\(\int \frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx\) [1439]

Optimal result

Integrand size = 28, antiderivative size = 249 \[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=-\frac {8787401429 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{40540500}-\frac {377529563 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{8108100}-\frac {1799981 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{162162}-\frac {67825 \sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}}{23166}-\frac {205}{429} \sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{3/2}-\frac {1}{13} \sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}-\frac {610627101631 E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )}{34749000 \sqrt {35}}+\frac {8787401429 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right )}{17374500 \sqrt {35}} \] Output:

-8787401429/40540500*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)-377529563/8 
108100*(1-2*x)^(1/2)*(2+3*x)^(3/2)*(3+5*x)^(1/2)-1799981/162162*(1-2*x)^(1 
/2)*(2+3*x)^(5/2)*(3+5*x)^(1/2)-67825/23166*(1-2*x)^(1/2)*(2+3*x)^(7/2)*(3 
+5*x)^(1/2)-205/429*(1-2*x)^(1/2)*(2+3*x)^(7/2)*(3+5*x)^(3/2)-1/13*(1-2*x) 
^(1/2)*(2+3*x)^(7/2)*(3+5*x)^(5/2)-610627101631/1216215000*EllipticE(1/11* 
55^(1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)[phone]/608107500*Elli 
pticF(1/11*55^(1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 8.26 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.45 \[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=\frac {-30 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (16001700059+19961825445 x+22592085750 x^2+18620894250 x^3+9351247500 x^4+2104987500 x^5\right )[phone] i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-628994862335 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{1216215000} \] Input:

Integrate[((2 + 3*x)^(7/2)*(3 + 5*x)^(5/2))/Sqrt[1 - 2*x],x]
 

Output:

(-30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(16001700059 + 19961825445* 
x + 22592085750*x^2 + 18620894250*x^3 + 9351247500*x^4 + 2104987500*x^5) + 
 (610627101631*I)*Sqrt[33]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - ( 
628994862335*I)*Sqrt[33]*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])/1216 
215000
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.14, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.536, Rules used = {112, 27, 171, 27, 171, 27, 171, 25, 171, 27, 171, 25, 176, 123, 129}

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.

