\(\int \frac {\sqrt {1+2 x^2} \sqrt {3+5 x^2}}{\sqrt {7+11 x^2}} \, dx\) [501]

Optimal result

Integrand size = 34, antiderivative size = 328 \[ \int \frac {\sqrt {1+2 x^2} \sqrt {3+5 x^2}}{\sqrt {7+11 x^2}} \, dx=\frac {x \sqrt {3+5 x^2} \sqrt {7+11 x^2}}{11 \sqrt {1+2 x^2}}-\frac {\sqrt {7} \sqrt {3+5 x^2} \sqrt {\frac {7+11 x^2}{1+2 x^2}} E\left (\arcsin \left (\frac {x}{\sqrt {3} \sqrt {1+2 x^2}}\right )|\frac {9}{7}\right )}{22 \sqrt {\frac {3+5 x^2}{1+2 x^2}} \sqrt {7+11 x^2}}+\frac {\sqrt {3+5 x^2} \sqrt {\frac {7+11 x^2}{1+2 x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {3} \sqrt {1+2 x^2}}\right ),\frac {9}{7}\right )}{4 \sqrt {7} \sqrt {\frac {3+5 x^2}{1+2 x^2}} \sqrt {7+11 x^2}}+\frac {51 \sqrt {3+5 x^2} \sqrt {\frac {7+11 x^2}{1+2 x^2}} \operatorname {EllipticPi}\left (6,\arcsin \left (\frac {x}{\sqrt {3} \sqrt {1+2 x^2}}\right ),\frac {9}{7}\right )}{44 \sqrt {7} \sqrt {\frac {3+5 x^2}{1+2 x^2}} \sqrt {7+11 x^2}} \] Output:

1/11*x*(5*x^2+3)^(1/2)*(11*x^2+7)^(1/2)/(2*x^2+1)^(1/2)-1/22*7^(1/2)*(5*x^ 
2+3)^(1/2)*((11*x^2+7)/(2*x^2+1))^(1/2)*EllipticE(1/3*x*3^(1/2)/(2*x^2+1)^ 
(1/2),3/7*7^(1/2))/((5*x^2+3)/(2*x^2+1))^(1/2)/(11*x^2+7)^(1/2)+1/28*(5*x^ 
2+3)^(1/2)*((11*x^2+7)/(2*x^2+1))^(1/2)*EllipticF(1/3*x*3^(1/2)/(2*x^2+1)^ 
(1/2),3/7*7^(1/2))*7^(1/2)/((5*x^2+3)/(2*x^2+1))^(1/2)/(11*x^2+7)^(1/2)+51 
/308*(5*x^2+3)^(1/2)*((11*x^2+7)/(2*x^2+1))^(1/2)*EllipticPi(1/3*x*3^(1/2) 
/(2*x^2+1)^(1/2),6,3/7*7^(1/2))*7^(1/2)/((5*x^2+3)/(2*x^2+1))^(1/2)/(11*x^ 
2+7)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.02 (sec) , antiderivative size = 301, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {1+2 x^2} \sqrt {3+5 x^2}}{\sqrt {7+11 x^2}} \, dx=\frac {\frac {847 x \sqrt {1+2 x^2} \left (3+5 x^2\right )}{\sqrt {7+11 x^2}}-\frac {77 i \sqrt {2+4 x^2} \sqrt {\frac {3+5 x^2}{7+11 x^2}} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}} x}{\sqrt {7+11 x^2}}\right )|\frac {9}{2}\right )}{\sqrt {\frac {1+2 x^2}{7+11 x^2}}}-\frac {16 \left (1+2 x^2\right ) \sqrt {\frac {3+5 x^2}{1+2 x^2}} \sqrt {\frac {49+77 x^2}{1+2 x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {3+6 x^2}}\right ),\frac {9}{7}\right )}{\sqrt {7+11 x^2}}-\frac {833 i \sqrt {1+2 x^2} \sqrt {\frac {3+5 x^2}{7+11 x^2}} \operatorname {EllipticPi}\left (-\frac {11}{3},i \text {arcsinh}\left (\frac {\sqrt {3} x}{\sqrt {7+11 x^2}}\right ),\frac {2}{9}\right )}{\sqrt {\frac {1+2 x^2}{7+11 x^2}}}}{1694 \sqrt {3+5 x^2}} \] Input:

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

Output:

((847*x*Sqrt[1 + 2*x^2]*(3 + 5*x^2))/Sqrt[7 + 11*x^2] - ((77*I)*Sqrt[2 + 4 
*x^2]*Sqrt[(3 + 5*x^2)/(7 + 11*x^2)]*EllipticE[I*ArcSinh[(Sqrt[2/3]*x)/Sqr 
t[7 + 11*x^2]], 9/2])/Sqrt[(1 + 2*x^2)/(7 + 11*x^2)] - (16*(1 + 2*x^2)*Sqr 
t[(3 + 5*x^2)/(1 + 2*x^2)]*Sqrt[(49 + 77*x^2)/(1 + 2*x^2)]*EllipticF[ArcSi 
n[x/Sqrt[3 + 6*x^2]], 9/7])/Sqrt[7 + 11*x^2] - ((833*I)*Sqrt[1 + 2*x^2]*Sq 
rt[(3 + 5*x^2)/(7 + 11*x^2)]*EllipticPi[-11/3, I*ArcSinh[(Sqrt[3]*x)/Sqrt[ 
7 + 11*x^2]], 2/9])/Sqrt[(1 + 2*x^2)/(7 + 11*x^2)])/(1694*Sqrt[3 + 5*x^2])
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.02, 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.

