\(\int \frac {\sqrt {3-5 x^2} \sqrt {1+2 x^2}}{\sqrt {7-11 x^2}} \, dx\) [507]

Optimal result

Integrand size = 34, antiderivative size = 340 \[ \int \frac {\sqrt {3-5 x^2} \sqrt {1+2 x^2}}{\sqrt {7-11 x^2}} \, dx=-\frac {x \sqrt {7-11 x^2} \sqrt {3-5 x^2}}{11 \sqrt {1+2 x^2}}+\frac {\sqrt {\frac {7}{11}} \sqrt {7-11 x^2} \sqrt {\frac {3-5 x^2}{1+2 x^2}} E\left (\arcsin \left (\frac {\sqrt {\frac {11}{3}} x}{\sqrt {1+2 x^2}}\right )|\frac {75}{77}\right )}{2 \sqrt {3-5 x^2} \sqrt {\frac {7-11 x^2}{1+2 x^2}}}+\frac {\sqrt {\frac {11}{7}} \sqrt {7-11 x^2} \sqrt {\frac {3-5 x^2}{1+2 x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {11}{3}} x}{\sqrt {1+2 x^2}}\right ),\frac {75}{77}\right )}{4 \sqrt {3-5 x^2} \sqrt {\frac {7-11 x^2}{1+2 x^2}}}-\frac {59 \sqrt {7-11 x^2} \sqrt {\frac {3-5 x^2}{1+2 x^2}} \operatorname {EllipticPi}\left (\frac {6}{11},\arcsin \left (\frac {\sqrt {\frac {11}{3}} x}{\sqrt {1+2 x^2}}\right ),\frac {75}{77}\right )}{44 \sqrt {77} \sqrt {3-5 x^2} \sqrt {\frac {7-11 x^2}{1+2 x^2}}} \] Output:

-1/11*x*(-11*x^2+7)^(1/2)*(-5*x^2+3)^(1/2)/(2*x^2+1)^(1/2)+1/22*77^(1/2)*( 
-11*x^2+7)^(1/2)*((-5*x^2+3)/(2*x^2+1))^(1/2)*EllipticE(1/3*33^(1/2)*x/(2* 
x^2+1)^(1/2),5/77*231^(1/2))/(-5*x^2+3)^(1/2)/((-11*x^2+7)/(2*x^2+1))^(1/2 
)+1/28*77^(1/2)*(-11*x^2+7)^(1/2)*((-5*x^2+3)/(2*x^2+1))^(1/2)*EllipticF(1 
/3*33^(1/2)*x/(2*x^2+1)^(1/2),5/77*231^(1/2))/(-5*x^2+3)^(1/2)/((-11*x^2+7 
)/(2*x^2+1))^(1/2)-59/3388*(-11*x^2+7)^(1/2)*((-5*x^2+3)/(2*x^2+1))^(1/2)* 
EllipticPi(1/3*33^(1/2)*x/(2*x^2+1)^(1/2),6/11,5/77*231^(1/2))*77^(1/2)/(- 
5*x^2+3)^(1/2)/((-11*x^2+7)/(2*x^2+1))^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.52 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {3-5 x^2} \sqrt {1+2 x^2}}{\sqrt {7-11 x^2}} \, dx=\frac {\sqrt {7-11 x^2} \sqrt {\frac {3-5 x^2}{7-11 x^2}} \left (242 x \sqrt {\frac {3-5 x^2}{7-11 x^2}}+484 x^3 \sqrt {\frac {3-5 x^2}{7-11 x^2}}+110 i \sqrt {21-33 x^2} \sqrt {\frac {1+2 x^2}{7-11 x^2}} E\left (i \text {arcsinh}\left (\frac {5 x}{\sqrt {7-11 x^2}}\right )|-\frac {2}{75}\right )+775 \sqrt {14-22 x^2} \sqrt {\frac {1+2 x^2}{7-11 x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {2}{3}} x}{\sqrt {7-11 x^2}}\right ),-\frac {75}{2}\right )+413 \sqrt {14-22 x^2} \sqrt {\frac {1+2 x^2}{7-11 x^2}} \operatorname {EllipticPi}\left (-\frac {33}{2},\arcsin \left (\frac {\sqrt {\frac {2}{3}} x}{\sqrt {7-11 x^2}}\right ),-\frac {75}{2}\right )\right )}{484 \sqrt {3-5 x^2} \sqrt {1+2 x^2}} \] Input:

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

Output:

(Sqrt[7 - 11*x^2]*Sqrt[(3 - 5*x^2)/(7 - 11*x^2)]*(242*x*Sqrt[(3 - 5*x^2)/( 
7 - 11*x^2)] + 484*x^3*Sqrt[(3 - 5*x^2)/(7 - 11*x^2)] + (110*I)*Sqrt[21 - 
33*x^2]*Sqrt[(1 + 2*x^2)/(7 - 11*x^2)]*EllipticE[I*ArcSinh[(5*x)/Sqrt[7 - 
11*x^2]], -2/75] + 775*Sqrt[14 - 22*x^2]*Sqrt[(1 + 2*x^2)/(7 - 11*x^2)]*El 
lipticF[ArcSin[(Sqrt[2/3]*x)/Sqrt[7 - 11*x^2]], -75/2] + 413*Sqrt[14 - 22* 
x^2]*Sqrt[(1 + 2*x^2)/(7 - 11*x^2)]*EllipticPi[-33/2, ArcSin[(Sqrt[2/3]*x) 
/Sqrt[7 - 11*x^2]], -75/2]))/(484*Sqrt[3 - 5*x^2]*Sqrt[1 + 2*x^2])
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 334, normalized size of antiderivative = 0.98, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {430, 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.

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

\(\Big \downarrow \) 430

\(\displaystyle \frac {25}{22} \int \frac {\sqrt {3-5 x^2}}{\sqrt {7-11 x^2} \left (2 x^2+1\right )^{3/2}}dx+\frac {125}{44} \int \frac {1}{\sqrt {7-11 x^2} \sqrt {3-5 x^2} \sqrt {2 x^2+1}}dx-\frac {59}{44} \int \frac {\sqrt {2 x^2+1}}{\sqrt {7-11 x^2} \sqrt {3-5 x^2}}dx-\frac {\sqrt {7-11 x^2} \sqrt {3-5 x^2} x}{11 \sqrt {2 x^2+1}}\)

\(\Big \downarrow \) 427

\(\displaystyle \frac {25}{22} \int \frac {\sqrt {3-5 x^2}}{\sqrt {7-11 x^2} \left (2 x^2+1\right )^{3/2}}dx-\frac {59}{44} \int \frac {\sqrt {2 x^2+1}}{\sqrt {7-11 x^2} \sqrt {3-5 x^2}}dx+\frac {125 \sqrt {3-5 x^2} \sqrt {\frac {2 x^2+1}{7-11 x^2}} \int \frac {\sqrt {3}}{\sqrt {3-\frac {2 x^2}{7-11 x^2}} \sqrt {\frac {25 x^2}{7-11 x^2}+1}}d\frac {x}{\sqrt {7-11 x^2}}}{44 \sqrt {3} \sqrt {\frac {3-5 x^2}{7-11 x^2}} \sqrt {2 x^2+1}}-\frac {\sqrt {7-11 x^2} \sqrt {3-5 x^2} x}{11 \sqrt {2 x^2+1}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {25}{22} \int \frac {\sqrt {3-5 x^2}}{\sqrt {7-11 x^2} \left (2 x^2+1\right )^{3/2}}dx-\frac {59}{44} \int \frac {\sqrt {2 x^2+1}}{\sqrt {7-11 x^2} \sqrt {3-5 x^2}}dx+\frac {125 \sqrt {3-5 x^2} \sqrt {\frac {2 x^2+1}{7-11 x^2}} \int \frac {1}{\sqrt {3-\frac {2 x^2}{7-11 x^2}} \sqrt {\frac {25 x^2}{7-11 x^2}+1}}d\frac {x}{\sqrt {7-11 x^2}}}{44 \sqrt {\frac {3-5 x^2}{7-11 x^2}} \sqrt {2 x^2+1}}-\frac {\sqrt {7-11 x^2} \sqrt {3-5 x^2} x}{11 \sqrt {2 x^2+1}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {25}{22} \int \frac {\sqrt {3-5 x^2}}{\sqrt {7-11 x^2} \left (2 x^2+1\right )^{3/2}}dx-\frac {59}{44} \int \frac {\sqrt {2 x^2+1}}{\sqrt {7-11 x^2} \sqrt {3-5 x^2}}dx+\frac {125 \sqrt {3-5 x^2} \sqrt {\frac {2 x^2+1}{7-11 x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {2}{3}} x}{\sqrt {7-11 x^2}}\right ),-\frac {75}{2}\right )}{44 \sqrt {2} \sqrt {\frac {3-5 x^2}{7-11 x^2}} \sqrt {2 x^2+1}}-\frac {\sqrt {7-11 x^2} \sqrt {3-5 x^2} x}{11 \sqrt {2 x^2+1}}\)

