\(\int \frac {\sqrt {2+3 x}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\) [1461]

Optimal result

Integrand size = 28, antiderivative size = 98 \[ \int \frac {\sqrt {2+3 x}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx=-\frac {2 \sqrt {1-2 x} \sqrt {2+3 x}}{11 \sqrt {3+5 x}}+\frac {2}{5} \sqrt {\frac {3}{11}} E\left (\arccos \left (\sqrt {\frac {2}{7}} \sqrt {2+3 x}\right )|\frac {35}{33}\right )-\frac {2}{5} \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\arccos \left (\sqrt {\frac {2}{7}} \sqrt {2+3 x}\right ),\frac {35}{33}\right ) \] Output:

-2/11*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(1/2)+2/55*33^(1/2)*EllipticE(1/ 
7*(21-42*x)^(1/2),1/33*1155^(1/2))-2/55*33^(1/2)*InverseJacobiAM(arccos(1/ 
7*14^(1/2)*(2+3*x)^(1/2)),1/33*1155^(1/2))
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.58 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.62 \[ \int \frac {\sqrt {2+3 x}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx=\frac {2}{55} \left (-\frac {5 \sqrt {1-2 x} \sqrt {2+3 x}}{\sqrt {3+5 x}}-i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )\right ) \] Input:

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

Output:

(2*((-5*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/Sqrt[3 + 5*x] - I*Sqrt[33]*EllipticE[ 
I*ArcSinh[Sqrt[9 + 15*x]], -2/33]))/55
 

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.83, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {110, 27, 124, 27, 123}

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.

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

\(\Big \downarrow \) 110

\(\displaystyle \frac {2}{11} \int \frac {3 \sqrt {1-2 x}}{2 \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {2 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{11} \int \frac {\sqrt {1-2 x}}{\sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {2 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 124

\(\displaystyle \frac {3 \sqrt {7} \sqrt {-5 x-3} \int \frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {-5 x-3} \sqrt {3 x+2}}dx}{11 \sqrt {5 x+3}}-\frac {2 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3 \sqrt {-5 x-3} \int \frac {\sqrt {1-2 x}}{\sqrt {-5 x-3} \sqrt {3 x+2}}dx}{11 \sqrt {5 x+3}}-\frac {2 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 123

\(\displaystyle \frac {2 \sqrt {\frac {7}{5}} \sqrt {-5 x-3} E\left (\arcsin \left (\sqrt {5} \sqrt {3 x+2}\right )|\frac {2}{35}\right )}{11 \sqrt {5 x+3}}-\frac {2 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\)

Input:

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

Output:

(-2*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(11*Sqrt[3 + 5*x]) + (2*Sqrt[7/5]*Sqrt[-3 
 - 5*x]*EllipticE[ArcSin[Sqrt[5]*Sqrt[2 + 3*x]], 2/35])/(11*Sqrt[3 + 5*x])
 

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 124

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

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.36

method	result	size
default	\(-\frac {\sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (33 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+2 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+60 x^{2}+10 x -20\right )}{55 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\)	\(133\)
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {\sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{77 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{77 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2 \left (-30 x^{2}-5 x +10\right )}{55 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(201\)

Input:

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

Output:

-1/55*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*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))+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))+60*x^2+10*x-20)/(30*x^3+23*x^2-7*x-6)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {2+3 x}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx=-\frac {2 \, {\left (34 \, \sqrt {-30} {\left (5 \, x + 3\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 45 \, \sqrt {-30} {\left (5 \, x + 3\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) + 225 \, \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}\right )}}{2475 \, {\left (5 \, x + 3\right )}} \] Input:

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

Output:

-2/2475*(34*sqrt(-30)*(5*x + 3)*weierstrassPInverse(1159/675, 38998/91125, 
 x + 23/90) + 45*sqrt(-30)*(5*x + 3)*weierstrassZeta(1159/675, 38998/91125 
, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)) + 225*sqrt(5*x + 
3)*sqrt(3*x + 2)*sqrt(-2*x + 1))/(5*x + 3)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {2+3 x}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx=-\left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{50 x^{3}+35 x^{2}-12 x -9}d x \right ) \] Input:

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

Output:

 - int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(50*x**3 + 35*x**2 - 
 12*x - 9),x)