3.23.74 \(\int \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{3/2} \, dx\) [2274]

3.23.74.1 Optimal result

Integrand size = 26, antiderivative size = 179 \[ \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{3/2} \, dx=\frac {6384004649 \sqrt {1-2 x} \sqrt {3+5 x}}{204800000}-\frac {580364059 (1-2 x)^{3/2} \sqrt {3+5 x}}{20480000}-\frac {52760369 (1-2 x)^{3/2} (3+5 x)^{3/2}}{7680000}-\frac {403 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}}{2800}-\frac {3}{70} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{5/2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2} (1480103+874608 x)}{640000}+\frac {70224051139 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{204800000 \sqrt {10}} \]

output

-52760369/7680000*(1-2*x)^(3/2)*(3+5*x)^(3/2)-403/2800*(1-2*x)^(3/2)*(2+3* 
x)^2*(3+5*x)^(5/2)-3/70*(1-2*x)^(3/2)*(2+3*x)^3*(3+5*x)^(5/2)-1/640000*(1- 
2*x)^(3/2)*(3+5*x)^(5/2)*(1480103+874608*x)[phone]/2048000000*arcsin( 
1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-580364059/20480000*(1-2*x)^(3/2)*(3+ 
5*x)^(1/2)[phone]/204800000*(1-2*x)^(1/2)*(3+5*x)^(1/2)
 

3.23.74.2 Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.52 \[ \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{3/2} \, dx=\frac {10 \sqrt {1-2 x} \left (-604565196363-1226406863625 x+843850516940 x^2+5499280071200 x^3+10250528112000 x^4+10254608640000 x^5+5498496000000 x^6+1244160000000 x^7\right )-1474705073919 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{43008000000 \sqrt {3+5 x}} \]

input

Integrate[Sqrt[1 - 2*x]*(2 + 3*x)^4*(3 + 5*x)^(3/2),x]
 

output

(10*Sqrt[1 - 2*x]*(-604565196363 - 1226406863625*x + 843850516940*x^2 + 54 
99280071200*x^3 + 10250528112000*x^4 + 10254608640000*x^5 + 5498496000000* 
x^6 + 1244160000000*x^7) - 1474705073919*Sqrt[30 + 50*x]*ArcTan[Sqrt[5/2 - 
 5*x]/Sqrt[3 + 5*x]])/(43008000000*Sqrt[3 + 5*x])
 

3.23.74.3 Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.14, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {111, 27, 170, 27, 164, 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.

\(\displaystyle \int \sqrt {1-2 x} (3 x+2)^4 (5 x+3)^{3/2} \, dx\)

\(\Big \downarrow \) 111

\(\displaystyle -\frac {1}{70} \int -\frac {1}{2} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{3/2} (1209 x+764)dx-\frac {3}{70} (1-2 x)^{3/2} (5 x+3)^{5/2} (3 x+2)^3\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{140} \int \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{3/2} (1209 x+764)dx-\frac {3}{70} (1-2 x)^{3/2} (3 x+2)^3 (5 x+3)^{5/2}\)

\(\Big \downarrow \) 170

\(\displaystyle \frac {1}{140} \left (-\frac {1}{60} \int -\frac {21}{2} \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2} (18221 x+11610)dx-\frac {403}{20} (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{5/2}\right )-\frac {3}{70} (1-2 x)^{3/2} (3 x+2)^3 (5 x+3)^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{140} \left (\frac {7}{40} \int \sqrt {1-2 x} (3 x+2) (5 x+3)^{3/2} (18221 x+11610)dx-\frac {403}{20} (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{5/2}\right )-\frac {3}{70} (1-2 x)^{3/2} (3 x+2)^3 (5 x+3)^{5/2}\)

