\(\int \frac {1}{\sqrt {2+3 x} \sqrt {4+5 x} \sqrt {1+x^2}} \, dx\) [143]

Optimal result

Integrand size = 28, antiderivative size = 196 \[ \int \frac {1}{\sqrt {2+3 x} \sqrt {4+5 x} \sqrt {1+x^2}} \, dx=-\frac {(2+3 x) \sqrt {\frac {1+x^2}{(2+3 x)^2}} \left (41+\frac {\sqrt {533} (4+5 x)}{2+3 x}\right ) \sqrt {\frac {41-\frac {46 (4+5 x)}{2+3 x}+\frac {13 (4+5 x)^2}{(2+3 x)^2}}{\left (41+\frac {\sqrt {533} (4+5 x)}{2+3 x}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{\frac {13}{41}} \sqrt {4+5 x}}{\sqrt {2+3 x}}\right ),\frac {533+23 \sqrt {533}}{1066}\right )}{\sqrt [4]{533} \sqrt {1+x^2} \sqrt {41-\frac {46 (4+5 x)}{2+3 x}+\frac {13 (4+5 x)^2}{(2+3 x)^2}}} \] Output:

-1/533*(2+3*x)*((x^2+1)/(2+3*x)^2)^(1/2)*(41+533^(1/2)*(4+5*x)/(2+3*x))*(( 
41-46*(4+5*x)/(2+3*x)+13*(4+5*x)^2/(2+3*x)^2)/(41+533^(1/2)*(4+5*x)/(2+3*x 
))^2)^(1/2)*InverseJacobiAM(2*arctan(1/41*13^(1/4)*41^(3/4)*(4+5*x)^(1/2)/ 
(2+3*x)^(1/2)),1/1066*(568178+24518*533^(1/2))^(1/2))*533^(3/4)/(x^2+1)^(1 
/2)/(41-46*(4+5*x)/(2+3*x)+13*(4+5*x)^2/(2+3*x)^2)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 35.28 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.64 \[ \int \frac {1}{\sqrt {2+3 x} \sqrt {4+5 x} \sqrt {1+x^2}} \, dx=-\frac {2 \sqrt {\frac {(4+5 i) (i+x)}{2+3 x}} \sqrt {2+3 x} \sqrt {4+5 x} \sqrt {-\frac {(5+4 i)-(4-5 i) x}{82+123 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(23+2 i) (4+5 x)}{2+3 x}}}{\sqrt {41}}\right ),\frac {525}{533}-\frac {92 i}{533}\right )}{\sqrt {\frac {(23+2 i) (4+5 x)}{2+3 x}} \sqrt {1+x^2}} \] Input:

Integrate[1/(Sqrt[2 + 3*x]*Sqrt[4 + 5*x]*Sqrt[1 + x^2]),x]
 

Output:

(-2*Sqrt[((4 + 5*I)*(I + x))/(2 + 3*x)]*Sqrt[2 + 3*x]*Sqrt[4 + 5*x]*Sqrt[- 
(((5 + 4*I) - (4 - 5*I)*x)/(82 + 123*x))]*EllipticF[ArcSin[Sqrt[((23 + 2*I 
)*(4 + 5*x))/(2 + 3*x)]/Sqrt[41]], 525/533 - (92*I)/533])/(Sqrt[((23 + 2*I 
)*(4 + 5*x))/(2 + 3*x)]*Sqrt[1 + x^2])
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {732, 1416}

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.

\(\displaystyle \int \frac {1}{\sqrt {3 x+2} \sqrt {5 x+4} \sqrt {x^2+1}} \, dx\)

\(\Big \downarrow \) 732

\(\displaystyle -\frac {2 (3 x+2) \sqrt {\frac {x^2+1}{(3 x+2)^2}} \int \frac {1}{\sqrt {\frac {13 (5 x+4)^2}{41 (3 x+2)^2}-\frac {46 (5 x+4)}{41 (3 x+2)}+1}}d\frac {\sqrt {5 x+4}}{\sqrt {3 x+2}}}{\sqrt {41} \sqrt {x^2+1}}\)

\(\Big \downarrow \) 1416

\(\displaystyle -\frac {(3 x+2) \sqrt {\frac {x^2+1}{(3 x+2)^2}} \left (\frac {\sqrt {533} (5 x+4)}{3 x+2}+41\right ) \sqrt {\frac {\frac {13 (5 x+4)^2}{(3 x+2)^2}-\frac {46 (5 x+4)}{3 x+2}+41}{\left (\frac {\sqrt {533} (5 x+4)}{3 x+2}+41\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{\frac {13}{41}} \sqrt {5 x+4}}{\sqrt {3 x+2}}\right ),\frac {533+23 \sqrt {533}}{1066}\right )}{\sqrt [4]{533} \sqrt {x^2+1} \sqrt {\frac {13 (5 x+4)^2}{(3 x+2)^2}-\frac {46 (5 x+4)}{3 x+2}+41}}\)

