\(\int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\) [1434]

Optimal result

Integrand size = 28, antiderivative size = 94 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx=-\frac {5}{9} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {31}{27} \sqrt {35} E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )+\frac {29 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right )}{27 \sqrt {35}} \] Output:

-5/9*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)-31/27*EllipticE(1/11*55^(1/ 
2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)+29/945*EllipticF(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 = 2.56 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.94 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx=\frac {1}{27} \left (-15 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+31 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-32 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right ) \] Input:

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

Output:

(-15*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x] + (31*I)*Sqrt[33]*EllipticE 
[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - (32*I)*Sqrt[33]*EllipticF[I*ArcSinh[S 
qrt[9 + 15*x]], -2/33])/27
 

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {113, 27, 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.

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

\(\Big \downarrow \) 113

\(\displaystyle -\frac {1}{9} \int -\frac {310 x+197}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {5}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{18} \int \frac {310 x+197}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {5}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\)

\(\Big \downarrow \) 176

\(\displaystyle \frac {1}{18} \left (11 \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+62 \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {5}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\)

\(\Big \downarrow \) 123

\(\displaystyle \frac {1}{18} \left (11 \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-62 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {5}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\)

\(\Big \downarrow \) 129

\(\displaystyle \frac {1}{18} \left (-2 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-62 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {5}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\)

Input:

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

Output:

(-5*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/9 + (-62*Sqrt[11/3]*Ellipti 
cE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33] - 2*Sqrt[11/3]*EllipticF[ArcSin 
[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/18
 

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 113

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

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 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 = 0.55 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.47

method	result	size
default	\(-\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \sqrt {2+3 x}\, \left (33 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-62 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+900 x^{3}+690 x^{2}-210 x -180\right )}{54 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\)	\(138\)
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {5 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{9}+\frac {197 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{378 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {155 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{189 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(192\)
risch	\(\frac {5 \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{9 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}+\frac {\left (\frac {197 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \operatorname {EllipticF}\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{990 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {31 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \operatorname {EllipticF}\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{99 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(241\)

Input:

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

Output:

-1/54*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(33*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))-6 
2*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))+900*x^3+690*x^2-210*x-180)/(30*x^3+23*x^2-7*x-6)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.52 \[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx=-\frac {5}{9} \, \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - \frac {106}{243} \, \sqrt {-30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + \frac {31}{27} \, \sqrt {-30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \] Input:

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

Output:

-5/9*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 106/243*sqrt(-30)*weiers 
trassPInverse(1159/675, 38998/91125, x + 23/90) + 31/27*sqrt(-30)*weierstr 
assZeta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, 
x + 23/90))
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx=-\frac {30 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{23}+\frac {775 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{30 x^{3}+23 x^{2}-7 x -6}d x \right )}{23}-\frac {312 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{30 x^{3}+23 x^{2}-7 x -6}d x \right )}{23} \] Input:

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

Output:

( - 30*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1) + 775*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) - 
 312*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(30*x**3 + 23*x**2 
 - 7*x - 6),x))/23