\(\Big \downarrow \) 164

\(\displaystyle \frac {1}{140} \left (\frac {7}{40} \left (\frac {52760369 \int \sqrt {1-2 x} (5 x+3)^{3/2}dx}{1600}-\frac {1}{800} (1-2 x)^{3/2} (5 x+3)^{5/2} (874608 x+1480103)\right )-\frac {403}{20} (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{5/2}\right )-\frac {3}{70} (1-2 x)^{3/2} (3 x+2)^3 (5 x+3)^{5/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{140} \left (\frac {7}{40} \left (\frac {52760369 \left (\frac {11}{4} \int \sqrt {1-2 x} \sqrt {5 x+3}dx-\frac {1}{6} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )}{1600}-\frac {1}{800} (1-2 x)^{3/2} (5 x+3)^{5/2} (874608 x+1480103)\right )-\frac {403}{20} (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{5/2}\right )-\frac {3}{70} (1-2 x)^{3/2} (3 x+2)^3 (5 x+3)^{5/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{140} \left (\frac {7}{40} \left (\frac {52760369 \left (\frac {11}{4} \left (\frac {11}{8} \int \frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}dx-\frac {1}{4} (1-2 x)^{3/2} \sqrt {5 x+3}\right )-\frac {1}{6} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )}{1600}-\frac {1}{800} (1-2 x)^{3/2} (5 x+3)^{5/2} (874608 x+1480103)\right )-\frac {403}{20} (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{5/2}\right )-\frac {3}{70} (1-2 x)^{3/2} (3 x+2)^3 (5 x+3)^{5/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{140} \left (\frac {7}{40} \left (\frac {52760369 \left (\frac {11}{4} \left (\frac {11}{8} \left (\frac {11}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} (1-2 x)^{3/2} \sqrt {5 x+3}\right )-\frac {1}{6} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )}{1600}-\frac {1}{800} (1-2 x)^{3/2} (5 x+3)^{5/2} (874608 x+1480103)\right )-\frac {403}{20} (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{5/2}\right )-\frac {3}{70} (1-2 x)^{3/2} (3 x+2)^3 (5 x+3)^{5/2}\)

\(\Big \downarrow \) 64

\(\displaystyle \frac {1}{140} \left (\frac {7}{40} \left (\frac {52760369 \left (\frac {11}{4} \left (\frac {11}{8} \left (\frac {11}{25} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} (1-2 x)^{3/2} \sqrt {5 x+3}\right )-\frac {1}{6} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )}{1600}-\frac {1}{800} (1-2 x)^{3/2} (5 x+3)^{5/2} (874608 x+1480103)\right )-\frac {403}{20} (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{5/2}\right )-\frac {3}{70} (1-2 x)^{3/2} (3 x+2)^3 (5 x+3)^{5/2}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {1}{140} \left (\frac {7}{40} \left (\frac {52760369 \left (\frac {11}{4} \left (\frac {11}{8} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5 \sqrt {10}}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} (1-2 x)^{3/2} \sqrt {5 x+3}\right )-\frac {1}{6} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )}{1600}-\frac {1}{800} (1-2 x)^{3/2} (5 x+3)^{5/2} (874608 x+1480103)\right )-\frac {403}{20} (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{5/2}\right )-\frac {3}{70} (1-2 x)^{3/2} (3 x+2)^3 (5 x+3)^{5/2}\)

input

Int[Sqrt[1 - 2*x]*(2 + 3*x)^4*(3 + 5*x)^(3/2),x]
 

output

(-3*(1 - 2*x)^(3/2)*(2 + 3*x)^3*(3 + 5*x)^(5/2))/70 + ((-403*(1 - 2*x)^(3/ 
2)*(2 + 3*x)^2*(3 + 5*x)^(5/2))/20 + (7*(-1/800*((1 - 2*x)^(3/2)*(3 + 5*x) 
^(5/2)*(1480103 + 874608*x)) + (52760369*(-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]*S 
qrt[3 + 5*x])/5 + (11*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(5*Sqrt[10])))/8)) 
/4))/1600))/40)/140
 

3.23.74.3.1 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 111

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

rule 164

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ 
))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - 
 b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( 
c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h 
*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 
3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + 
d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3))   Int[( 
a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] 
&& NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
 

rule 170

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( 
e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) 
  Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 
) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) 
+ h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] 
 && IntegerQ[m]
 

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]
 

3.23.74.4 Maple [A] (verified)

Time = 1.13 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.66

method	result	size
risch	\(-\frac {\left (248832000000 x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}-72932734340 x -201521732121\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{4300800000 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {70224051139 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{4096000000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(118\)
default	\(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (4976640000000 \sqrt {-10 x^{2}-x +3}\, x^{6}[phone] x^{5} \sqrt {-10 x^{2}-x +3}[phone] x^{4} \sqrt {-10 x^{2}-x +3}[phone] x^{3} \sqrt {-10 x^{2}-x +3}[phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-1458654686800 x \sqrt {-10 x^{2}-x +3}-4030434642420 \sqrt {-10 x^{2}-x +3}\right )}{86016000000 \sqrt {-10 x^{2}-x +3}}\)	\(155\)

input

int((2+3*x)^4*(3+5*x)^(3/2)*(1-2*x)^(1/2),x,method=_RETURNVERBOSE)
 

output

-1/4300800000*(248832000000*x^6+950400000000*x^5+1480681728000*x^4+1161696 
585600*x^3+402838062880*x^2-72932734340*x-201521732121)*(-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)+70224 
051139/4096000000*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)
 

