3.24.9 \(\int \frac {\sqrt {1-2 x}}{(2+3 x)^5 \sqrt {3+5 x}} \, dx\) [2309]

Optimal. Leaf size=151 \[ \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{4 (2+3 x)^4}+\frac {81 \sqrt {1-2 x} \sqrt {3+5 x}}{56 (2+3 x)^3}+\frac {14145 \sqrt {1-2 x} \sqrt {3+5 x}}{1568 (2+3 x)^2}+\frac {1479375 \sqrt {1-2 x} \sqrt {3+5 x}}{21952 (2+3 x)}-\frac {16925425 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{21952 \sqrt {7}} \]

[Out]

-16925425/153664*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+1/4*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*
x)^4+81/56*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3+14145/1568*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2+1479375/2195
2*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)

________________________________________________________________________________________

Rubi [A]
time = 0.04, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {101, 156, 12, 95, 210} \begin {gather*} -\frac {16925425 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{21952 \sqrt {7}}+\frac {1479375 \sqrt {1-2 x} \sqrt {5 x+3}}{21952 (3 x+2)}+\frac {14145 \sqrt {1-2 x} \sqrt {5 x+3}}{1568 (3 x+2)^2}+\frac {81 \sqrt {1-2 x} \sqrt {5 x+3}}{56 (3 x+2)^3}+\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{4 (3 x+2)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(4*(2 + 3*x)^4) + (81*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(56*(2 + 3*x)^3) + (14145*Sqr
t[1 - 2*x]*Sqrt[3 + 5*x])/(1568*(2 + 3*x)^2) + (1479375*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21952*(2 + 3*x)) - (1692
5425*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(21952*Sqrt[7])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 101

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 + 1)/((m + 1)*(b*e - a*f))), x] - Dist[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 156

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

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])

Rubi steps

\begin {align*} \int \frac {\sqrt {1-2 x}}{(2+3 x)^5 \sqrt {3+5 x}} \, dx &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{4 (2+3 x)^4}-\frac {1}{4} \int \frac {-\frac {41}{2}+30 x}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx\\ &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{4 (2+3 x)^4}+\frac {81 \sqrt {1-2 x} \sqrt {3+5 x}}{56 (2+3 x)^3}-\frac {1}{84} \int \frac {-\frac {7665}{4}+2430 x}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx\\ &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{4 (2+3 x)^4}+\frac {81 \sqrt {1-2 x} \sqrt {3+5 x}}{56 (2+3 x)^3}+\frac {14145 \sqrt {1-2 x} \sqrt {3+5 x}}{1568 (2+3 x)^2}-\frac {\int \frac {-\frac {913575}{8}+\frac {212175 x}{2}}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{1176}\\ &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{4 (2+3 x)^4}+\frac {81 \sqrt {1-2 x} \sqrt {3+5 x}}{56 (2+3 x)^3}+\frac {14145 \sqrt {1-2 x} \sqrt {3+5 x}}{1568 (2+3 x)^2}+\frac {1479375 \sqrt {1-2 x} \sqrt {3+5 x}}{21952 (2+3 x)}-\frac {\int -\frac {50776275}{16 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{8232}\\ &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{4 (2+3 x)^4}+\frac {81 \sqrt {1-2 x} \sqrt {3+5 x}}{56 (2+3 x)^3}+\frac {14145 \sqrt {1-2 x} \sqrt {3+5 x}}{1568 (2+3 x)^2}+\frac {1479375 \sqrt {1-2 x} \sqrt {3+5 x}}{21952 (2+3 x)}+\frac {16925425 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{43904}\\ &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{4 (2+3 x)^4}+\frac {81 \sqrt {1-2 x} \sqrt {3+5 x}}{56 (2+3 x)^3}+\frac {14145 \sqrt {1-2 x} \sqrt {3+5 x}}{1568 (2+3 x)^2}+\frac {1479375 \sqrt {1-2 x} \sqrt {3+5 x}}{21952 (2+3 x)}+\frac {16925425 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{21952}\\ &=\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{4 (2+3 x)^4}+\frac {81 \sqrt {1-2 x} \sqrt {3+5 x}}{56 (2+3 x)^3}+\frac {14145 \sqrt {1-2 x} \sqrt {3+5 x}}{1568 (2+3 x)^2}+\frac {1479375 \sqrt {1-2 x} \sqrt {3+5 x}}{21952 (2+3 x)}-\frac {16925425 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{21952 \sqrt {7}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains complex when optimal does not.
time = 0.55, size = 101, normalized size = 0.67 \begin {gather*} \frac {25 \left (\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (12696112+55729116 x+81668520 x^2+39943125 x^3\right )}{25 (2+3 x)^4}-677017 i \sqrt {7} \tanh ^{-1}\left (\frac {1}{7} \left (2 \sqrt {70}+3 \sqrt {70} x+3 i \sqrt {7-14 x} \sqrt {3+5 x}\right )\right )\right )}{153664} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(25*((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(12696112 + 55729116*x + 81668520*x^2 + 39943125*x^3))/(25*(2 + 3*x)^4) -
(677017*I)*Sqrt[7]*ArcTanh[(2*Sqrt[70] + 3*Sqrt[70]*x + (3*I)*Sqrt[7 - 14*x]*Sqrt[3 + 5*x])/7]))/153664

