-243-18 x) \sqrt {1-2 x} (2+3 x)^2}{3+5 x} \, dx \\ & = -\frac {2}{875} (1-2 x)^{3/2} (2+3 x)^2+\frac {11}{75} (1-2 x)^{3/2} (2+3 x)^3-\frac {(1-2 x)^{3/2} (2+3 x)^4}{5 (3+5 x)}+\frac {\int \frac {\sqrt {1-2 x} (2+3 x) (17010+25641 x)}{3+5 x} \, dx}{7875} \\ & = -\frac {2}{875} (1-2 x)^{3/2} (2+3 x)^2+\frac {11}{75} (1-2 x)^{3/2} (2+3 x)^3-\frac {(1-2 x)^{3/2} (2+3 x)^4}{5 (3+5 x)}-\frac {(1-2 x)^{3/2} (5678+3663 x)}{9375}+\frac {129 \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx}{3125} \\ & = \frac {258 \sqrt {1-2 x}}{15625}-\frac {2}{875} (1-2 x)^{3/2} (2+3 x)^2+\frac {11}{75} (1-2 x)^{3/2} (2+3 x)^3-\frac {(1-2 x)^{3/2} (2+3 x)^4}{5 (3+5 x)}-\frac {(1-2 x)^{3/2} (5678+3663 x)}{9375}+\frac {1419 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{15625} \\ & = \frac {258 \sqrt {1-2 x}}{15625}-\frac {2}{875} (1-2 x)^{3/2} (2+3 x)^2+\frac {11}{75} (1-2 x)^{3/2} (2+3 x)^3-\frac {(1-2 x)^{3/2} (2+3 x)^4}{5 (3+5 x)}-\frac {(1-2 x)^{3/2} (5678+3663 x)}{9375}-\frac {1419 \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{15625} \\ & = \frac {258 \sqrt {1-2 x}}{15625}-\frac {2}{875} (1-2 x)^{3/2} (2+3 x)^2+\frac {11}{75} (1-2 x)^{3/2} (2+3 x)^3-\frac {(1-2 x)^{3/2} (2+3 x)^4}{5 (3+5 x)}-\frac {(1-2 x)^{3/2} (5678+3663 x)}{9375}-\frac {258 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.61 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{(3+5 x)^2} \, dx=-\frac {5 \sqrt {1-2 x} \left (161312-143235 x-924335 x^2+157275 x^3+1395000 x^4+787500 x^5\right )+1806 \sqrt {55} (3+5 x) \text {arctanh}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{546875 (3+5 x)} \]

[In]

Integrate[((1 - 2*x)^(3/2)*(2 + 3*x)^4)/(3 + 5*x)^2,x]

[Out]

-1/546875*(5*Sqrt[1 - 2*x]*(161312 - 143235*x - 924335*x^2 + 157275*x^3 + 1395000*x^4 + 787500*x^5) + 1806*Sqr
t[55]*(3 + 5*x)*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(3 + 5*x)

Maple [A] (verified)

Time = 1.01 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.52

method	result	size
risch	\(\frac {1575000 x^{6}+2002500 x^{5}-1080450 x^{4}-2005945 x^{3}+637865 x^{2}+465859 x -161312}{109375 \left (3+5 x \right ) \sqrt {1-2 x}}-\frac {258 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{78125}\)	\(66\)
pseudoelliptic	\(\frac {-1806 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \left (3+5 x \right ) \sqrt {55}-5 \sqrt {1-2 x}\, \left (787500 x^{5}+1395000 x^{4}+157275 x^{3}-924335 x^{2}-143235 x +161312\right )}{1640625+2734375 x}\)	\(67\)
derivativedivides	\(-\frac {9 \left (1-2 x \right )^{\frac {9}{2}}}{100}+\frac {999 \left (1-2 x \right )^{\frac {7}{2}}}{1750}-\frac {12393 \left (1-2 x \right )^{\frac {5}{2}}}{12500}+\frac {8 \left (1-2 x \right )^{\frac {3}{2}}}{3125}+\frac {52 \sqrt {1-2 x}}{3125}+\frac {22 \sqrt {1-2 x}}{78125 \left (-\frac {6}{5}-2 x \right )}-\frac {258 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{78125}\)	\(81\)
default	\(-\frac {9 \left (1-2 x \right )^{\frac {9}{2}}}{100}+\frac {999 \left (1-2 x \right )^{\frac {7}{2}}}{1750}-\frac {12393 \left (1-2 x \right )^{\frac {5}{2}}}{12500}+\frac {8 \left (1-2 x \right )^{\frac {3}{2}}}{3125}+\frac {52 \sqrt {1-2 x}}{3125}+\frac {22 \sqrt {1-2 x}}{78125 \left (-\frac {6}{5}-2 x \right )}-\frac {258 \,\operatorname {arctanh}\left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{78125}\)	\(81\)
trager	\(-\frac {\left (787500 x^{5}+1395000 x^{4}+157275 x^{3}-924335 x^{2}-143235 x +161312\right ) \sqrt {1-2 x}}{109375 \left (3+5 x \right )}-\frac {129 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{78125}\)	\(87\)

[In]

int((1-2*x)^(3/2)*(2+3*x)^4/(3+5*x)^2,x,method=_RETURNVERBOSE)

[Out]

1/109375*(1575000*x^6+2002500*x^5-1080450*x^4-2005945*x^3+637865*x^2+465859*x-161312)/(3+5*x)/(1-2*x)^(1/2)-25
8/78125*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.66 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{(3+5 x)^2} \, dx=\frac {903 \, \sqrt {11} \sqrt {5} {\left (5 \, x + 3\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - 5 \, {\left (787500 \, x^{5} + 1395000 \, x^{4} + 157275 \, x^{3} - 924335 \, x^{2} - 143235 \, x + 161312\right )} \sqrt {-2 \, x + 1}}{546875 \, {\left (5 \, x + 3\right )}} \]