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

\(\Big \downarrow \) 112

\(\displaystyle \frac {1}{13} \int \frac {(3 x+2)^{5/2} (5 x+3)^{3/2} (410 x+257)}{2 \sqrt {1-2 x}}dx-\frac {1}{13} \sqrt {1-2 x} (3 x+2)^{7/2} (5 x+3)^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{26} \int \frac {(3 x+2)^{5/2} (5 x+3)^{3/2} (410 x+257)}{\sqrt {1-2 x}}dx-\frac {1}{13} \sqrt {1-2 x} (3 x+2)^{7/2} (5 x+3)^{5/2}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {1}{26} \left (-\frac {1}{55} \int -\frac {5 (3 x+2)^{3/2} (5 x+3)^{3/2} (14303 x+9057)}{\sqrt {1-2 x}}dx-\frac {82}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )-\frac {1}{13} \sqrt {1-2 x} (3 x+2)^{7/2} (5 x+3)^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{26} \left (\frac {1}{11} \int \frac {(3 x+2)^{3/2} (5 x+3)^{3/2} (14303 x+9057)}{\sqrt {1-2 x}}dx-\frac {82}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )-\frac {1}{13} \sqrt {1-2 x} (3 x+2)^{7/2} (5 x+3)^{5/2}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {1}{26} \left (\frac {1}{11} \left (-\frac {1}{45} \int -\frac {\sqrt {3 x+2} (5 x+3)^{3/2} (3990114 x+2559955)}{2 \sqrt {1-2 x}}dx-\frac {14303}{45} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )-\frac {82}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )-\frac {1}{13} \sqrt {1-2 x} (3 x+2)^{7/2} (5 x+3)^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{26} \left (\frac {1}{11} \left (\frac {1}{90} \int \frac {\sqrt {3 x+2} (5 x+3)^{3/2} (3990114 x+2559955)}{\sqrt {1-2 x}}dx-\frac {14303}{45} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )-\frac {82}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )-\frac {1}{13} \sqrt {1-2 x} (3 x+2)^{7/2} (5 x+3)^{5/2}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {1}{26} \left (\frac {1}{11} \left (\frac {1}{90} \left (-\frac {1}{35} \int -\frac {(5 x+3)^{3/2} (416429493 x+272964529)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {3990114}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {14303}{45} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )-\frac {82}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )-\frac {1}{13} \sqrt {1-2 x} (3 x+2)^{7/2} (5 x+3)^{5/2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{26} \left (\frac {1}{11} \left (\frac {1}{90} \left (\frac {1}{35} \int \frac {(5 x+3)^{3/2} (416429493 x+272964529)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {3990114}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {14303}{45} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )-\frac {82}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )-\frac {1}{13} \sqrt {1-2 x} (3 x+2)^{7/2} (5 x+3)^{5/2}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {1}{26} \left (\frac {1}{11} \left (\frac {1}{90} \left (\frac {1}{35} \left (-\frac {1}{15} \int -\frac {3 \sqrt {5 x+3} (18367760704 x+11936801307)}{2 \sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {138809831}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {3990114}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {14303}{45} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )-\frac {82}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )-\frac {1}{13} \sqrt {1-2 x} (3 x+2)^{7/2} (5 x+3)^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{26} \left (\frac {1}{11} \left (\frac {1}{90} \left (\frac {1}{35} \left (\frac {1}{10} \int \frac {\sqrt {5 x+3} (18367760704 x+11936801307)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {138809831}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {3990114}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {14303}{45} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )-\frac {82}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )-\frac {1}{13} \sqrt {1-2 x} (3 x+2)^{7/2} (5 x+3)^{5/2}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {1}{26} \left (\frac {1}{11} \left (\frac {1}{90} \left (\frac {1}{35} \left (\frac {1}{10} \left (-\frac {1}{9} \int -\frac {610627101631 x+386580797753}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {18367760704}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {138809831}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {3990114}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {14303}{45} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )-\frac {82}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )-\frac {1}{13} \sqrt {1-2 x} (3 x+2)^{7/2} (5 x+3)^{5/2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{26} \left (\frac {1}{11} \left (\frac {1}{90} \left (\frac {1}{35} \left (\frac {1}{10} \left (\frac {1}{9} \int \frac {610627101631 x+386580797753}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {18367760704}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {138809831}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {3990114}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {14303}{45} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )-\frac {82}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )-\frac {1}{13} \sqrt {1-2 x} (3 x+2)^{7/2} (5 x+3)^{5/2}\)

\(\Big \downarrow \) 176

\(\displaystyle \frac {1}{26} \left (\frac {1}{11} \left (\frac {1}{90} \left (\frac {1}{35} \left (\frac {1}{10} \left (\frac {1}{9} \left (\frac {101022683872}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {610627101631}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {18367760704}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {138809831}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {3990114}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {14303}{45} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )-\frac {82}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )-\frac {1}{13} \sqrt {1-2 x} (3 x+2)^{7/2} (5 x+3)^{5/2}\)

\(\Big \downarrow \) 123

\(\displaystyle \frac {1}{26} \left (\frac {1}{11} \left (\frac {1}{90} \left (\frac {1}{35} \left (\frac {1}{10} \left (\frac {1}{9} \left (\frac {101022683872}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {610627101631}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {18367760704}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {138809831}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {3990114}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {14303}{45} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )-\frac {82}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )-\frac {1}{13} \sqrt {1-2 x} (3 x+2)^{7/2} (5 x+3)^{5/2}\)

\(\Big \downarrow \) 129

\(\displaystyle \frac {1}{26} \left (\frac {1}{11} \left (\frac {1}{90} \left (\frac {1}{35} \left (\frac {1}{10} \left (\frac {1}{9} \left (-\frac {18367760704}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {610627101631}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {18367760704}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {138809831}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {3990114}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {14303}{45} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )-\frac {82}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )-\frac {1}{13} \sqrt {1-2 x} (3 x+2)^{7/2} (5 x+3)^{5/2}\)