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(249\) vs. \(2(118)=236\).
time = 0.15, size = 250, normalized size = 1.66

method	result	size
risch	\(-\frac {\sqrt {3+5 x}\, \left (-1+2 x \right ) \left (39943125 x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{21952 \left (2+3 x \right )^{4} \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {16925425 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{307328 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(129\)
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (1370959425 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}[phone] x^{3} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x [phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] x \sqrt {-10 x^{2}-x +3}[phone] \sqrt {-10 x^{2}-x +3}\right )}{307328 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{4}}\)	\(250\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/307328*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(1370959425*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^
4+3655891800*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+3655891800*7^(1/2)*arctan(1/14*(37
*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+559203750*x^3*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x
+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+1143359280*x^2*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+20)
*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/
2)/(2+3*x)^4

________________________________________________________________________________________

Maxima [A]
time = 0.53, size = 143, normalized size = 0.95 \begin {gather*} \frac {16925425}{307328} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {\sqrt {-10 \, x^{2} - x + 3}}{4 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {81 \, \sqrt {-10 \, x^{2} - x + 3}}{56 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {14145 \, \sqrt {-10 \, x^{2} - x + 3}}{1568 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {1479375 \, \sqrt {-10 \, x^{2} - x + 3}}{21952 \, {\left (3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

16925425/307328*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 1/4*sqrt(-10*x^2 - x + 3)/(81*x^4
+ 216*x^3 + 216*x^2 + 96*x + 16) + 81/56*sqrt(-10*x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) + 14145/1568*sqrt(
-10*x^2 - x + 3)/(9*x^2 + 12*x + 4) + 1479375/21952*sqrt(-10*x^2 - x + 3)/(3*x + 2)

________________________________________________________________________________________

Fricas [A]
time = 0.93, size = 116, normalized size = 0.77 \begin {gather*} -\frac {16925425 \, \sqrt {7} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (39943125 \, x^{3} + 81668520 \, x^{2} + 55729116 \, x + 12696112\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{307328 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/307328*(16925425*sqrt(7)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x
+ 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(39943125*x^3 + 81668520*x^2 + 55729116*x + 12696112)*sqrt(5*x + 3)
*sqrt(-2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 373 vs. \(2 (118) = 236\).
time = 1.01, size = 373, normalized size = 2.47 \begin {gather*} \frac {55}{614656} \, \sqrt {5} {\left (61547 \, \sqrt {70} \sqrt {2} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {280 \, \sqrt {2} {\left (157973 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 83743800 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 17691640512 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {1351079744000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {5404318976000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{{\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{4}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