[In]

integrate((1-2*x)^(3/2)*(2+3*x)^4/(3+5*x)^2,x, algorithm="fricas")

[Out]

1/546875*(903*sqrt(11)*sqrt(5)*(5*x + 3)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) - 5*(78750
0*x^5 + 1395000*x^4 + 157275*x^3 - 924335*x^2 - 143235*x + 161312)*sqrt(-2*x + 1))/(5*x + 3)

Sympy [A] (verification not implemented)

Time = 42.42 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.73 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{(3+5 x)^2} \, dx=- \frac {9 \left (1 - 2 x\right )^{\frac {9}{2}}}{100} + \frac {999 \left (1 - 2 x\right )^{\frac {7}{2}}}{1750} - \frac {12393 \left (1 - 2 x\right )^{\frac {5}{2}}}{12500} + \frac {8 \left (1 - 2 x\right )^{\frac {3}{2}}}{3125} + \frac {52 \sqrt {1 - 2 x}}{3125} + \frac {128 \sqrt {55} \left (\log {\left (\sqrt {1 - 2 x} - \frac {\sqrt {55}}{5} \right )} - \log {\left (\sqrt {1 - 2 x} + \frac {\sqrt {55}}{5} \right )}\right )}{78125} - \frac {484 \left (\begin {cases} \frac {\sqrt {55} \left (- \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} - 1\right )}\right )}{605} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{15625} \]

[In]

integrate((1-2*x)**(3/2)*(2+3*x)**4/(3+5*x)**2,x)

[Out]

-9*(1 - 2*x)**(9/2)/100 + 999*(1 - 2*x)**(7/2)/1750 - 12393*(1 - 2*x)**(5/2)/12500 + 8*(1 - 2*x)**(3/2)/3125 +
52*sqrt(1 - 2*x)/3125 + 128*sqrt(55)*(log(sqrt(1 - 2*x) - sqrt(55)/5) - log(sqrt(1 - 2*x) + sqrt(55)/5))/7812
5 - 484*Piecewise((sqrt(55)*(-log(sqrt(55)*sqrt(1 - 2*x)/11 - 1)/4 + log(sqrt(55)*sqrt(1 - 2*x)/11 + 1)/4 - 1/
(4*(sqrt(55)*sqrt(1 - 2*x)/11 + 1)) - 1/(4*(sqrt(55)*sqrt(1 - 2*x)/11 - 1)))/605, (sqrt(1 - 2*x) > -sqrt(55)/5
) & (sqrt(1 - 2*x) < sqrt(55)/5)))/15625

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.77 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{(3+5 x)^2} \, dx=-\frac {9}{100} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {999}{1750} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {12393}{12500} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {8}{3125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {129}{78125} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {52}{3125} \, \sqrt {-2 \, x + 1} - \frac {11 \, \sqrt {-2 \, x + 1}}{15625 \, {\left (5 \, x + 3\right )}} \]

[In]

integrate((1-2*x)^(3/2)*(2+3*x)^4/(3+5*x)^2,x, algorithm="maxima")

[Out]

-9/100*(-2*x + 1)^(9/2) + 999/1750*(-2*x + 1)^(7/2) - 12393/12500*(-2*x + 1)^(5/2) + 8/3125*(-2*x + 1)^(3/2) +
129/78125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 52/3125*sqrt(-2*x + 1)
- 11/15625*sqrt(-2*x + 1)/(5*x + 3)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.95 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{(3+5 x)^2} \, dx=-\frac {9}{100} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {999}{1750} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {12393}{12500} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {8}{3125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {129}{78125} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {52}{3125} \, \sqrt {-2 \, x + 1} - \frac {11 \, \sqrt {-2 \, x + 1}}{15625 \, {\left (5 \, x + 3\right )}} \]

[In]

integrate((1-2*x)^(3/2)*(2+3*x)^4/(3+5*x)^2,x, algorithm="giac")

[Out]

-9/100*(2*x - 1)^4*sqrt(-2*x + 1) - 999/1750*(2*x - 1)^3*sqrt(-2*x + 1) - 12393/12500*(2*x - 1)^2*sqrt(-2*x +
1) + 8/3125*(-2*x + 1)^(3/2) + 129/78125*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*s
qrt(-2*x + 1))) + 52/3125*sqrt(-2*x + 1) - 11/15625*sqrt(-2*x + 1)/(5*x + 3)

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.64 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{(3+5 x)^2} \, dx=\frac {52\,\sqrt {1-2\,x}}{3125}-\frac {22\,\sqrt {1-2\,x}}{78125\,\left (2\,x+\frac {6}{5}\right )}+\frac {8\,{\left (1-2\,x\right )}^{3/2}}{3125}-\frac {12393\,{\left (1-2\,x\right )}^{5/2}}{12500}+\frac {999\,{\left (1-2\,x\right )}^{7/2}}{1750}-\frac {9\,{\left (1-2\,x\right )}^{9/2}}{100}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,258{}\mathrm {i}}{78125} \]

[In]

int(((1 - 2*x)^(3/2)*(3*x + 2)^4)/(5*x + 3)^2,x)

[Out]

(55^(1/2)*atan((55^(1/2)*(1 - 2*x)^(1/2)*1i)/11)*258i)/78125 - (22*(1 - 2*x)^(1/2))/(78125*(2*x + 6/5)) + (52*
(1 - 2*x)^(1/2))/3125 + (8*(1 - 2*x)^(3/2))/3125 - (12393*(1 - 2*x)^(5/2))/12500 + (999*(1 - 2*x)^(7/2))/1750
- (9*(1 - 2*x)^(9/2))/100