Input:

Int[((2 + 3*x)^(7/2)*(3 + 5*x)^(5/2))/Sqrt[1 - 2*x],x]
 

Output:

-1/13*(Sqrt[1 - 2*x]*(2 + 3*x)^(7/2)*(3 + 5*x)^(5/2)) + ((-82*Sqrt[1 - 2*x 
]*(2 + 3*x)^(5/2)*(3 + 5*x)^(5/2))/11 + ((-14303*Sqrt[1 - 2*x]*(2 + 3*x)^( 
3/2)*(3 + 5*x)^(5/2))/45 + ((-3990114*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x 
)^(5/2))/35 + ((-138809831*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/5 
+ ((-18367760704*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/9 + ((-6106271 
01631*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (1 
8367760704*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 
)/9)/10)/35)/90)/11)/26
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 112

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

rule 123

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ 
)]), x_] :> Simp[(2/b)*Rt[-(b*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x] 
/Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, 
b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !L 
tQ[-(b*c - a*d)/d, 0] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d 
), 0] && GtQ[d/(d*e - c*f), 0] &&  !LtQ[(b*c - a*d)/b, 0])
 

rule 129

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x 
_)]), x_] :> Simp[2*(Rt[-b/d, 2]/(b*Sqrt[(b*e - a*f)/b]))*EllipticF[ArcSin[ 
Sqrt[a + b*x]/(Rt[-b/d, 2]*Sqrt[(b*c - a*d)/b])], f*((b*c - a*d)/(d*(b*e - 
a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ 
[(b*e - a*f)/b, 0] && PosQ[-b/d] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[(d 
*e - c*f)/d, 0] && GtQ[-d/b, 0]) &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[(( 
-b)*e + a*f)/f, 0] && GtQ[-f/b, 0]) &&  !(SimplerQ[e + f*x, a + b*x] && GtQ 
[((-d)*e + c*f)/f, 0] && GtQ[((-b)*e + a*f)/f, 0] && (PosQ[-f/d] || PosQ[-f 
/b]))
 

rule 171

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] 
 && IntegersQ[2*m, 2*n, 2*p]
 

rule 176

Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]* 
Sqrt[(e_) + (f_.)*(x_)]), x_] :> Simp[h/f   Int[Sqrt[e + f*x]/(Sqrt[a + b*x 
]*Sqrt[c + d*x]), x], x] + Simp[(f*g - e*h)/f   Int[1/(Sqrt[a + b*x]*Sqrt[c 
 + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && Sim 
plerQ[a + b*x, e + f*x] && SimplerQ[c + d*x, e + f*x]
 

Maple [A] (verified)

Time = 0.89 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.65

method	result	size
default	\(-\frac {\sqrt {2+3 x}\, \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (1894488750000 x^{8}[phone] x^{7}[phone] x^{6}[phone] \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-610627101631 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}-6953485592490 x -2880306010620\right )}{1216215000 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\)	\(163\)
risch	\(\frac {\left (2104987500 x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{40540500 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}+\frac {\left (\frac {386580797753 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \operatorname {EllipticF}\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{4459455000 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {610627101631 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \operatorname {EllipticF}\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{4459455000 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(266\)
elliptic	\(\frac {\sqrt {3+5 x}\, \sqrt {2+3 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {443596121 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{900900}-\frac {16001700059 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{40540500}+\frac {386580797753 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{1702701000 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {610627101631 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{1702701000 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {772379 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1386}-\frac {394093 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{858}-\frac {32985 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{4}}{143}-\frac {675 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{5}}{13}\right )}{\left (15 x^{2}+19 x +6\right ) \sqrt {1-2 x}}\)	\(312\)