\(\Big \downarrow \) 428

\(\displaystyle \frac {25}{22} \int \frac {\sqrt {3-5 x^2}}{\sqrt {7-11 x^2} \left (2 x^2+1\right )^{3/2}}dx-\frac {59 \sqrt {7-11 x^2} \sqrt {\frac {3-5 x^2}{2 x^2+1}} \int \frac {\sqrt {21}}{\sqrt {7-\frac {25 x^2}{2 x^2+1}} \sqrt {3-\frac {11 x^2}{2 x^2+1}} \left (1-\frac {2 x^2}{2 x^2+1}\right )}d\frac {x}{\sqrt {2 x^2+1}}}{44 \sqrt {21} \sqrt {3-5 x^2} \sqrt {\frac {7-11 x^2}{2 x^2+1}}}+\frac {125 \sqrt {3-5 x^2} \sqrt {\frac {2 x^2+1}{7-11 x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {2}{3}} x}{\sqrt {7-11 x^2}}\right ),-\frac {75}{2}\right )}{44 \sqrt {2} \sqrt {\frac {3-5 x^2}{7-11 x^2}} \sqrt {2 x^2+1}}-\frac {\sqrt {7-11 x^2} \sqrt {3-5 x^2} x}{11 \sqrt {2 x^2+1}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {25}{22} \int \frac {\sqrt {3-5 x^2}}{\sqrt {7-11 x^2} \left (2 x^2+1\right )^{3/2}}dx-\frac {59 \sqrt {7-11 x^2} \sqrt {\frac {3-5 x^2}{2 x^2+1}} \int \frac {1}{\sqrt {7-\frac {25 x^2}{2 x^2+1}} \sqrt {3-\frac {11 x^2}{2 x^2+1}} \left (1-\frac {2 x^2}{2 x^2+1}\right )}d\frac {x}{\sqrt {2 x^2+1}}}{44 \sqrt {3-5 x^2} \sqrt {\frac {7-11 x^2}{2 x^2+1}}}+\frac {125 \sqrt {3-5 x^2} \sqrt {\frac {2 x^2+1}{7-11 x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {2}{3}} x}{\sqrt {7-11 x^2}}\right ),-\frac {75}{2}\right )}{44 \sqrt {2} \sqrt {\frac {3-5 x^2}{7-11 x^2}} \sqrt {2 x^2+1}}-\frac {\sqrt {7-11 x^2} \sqrt {3-5 x^2} x}{11 \sqrt {2 x^2+1}}\)

