\(\int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx\) [1462]

Optimal result

Integrand size = 28, antiderivative size = 63 \[ \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx=-\frac {10 \sqrt {1-2 x} \sqrt {2+3 x}}{11 \sqrt {3+5 x}}+2 \sqrt {\frac {3}{11}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \] Output:

-10/11*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(1/2)+2/11*33^(1/2)*EllipticE(1 
/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.57 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.62 \[ \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx=\frac {-10 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-2 i \sqrt {33} (3+5 x) E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+2 i \sqrt {33} (3+5 x) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{33+55 x} \] Input:

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

Output:

(-10*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x] - (2*I)*Sqrt[33]*(3 + 5*x)* 
EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] + (2*I)*Sqrt[33]*(3 + 5*x)*Ell 
ipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])/(33 + 55*x)
 

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {115, 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.

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

\(\Big \downarrow \) 115

\(\displaystyle -\frac {2}{11} \int \frac {3 \sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {10 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {6}{11} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {10 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 123

\(\displaystyle 2 \sqrt {\frac {3}{11}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {10 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\)

Input:

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

Output:

(-10*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(11*Sqrt[3 + 5*x]) + 2*Sqrt[3/11]*Ellipt 
icE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33]
 

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 115

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 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[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*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] && 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])
 

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.44

method	result	size
default	\(-\frac {2 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (\sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+30 x^{2}+5 x -10\right )}{11 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\)	\(91\)
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {6 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{77 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {10 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{77 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2 \left (-30 x^{2}-5 x +10\right )}{11 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(201\)

Input:

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

Output:

-2/11*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(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))+30*x 
^2+5*x-10)/(30*x^3+23*x^2-7*x-6)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx=\frac {31 \, \sqrt {-30} {\left (5 \, x + 3\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 90 \, \sqrt {-30} {\left (5 \, x + 3\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) - 450 \, \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{495 \, {\left (5 \, x + 3\right )}} \] Input:

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

Output:

1/495*(31*sqrt(-30)*(5*x + 3)*weierstrassPInverse(1159/675, 38998/91125, x 
 + 23/90) - 90*sqrt(-30)*(5*x + 3)*weierstrassZeta(1159/675, 38998/91125, 
weierstrassPInverse(1159/675, 38998/91125, x + 23/90)) - 450*sqrt(5*x + 3) 
*sqrt(3*x + 2)*sqrt(-2*x + 1))/(5*x + 3)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx=-\left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{150 x^{4}+205 x^{3}+34 x^{2}-51 x -18}d x \right ) \] Input:

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

Output:

 - int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(150*x**4 + 205*x**3 
 + 34*x**2 - 51*x - 18),x)