Input:

int((2+3*x)^(7/2)*(3+5*x)^(5/2)/(1-2*x)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/1216215000*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(1894488750000*x^8 
[phone]*x^7+22769118225000*x^6+303068051616*2^(1/2)*(2+3*x)^(1/2)*( 
-3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))-61 
0627101631*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticE(1/ 
7*(28+42*x)^(1/2),1/2*70^(1/2))[phone]*x^5+27960569725500*x^4+2007 
9090637650*x^3+2782614262260*x^2-6953485592490*x-2880306010620)/(30*x^3+23 
*x^2-7*x-6)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.30 \[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=-\frac {1}{40540500} \, {\left (2104987500 \, x^{5} + 9351247500 \, x^{4} + 18620894250 \, x^{3} + 22592085750 \, x^{2} + 19961825445 \, x + 16001700059\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - \frac {20747848460257}{109459350000} \, \sqrt {-30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + \frac {610627101631}{1216215000} \, \sqrt {-30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \] Input:

integrate((2+3*x)^(7/2)*(3+5*x)^(5/2)/(1-2*x)^(1/2),x, algorithm="fricas")
 

Output:

-1/40540500*(2104987500*x^5 + 9351247500*x^4 + 18620894250*x^3 + 225920857 
50*x^2 + 19961825445*x + 16001700059)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2* 
x + 1) - 20747848460257/109459350000*sqrt(-30)*weierstrassPInverse(1159/67 
5, 38998/91125, x + 23/90) + 610627101631/1216215000*sqrt(-30)*weierstrass 
Zeta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 
 23/90))
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=\text {Timed out} \] Input:

integrate((2+3*x)**(7/2)*(3+5*x)**(5/2)/(1-2*x)**(1/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}}}{\sqrt {-2 \, x + 1}} \,d x } \] Input:

integrate((2+3*x)^(7/2)*(3+5*x)^(5/2)/(1-2*x)^(1/2),x, algorithm="maxima")
 

Output:

integrate((5*x + 3)^(5/2)*(3*x + 2)^(7/2)/sqrt(-2*x + 1), x)
 

Giac [F]

\[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}}}{\sqrt {-2 \, x + 1}} \,d x } \] Input:

integrate((2+3*x)^(7/2)*(3+5*x)^(5/2)/(1-2*x)^(1/2),x, algorithm="giac")
 

Output:

integrate((5*x + 3)^(5/2)*(3*x + 2)^(7/2)/sqrt(-2*x + 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=\int \frac {{\left (3\,x+2\right )}^{7/2}\,{\left (5\,x+3\right )}^{5/2}}{\sqrt {1-2\,x}} \,d x \] Input:

int(((3*x + 2)^(7/2)*(5*x + 3)^(5/2))/(1 - 2*x)^(1/2),x)
 

Output:

int(((3*x + 2)^(7/2)*(5*x + 3)^(5/2))/(1 - 2*x)^(1/2), x)
 

Reduce [F]

\[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=-\frac {675 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{5}}{13}-\frac {32985 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{4}}{143}-\frac {394093 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{3}}{858}-\frac {772379 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{1386}-\frac {443596121 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x}{900900}-\frac {9975594847 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{13813800}+\frac {610627101631 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{30 x^{3}+23 x^{2}-7 x -6}d x \right )}{41441400}-\frac {54461991131 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{30 x^{3}+23 x^{2}-7 x -6}d x \right )}{9209200} \] Input:

int((2+3*x)^(7/2)*(3+5*x)^(5/2)/(1-2*x)^(1/2),x)
 

Output:

( - 4303530000*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**5 - 1911810 
6000*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**4 - 38069383800*sqrt( 
3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**3 - 46188264200*sqrt(3*x + 2)*s 
qrt(5*x + 3)*sqrt( - 2*x + 1)*x**2 - 40810843132*sqrt(3*x + 2)*sqrt(5*x + 
3)*sqrt( - 2*x + 1)*x - 59853569082*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2* 
x + 1) + 1221254203262*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x 
**2)/(30*x**3 + 23*x**2 - 7*x - 6),x) - 490157920179*int((sqrt(3*x + 2)*sq 
rt(5*x + 3)*sqrt( - 2*x + 1))/(30*x**3 + 23*x**2 - 7*x - 6),x))/82882800