\(\int \frac {\sqrt {3-5 x^2} \sqrt {1-2 x^2}}{\sqrt {7+11 x^2}} \, dx\) [504]

Optimal result

Integrand size = 34, antiderivative size = 313 \[ \int \frac {\sqrt {3-5 x^2} \sqrt {1-2 x^2}}{\sqrt {7+11 x^2}} \, dx=-\frac {5 x \sqrt {1-2 x^2} \sqrt {7+11 x^2}}{22 \sqrt {3-5 x^2}}-\frac {\sqrt {7} \sqrt {1-2 x^2} \sqrt {\frac {7+11 x^2}{3-5 x^2}} E\left (\arcsin \left (\frac {x}{\sqrt {3-5 x^2}}\right )|-\frac {68}{7}\right )}{22 \sqrt {\frac {1-2 x^2}{3-5 x^2}} \sqrt {7+11 x^2}}-\frac {3 \sqrt {1-2 x^2} \sqrt {\frac {7+11 x^2}{3-5 x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {3-5 x^2}}\right ),-\frac {68}{7}\right )}{10 \sqrt {7} \sqrt {\frac {1-2 x^2}{3-5 x^2}} \sqrt {7+11 x^2}}+\frac {573 \sqrt {1-2 x^2} \sqrt {\frac {7+11 x^2}{3-5 x^2}} \operatorname {EllipticPi}\left (-5,\arcsin \left (\frac {x}{\sqrt {3-5 x^2}}\right ),-\frac {68}{7}\right )}{110 \sqrt {7} \sqrt {\frac {1-2 x^2}{3-5 x^2}} \sqrt {7+11 x^2}} \] Output:

-5/22*x*(-2*x^2+1)^(1/2)*(11*x^2+7)^(1/2)/(-5*x^2+3)^(1/2)-1/22*7^(1/2)*(- 
2*x^2+1)^(1/2)*((11*x^2+7)/(-5*x^2+3))^(1/2)*EllipticE(x/(-5*x^2+3)^(1/2), 
2/7*I*119^(1/2))/((-2*x^2+1)/(-5*x^2+3))^(1/2)/(11*x^2+7)^(1/2)-3/70*(-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)+573/7 
70*(-2*x^2+1)^(1/2)*((11*x^2+7)/(-5*x^2+3))^(1/2)*EllipticPi(x/(-5*x^2+3)^ 
(1/2),-5,2/7*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 = 2.15 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {3-5 x^2} \sqrt {1-2 x^2}}{\sqrt {7+11 x^2}} \, dx=\frac {\frac {12705 x \sqrt {3-5 x^2} \left (1-2 x^2\right )}{\sqrt {7+11 x^2}}-\frac {5775 \sqrt {9-15 x^2} \sqrt {\frac {1-2 x^2}{7+11 x^2}} E\left (\arcsin \left (\frac {5 x}{\sqrt {7+11 x^2}}\right )|\frac {68}{75}\right )}{\sqrt {\frac {3-5 x^2}{7+11 x^2}}}-\frac {37875 \left (-1+2 x^2\right ) \sqrt {\frac {49+77 x^2}{3-5 x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {3-5 x^2}}\right ),-\frac {68}{7}\right )}{\sqrt {\frac {1-2 x^2}{3-5 x^2}} \sqrt {7+11 x^2}}-\frac {9359 \sqrt {9-15 x^2} \sqrt {\frac {1-2 x^2}{7+11 x^2}} \operatorname {EllipticPi}\left (\frac {11}{25},\arcsin \left (\frac {5 x}{\sqrt {7+11 x^2}}\right ),\frac {68}{75}\right )}{\sqrt {\frac {3-5 x^2}{7+11 x^2}}}}{25410 \sqrt {1-2 x^2}} \] Input:

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

Output:

((12705*x*Sqrt[3 - 5*x^2]*(1 - 2*x^2))/Sqrt[7 + 11*x^2] - (5775*Sqrt[9 - 1 
5*x^2]*Sqrt[(1 - 2*x^2)/(7 + 11*x^2)]*EllipticE[ArcSin[(5*x)/Sqrt[7 + 11*x 
^2]], 68/75])/Sqrt[(3 - 5*x^2)/(7 + 11*x^2)] - (37875*(-1 + 2*x^2)*Sqrt[(4 
9 + 77*x^2)/(3 - 5*x^2)]*EllipticF[ArcSin[x/Sqrt[3 - 5*x^2]], -68/7])/(Sqr 
t[(1 - 2*x^2)/(3 - 5*x^2)]*Sqrt[7 + 11*x^2]) - (9359*Sqrt[9 - 15*x^2]*Sqrt 
[(1 - 2*x^2)/(7 + 11*x^2)]*EllipticPi[11/25, ArcSin[(5*x)/Sqrt[7 + 11*x^2] 
], 68/75])/Sqrt[(3 - 5*x^2)/(7 + 11*x^2)])/(25410*Sqrt[1 - 2*x^2])
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.04, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {431, 427, 27, 321, 428, 27, 412, 429, 27, 327}

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.

