\(\int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} \sqrt {2+3 x}} \, dx\) [1563]

Optimal result

Integrand size = 28, antiderivative size = 129 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} \sqrt {2+3 x}} \, dx=\frac {2 \sqrt {2+3 x} \sqrt {3+5 x}}{21 (1-2 x)^{3/2}}+\frac {62 \sqrt {2+3 x} \sqrt {3+5 x}}{1617 \sqrt {1-2 x}}+\frac {31}{231} \sqrt {\frac {5}{7}} E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )+\frac {2}{231} \sqrt {\frac {5}{7}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right ) \] Output:

2/21*(2+3*x)^(1/2)*(3+5*x)^(1/2)/(1-2*x)^(3/2)+62/1617*(2+3*x)^(1/2)*(3+5* 
x)^(1/2)/(1-2*x)^(1/2)+31/1617*EllipticE(1/11*55^(1/2)*(1-2*x)^(1/2),1/35* 
1155^(1/2))*35^(1/2)+2/1617*EllipticF(1/11*55^(1/2)*(1-2*x)^(1/2),1/35*115 
5^(1/2))*35^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.66 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} \sqrt {2+3 x}} \, dx=-\frac {4 \sqrt {2+3 x} \sqrt {3+5 x} (-54+31 x)-31 i \sqrt {33-66 x} (-1+2 x) E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+35 i \sqrt {33-66 x} (-1+2 x) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{1617 (1-2 x)^{3/2}} \] Input:

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

Output:

-1/1617*(4*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-54 + 31*x) - (31*I)*Sqrt[33 - 66* 
x]*(-1 + 2*x)*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] + (35*I)*Sqrt[33 
 - 66*x]*(-1 + 2*x)*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])/(1 - 2*x) 
^(3/2)
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {110, 27, 169, 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.

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

\(\Big \downarrow \) 110

\(\displaystyle \frac {2 \sqrt {3 x+2} \sqrt {5 x+3}}{21 (1-2 x)^{3/2}}-\frac {2}{21} \int -\frac {15 x+8}{2 (1-2 x)^{3/2} \sqrt {3 x+2} \sqrt {5 x+3}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{21} \int \frac {15 x+8}{(1-2 x)^{3/2} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {2 \sqrt {3 x+2} \sqrt {5 x+3}}{21 (1-2 x)^{3/2}}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {1}{21} \left (\frac {62 \sqrt {3 x+2} \sqrt {5 x+3}}{77 \sqrt {1-2 x}}-\frac {2}{77} \int \frac {15 (31 x+23)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx\right )+\frac {2 \sqrt {3 x+2} \sqrt {5 x+3}}{21 (1-2 x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{21} \left (\frac {62 \sqrt {3 x+2} \sqrt {5 x+3}}{77 \sqrt {1-2 x}}-\frac {15}{77} \int \frac {31 x+23}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx\right )+\frac {2 \sqrt {3 x+2} \sqrt {5 x+3}}{21 (1-2 x)^{3/2}}\)

\(\Big \downarrow \) 176

\(\displaystyle \frac {1}{21} \left (\frac {62 \sqrt {3 x+2} \sqrt {5 x+3}}{77 \sqrt {1-2 x}}-\frac {15}{77} \left (\frac {22}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {31}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )\right )+\frac {2 \sqrt {3 x+2} \sqrt {5 x+3}}{21 (1-2 x)^{3/2}}\)

\(\Big \downarrow \) 123

\(\displaystyle \frac {1}{21} \left (\frac {62 \sqrt {3 x+2} \sqrt {5 x+3}}{77 \sqrt {1-2 x}}-\frac {15}{77} \left (\frac {22}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {31}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )\right )+\frac {2 \sqrt {3 x+2} \sqrt {5 x+3}}{21 (1-2 x)^{3/2}}\)

\(\Big \downarrow \) 129

\(\displaystyle \frac {1}{21} \left (\frac {62 \sqrt {3 x+2} \sqrt {5 x+3}}{77 \sqrt {1-2 x}}-\frac {15}{77} \left (-\frac {4}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {31}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )\right )+\frac {2 \sqrt {3 x+2} \sqrt {5 x+3}}{21 (1-2 x)^{3/2}}\)

Input:

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

Output:

(2*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)) + ((62*Sqrt[2 + 3*x]* 
Sqrt[3 + 5*x])/(77*Sqrt[1 - 2*x]) - (15*((-31*Sqrt[11/3]*EllipticE[ArcSin[ 
Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (4*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[ 
3/7]*Sqrt[1 - 2*x]], 35/33])/5))/77)/21
 

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 110

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 
1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f))   Int[(a + b*x)^(m + 1) 
*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + 
p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt 
Q[n, 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 169

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

Leaf count of result is larger than twice the leaf count of optimal. \(216\) vs. \(2(93)=186\).

Time = 0.60 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.68

method	result	size
default	\(\frac {\left (132 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-62 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-66 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+31 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-1860 x^{3}+884 x^{2}+3360 x +1296\right ) \sqrt {2+3 x}\, \sqrt {3+5 x}}{1617 \left (1-2 x \right )^{\frac {3}{2}} \left (15 x^{2}+19 x +6\right )}\)	\(217\)
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {\sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{42 \left (x -\frac {1}{2}\right )^{2}}-\frac {31 \left (-30 x^{2}-38 x -12\right )}{1617 \sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}-\frac {115 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{11319 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {155 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{11319 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(225\)

Input:

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

Output:

1/1617*(132*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x*(2+3*x)^ 
(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)-62*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2 
),1/2*70^(1/2))*x*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)-66*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))+31*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))-1860*x^3+884*x^2+3360*x+1296)*(2+3*x)^( 
1/2)*(3+5*x)^(1/2)/(1-2*x)^(3/2)/(15*x^2+19*x+6)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} \sqrt {2+3 x}} \, dx=-\frac {360 \, {\left (31 \, x - 54\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 1357 \, \sqrt {-30} {\left (4 \, x^{2} - 4 \, x + 1\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 2790 \, \sqrt {-30} {\left (4 \, x^{2} - 4 \, x + 1\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{145530 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \] Input:

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

Output:

-1/145530*(360*(31*x - 54)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 13 
57*sqrt(-30)*(4*x^2 - 4*x + 1)*weierstrassPInverse(1159/675, 38998/91125, 
x + 23/90) + 2790*sqrt(-30)*(4*x^2 - 4*x + 1)*weierstrassZeta(1159/675, 38 
998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(4*x^2 
- 4*x + 1)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} \sqrt {2+3 x}} \, dx=-\left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{24 x^{4}-20 x^{3}-6 x^{2}+9 x -2}d x \right ) \] Input:

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

Output:

 - int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(24*x**4 - 20*x**3 - 
 6*x**2 + 9*x - 2),x)