\(\int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 \sqrt {3+5 x}} \, dx\) [1162]

Optimal result

Integrand size = 26, antiderivative size = 115 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 \sqrt {3+5 x}} \, dx=-\frac {3895 \sqrt {3+5 x}}{7546 \sqrt {1-2 x}}+\frac {3 \sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}+\frac {345 \sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)}-\frac {12465 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1372 \sqrt {7}} \] Output:

-3895/7546*(3+5*x)^(1/2)/(1-2*x)^(1/2)+3/14*(3+5*x)^(1/2)/(1-2*x)^(1/2)/(2 
+3*x)^2+345/196*(3+5*x)^(1/2)/(1-2*x)^(1/2)/(2+3*x)-12465/9604*7^(1/2)*arc 
tan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.74 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 \sqrt {3+5 x}} \, dx=\frac {-7 \sqrt {3+5 x} \left (-25204+13785 x+70110 x^2\right )-137115 \sqrt {7-14 x} (2+3 x)^2 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{105644 \sqrt {1-2 x} (2+3 x)^2} \] Input:

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

Output:

(-7*Sqrt[3 + 5*x]*(-25204 + 13785*x + 70110*x^2) - 137115*Sqrt[7 - 14*x]*( 
2 + 3*x)^2*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(105644*Sqrt[1 - 
 2*x]*(2 + 3*x)^2)
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {114, 27, 168, 27, 169, 27, 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.

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

\(\Big \downarrow \) 114

\(\displaystyle \frac {1}{14} \int \frac {5 (7-24 x)}{2 (1-2 x)^{3/2} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {3 \sqrt {5 x+3}}{14 \sqrt {1-2 x} (3 x+2)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {5}{28} \int \frac {7-24 x}{(1-2 x)^{3/2} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {3 \sqrt {5 x+3}}{14 \sqrt {1-2 x} (3 x+2)^2}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {5}{28} \left (\frac {1}{7} \int -\frac {1380 x+89}{2 (1-2 x)^{3/2} (3 x+2) \sqrt {5 x+3}}dx+\frac {69 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )+\frac {3 \sqrt {5 x+3}}{14 \sqrt {1-2 x} (3 x+2)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {5}{28} \left (\frac {69 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}-\frac {1}{14} \int \frac {1380 x+89}{(1-2 x)^{3/2} (3 x+2) \sqrt {5 x+3}}dx\right )+\frac {3 \sqrt {5 x+3}}{14 \sqrt {1-2 x} (3 x+2)^2}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {5}{28} \left (\frac {1}{14} \left (\frac {2}{77} \int \frac {27423}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {3116 \sqrt {5 x+3}}{77 \sqrt {1-2 x}}\right )+\frac {69 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )+\frac {3 \sqrt {5 x+3}}{14 \sqrt {1-2 x} (3 x+2)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {5}{28} \left (\frac {1}{14} \left (\frac {2493}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {3116 \sqrt {5 x+3}}{77 \sqrt {1-2 x}}\right )+\frac {69 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )+\frac {3 \sqrt {5 x+3}}{14 \sqrt {1-2 x} (3 x+2)^2}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {5}{28} \left (\frac {1}{14} \left (\frac {4986}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {3116 \sqrt {5 x+3}}{77 \sqrt {1-2 x}}\right )+\frac {69 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )+\frac {3 \sqrt {5 x+3}}{14 \sqrt {1-2 x} (3 x+2)^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {5}{28} \left (\frac {1}{14} \left (-\frac {4986 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}-\frac {3116 \sqrt {5 x+3}}{77 \sqrt {1-2 x}}\right )+\frac {69 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )+\frac {3 \sqrt {5 x+3}}{14 \sqrt {1-2 x} (3 x+2)^2}\)

Input:

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

Output:

(3*Sqrt[3 + 5*x])/(14*Sqrt[1 - 2*x]*(2 + 3*x)^2) + (5*((69*Sqrt[3 + 5*x])/ 
(7*Sqrt[1 - 2*x]*(2 + 3*x)) + ((-3116*Sqrt[3 + 5*x])/(77*Sqrt[1 - 2*x]) - 
(4986*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7]))/14))/28
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 114

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[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*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
 

rule 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> 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] + S 
imp[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 169

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> 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] + S 
imp[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] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n, 2*p]
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(201\) vs. \(2(88)=176\).

Time = 0.24 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.76

method	result	size
default	\(-\frac {\left (2468070 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+2056725 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}-548460 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +981540 x^{2} \sqrt {-10 x^{2}-x +3}-548460 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+192990 x \sqrt {-10 x^{2}-x +3}-352856 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {3+5 x}}{211288 \left (2+3 x \right )^{2} \sqrt {1-2 x}\, \sqrt {-10 x^{2}-x +3}}\)	\(202\)

Input:

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

Output:

-1/211288*(2468070*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/ 
2))*x^3+2056725*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2)) 
*x^2-548460*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+9 
81540*x^2*(-10*x^2-x+3)^(1/2)-548460*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2) 
/(-10*x^2-x+3)^(1/2))+192990*x*(-10*x^2-x+3)^(1/2)-352856*(-10*x^2-x+3)^(1 
/2))*(3+5*x)^(1/2)/(2+3*x)^2/(1-2*x)^(1/2)/(-10*x^2-x+3)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 \sqrt {3+5 x}} \, dx=-\frac {137115 \, \sqrt {7} {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (70110 \, x^{2} + 13785 \, x - 25204\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{211288 \, {\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )}} \] Input:

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

Output:

-1/211288*(137115*sqrt(7)*(18*x^3 + 15*x^2 - 4*x - 4)*arctan(1/14*sqrt(7)* 
(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(70110*x^2 
 + 13785*x - 25204)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(18*x^3 + 15*x^2 - 4*x - 
 4)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 278 vs. \(2 (88) = 176\).

Time = 0.25 (sec) , antiderivative size = 278, normalized size of antiderivative = 2.42 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 \sqrt {3+5 x}} \, dx=\frac {2493}{38416} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {16 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{18865 \, {\left (2 \, x - 1\right )}} + \frac {297 \, \sqrt {10} {\left (9 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {1640 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {6560 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{98 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{2}} \] Input:

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

Output:

2493/38416*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)))) - 16/18865*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x 
- 1) + 297/98*sqrt(10)*(9*((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 + 1640*(sqrt 
(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 6560*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) 
^2
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 296, normalized size of antiderivative = 2.57 \[ \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3 \sqrt {3+5 x}} \, dx=\frac {1234035 \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}-\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x^{2}+1645380 \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}-\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x +548460 \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}-\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right )-1234035 \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}+\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x^{2}-1645380 \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}+\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x -548460 \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}+\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right )-490770 \sqrt {5 x +3}\, x^{2}-96495 \sqrt {5 x +3}\, x +176428 \sqrt {5 x +3}}{105644 \sqrt {-2 x +1}\, \left (9 x^{2}+12 x +4\right )} \] Input:

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

Output:

(1234035*sqrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt 
( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**2 + 1645380*sqrt( - 2*x + 
1)*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/s 
qrt(11))/2))/sqrt(2))*x + 548460*sqrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) - 
 sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2)) - 123 
4035*sqrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 
2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**2 - 1645380*sqrt( - 2*x + 1)*s 
qrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt( 
11))/2))/sqrt(2))*x - 548460*sqrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) + sqr 
t(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2)) - 490770* 
sqrt(5*x + 3)*x**2 - 96495*sqrt(5*x + 3)*x + 176428*sqrt(5*x + 3))/(105644 
*sqrt( - 2*x + 1)*(9*x**2 + 12*x + 4))