\(\int \frac {1}{\sqrt {3-5 x^2} \sqrt {1+2 x^2} \sqrt {7+11 x^2}} \, dx\) [488]

Optimal result

Integrand size = 34, antiderivative size = 95 \[ \int \frac {1}{\sqrt {3-5 x^2} \sqrt {1+2 x^2} \sqrt {7+11 x^2}} \, dx=\frac {\sqrt {3-5 x^2} \sqrt {\frac {7+11 x^2}{1+2 x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {11}{3}} x}{\sqrt {1+2 x^2}}\right ),\frac {9}{77}\right )}{\sqrt {77} \sqrt {\frac {3-5 x^2}{1+2 x^2}} \sqrt {7+11 x^2}} \] Output:

1/77*(-5*x^2+3)^(1/2)*((11*x^2+7)/(2*x^2+1))^(1/2)*EllipticF(1/3*33^(1/2)* 
x/(2*x^2+1)^(1/2),3/77*77^(1/2))*77^(1/2)/((-5*x^2+3)/(2*x^2+1))^(1/2)/(11 
*x^2+7)^(1/2)
 

Mathematica [A] (verified)

Time = 1.19 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.37 \[ \int \frac {1}{\sqrt {3-5 x^2} \sqrt {1+2 x^2} \sqrt {7+11 x^2}} \, dx=-\frac {x \sqrt {1+2 x^2} \sqrt {\frac {-3+5 x^2}{1+2 x^2}} \sqrt {\frac {7+11 x^2}{1+2 x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {\frac {77+121 x^2}{17+34 x^2}}\right ),\frac {68}{77}\right )}{\sqrt {77} \sqrt {3-5 x^2} \sqrt {\frac {x^2}{1+2 x^2}} \sqrt {7+11 x^2}} \] Input:

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

Output:

-((x*Sqrt[1 + 2*x^2]*Sqrt[(-3 + 5*x^2)/(1 + 2*x^2)]*Sqrt[(7 + 11*x^2)/(1 + 
 2*x^2)]*EllipticF[ArcSin[Sqrt[(77 + 121*x^2)/(17 + 34*x^2)]/2], 68/77])/( 
Sqrt[77]*Sqrt[3 - 5*x^2]*Sqrt[x^2/(1 + 2*x^2)]*Sqrt[7 + 11*x^2]))
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.81, 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.

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

\(\Big \downarrow \) 427

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 320

\(\displaystyle \frac {\sqrt {2 x^2+1} \sqrt {\frac {11 x^2+7}{3-5 x^2}} \sqrt {\frac {68 x^2}{3-5 x^2}+7} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {11} x}{\sqrt {3-5 x^2}}\right ),\frac {9}{77}\right )}{\sqrt {77} \sqrt {\frac {2 x^2+1}{3-5 x^2}} \sqrt {11 x^2+7} \sqrt {\frac {11 x^2}{3-5 x^2}+1} \sqrt {\frac {\frac {68 x^2}{3-5 x^2}+7}{\frac {11 x^2}{3-5 x^2}+1}}}\)

Input:

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

Output:

(Sqrt[1 + 2*x^2]*Sqrt[(7 + 11*x^2)/(3 - 5*x^2)]*Sqrt[7 + (68*x^2)/(3 - 5*x 
^2)]*EllipticF[ArcTan[(Sqrt[11]*x)/Sqrt[3 - 5*x^2]], 9/77])/(Sqrt[77]*Sqrt 
[(1 + 2*x^2)/(3 - 5*x^2)]*Sqrt[7 + 11*x^2]*Sqrt[1 + (11*x^2)/(3 - 5*x^2)]* 
Sqrt[(7 + (68*x^2)/(3 - 5*x^2))/(1 + (11*x^2)/(3 - 5*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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(-sqrt(11*x^2 + 7)*sqrt(2*x^2 + 1)*sqrt(-5*x^2 + 3)/(110*x^6 + 59* 
x^4 - 40*x^2 - 21), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\sqrt {3-5 x^2} \sqrt {1+2 x^2} \sqrt {7+11 x^2}} \, dx=-\left (\int \frac {\sqrt {2 x^{2}+1}\, \sqrt {11 x^{2}+7}\, \sqrt {-5 x^{2}+3}}{110 x^{6}+59 x^{4}-40 x^{2}-21}d x \right ) \] Input:

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

Output:

 - int((sqrt(2*x**2 + 1)*sqrt(11*x**2 + 7)*sqrt( - 5*x**2 + 3))/(110*x**6 
+ 59*x**4 - 40*x**2 - 21),x)