\(\Big \downarrow \) 412

\(\displaystyle \frac {25}{22} \int \frac {\sqrt {3-5 x^2}}{\sqrt {7-11 x^2} \left (2 x^2+1\right )^{3/2}}dx+\frac {125 \sqrt {3-5 x^2} \sqrt {\frac {2 x^2+1}{7-11 x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {2}{3}} x}{\sqrt {7-11 x^2}}\right ),-\frac {75}{2}\right )}{44 \sqrt {2} \sqrt {\frac {3-5 x^2}{7-11 x^2}} \sqrt {2 x^2+1}}-\frac {59 \sqrt {7-11 x^2} \sqrt {\frac {3-5 x^2}{2 x^2+1}} \operatorname {EllipticPi}\left (\frac {14}{25},\arcsin \left (\frac {5 x}{\sqrt {7} \sqrt {2 x^2+1}}\right ),\frac {77}{75}\right )}{220 \sqrt {3} \sqrt {3-5 x^2} \sqrt {\frac {7-11 x^2}{2 x^2+1}}}-\frac {\sqrt {7-11 x^2} \sqrt {3-5 x^2} x}{11 \sqrt {2 x^2+1}}\)

\(\Big \downarrow \) 429

\(\displaystyle \frac {25 \sqrt {\frac {3}{7}} \sqrt {3-5 x^2} \sqrt {\frac {7-11 x^2}{2 x^2+1}} \int \frac {\sqrt {\frac {7}{3}} \sqrt {3-\frac {11 x^2}{2 x^2+1}}}{\sqrt {7-\frac {25 x^2}{2 x^2+1}}}d\frac {x}{\sqrt {2 x^2+1}}}{22 \sqrt {7-11 x^2} \sqrt {\frac {3-5 x^2}{2 x^2+1}}}+\frac {125 \sqrt {3-5 x^2} \sqrt {\frac {2 x^2+1}{7-11 x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {2}{3}} x}{\sqrt {7-11 x^2}}\right ),-\frac {75}{2}\right )}{44 \sqrt {2} \sqrt {\frac {3-5 x^2}{7-11 x^2}} \sqrt {2 x^2+1}}-\frac {59 \sqrt {7-11 x^2} \sqrt {\frac {3-5 x^2}{2 x^2+1}} \operatorname {EllipticPi}\left (\frac {14}{25},\arcsin \left (\frac {5 x}{\sqrt {7} \sqrt {2 x^2+1}}\right ),\frac {77}{75}\right )}{220 \sqrt {3} \sqrt {3-5 x^2} \sqrt {\frac {7-11 x^2}{2 x^2+1}}}-\frac {\sqrt {7-11 x^2} \sqrt {3-5 x^2} x}{11 \sqrt {2 x^2+1}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {25 \sqrt {3-5 x^2} \sqrt {\frac {7-11 x^2}{2 x^2+1}} \int \frac {\sqrt {3-\frac {11 x^2}{2 x^2+1}}}{\sqrt {7-\frac {25 x^2}{2 x^2+1}}}d\frac {x}{\sqrt {2 x^2+1}}}{22 \sqrt {7-11 x^2} \sqrt {\frac {3-5 x^2}{2 x^2+1}}}+\frac {125 \sqrt {3-5 x^2} \sqrt {\frac {2 x^2+1}{7-11 x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {2}{3}} x}{\sqrt {7-11 x^2}}\right ),-\frac {75}{2}\right )}{44 \sqrt {2} \sqrt {\frac {3-5 x^2}{7-11 x^2}} \sqrt {2 x^2+1}}-\frac {59 \sqrt {7-11 x^2} \sqrt {\frac {3-5 x^2}{2 x^2+1}} \operatorname {EllipticPi}\left (\frac {14}{25},\arcsin \left (\frac {5 x}{\sqrt {7} \sqrt {2 x^2+1}}\right ),\frac {77}{75}\right )}{220 \sqrt {3} \sqrt {3-5 x^2} \sqrt {\frac {7-11 x^2}{2 x^2+1}}}-\frac {\sqrt {7-11 x^2} \sqrt {3-5 x^2} x}{11 \sqrt {2 x^2+1}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {125 \sqrt {3-5 x^2} \sqrt {\frac {2 x^2+1}{7-11 x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {2}{3}} x}{\sqrt {7-11 x^2}}\right ),-\frac {75}{2}\right )}{44 \sqrt {2} \sqrt {\frac {3-5 x^2}{7-11 x^2}} \sqrt {2 x^2+1}}+\frac {5 \sqrt {3} \sqrt {3-5 x^2} \sqrt {\frac {7-11 x^2}{2 x^2+1}} E\left (\arcsin \left (\frac {5 x}{\sqrt {7} \sqrt {2 x^2+1}}\right )|\frac {77}{75}\right )}{22 \sqrt {7-11 x^2} \sqrt {\frac {3-5 x^2}{2 x^2+1}}}-\frac {59 \sqrt {7-11 x^2} \sqrt {\frac {3-5 x^2}{2 x^2+1}} \operatorname {EllipticPi}\left (\frac {14}{25},\arcsin \left (\frac {5 x}{\sqrt {7} \sqrt {2 x^2+1}}\right ),\frac {77}{75}\right )}{220 \sqrt {3} \sqrt {3-5 x^2} \sqrt {\frac {7-11 x^2}{2 x^2+1}}}-\frac {\sqrt {7-11 x^2} \sqrt {3-5 x^2} x}{11 \sqrt {2 x^2+1}}\)