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

\(\Big \downarrow \) 431

\(\displaystyle -\frac {238}{11} \int \frac {\sqrt {1-2 x^2}}{\sqrt {3-5 x^2} \left (11 x^2+7\right )^{3/2}}dx+\frac {1224}{121} \int \frac {1}{\sqrt {3-5 x^2} \sqrt {1-2 x^2} \sqrt {11 x^2+7}}dx-\frac {191}{242} \int \frac {\sqrt {11 x^2+7}}{\sqrt {3-5 x^2} \sqrt {1-2 x^2}}dx+\frac {\sqrt {3-5 x^2} \sqrt {1-2 x^2} x}{2 \sqrt {11 x^2+7}}\)

\(\Big \downarrow \) 427

\(\displaystyle -\frac {238}{11} \int \frac {\sqrt {1-2 x^2}}{\sqrt {3-5 x^2} \left (11 x^2+7\right )^{3/2}}dx-\frac {191}{242} \int \frac {\sqrt {11 x^2+7}}{\sqrt {3-5 x^2} \sqrt {1-2 x^2}}dx+\frac {1224 \sqrt {1-2 x^2} \sqrt {\frac {11 x^2+7}{3-5 x^2}} \int \frac {\sqrt {7}}{\sqrt {1-\frac {x^2}{3-5 x^2}} \sqrt {\frac {68 x^2}{3-5 x^2}+7}}d\frac {x}{\sqrt {3-5 x^2}}}{121 \sqrt {7} \sqrt {\frac {1-2 x^2}{3-5 x^2}} \sqrt {11 x^2+7}}+\frac {\sqrt {3-5 x^2} \sqrt {1-2 x^2} x}{2 \sqrt {11 x^2+7}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {238}{11} \int \frac {\sqrt {1-2 x^2}}{\sqrt {3-5 x^2} \left (11 x^2+7\right )^{3/2}}dx-\frac {191}{242} \int \frac {\sqrt {11 x^2+7}}{\sqrt {3-5 x^2} \sqrt {1-2 x^2}}dx+\frac {1224 \sqrt {1-2 x^2} \sqrt {\frac {11 x^2+7}{3-5 x^2}} \int \frac {1}{\sqrt {1-\frac {x^2}{3-5 x^2}} \sqrt {\frac {68 x^2}{3-5 x^2}+7}}d\frac {x}{\sqrt {3-5 x^2}}}{121 \sqrt {\frac {1-2 x^2}{3-5 x^2}} \sqrt {11 x^2+7}}+\frac {\sqrt {3-5 x^2} \sqrt {1-2 x^2} x}{2 \sqrt {11 x^2+7}}\)

\(\Big \downarrow \) 321

\(\displaystyle -\frac {238}{11} \int \frac {\sqrt {1-2 x^2}}{\sqrt {3-5 x^2} \left (11 x^2+7\right )^{3/2}}dx-\frac {191}{242} \int \frac {\sqrt {11 x^2+7}}{\sqrt {3-5 x^2} \sqrt {1-2 x^2}}dx+\frac {1224 \sqrt {1-2 x^2} \sqrt {\frac {11 x^2+7}{3-5 x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {3-5 x^2}}\right ),-\frac {68}{7}\right )}{121 \sqrt {7} \sqrt {\frac {1-2 x^2}{3-5 x^2}} \sqrt {11 x^2+7}}+\frac {\sqrt {3-5 x^2} \sqrt {1-2 x^2} x}{2 \sqrt {11 x^2+7}}\)