Input:

Int[1/(Sqrt[2 + 3*x]*Sqrt[4 + 5*x]*Sqrt[1 + x^2]),x]
 

Output:

-(((2 + 3*x)*Sqrt[(1 + x^2)/(2 + 3*x)^2]*(41 + (Sqrt[533]*(4 + 5*x))/(2 + 
3*x))*Sqrt[(41 - (46*(4 + 5*x))/(2 + 3*x) + (13*(4 + 5*x)^2)/(2 + 3*x)^2)/ 
(41 + (Sqrt[533]*(4 + 5*x))/(2 + 3*x))^2]*EllipticF[2*ArcTan[((13/41)^(1/4 
)*Sqrt[4 + 5*x])/Sqrt[2 + 3*x]], (533 + 23*Sqrt[533])/1066])/(533^(1/4)*Sq 
rt[1 + x^2]*Sqrt[41 - (46*(4 + 5*x))/(2 + 3*x) + (13*(4 + 5*x)^2)/(2 + 3*x 
)^2]))
 

Defintions of rubi rules used

rule 732

Int[1/(Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(a_) + (b_.)* 
(x_)^2]), x_Symbol] :> Simp[-2*(c + d*x)*(Sqrt[(d*e - c*f)^2*((a + b*x^2)/( 
(b*e^2 + a*f^2)*(c + d*x)^2))]/((d*e - c*f)*Sqrt[a + b*x^2]))   Subst[Int[1 
/Sqrt[Simp[1 - (2*b*c*e + 2*a*d*f)*(x^2/(b*e^2 + a*f^2)) + (b*c^2 + a*d^2)* 
(x^4/(b*e^2 + a*f^2)), x]], x], x, Sqrt[e + f*x]/Sqrt[c + d*x]], x] /; Free 
Q[{a, b, c, d, e, f}, x]
 

rule 1416

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c 
/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.21 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.71

method	result	size
elliptic	\(\frac {\left (\frac {115}{123}-\frac {10 i}{123}\right ) \sqrt {\left (3 x +2\right ) \left (5 x +4\right ) \left (x^{2}+1\right )}\, \sqrt {\frac {\left (\frac {69}{65}+\frac {6 i}{65}\right ) \left (x +\frac {2}{3}\right )}{x +\frac {4}{5}}}\, \left (x +\frac {4}{5}\right )^{2} \sqrt {\frac {\left (-\frac {4}{65}+\frac {6 i}{65}\right ) \left (x -i\right )}{x +\frac {4}{5}}}\, \sqrt {\frac {\left (-\frac {4}{65}-\frac {6 i}{65}\right ) \left (x +i\right )}{x +\frac {4}{5}}}\, \sqrt {15}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (\frac {69}{65}+\frac {6 i}{65}\right ) \left (x +\frac {2}{3}\right )}{x +\frac {4}{5}}}, \frac {23 \sqrt {533}}{533}-\frac {2 i \sqrt {533}}{533}\right )}{\sqrt {3 x +2}\, \sqrt {5 x +4}\, \sqrt {x^{2}+1}\, \sqrt {\left (x +\frac {2}{3}\right ) \left (x +\frac {4}{5}\right ) \left (x -i\right ) \left (x +i\right )}}\)	\(140\)
default	\(\frac {\left (\frac {23}{41}-\frac {2 i}{41}\right ) \sqrt {3 x +2}\, \left (5 x +4\right )^{\frac {5}{2}} \sqrt {x^{2}+1}\, \sqrt {\frac {\left (\frac {23}{13}+\frac {2 i}{13}\right ) \left (3 x +2\right )}{5 x +4}}\, \sqrt {\frac {\left (\frac {4}{13}-\frac {6 i}{13}\right ) \left (-x +i\right )}{5 x +4}}\, \sqrt {\frac {\left (-\frac {4}{13}-\frac {6 i}{13}\right ) \left (x +i\right )}{5 x +4}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (\frac {23}{13}+\frac {2 i}{13}\right ) \left (3 x +2\right )}{5 x +4}}, \frac {23 \sqrt {533}}{533}-\frac {2 i \sqrt {533}}{533}\right )}{\sqrt {15 x^{4}+22 x^{3}+23 x^{2}+22 x +8}\, \sqrt {-\left (3 x +2\right ) \left (5 x +4\right ) \left (-x +i\right ) \left (x +i\right )}}\)	\(157\)