3.23.74.5 Fricas [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.49 \[ \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{3/2} \, dx=\frac {1}{4300800000} \, {\left (248832000000 \, x^{6} + 950400000000 \, x^{5} + 1480681728000 \, x^{4} + 1161696585600 \, x^{3} + 402838062880 \, x^{2} - 72932734340 \, x - 201521732121\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {70224051139}{4096000000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]

input

integrate((2+3*x)^4*(3+5*x)^(3/2)*(1-2*x)^(1/2),x, algorithm="fricas")
 

output

1/4300800000*(248832000000*x^6 + 950400000000*x^5 + 1480681728000*x^4 + 11 
61696585600*x^3 + 402838062880*x^2 - 72932734340*x - 201521732121)*sqrt(5* 
x + 3)*sqrt(-2*x + 1) - 70224051139/4096000000*sqrt(10)*arctan(1/20*sqrt(1 
0)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))
 

3.23.74.6 Sympy [F]

\[ \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{3/2} \, dx=\int \sqrt {1 - 2 x} \left (3 x + 2\right )^{4} \left (5 x + 3\right )^{\frac {3}{2}}\, dx \]

input

integrate((2+3*x)**4*(3+5*x)**(3/2)*(1-2*x)**(1/2),x)
 

output

Integral(sqrt(1 - 2*x)*(3*x + 2)**4*(5*x + 3)**(3/2), x)
 

3.23.74.7 Maxima [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.68 \[ \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{3/2} \, dx=-\frac {81}{14} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{4} - \frac {12051}{560} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} - \frac {1904661}{56000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} - \frac {134695173}{4480000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {890455739}{53760000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {580364059}{10240000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {70224051139}{4096000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {580364059}{204800000} \, \sqrt {-10 \, x^{2} - x + 3} \]

input

integrate((2+3*x)^4*(3+5*x)^(3/2)*(1-2*x)^(1/2),x, algorithm="maxima")
 

output

-81/14*(-10*x^2 - x + 3)^(3/2)*x^4 - 12051/560*(-10*x^2 - x + 3)^(3/2)*x^3 
 - 1904661/56000*(-10*x^2 - x + 3)^(3/2)*x^2 - 134695173/4480000*(-10*x^2 
- x + 3)^(3/2)*x - 890455739/53760000*(-10*x^2 - x + 3)^(3/2) + 580364059/ 
10240000*sqrt(-10*x^2 - x + 3)*x - 70224051139/4096000000*sqrt(10)*arcsin( 
-20/11*x - 1/11) + 580364059/204800000*sqrt(-10*x^2 - x + 3)
 

3.23.74.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 446 vs. \(2 (134) = 268\).

Time = 0.37 (sec) , antiderivative size = 446, normalized size of antiderivative = 2.49 \[ \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{3/2} \, dx=\frac {27}{71680000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (20 \, {\left (120 \, x - 443\right )} {\left (5 \, x + 3\right )} + 94933\right )} {\left (5 \, x + 3\right )} - 7838433\right )} {\left (5 \, x + 3\right )} + 98794353\right )} {\left (5 \, x + 3\right )} - 1568443065\right )} {\left (5 \, x + 3\right )} + 8438816295\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 17534989395 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {261}{1280000000} \, \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 {4203}{320000000} \, \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 {451}{400000} \, \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 {653}{15000} \, \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 {84}{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 {72}{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)^4*(3+5*x)^(3/2)*(1-2*x)^(1/2),x, algorithm="giac")
 

output

27/71680000000*sqrt(5)*(2*(4*(8*(4*(16*(20*(120*x - 443)*(5*x + 3) + 94933 
)*(5*x + 3) - 7838433)*(5*x + 3) + 98794353)*(5*x + 3) - 1568443065)*(5*x 
+ 3) + 8438816295)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 17534989395*sqrt(2)*arc 
sin(1/11*sqrt(22)*sqrt(5*x + 3))) + 261/1280000000*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)*sqrt(-10*x + 5) - 220189365*sqrt(2)* 
arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 4203/320000000*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*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt 
(5*x + 3))) + 451/400000*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))) + 653/15000*sqrt(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))) + 84/125*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))) + 72/25*sqrt(5)*( 
11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 2*sqrt(5*x + 3)*sqrt(-10* 
x + 5))
 

3.23.74.9 Mupad [F(-1)]

Timed out. \[ \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{3/2} \, dx=\int \sqrt {1-2\,x}\,{\left (3\,x+2\right )}^4\,{\left (5\,x+3\right )}^{3/2} \,d x \]

input

int((1 - 2*x)^(1/2)*(3*x + 2)^4*(5*x + 3)^(3/2),x)
 

output

int((1 - 2*x)^(1/2)*(3*x + 2)^4*(5*x + 3)^(3/2), x)