55/614656*sqrt(5)*(61547*sqrt(70)*sqrt(2)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x +
5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) + 280*sqrt(2)*(157973*((sqrt(2)*sqrt(-1
0*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^7 + 83743800*((sqrt
(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^5 + 1769
1640512*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(
22)))^3 + 1351079744000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 5404318976000*sqrt(5*x + 3)/(sqrt
(2)*sqrt(-10*x + 5) - sqrt(22)))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(
2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^4)

________________________________________________________________________________________

Mupad [B]
time = 18.15, size = 1509, normalized size = 9.99 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((51771207139*((1 - 2*x)^(1/2) - 1)^7)/(76562500*(3^(1/2) - (5*x + 3)^(1/2))^7) - (90265424*((1 - 2*x)^(1/2) -
1)^3)/(2734375*(3^(1/2) - (5*x + 3)^(1/2))^3) - (622608669*((1 - 2*x)^(1/2) - 1)^5)/(7656250*(3^(1/2) - (5*x
+ 3)^(1/2))^5) - (33845362*((1 - 2*x)^(1/2) - 1))/(133984375*(3^(1/2) - (5*x + 3)^(1/2))) - (51771207139*((1 -
2*x)^(1/2) - 1)^9)/(30625000*(3^(1/2) - (5*x + 3)^(1/2))^9) + (622608669*((1 - 2*x)^(1/2) - 1)^11)/(490000*(3
^(1/2) - (5*x + 3)^(1/2))^11) + (5641589*((1 - 2*x)^(1/2) - 1)^13)/(1750*(3^(1/2) - (5*x + 3)^(1/2))^13) + (16
922681*((1 - 2*x)^(1/2) - 1)^15)/(109760*(3^(1/2) - (5*x + 3)^(1/2))^15) + (10154863*3^(1/2)*((1 - 2*x)^(1/2)
- 1)^2)/(3828125*(3^(1/2) - (5*x + 3)^(1/2))^2) + (169254138*3^(1/2)*((1 - 2*x)^(1/2) - 1)^4)/(2734375*(3^(1/2
) - (5*x + 3)^(1/2))^4) - (8594094207*3^(1/2)*((1 - 2*x)^(1/2) - 1)^6)/(38281250*(3^(1/2) - (5*x + 3)^(1/2))^6
) + (143880176831*3^(1/2)*((1 - 2*x)^(1/2) - 1)^8)/(267968750*(3^(1/2) - (5*x + 3)^(1/2))^8) - (8594094207*3^(
1/2)*((1 - 2*x)^(1/2) - 1)^10)/(6125000*(3^(1/2) - (5*x + 3)^(1/2))^10) + (84627069*3^(1/2)*((1 - 2*x)^(1/2) -
1)^12)/(35000*(3^(1/2) - (5*x + 3)^(1/2))^12) + (10154863*3^(1/2)*((1 - 2*x)^(1/2) - 1)^14)/(15680*(3^(1/2) -
(5*x + 3)^(1/2))^14))/((45056*((1 - 2*x)^(1/2) - 1)^2)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^2) + (294784*((1 -
2*x)^(1/2) - 1)^4)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^4) - (1921024*((1 - 2*x)^(1/2) - 1)^6)/(390625*(3^(1/2
) - (5*x + 3)^(1/2))^6) + (5828656*((1 - 2*x)^(1/2) - 1)^8)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^8) - (480256*(
(1 - 2*x)^(1/2) - 1)^10)/(15625*(3^(1/2) - (5*x + 3)^(1/2))^10) + (18424*((1 - 2*x)^(1/2) - 1)^12)/(625*(3^(1/
2) - (5*x + 3)^(1/2))^12) + (704*((1 - 2*x)^(1/2) - 1)^14)/(25*(3^(1/2) - (5*x + 3)^(1/2))^14) + ((1 - 2*x)^(1
