\(\int (1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2} \, dx\) [954]

Optimal result

Integrand size = 24, antiderivative size = 160 \[ \int (1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2} \, dx=\frac {746691 \sqrt {1-2 x} \sqrt {3+5 x}}{204800}+\frac {22627 (1-2 x)^{3/2} \sqrt {3+5 x}}{20480}-\frac {2057 (1-2 x)^{5/2} \sqrt {3+5 x}}{1024}-\frac {187}{256} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac {17}{80} (1-2 x)^{5/2} (3+5 x)^{5/2}-\frac {1}{20} (1-2 x)^{5/2} (3+5 x)^{7/2}+\frac {8213601 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{204800 \sqrt {10}} \] Output:

746691/204800*(1-2*x)^(1/2)*(3+5*x)^(1/2)+22627/20480*(1-2*x)^(3/2)*(3+5*x 
)^(1/2)-2057/1024*(1-2*x)^(5/2)*(3+5*x)^(1/2)-187/256*(1-2*x)^(5/2)*(3+5*x 
)^(3/2)-17/80*(1-2*x)^(5/2)*(3+5*x)^(5/2)-1/20*(1-2*x)^(5/2)*(3+5*x)^(7/2) 
+8213601/2048000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.55 \[ \int (1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2} \, dx=\frac {-10 \sqrt {1-2 x} \left (1666197-3897705 x-23840180 x^2-16824800 x^3+32624000 x^4+57600000 x^5+25600000 x^6\right )-8213601 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{2048000 \sqrt {3+5 x}} \] Input:

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

Output:

(-10*Sqrt[1 - 2*x]*(1666197 - 3897705*x - 23840180*x^2 - 16824800*x^3 + 32 
624000*x^4 + 57600000*x^5 + 25600000*x^6) - 8213601*Sqrt[30 + 50*x]*ArcTan 
[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/(2048000*Sqrt[3 + 5*x])
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.16, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {90, 60, 60, 60, 60, 60, 64, 223}

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.

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

\(\Big \downarrow \) 90

\(\displaystyle \frac {17}{8} \int (1-2 x)^{3/2} (5 x+3)^{5/2}dx-\frac {1}{20} (1-2 x)^{5/2} (5 x+3)^{7/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {17}{8} \left (\frac {11}{4} \int (1-2 x)^{3/2} (5 x+3)^{3/2}dx-\frac {1}{10} (1-2 x)^{5/2} (5 x+3)^{5/2}\right )-\frac {1}{20} (1-2 x)^{5/2} (5 x+3)^{7/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {17}{8} \left (\frac {11}{4} \left (\frac {33}{16} \int (1-2 x)^{3/2} \sqrt {5 x+3}dx-\frac {1}{8} (1-2 x)^{5/2} (5 x+3)^{3/2}\right )-\frac {1}{10} (1-2 x)^{5/2} (5 x+3)^{5/2}\right )-\frac {1}{20} (1-2 x)^{5/2} (5 x+3)^{7/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {17}{8} \left (\frac {11}{4} \left (\frac {33}{16} \left (\frac {11}{12} \int \frac {(1-2 x)^{3/2}}{\sqrt {5 x+3}}dx-\frac {1}{6} (1-2 x)^{5/2} \sqrt {5 x+3}\right )-\frac {1}{8} (1-2 x)^{5/2} (5 x+3)^{3/2}\right )-\frac {1}{10} (1-2 x)^{5/2} (5 x+3)^{5/2}\right )-\frac {1}{20} (1-2 x)^{5/2} (5 x+3)^{7/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {17}{8} \left (\frac {11}{4} \left (\frac {33}{16} \left (\frac {11}{12} \left (\frac {33}{20} \int \frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}dx+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )-\frac {1}{6} (1-2 x)^{5/2} \sqrt {5 x+3}\right )-\frac {1}{8} (1-2 x)^{5/2} (5 x+3)^{3/2}\right )-\frac {1}{10} (1-2 x)^{5/2} (5 x+3)^{5/2}\right )-\frac {1}{20} (1-2 x)^{5/2} (5 x+3)^{7/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {17}{8} \left (\frac {11}{4} \left (\frac {33}{16} \left (\frac {11}{12} \left (\frac {33}{20} \left (\frac {11}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )-\frac {1}{6} (1-2 x)^{5/2} \sqrt {5 x+3}\right )-\frac {1}{8} (1-2 x)^{5/2} (5 x+3)^{3/2}\right )-\frac {1}{10} (1-2 x)^{5/2} (5 x+3)^{5/2}\right )-\frac {1}{20} (1-2 x)^{5/2} (5 x+3)^{7/2}\)