Input:

int(1/(3*x+2)^(1/2)/(5*x+4)^(1/2)/(x^2+1)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

(115/123-10/123*I)*((3*x+2)*(5*x+4)*(x^2+1))^(1/2)/(3*x+2)^(1/2)/(5*x+4)^( 
1/2)/(x^2+1)^(1/2)*((69/65+6/65*I)*(x+2/3)/(x+4/5))^(1/2)*(x+4/5)^2*((-4/6 
5+6/65*I)*(x-I)/(x+4/5))^(1/2)*((-4/65-6/65*I)*(x+I)/(x+4/5))^(1/2)*15^(1/ 
2)/((x+2/3)*(x+4/5)*(x-I)*(x+I))^(1/2)*EllipticF(((69/65+6/65*I)*(x+2/3)/( 
x+4/5))^(1/2),23/533*533^(1/2)-2/533*I*533^(1/2))
 

Fricas [F]

\[ \int \frac {1}{\sqrt {2+3 x} \sqrt {4+5 x} \sqrt {1+x^2}} \, dx=\int { \frac {1}{\sqrt {x^{2} + 1} \sqrt {5 \, x + 4} \sqrt {3 \, x + 2}} \,d x } \] Input:

integrate(1/(2+3*x)^(1/2)/(4+5*x)^(1/2)/(x^2+1)^(1/2),x, algorithm="fricas 
")
 

Output:

integral(sqrt(x^2 + 1)*sqrt(5*x + 4)*sqrt(3*x + 2)/(15*x^4 + 22*x^3 + 23*x 
^2 + 22*x + 8), x)
 

Sympy [F]

\[ \int \frac {1}{\sqrt {2+3 x} \sqrt {4+5 x} \sqrt {1+x^2}} \, dx=\int \frac {1}{\sqrt {3 x + 2} \sqrt {5 x + 4} \sqrt {x^{2} + 1}}\, dx \] Input:

integrate(1/(2+3*x)**(1/2)/(4+5*x)**(1/2)/(x**2+1)**(1/2),x)
 

Output:

Integral(1/(sqrt(3*x + 2)*sqrt(5*x + 4)*sqrt(x**2 + 1)), x)
 

Maxima [F]

\[ \int \frac {1}{\sqrt {2+3 x} \sqrt {4+5 x} \sqrt {1+x^2}} \, dx=\int { \frac {1}{\sqrt {x^{2} + 1} \sqrt {5 \, x + 4} \sqrt {3 \, x + 2}} \,d x } \] Input:

integrate(1/(2+3*x)^(1/2)/(4+5*x)^(1/2)/(x^2+1)^(1/2),x, algorithm="maxima 
")
 

Output:

integrate(1/(sqrt(x^2 + 1)*sqrt(5*x + 4)*sqrt(3*x + 2)), x)
 

Giac [F]

\[ \int \frac {1}{\sqrt {2+3 x} \sqrt {4+5 x} \sqrt {1+x^2}} \, dx=\int { \frac {1}{\sqrt {x^{2} + 1} \sqrt {5 \, x + 4} \sqrt {3 \, x + 2}} \,d x } \] Input:

integrate(1/(2+3*x)^(1/2)/(4+5*x)^(1/2)/(x^2+1)^(1/2),x, algorithm="giac")
 

Output:

integrate(1/(sqrt(x^2 + 1)*sqrt(5*x + 4)*sqrt(3*x + 2)), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {2+3 x} \sqrt {4+5 x} \sqrt {1+x^2}} \, dx=\int \frac {1}{\sqrt {3\,x+2}\,\sqrt {5\,x+4}\,\sqrt {x^2+1}} \,d x \] Input:

int(1/((3*x + 2)^(1/2)*(5*x + 4)^(1/2)*(x^2 + 1)^(1/2)),x)
 

Output:

int(1/((3*x + 2)^(1/2)*(5*x + 4)^(1/2)*(x^2 + 1)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {1}{\sqrt {2+3 x} \sqrt {4+5 x} \sqrt {1+x^2}} \, dx=\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +4}\, \sqrt {x^{2}+1}}{15 x^{4}+22 x^{3}+23 x^{2}+22 x +8}d x \] Input:

int(1/(2+3*x)^(1/2)/(4+5*x)^(1/2)/(x^2+1)^(1/2),x)
 

Output:

int((sqrt(3*x + 2)*sqrt(5*x + 4)*sqrt(x**2 + 1))/(15*x**4 + 22*x**3 + 23*x 
**2 + 22*x + 8),x)