\(\int \frac {1}{\sqrt {7-11 x^2} \sqrt {1+2 x^2} \sqrt {3+5 x^2}} \, dx\) [492]

Optimal result

Integrand size = 34, antiderivative size = 93 \[ \int \frac {1}{\sqrt {7-11 x^2} \sqrt {1+2 x^2} \sqrt {3+5 x^2}} \, dx=\frac {\sqrt {7-11 x^2} \sqrt {\frac {3+5 x^2}{1+2 x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {3} \sqrt {1+2 x^2}}\right ),\frac {75}{7}\right )}{\sqrt {7} \sqrt {\frac {7-11 x^2}{1+2 x^2}} \sqrt {3+5 x^2}} \] Output:

1/7*(-11*x^2+7)^(1/2)*((5*x^2+3)/(2*x^2+1))^(1/2)*EllipticF(1/3*x*3^(1/2)/ 
(2*x^2+1)^(1/2),5/7*21^(1/2))*7^(1/2)/((-11*x^2+7)/(2*x^2+1))^(1/2)/(5*x^2 
+3)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 1.16 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.22 \[ \int \frac {1}{\sqrt {7-11 x^2} \sqrt {1+2 x^2} \sqrt {3+5 x^2}} \, dx=-\frac {x \sqrt {3+5 x^2} \sqrt {\frac {-7+11 x^2}{1+2 x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {5}{2} \sqrt {\frac {3+5 x^2}{17+34 x^2}}\right ),\frac {68}{75}\right )}{5 \sqrt {21-33 x^2} \sqrt {1+2 x^2} \sqrt {\frac {x^2 \left (3+5 x^2\right )}{\left (1+2 x^2\right )^2}}} \] Input:

Integrate[1/(Sqrt[7 - 11*x^2]*Sqrt[1 + 2*x^2]*Sqrt[3 + 5*x^2]),x]
 

Output:

-1/5*(x*Sqrt[3 + 5*x^2]*Sqrt[(-7 + 11*x^2)/(1 + 2*x^2)]*EllipticF[ArcSin[( 
5*Sqrt[(3 + 5*x^2)/(17 + 34*x^2)])/2], 68/75])/(Sqrt[21 - 33*x^2]*Sqrt[1 + 
 2*x^2]*Sqrt[(x^2*(3 + 5*x^2))/(1 + 2*x^2)^2])
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.84, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {427, 27, 320}

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.

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

\(\Big \downarrow \) 427

\(\displaystyle \frac {\sqrt {2 x^2+1} \sqrt {\frac {5 x^2+3}{7-11 x^2}} \int \frac {\sqrt {3}}{\sqrt {\frac {25 x^2}{7-11 x^2}+1} \sqrt {\frac {68 x^2}{7-11 x^2}+3}}d\frac {x}{\sqrt {7-11 x^2}}}{\sqrt {3} \sqrt {\frac {2 x^2+1}{7-11 x^2}} \sqrt {5 x^2+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {2 x^2+1} \sqrt {\frac {5 x^2+3}{7-11 x^2}} \int \frac {1}{\sqrt {\frac {25 x^2}{7-11 x^2}+1} \sqrt {\frac {68 x^2}{7-11 x^2}+3}}d\frac {x}{\sqrt {7-11 x^2}}}{\sqrt {\frac {2 x^2+1}{7-11 x^2}} \sqrt {5 x^2+3}}\)

\(\Big \downarrow \) 320

\(\displaystyle \frac {\sqrt {2 x^2+1} \sqrt {\frac {5 x^2+3}{7-11 x^2}} \sqrt {\frac {68 x^2}{7-11 x^2}+3} \operatorname {EllipticF}\left (\arctan \left (\frac {5 x}{\sqrt {7-11 x^2}}\right ),\frac {7}{75}\right )}{5 \sqrt {3} \sqrt {\frac {2 x^2+1}{7-11 x^2}} \sqrt {5 x^2+3} \sqrt {\frac {25 x^2}{7-11 x^2}+1} \sqrt {\frac {\frac {68 x^2}{7-11 x^2}+3}{\frac {25 x^2}{7-11 x^2}+1}}}\)

Input:

Int[1/(Sqrt[7 - 11*x^2]*Sqrt[1 + 2*x^2]*Sqrt[3 + 5*x^2]),x]
 

Output:

(Sqrt[1 + 2*x^2]*Sqrt[(3 + 5*x^2)/(7 - 11*x^2)]*Sqrt[3 + (68*x^2)/(7 - 11* 
x^2)]*EllipticF[ArcTan[(5*x)/Sqrt[7 - 11*x^2]], 7/75])/(5*Sqrt[3]*Sqrt[(1 
+ 2*x^2)/(7 - 11*x^2)]*Sqrt[3 + 5*x^2]*Sqrt[1 + (25*x^2)/(7 - 11*x^2)]*Sqr 
t[(3 + (68*x^2)/(7 - 11*x^2))/(1 + (25*x^2)/(7 - 11*x^2))])
 

