3.23.88 \(\int \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2} \, dx\) [2288]

3.23.88.1 Optimal result

Integrand size = 26, antiderivative size = 165 \[ \int \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2} \, dx=\frac {1422839 \sqrt {1-2 x} \sqrt {3+5 x}}{81920}-\frac {129349 (1-2 x)^{3/2} \sqrt {3+5 x}}{8192}-\frac {11759 (1-2 x)^{3/2} (3+5 x)^{3/2}}{3072}-\frac {1069 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1280}-\frac {13}{80} (1-2 x)^{3/2} (3+5 x)^{7/2}-\frac {1}{20} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{7/2}+\frac {15651229 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{81920 \sqrt {10}} \]

output

-11759/3072*(1-2*x)^(3/2)*(3+5*x)^(3/2)-1069/1280*(1-2*x)^(3/2)*(3+5*x)^(5 
/2)-13/80*(1-2*x)^(3/2)*(3+5*x)^(7/2)-1/20*(1-2*x)^(3/2)*(2+3*x)*(3+5*x)^( 
7/2)[phone]/819200*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-129349/8 
192*(1-2*x)^(3/2)*(3+5*x)^(1/2)+1422839/81920*(1-2*x)^(1/2)*(3+5*x)^(1/2)
 

3.23.88.2 Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.53 \[ \int \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2} \, dx=\frac {10 \sqrt {1-2 x} \left (-18069507-30062193 x+49613228 x^2+181202720 x^3+248764800 x^4+168192000 x^5+46080000 x^6\right )-46953687 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{2457600 \sqrt {3+5 x}} \]

input

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

output

(10*Sqrt[1 - 2*x]*(-18069507 - 30062193*x + 49613228*x^2 + 181202720*x^3 + 
 248764800*x^4 + 168192000*x^5 + 46080000*x^6) - 46953687*Sqrt[30 + 50*x]* 
ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/(2457600*Sqrt[3 + 5*x])
 

3.23.88.3 Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.15, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {101, 27, 90, 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.

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

\(\Big \downarrow \) 101

\(\displaystyle -\frac {1}{60} \int -\frac {3}{2} \sqrt {1-2 x} (5 x+3)^{5/2} (325 x+212)dx-\frac {1}{20} (1-2 x)^{3/2} (3 x+2) (5 x+3)^{7/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{40} \int \sqrt {1-2 x} (5 x+3)^{5/2} (325 x+212)dx-\frac {1}{20} (1-2 x)^{3/2} (3 x+2) (5 x+3)^{7/2}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {1}{40} \left (\frac {1069}{4} \int \sqrt {1-2 x} (5 x+3)^{5/2}dx-\frac {13}{2} (1-2 x)^{3/2} (5 x+3)^{7/2}\right )-\frac {1}{20} (1-2 x)^{3/2} (3 x+2) (5 x+3)^{7/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{40} \left (\frac {1069}{4} \left (\frac {55}{16} \int \sqrt {1-2 x} (5 x+3)^{3/2}dx-\frac {1}{8} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )-\frac {13}{2} (1-2 x)^{3/2} (5 x+3)^{7/2}\right )-\frac {1}{20} (1-2 x)^{3/2} (3 x+2) (5 x+3)^{7/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{40} \left (\frac {1069}{4} \left (\frac {55}{16} \left (\frac {11}{4} \int \sqrt {1-2 x} \sqrt {5 x+3}dx-\frac {1}{6} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )-\frac {1}{8} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )-\frac {13}{2} (1-2 x)^{3/2} (5 x+3)^{7/2}\right )-\frac {1}{20} (1-2 x)^{3/2} (3 x+2) (5 x+3)^{7/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{40} \left (\frac {1069}{4} \left (\frac {55}{16} \left (\frac {11}{4} \left (\frac {11}{8} \int \frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}dx-\frac {1}{4} (1-2 x)^{3/2} \sqrt {5 x+3}\right )-\frac {1}{6} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )-\frac {1}{8} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )-\frac {13}{2} (1-2 x)^{3/2} (5 x+3)^{7/2}\right )-\frac {1}{20} (1-2 x)^{3/2} (3 x+2) (5 x+3)^{7/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{40} \left (\frac {1069}{4} \left (\frac {55}{16} \left (\frac {11}{4} \left (\frac {11}{8} \left (\frac {11}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} (1-2 x)^{3/2} \sqrt {5 x+3}\right )-\frac {1}{6} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )-\frac {1}{8} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )-\frac {13}{2} (1-2 x)^{3/2} (5 x+3)^{7/2}\right )-\frac {1}{20} (1-2 x)^{3/2} (3 x+2) (5 x+3)^{7/2}\)

