\(\int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^5} \, dx\) [1090]

Optimal result

Integrand size = 26, antiderivative size = 151 \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^5} \, dx=\frac {53 \sqrt {1-2 x} \sqrt {3+5 x}}{1512 (2+3 x)^3}-\frac {1355 \sqrt {1-2 x} \sqrt {3+5 x}}{6048 (2+3 x)^2}+\frac {4315 \sqrt {1-2 x} \sqrt {3+5 x}}{84672 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{84 (2+3 x)^4}-\frac {6655 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{3136 \sqrt {7}} \] Output:

53/1512*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3-1355/6048*(1-2*x)^(1/2)*(3+5 
*x)^(1/2)/(2+3*x)^2+4315*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(169344+254016*x)+1/8 
4*(1-2*x)^(1/2)*(3+5*x)^(3/2)/(2+3*x)^4-6655/21952*7^(1/2)*arctan(1/7*(1-2 
*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.52 \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^5} \, dx=\frac {\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (-3600-6484 x+6920 x^2+12945 x^3\right )}{(2+3 x)^4}-19965 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{65856} \] Input:

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

Output:

((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-3600 - 6484*x + 6920*x^2 + 12945*x^3))/( 
2 + 3*x)^4 - 19965*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/ 
65856
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.10, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {107, 105, 105, 105, 104, 217}

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.

\(\displaystyle \int \frac {(5 x+3)^{5/2}}{\sqrt {1-2 x} (3 x+2)^5} \, dx\)

\(\Big \downarrow \) 107

\(\displaystyle \frac {7}{8} \int \frac {(5 x+3)^{5/2}}{\sqrt {1-2 x} (3 x+2)^4}dx+\frac {3 \sqrt {1-2 x} (5 x+3)^{7/2}}{28 (3 x+2)^4}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {7}{8} \left (\frac {55}{42} \int \frac {(5 x+3)^{3/2}}{\sqrt {1-2 x} (3 x+2)^3}dx-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^3}\right )+\frac {3 \sqrt {1-2 x} (5 x+3)^{7/2}}{28 (3 x+2)^4}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {7}{8} \left (\frac {55}{42} \left (\frac {33}{28} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^3}\right )+\frac {3 \sqrt {1-2 x} (5 x+3)^{7/2}}{28 (3 x+2)^4}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {7}{8} \left (\frac {55}{42} \left (\frac {33}{28} \left (\frac {11}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^3}\right )+\frac {3 \sqrt {1-2 x} (5 x+3)^{7/2}}{28 (3 x+2)^4}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {7}{8} \left (\frac {55}{42} \left (\frac {33}{28} \left (\frac {11}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^3}\right )+\frac {3 \sqrt {1-2 x} (5 x+3)^{7/2}}{28 (3 x+2)^4}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {7}{8} \left (\frac {55}{42} \left (\frac {33}{28} \left (-\frac {11 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^3}\right )+\frac {3 \sqrt {1-2 x} (5 x+3)^{7/2}}{28 (3 x+2)^4}\)

Input:

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

Output:

(3*Sqrt[1 - 2*x]*(3 + 5*x)^(7/2))/(28*(2 + 3*x)^4) + (7*(-1/21*(Sqrt[1 - 2 
*x]*(3 + 5*x)^(5/2))/(2 + 3*x)^3 + (55*(-1/14*(Sqrt[1 - 2*x]*(3 + 5*x)^(3/ 
2))/(2 + 3*x)^2 + (33*(-1/7*(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2 + 3*x) - (11* 
ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7])))/28))/42))/8
 

Defintions of rubi rules used

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[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] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 105

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

rule 107

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(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] + Simp[(a*d*f*(m + 1) + b*c*f*(n + 
 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x 
] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 1])
 

rule 217

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])
 