\(\Big \downarrow \) 428

\(\displaystyle -\frac {238}{11} \int \frac {\sqrt {1-2 x^2}}{\sqrt {3-5 x^2} \left (11 x^2+7\right )^{3/2}}dx-\frac {1337 \sqrt {3-5 x^2} \sqrt {\frac {1-2 x^2}{11 x^2+7}} \int \frac {\sqrt {3}}{\sqrt {3-\frac {68 x^2}{11 x^2+7}} \sqrt {1-\frac {25 x^2}{11 x^2+7}} \left (1-\frac {11 x^2}{11 x^2+7}\right )}d\frac {x}{\sqrt {11 x^2+7}}}{242 \sqrt {3} \sqrt {1-2 x^2} \sqrt {\frac {3-5 x^2}{11 x^2+7}}}+\frac {1224 \sqrt {1-2 x^2} \sqrt {\frac {11 x^2+7}{3-5 x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {3-5 x^2}}\right ),-\frac {68}{7}\right )}{121 \sqrt {7} \sqrt {\frac {1-2 x^2}{3-5 x^2}} \sqrt {11 x^2+7}}+\frac {\sqrt {3-5 x^2} \sqrt {1-2 x^2} x}{2 \sqrt {11 x^2+7}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {238}{11} \int \frac {\sqrt {1-2 x^2}}{\sqrt {3-5 x^2} \left (11 x^2+7\right )^{3/2}}dx-\frac {1337 \sqrt {3-5 x^2} \sqrt {\frac {1-2 x^2}{11 x^2+7}} \int \frac {1}{\sqrt {3-\frac {68 x^2}{11 x^2+7}} \sqrt {1-\frac {25 x^2}{11 x^2+7}} \left (1-\frac {11 x^2}{11 x^2+7}\right )}d\frac {x}{\sqrt {11 x^2+7}}}{242 \sqrt {1-2 x^2} \sqrt {\frac {3-5 x^2}{11 x^2+7}}}+\frac {1224 \sqrt {1-2 x^2} \sqrt {\frac {11 x^2+7}{3-5 x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {3-5 x^2}}\right ),-\frac {68}{7}\right )}{121 \sqrt {7} \sqrt {\frac {1-2 x^2}{3-5 x^2}} \sqrt {11 x^2+7}}+\frac {\sqrt {3-5 x^2} \sqrt {1-2 x^2} x}{2 \sqrt {11 x^2+7}}\)

\(\Big \downarrow \) 412

\(\displaystyle -\frac {238}{11} \int \frac {\sqrt {1-2 x^2}}{\sqrt {3-5 x^2} \left (11 x^2+7\right )^{3/2}}dx+\frac {1224 \sqrt {1-2 x^2} \sqrt {\frac {11 x^2+7}{3-5 x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {3-5 x^2}}\right ),-\frac {68}{7}\right )}{121 \sqrt {7} \sqrt {\frac {1-2 x^2}{3-5 x^2}} \sqrt {11 x^2+7}}-\frac {1337 \sqrt {3-5 x^2} \sqrt {\frac {1-2 x^2}{11 x^2+7}} \operatorname {EllipticPi}\left (\frac {11}{25},\arcsin \left (\frac {5 x}{\sqrt {11 x^2+7}}\right ),\frac {68}{75}\right )}{1210 \sqrt {3} \sqrt {1-2 x^2} \sqrt {\frac {3-5 x^2}{11 x^2+7}}}+\frac {\sqrt {3-5 x^2} \sqrt {1-2 x^2} x}{2 \sqrt {11 x^2+7}}\)

\(\Big \downarrow \) 429

\(\displaystyle -\frac {34 \sqrt {1-2 x^2} \sqrt {\frac {3-5 x^2}{11 x^2+7}} \int \frac {\sqrt {3} \sqrt {1-\frac {25 x^2}{11 x^2+7}}}{\sqrt {3-\frac {68 x^2}{11 x^2+7}}}d\frac {x}{\sqrt {11 x^2+7}}}{11 \sqrt {3} \sqrt {3-5 x^2} \sqrt {\frac {1-2 x^2}{11 x^2+7}}}+\frac {1224 \sqrt {1-2 x^2} \sqrt {\frac {11 x^2+7}{3-5 x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {3-5 x^2}}\right ),-\frac {68}{7}\right )}{121 \sqrt {7} \sqrt {\frac {1-2 x^2}{3-5 x^2}} \sqrt {11 x^2+7}}-\frac {1337 \sqrt {3-5 x^2} \sqrt {\frac {1-2 x^2}{11 x^2+7}} \operatorname {EllipticPi}\left (\frac {11}{25},\arcsin \left (\frac {5 x}{\sqrt {11 x^2+7}}\right ),\frac {68}{75}\right )}{1210 \sqrt {3} \sqrt {1-2 x^2} \sqrt {\frac {3-5 x^2}{11 x^2+7}}}+\frac {\sqrt {3-5 x^2} \sqrt {1-2 x^2} x}{2 \sqrt {11 x^2+7}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {34 \sqrt {1-2 x^2} \sqrt {\frac {3-5 x^2}{11 x^2+7}} \int \frac {\sqrt {1-\frac {25 x^2}{11 x^2+7}}}{\sqrt {3-\frac {68 x^2}{11 x^2+7}}}d\frac {x}{\sqrt {11 x^2+7}}}{11 \sqrt {3-5 x^2} \sqrt {\frac {1-2 x^2}{11 x^2+7}}}+\frac {1224 \sqrt {1-2 x^2} \sqrt {\frac {11 x^2+7}{3-5 x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {3-5 x^2}}\right ),-\frac {68}{7}\right )}{121 \sqrt {7} \sqrt {\frac {1-2 x^2}{3-5 x^2}} \sqrt {11 x^2+7}}-\frac {1337 \sqrt {3-5 x^2} \sqrt {\frac {1-2 x^2}{11 x^2+7}} \operatorname {EllipticPi}\left (\frac {11}{25},\arcsin \left (\frac {5 x}{\sqrt {11 x^2+7}}\right ),\frac {68}{75}\right )}{1210 \sqrt {3} \sqrt {1-2 x^2} \sqrt {\frac {3-5 x^2}{11 x^2+7}}}+\frac {\sqrt {3-5 x^2} \sqrt {1-2 x^2} x}{2 \sqrt {11 x^2+7}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {1224 \sqrt {1-2 x^2} \sqrt {\frac {11 x^2+7}{3-5 x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {3-5 x^2}}\right ),-\frac {68}{7}\right )}{121 \sqrt {7} \sqrt {\frac {1-2 x^2}{3-5 x^2}} \sqrt {11 x^2+7}}-\frac {\sqrt {17} \sqrt {1-2 x^2} \sqrt {\frac {3-5 x^2}{11 x^2+7}} E\left (\arcsin \left (\frac {2 \sqrt {\frac {17}{3}} x}{\sqrt {11 x^2+7}}\right )|\frac {75}{68}\right )}{11 \sqrt {3-5 x^2} \sqrt {\frac {1-2 x^2}{11 x^2+7}}}-\frac {1337 \sqrt {3-5 x^2} \sqrt {\frac {1-2 x^2}{11 x^2+7}} \operatorname {EllipticPi}\left (\frac {11}{25},\arcsin \left (\frac {5 x}{\sqrt {11 x^2+7}}\right ),\frac {68}{75}\right )}{1210 \sqrt {3} \sqrt {1-2 x^2} \sqrt {\frac {3-5 x^2}{11 x^2+7}}}+\frac {\sqrt {3-5 x^2} \sqrt {1-2 x^2} x}{2 \sqrt {11 x^2+7}}\)