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 320

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( 
c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre 
eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 427

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

Maple [F]

\[\int \frac {1}{\sqrt {-11 x^{2}+7}\, \sqrt {2 x^{2}+1}\, \sqrt {5 x^{2}+3}}d x\]

Input:

int(1/(-11*x^2+7)^(1/2)/(2*x^2+1)^(1/2)/(5*x^2+3)^(1/2),x)
 

Output:

int(1/(-11*x^2+7)^(1/2)/(2*x^2+1)^(1/2)/(5*x^2+3)^(1/2),x)
 

Fricas [F]

\[ \int \frac {1}{\sqrt {7-11 x^2} \sqrt {1+2 x^2} \sqrt {3+5 x^2}} \, dx=\int { \frac {1}{\sqrt {5 \, x^{2} + 3} \sqrt {2 \, x^{2} + 1} \sqrt {-11 \, x^{2} + 7}} \,d x } \] Input:

integrate(1/(-11*x^2+7)^(1/2)/(2*x^2+1)^(1/2)/(5*x^2+3)^(1/2),x, algorithm 
="fricas")
 

Output:

integral(-sqrt(5*x^2 + 3)*sqrt(2*x^2 + 1)*sqrt(-11*x^2 + 7)/(110*x^6 + 51* 
x^4 - 44*x^2 - 21), x)
 

Sympy [F]

\[ \int \frac {1}{\sqrt {7-11 x^2} \sqrt {1+2 x^2} \sqrt {3+5 x^2}} \, dx=\int \frac {1}{\sqrt {7 - 11 x^{2}} \sqrt {2 x^{2} + 1} \sqrt {5 x^{2} + 3}}\, dx \] Input:

integrate(1/(-11*x**2+7)**(1/2)/(2*x**2+1)**(1/2)/(5*x**2+3)**(1/2),x)
 

Output:

Integral(1/(sqrt(7 - 11*x**2)*sqrt(2*x**2 + 1)*sqrt(5*x**2 + 3)), x)
 

Maxima [F]

\[ \int \frac {1}{\sqrt {7-11 x^2} \sqrt {1+2 x^2} \sqrt {3+5 x^2}} \, dx=\int { \frac {1}{\sqrt {5 \, x^{2} + 3} \sqrt {2 \, x^{2} + 1} \sqrt {-11 \, x^{2} + 7}} \,d x } \] Input:

integrate(1/(-11*x^2+7)^(1/2)/(2*x^2+1)^(1/2)/(5*x^2+3)^(1/2),x, algorithm 
="maxima")
 

Output:

integrate(1/(sqrt(5*x^2 + 3)*sqrt(2*x^2 + 1)*sqrt(-11*x^2 + 7)), x)
 

Giac [F]

\[ \int \frac {1}{\sqrt {7-11 x^2} \sqrt {1+2 x^2} \sqrt {3+5 x^2}} \, dx=\int { \frac {1}{\sqrt {5 \, x^{2} + 3} \sqrt {2 \, x^{2} + 1} \sqrt {-11 \, x^{2} + 7}} \,d x } \] Input:

integrate(1/(-11*x^2+7)^(1/2)/(2*x^2+1)^(1/2)/(5*x^2+3)^(1/2),x, algorithm 
="giac")
 

Output:

integrate(1/(sqrt(5*x^2 + 3)*sqrt(2*x^2 + 1)*sqrt(-11*x^2 + 7)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {7-11 x^2} \sqrt {1+2 x^2} \sqrt {3+5 x^2}} \, dx=\int \frac {1}{\sqrt {2\,x^2+1}\,\sqrt {5\,x^2+3}\,\sqrt {7-11\,x^2}} \,d x \] Input:

int(1/((2*x^2 + 1)^(1/2)*(5*x^2 + 3)^(1/2)*(7 - 11*x^2)^(1/2)),x)
 

Output:

int(1/((2*x^2 + 1)^(1/2)*(5*x^2 + 3)^(1/2)*(7 - 11*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {1}{\sqrt {7-11 x^2} \sqrt {1+2 x^2} \sqrt {3+5 x^2}} \, dx=-\left (\int \frac {\sqrt {2 x^{2}+1}\, \sqrt {5 x^{2}+3}\, \sqrt {-11 x^{2}+7}}{110 x^{6}+51 x^{4}-44 x^{2}-21}d x \right ) \] Input:

int(1/(-11*x^2+7)^(1/2)/(2*x^2+1)^(1/2)/(5*x^2+3)^(1/2),x)
 

Output:

 - int((sqrt(2*x**2 + 1)*sqrt(5*x**2 + 3)*sqrt( - 11*x**2 + 7))/(110*x**6 
+ 51*x**4 - 44*x**2 - 21),x)