\(\Big \downarrow \) 64

\(\displaystyle \frac {17}{8} \left (\frac {11}{4} \left (\frac {33}{16} \left (\frac {11}{12} \left (\frac {33}{20} \left (\frac {11}{25} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )-\frac {1}{6} (1-2 x)^{5/2} \sqrt {5 x+3}\right )-\frac {1}{8} (1-2 x)^{5/2} (5 x+3)^{3/2}\right )-\frac {1}{10} (1-2 x)^{5/2} (5 x+3)^{5/2}\right )-\frac {1}{20} (1-2 x)^{5/2} (5 x+3)^{7/2}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {17}{8} \left (\frac {11}{4} \left (\frac {33}{16} \left (\frac {11}{12} \left (\frac {33}{20} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5 \sqrt {10}}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )-\frac {1}{6} (1-2 x)^{5/2} \sqrt {5 x+3}\right )-\frac {1}{8} (1-2 x)^{5/2} (5 x+3)^{3/2}\right )-\frac {1}{10} (1-2 x)^{5/2} (5 x+3)^{5/2}\right )-\frac {1}{20} (1-2 x)^{5/2} (5 x+3)^{7/2}\)

Input:

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

Output:

-1/20*((1 - 2*x)^(5/2)*(3 + 5*x)^(7/2)) + (17*(-1/10*((1 - 2*x)^(5/2)*(3 + 
 5*x)^(5/2)) + (11*(-1/8*((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2)) + (33*(-1/6*((1 
 - 2*x)^(5/2)*Sqrt[3 + 5*x]) + (11*(((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/10 + ( 
33*((Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/5 + (11*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]] 
)/(5*Sqrt[10])))/20))/12))/16))/4))/8
 

Defintions of rubi rules used

rule 60

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] + Simp[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] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 64

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp 
[2/b   Subst[Int[1/Sqrt[c - a*(d/b) + d*(x^2/b)], x], x, Sqrt[a + b*x]], x] 
 /; FreeQ[{a, b, c, d}, x] && GtQ[c - a*(d/b), 0] && ( !GtQ[a - c*(b/d), 0] 
 || PosQ[b])
 

rule 90

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

rule 223

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

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.71

method	result	size
risch	\(\frac {\left (5120000 x^{5}+8448000 x^{4}+1456000 x^{3}-4238560 x^{2}-2224900 x +555399\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{204800 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {8213601 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{4096000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(113\)
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (-102400000 x^{5} \sqrt {-10 x^{2}-x +3}-168960000 x^{4} \sqrt {-10 x^{2}-x +3}-29120000 x^{3} \sqrt {-10 x^{2}-x +3}[phone] x^{2} \sqrt {-10 x^{2}-x +3}+8213601 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] x \sqrt {-10 x^{2}-x +3}-11107980 \sqrt {-10 x^{2}-x +3}\right )}{4096000 \sqrt {-10 x^{2}-x +3}}\)	\(138\)

Input:

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

Output:

1/204800*(5120000*x^5+8448000*x^4+1456000*x^3-4238560*x^2-2224900*x+555399 
)*(-1+2*x)*(3+5*x)^(1/2)/(-(-1+2*x)*(3+5*x))^(1/2)*((1-2*x)*(3+5*x))^(1/2) 
/(1-2*x)^(1/2)+8213601/4096000*10^(1/2)*arcsin(20/11*x+1/11)*((1-2*x)*(3+5 
*x))^(1/2)/(1-2*x)^(1/2)/(3+5*x)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.51 \[ \int (1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2} \, dx=-\frac {1}{204800} \, {\left (5120000 \, x^{5} + 8448000 \, x^{4} + 1456000 \, x^{3} - 4238560 \, x^{2} - 2224900 \, x + 555399\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {8213601}{4096000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \] Input:

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

Output:

-1/204800*(5120000*x^5 + 8448000*x^4 + 1456000*x^3 - 4238560*x^2 - 2224900 
*x + 555399)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 8213601/4096000*sqrt(10)*arcta 
n(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.62 \[ \int (1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2} \, dx=-\frac {1}{4} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x - \frac {29}{80} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {187}{128} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {187}{2560} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {67881}{10240} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {8213601}{4096000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {67881}{204800} \, \sqrt {-10 \, x^{2} - x + 3} \] Input:

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

Output:

-1/4*(-10*x^2 - x + 3)^(5/2)*x - 29/80*(-10*x^2 - x + 3)^(5/2) + 187/128*( 
-10*x^2 - x + 3)^(3/2)*x + 187/2560*(-10*x^2 - x + 3)^(3/2) + 67881/10240* 
sqrt(-10*x^2 - x + 3)*x - 8213601/4096000*sqrt(10)*arcsin(-20/11*x - 1/11) 
 + 67881/204800*sqrt(-10*x^2 - x + 3)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 356 vs. \(2 (115) = 230\).

Time = 0.16 (sec) , antiderivative size = 356, normalized size of antiderivative = 2.22 \[ \int (1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2} \, dx=-\frac {1}{51200000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (100 \, x - 311\right )} {\left (5 \, x + 3\right )} + 46071\right )} {\left (5 \, x + 3\right )} - 775911\right )} {\left (5 \, x + 3\right )} + 15385695\right )} {\left (5 \, x + 3\right )} - 99422145\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 220189365 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {59}{38400000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {157}{1920000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {51}{40000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {243}{2000} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {27}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \] Input:

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

Output:

-1/51200000*sqrt(5)*(2*(4*(8*(4*(16*(100*x - 311)*(5*x + 3) + 46071)*(5*x 
+ 3) - 775911)*(5*x + 3) + 15385695)*(5*x + 3) - 99422145)*sqrt(5*x + 3)*s 
qrt(-10*x + 5) - 220189365*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 
59/38400000*sqrt(5)*(2*(4*(8*(12*(80*x - 203)*(5*x + 3) + 19073)*(5*x + 3) 
 - 506185)*(5*x + 3) + 4031895)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 10392195*s 
qrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 157/1920000*sqrt(5)*(2*(4*(8 
*(60*x - 119)*(5*x + 3) + 6163)*(5*x + 3) - 66189)*sqrt(5*x + 3)*sqrt(-10* 
x + 5) - 184305*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 51/40000*sq 
rt(5)*(2*(4*(40*x - 59)*(5*x + 3) + 1293)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 
4785*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 243/2000*sqrt(5)*(2*(2 
0*x - 23)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 143*sqrt(2)*arcsin(1/11*sqrt(22) 
*sqrt(5*x + 3))) + 27/25*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x 
 + 3)) + 2*sqrt(5*x + 3)*sqrt(-10*x + 5))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.72 \[ \int (1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2} \, dx=-\frac {8213601 \sqrt {10}\, \mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2048000}-25 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{5}-\frac {165 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{4}}{4}-\frac {455 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{3}}{64}+\frac {26491 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{1280}+\frac {22249 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x}{2048}-\frac {555399 \sqrt {5 x +3}\, \sqrt {-2 x +1}}{204800} \] Input:

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

Output:

( - 8213601*sqrt(10)*asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11)) - 51200000* 
sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**5 - 84480000*sqrt(5*x + 3)*sqrt( - 2*x + 
 1)*x**4 - 14560000*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**3 + 42385600*sqrt(5* 
x + 3)*sqrt( - 2*x + 1)*x**2 + 22249000*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x - 
 5553990*sqrt(5*x + 3)*sqrt( - 2*x + 1))/2048000