\(\Big \downarrow \) 64

\(\displaystyle \frac {1}{40} \left (\frac {1069}{4} \left (\frac {55}{16} \left (\frac {11}{4} \left (\frac {11}{8} \left (\frac {11}{25} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} (1-2 x)^{3/2} \sqrt {5 x+3}\right )-\frac {1}{6} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )-\frac {1}{8} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )-\frac {13}{2} (1-2 x)^{3/2} (5 x+3)^{7/2}\right )-\frac {1}{20} (1-2 x)^{3/2} (3 x+2) (5 x+3)^{7/2}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {1}{40} \left (\frac {1069}{4} \left (\frac {55}{16} \left (\frac {11}{4} \left (\frac {11}{8} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5 \sqrt {10}}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} (1-2 x)^{3/2} \sqrt {5 x+3}\right )-\frac {1}{6} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )-\frac {1}{8} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )-\frac {13}{2} (1-2 x)^{3/2} (5 x+3)^{7/2}\right )-\frac {1}{20} (1-2 x)^{3/2} (3 x+2) (5 x+3)^{7/2}\)

input

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

output

-1/20*((1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*x)^(7/2)) + ((-13*(1 - 2*x)^(3/2)* 
(3 + 5*x)^(7/2))/2 + (1069*(-1/8*((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2)) + (55*( 
-1/6*((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + (11*(-1/4*((1 - 2*x)^(3/2)*Sqrt[3 
 + 5*x]) + (11*((Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/5 + (11*ArcSin[Sqrt[2/11]*Sq 
rt[3 + 5*x]])/(5*Sqrt[10])))/8))/4))/16))/4)/40
 

3.23.88.3.1 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 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 101

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + 
 p + 3))), x] + Simp[1/(d*f*(n + p + 3))   Int[(c + d*x)^n*(e + f*x)^p*Simp 
[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f 
*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, 
 c, d, e, f, n, p}, x] && NeQ[n + p + 3, 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]
 

3.23.88.4 Maple [A] (verified)

Time = 1.11 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.68

method	result	size
risch	\(-\frac {\left (9216000 x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}+17884 x -6023169\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{245760 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {15651229 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{1638400 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(113\)
default	\(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (184320000 x^{5} \sqrt {-10 x^{2}-x +3}[phone] x^{4} \sqrt {-10 x^{2}-x +3}[phone] x^{3} \sqrt {-10 x^{2}-x +3}[phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+357680 x \sqrt {-10 x^{2}-x +3}-120463380 \sqrt {-10 x^{2}-x +3}\right )}{4915200 \sqrt {-10 x^{2}-x +3}}\)	\(138\)

input

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

output

-1/245760*(9216000*x^5+28108800*x^4+32887680*x^3+16507936*x^2+17884*x-6023 
169)*(-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)[phone]/1638400*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)
 

3.23.88.5 Fricas [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.50 \[ \int \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2} \, dx=\frac {1}{245760} \, {\left (9216000 \, x^{5} + 28108800 \, x^{4} + 32887680 \, x^{3} + 16507936 \, x^{2} + 17884 \, x - 6023169\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {15651229}{1638400} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]

input

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

output

1/245760*(9216000*x^5 + 28108800*x^4 + 32887680*x^3 + 16507936*x^2 + 17884 
*x - 6023169)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 15651229/1638400*sqrt(10)*arc 
tan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3) 
)
 

3.23.88.6 Sympy [F]

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

input

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

output

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

3.23.88.7 Maxima [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.63 \[ \int \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2} \, dx=-\frac {15}{4} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} - \frac {177}{16} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} - \frac {17153}{1280} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {133567}{15360} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {129349}{4096} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {15651229}{1638400} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {129349}{81920} \, \sqrt {-10 \, x^{2} - x + 3} \]

input

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

output

-15/4*(-10*x^2 - x + 3)^(3/2)*x^3 - 177/16*(-10*x^2 - x + 3)^(3/2)*x^2 - 1 
7153/1280*(-10*x^2 - x + 3)^(3/2)*x - 133567/15360*(-10*x^2 - x + 3)^(3/2) 
 + 129349/4096*sqrt(-10*x^2 - x + 3)*x - 15651229/1638400*sqrt(10)*arcsin( 
-20/11*x - 1/11) + 129349/81920*sqrt(-10*x^2 - x + 3)
 

3.23.88.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 356 vs. \(2 (120) = 240\).

Time = 0.38 (sec) , antiderivative size = 356, normalized size of antiderivative = 2.16 \[ \int \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2} \, dx=\frac {3}{102400000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (100 \, x - 311\right )} {\left (5 \, x + 3\right )} + 46071\right )} {\left (5 \, x + 3\right )} - 775911\right )} {\left (5 \, x + 3\right )} + 15385695\right )} {\left (5 \, x + 3\right )} - 99422145\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 220189365 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {47}{12800000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {883}{1920000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {921}{40000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {54}{125} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {54}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]

input

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

output

3/102400000*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))) + 
47/12800000*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))) + 883/1920000*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))) + 921/40000*s 
qrt(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))) + 54/125*sqrt(5)*(2*(20 
*x - 23)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 143*sqrt(2)*arcsin(1/11*sqrt(22)* 
sqrt(5*x + 3))) + 54/25*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x 
+ 3)) + 2*sqrt(5*x + 3)*sqrt(-10*x + 5))
 

3.23.88.9 Mupad [F(-1)]

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

input

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

output

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