/2) - 1)^16/(3^(1/2) - (5*x + 3)^(1/2))^16 - (21504*3^(1/2)*((1 - 2*x)^(1/2) - 1)^3)/(78125*(3^(1/2) - (5*x +
3)^(1/2))^3) + (48384*3^(1/2)*((1 - 2*x)^(1/2) - 1)^5)/(78125*(3^(1/2) - (5*x + 3)^(1/2))^5) - (4992*3^(1/2)*(
(1 - 2*x)^(1/2) - 1)^7)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^7) + (2496*3^(1/2)*((1 - 2*x)^(1/2) - 1)^9)/(78125
*(3^(1/2) - (5*x + 3)^(1/2))^9) - (6048*3^(1/2)*((1 - 2*x)^(1/2) - 1)^11)/(625*(3^(1/2) - (5*x + 3)^(1/2))^11)
+ (672*3^(1/2)*((1 - 2*x)^(1/2) - 1)^13)/(25*(3^(1/2) - (5*x + 3)^(1/2))^13) + (24*3^(1/2)*((1 - 2*x)^(1/2) -
1)^15)/(5*(3^(1/2) - (5*x + 3)^(1/2))^15) - (3072*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(390625*(3^(1/2) - (5*x + 3)
^(1/2))) + 256/390625) - (16925425*7^(1/2)*atan(((16925425*7^(1/2)*((2031051*3^(1/2))/13720 + (2031051*((1 - 2
*x)^(1/2) - 1))/(27440*(3^(1/2) - (5*x + 3)^(1/2))) - (7^(1/2)*((212*((1 - 2*x)^(1/2) - 1)^2)/(25*(3^(1/2) - (
5*x + 3)^(1/2))^2) + (888*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(125*(3^(1/2) - (5*x + 3)^(1/2))) - 536/125)*16925425
i)/307328 - (2031051*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(5488*(3^(1/2) - (5*x + 3)^(1/2))^2)))/307328 + (1692542
5*7^(1/2)*((2031051*3^(1/2))/13720 + (2031051*((1 - 2*x)^(1/2) - 1))/(27440*(3^(1/2) - (5*x + 3)^(1/2))) + (7^
(1/2)*((212*((1 - 2*x)^(1/2) - 1)^2)/(25*(3^(1/2) - (5*x + 3)^(1/2))^2) + (888*3^(1/2)*((1 - 2*x)^(1/2) - 1))/
(125*(3^(1/2) - (5*x + 3)^(1/2))) - 536/125)*16925425i)/307328 - (2031051*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(54
88*(3^(1/2) - (5*x + 3)^(1/2))^2)))/307328)/((7^(1/2)*((2031051*3^(1/2))/13720 + (2031051*((1 - 2*x)^(1/2) - 1
))/(27440*(3^(1/2) - (5*x + 3)^(1/2))) - (7^(1/2)*((212*((1 - 2*x)^(1/2) - 1)^2)/(25*(3^(1/2) - (5*x + 3)^(1/2
))^2) + (888*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(125*(3^(1/2) - (5*x + 3)^(1/2))) - 536/125)*16925425i)/307328 - (
2031051*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(5488*(3^(1/2) - (5*x + 3)^(1/2))^2))*16925425i)/307328 - (7^(1/2)*((
2031051*3^(1/2))/13720 + (2031051*((1 - 2*x)^(1/2) - 1))/(27440*(3^(1/2) - (5*x + 3)^(1/2))) + (7^(1/2)*((212*
((1 - 2*x)^(1/2) - 1)^2)/(25*(3^(1/2) - (5*x + 3)^(1/2))^2) + (888*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(125*(3^(1/2
) - (5*x + 3)^(1/2))) - 536/125)*16925425i)/307328 - (2031051*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(5488*(3^(1/2)
- (5*x + 3)^(1/2))^2))*16925425i)/307328 + (11458800457225*((1 - 2*x)^(1/2) - 1)^2)/(240945152*(3^(1/2) - (5*x
+ 3)^(1/2))^2) + 2291760091445/120472576)))/153664

________________________________________________________________________________________