\(\int \frac {1}{\sqrt {3-5 x^2} \sqrt {1-2 x^2} \sqrt {7+11 x^2}} \, dx\) [487]

Optimal result

Integrand size = 34, antiderivative size = 88 \[ \int \frac {1}{\sqrt {3-5 x^2} \sqrt {1-2 x^2} \sqrt {7+11 x^2}} \, dx=\frac {\sqrt {1-2 x^2} \sqrt {\frac {7+11 x^2}{3-5 x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {3-5 x^2}}\right ),-\frac {68}{7}\right )}{\sqrt {7} \sqrt {\frac {1-2 x^2}{3-5 x^2}} \sqrt {7+11 x^2}} \] Output:

1/7*(-2*x^2+1)^(1/2)*((11*x^2+7)/(-5*x^2+3))^(1/2)*EllipticF(x/(-5*x^2+3)^ 
(1/2),2/7*I*119^(1/2))*7^(1/2)/((-2*x^2+1)/(-5*x^2+3))^(1/2)/(11*x^2+7)^(1 
/2)
 

Mathematica [A] (verified)

Time = 1.12 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\sqrt {3-5 x^2} \sqrt {1-2 x^2} \sqrt {7+11 x^2}} \, dx=\frac {\sqrt {1-2 x^2} \sqrt {\frac {7+11 x^2}{3-5 x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {3-5 x^2}}\right ),-\frac {68}{7}\right )}{\sqrt {\frac {7-14 x^2}{3-5 x^2}} \sqrt {7+11 x^2}} \] Input:

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

Output:

(Sqrt[1 - 2*x^2]*Sqrt[(7 + 11*x^2)/(3 - 5*x^2)]*EllipticF[ArcSin[x/Sqrt[3 
- 5*x^2]], -68/7])/(Sqrt[(7 - 14*x^2)/(3 - 5*x^2)]*Sqrt[7 + 11*x^2])
 

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {427, 27, 321}

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.

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

\(\Big \downarrow \) 427

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 321

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

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)]*EllipticF[ArcSin[x/Sqrt[3 
- 5*x^2]], -68/7])/(Sqrt[7]*Sqrt[(1 - 2*x^2)/(3 - 5*x^2)]*Sqrt[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 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[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, algorith 
m="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {1}{\sqrt {3-5 x^2} \sqrt {1-2 x^2} \sqrt {7+11 x^2}} \, dx=\int \frac {1}{\sqrt {1 - 2 x^{2}} \sqrt {3 - 5 x^{2}} \sqrt {11 x^{2} + 7}}\, dx \] 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(1 - 2*x**2)*sqrt(3 - 5*x**2)*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, algorith 
m="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, algorith 
m="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. \[ \int \frac {1}{\sqrt {3-5 x^2} \sqrt {1-2 x^2} \sqrt {7+11 x^2}} \, dx=\int \frac {1}{\sqrt {1-2\,x^2}\,\sqrt {3-5\,x^2}\,\sqrt {11\,x^2+7}} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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