Input:

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

Output:

(x*Sqrt[3 - 5*x^2]*Sqrt[1 - 2*x^2])/(2*Sqrt[7 + 11*x^2]) - (Sqrt[17]*Sqrt[ 
1 - 2*x^2]*Sqrt[(3 - 5*x^2)/(7 + 11*x^2)]*EllipticE[ArcSin[(2*Sqrt[17/3]*x 
)/Sqrt[7 + 11*x^2]], 75/68])/(11*Sqrt[3 - 5*x^2]*Sqrt[(1 - 2*x^2)/(7 + 11* 
x^2)]) + (1224*Sqrt[1 - 2*x^2]*Sqrt[(7 + 11*x^2)/(3 - 5*x^2)]*EllipticF[Ar 
cSin[x/Sqrt[3 - 5*x^2]], -68/7])/(121*Sqrt[7]*Sqrt[(1 - 2*x^2)/(3 - 5*x^2) 
]*Sqrt[7 + 11*x^2]) - (1337*Sqrt[3 - 5*x^2]*Sqrt[(1 - 2*x^2)/(7 + 11*x^2)] 
*EllipticPi[11/25, ArcSin[(5*x)/Sqrt[7 + 11*x^2]], 68/75])/(1210*Sqrt[3]*S 
qrt[1 - 2*x^2]*Sqrt[(3 - 5*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 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 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[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]
 

rule 412

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* 
(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, 
 f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && S 
implerSqrtQ[-f/e, -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]
 

rule 428

Int[Sqrt[(a_) + (b_.)*(x_)^2]/(Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)* 
(x_)^2]), x_Symbol] :> Simp[a*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/((1 - b*x^2)*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]
 

rule 429

Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)^(3/2)*Sqrt[(e_) + (f_. 
)*(x_)^2]), x_Symbol] :> Simp[Sqrt[c + d*x^2]*(Sqrt[a*((e + f*x^2)/(e*(a + 
b*x^2)))]/(a*Sqrt[e + f*x^2]*Sqrt[a*((c + d*x^2)/(c*(a + b*x^2)))]))   Subs 
t[Int[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]
 

rule 431

Int[(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2])/Sqrt[(e_) + (f_.) 
*(x_)^2], x_Symbol] :> Simp[x*Sqrt[a + b*x^2]*(Sqrt[c + d*x^2]/(2*Sqrt[e + 
f*x^2])), x] + (Simp[e*((b*e - a*f)/(2*f))   Int[Sqrt[c + d*x^2]/(Sqrt[a + 
b*x^2]*(e + f*x^2)^(3/2)), x], x] - Simp[(b*d*e - b*c*f - a*d*f)/(2*f^2) 
Int[Sqrt[e + f*x^2]/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] + Simp[(b*e - 
 a*f)*((d*e - 2*c*f)/(2*f^2))   Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*Sqrt 
[e + f*x^2]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && NegQ[(d*e - c*f)/c 
]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

int((-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))/(11*x**2 + 
 7),x)