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.85

method	result	size
risch	\(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (12945 x^{3}+6920 x^{2}-6484 x -3600\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{9408 \left (2+3 x \right )^{4} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {6655 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{43904 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(129\)
default	\(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (1617165 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+4312440 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+4312440 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+181230 x^{3} \sqrt {-10 x^{2}-x +3}+1916640 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +96880 x^{2} \sqrt {-10 x^{2}-x +3}+319440 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-90776 x \sqrt {-10 x^{2}-x +3}-50400 \sqrt {-10 x^{2}-x +3}\right )}{131712 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{4}}\)	\(250\)

Input:

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

Output:

-1/9408*(-1+2*x)*(3+5*x)^(1/2)*(12945*x^3+6920*x^2-6484*x-3600)/(2+3*x)^4/ 
(-(-1+2*x)*(3+5*x))^(1/2)*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)+6655/43904 
*7^(1/2)*arctan(9/14*(20/3+37/3*x)*7^(1/2)/(-90*(2/3+x)^2+67+111*x)^(1/2)) 
*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)/(3+5*x)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.77 \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^5} \, dx=-\frac {19965 \, \sqrt {7} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (12945 \, x^{3} + 6920 \, x^{2} - 6484 \, x - 3600\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{131712 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \] Input:

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

Output:

-1/131712*(19965*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)) - 1 
4*(12945*x^3 + 6920*x^2 - 6484*x - 3600)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(81 
*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.95 \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^5} \, dx=\frac {6655}{43904} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {\sqrt {-10 \, x^{2} - x + 3}}{252 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {83 \, \sqrt {-10 \, x^{2} - x + 3}}{1512 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} - \frac {1355 \, \sqrt {-10 \, x^{2} - x + 3}}{6048 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {4315 \, \sqrt {-10 \, x^{2} - x + 3}}{84672 \, {\left (3 \, x + 2\right )}} \] Input:

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

Output:

6655/43904*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 1/2 
52*sqrt(-10*x^2 - x + 3)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 83/151 
2*sqrt(-10*x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) - 1355/6048*sqrt(-10* 
x^2 - x + 3)/(9*x^2 + 12*x + 4) + 4315/84672*sqrt(-10*x^2 - x + 3)/(3*x + 
2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 368 vs. \(2 (118) = 236\).

Time = 0.34 (sec) , antiderivative size = 368, normalized size of antiderivative = 2.44 \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^5} \, dx=\frac {1331}{87808} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {6655 \, \sqrt {10} {\left (3 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 3080 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 1144640 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {2956800 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {11827200 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{4704 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{4}} \] Input:

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

Output:

1331/87808*sqrt(70)*sqrt(10)*(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)))) - 6655/4704*sqrt(10)*(3*((sqrt(2)*sqrt(-10*x + 5) - sq 
rt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22 
)))^7 + 3080*((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 + 1144640*((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 + 2956800*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5 
*x + 3) - 11827200*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 [F(-1)]

Timed out. \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^5} \, dx=\int \frac {{\left (5\,x+3\right )}^{5/2}}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^5} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 442, normalized size of antiderivative = 2.93 \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^5} \, dx =\text {Too large to display} \] Input:

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

Output:

(1617165*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt 
(5))/sqrt(11))/2))/sqrt(2))*x**4 + 4312440*sqrt(7)*atan((sqrt(33) - sqrt(3 
5)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**3 + 43124 
40*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/s 
qrt(11))/2))/sqrt(2))*x**2 + 1916640*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan 
(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x + 319440*sqrt(7) 
*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2 
))/sqrt(2)) - 1617165*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 
2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**4 - 4312440*sqrt(7)*atan((sqrt 
(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2)) 
*x**3 - 4312440*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 
1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**2 - 1916640*sqrt(7)*atan((sqrt(33) + 
 sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x - 3 
19440*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5) 
)/sqrt(11))/2))/sqrt(2)) + 90615*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**3 + 484 
40*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2 - 45388*sqrt(5*x + 3)*sqrt( - 2*x + 
 1)*x - 25200*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(65856*(81*x**4 + 216*x**3 + 
 216*x**2 + 96*x + 16))