Input:

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

Output:

-1/11*(x*Sqrt[7 - 11*x^2]*Sqrt[3 - 5*x^2])/Sqrt[1 + 2*x^2] + (5*Sqrt[3]*Sq 
rt[3 - 5*x^2]*Sqrt[(7 - 11*x^2)/(1 + 2*x^2)]*EllipticE[ArcSin[(5*x)/(Sqrt[ 
7]*Sqrt[1 + 2*x^2])], 77/75])/(22*Sqrt[7 - 11*x^2]*Sqrt[(3 - 5*x^2)/(1 + 2 
*x^2)]) + (125*Sqrt[3 - 5*x^2]*Sqrt[(1 + 2*x^2)/(7 - 11*x^2)]*EllipticF[Ar 
cSin[(Sqrt[2/3]*x)/Sqrt[7 - 11*x^2]], -75/2])/(44*Sqrt[2]*Sqrt[(3 - 5*x^2) 
/(7 - 11*x^2)]*Sqrt[1 + 2*x^2]) - (59*Sqrt[7 - 11*x^2]*Sqrt[(3 - 5*x^2)/(1 
 + 2*x^2)]*EllipticPi[14/25, ArcSin[(5*x)/(Sqrt[7]*Sqrt[1 + 2*x^2])], 77/7 
5])/(220*Sqrt[3]*Sqrt[3 - 5*x^2]*Sqrt[(7 - 11*x^2)/(1 + 2*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 430

Int[(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2])/Sqrt[(e_) + (f_.) 
*(x_)^2], x_Symbol] :> Simp[d*x*Sqrt[a + b*x^2]*(Sqrt[e + f*x^2]/(2*f*Sqrt[ 
c + d*x^2])), x] + (-Simp[c*((d*e - c*f)/(2*f))   Int[Sqrt[a + b*x^2]/((c + 
 d*x^2)^(3/2)*Sqrt[e + f*x^2]), x], x] - Simp[(b*d*e - b*c*f - a*d*f)/(2*d* 
f)   Int[Sqrt[c + d*x^2]/(Sqrt[a + b*x^2]*Sqrt[e + f*x^2]), x], x] + Simp[b 
*c*((d*e - c*f)/(2*d*f))   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] && PosQ[(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)/(11*x^2 - 7), 
 x)
 

Sympy [F]

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

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

Output:

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

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

Output:

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

Reduce [F]

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

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