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

\(\Big \downarrow \) 430

\(\displaystyle -\frac {3}{11} \int \frac {\sqrt {2 x^2+1}}{\left (5 x^2+3\right )^{3/2} \sqrt {11 x^2+7}}dx+\frac {6}{55} \int \frac {1}{\sqrt {2 x^2+1} \sqrt {5 x^2+3} \sqrt {11 x^2+7}}dx+\frac {51}{110} \int \frac {\sqrt {5 x^2+3}}{\sqrt {2 x^2+1} \sqrt {11 x^2+7}}dx+\frac {5 \sqrt {2 x^2+1} \sqrt {11 x^2+7} x}{22 \sqrt {5 x^2+3}}\)

\(\Big \downarrow \) 427

\(\displaystyle -\frac {3}{11} \int \frac {\sqrt {2 x^2+1}}{\left (5 x^2+3\right )^{3/2} \sqrt {11 x^2+7}}dx+\frac {51}{110} \int \frac {\sqrt {5 x^2+3}}{\sqrt {2 x^2+1} \sqrt {11 x^2+7}}dx+\frac {2 \sqrt {\frac {3}{7}} \sqrt {5 x^2+3} \sqrt {\frac {11 x^2+7}{2 x^2+1}} \int \frac {\sqrt {21}}{\sqrt {7-\frac {3 x^2}{2 x^2+1}} \sqrt {3-\frac {x^2}{2 x^2+1}}}d\frac {x}{\sqrt {2 x^2+1}}}{55 \sqrt {\frac {5 x^2+3}{2 x^2+1}} \sqrt {11 x^2+7}}+\frac {5 \sqrt {2 x^2+1} \sqrt {11 x^2+7} x}{22 \sqrt {5 x^2+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {3}{11} \int \frac {\sqrt {2 x^2+1}}{\left (5 x^2+3\right )^{3/2} \sqrt {11 x^2+7}}dx+\frac {51}{110} \int \frac {\sqrt {5 x^2+3}}{\sqrt {2 x^2+1} \sqrt {11 x^2+7}}dx+\frac {6 \sqrt {5 x^2+3} \sqrt {\frac {11 x^2+7}{2 x^2+1}} \int \frac {1}{\sqrt {7-\frac {3 x^2}{2 x^2+1}} \sqrt {3-\frac {x^2}{2 x^2+1}}}d\frac {x}{\sqrt {2 x^2+1}}}{55 \sqrt {\frac {5 x^2+3}{2 x^2+1}} \sqrt {11 x^2+7}}+\frac {5 \sqrt {2 x^2+1} \sqrt {11 x^2+7} x}{22 \sqrt {5 x^2+3}}\)

\(\Big \downarrow \) 321

\(\displaystyle -\frac {3}{11} \int \frac {\sqrt {2 x^2+1}}{\left (5 x^2+3\right )^{3/2} \sqrt {11 x^2+7}}dx+\frac {51}{110} \int \frac {\sqrt {5 x^2+3}}{\sqrt {2 x^2+1} \sqrt {11 x^2+7}}dx+\frac {6 \sqrt {5 x^2+3} \sqrt {\frac {11 x^2+7}{2 x^2+1}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {3} \sqrt {2 x^2+1}}\right ),\frac {9}{7}\right )}{55 \sqrt {7} \sqrt {\frac {5 x^2+3}{2 x^2+1}} \sqrt {11 x^2+7}}+\frac {5 \sqrt {2 x^2+1} \sqrt {11 x^2+7} x}{22 \sqrt {5 x^2+3}}\)

\(\Big \downarrow \) 428

\(\displaystyle -\frac {3}{11} \int \frac {\sqrt {2 x^2+1}}{\left (5 x^2+3\right )^{3/2} \sqrt {11 x^2+7}}dx+\frac {153 \sqrt {2 x^2+1} \sqrt {\frac {11 x^2+7}{5 x^2+3}} \int \frac {\sqrt {7}}{\left (1-\frac {5 x^2}{5 x^2+3}\right ) \sqrt {7-\frac {2 x^2}{5 x^2+3}} \sqrt {\frac {x^2}{5 x^2+3}+1}}d\frac {x}{\sqrt {5 x^2+3}}}{110 \sqrt {7} \sqrt {\frac {2 x^2+1}{5 x^2+3}} \sqrt {11 x^2+7}}+\frac {6 \sqrt {5 x^2+3} \sqrt {\frac {11 x^2+7}{2 x^2+1}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {3} \sqrt {2 x^2+1}}\right ),\frac {9}{7}\right )}{55 \sqrt {7} \sqrt {\frac {5 x^2+3}{2 x^2+1}} \sqrt {11 x^2+7}}+\frac {5 \sqrt {2 x^2+1} \sqrt {11 x^2+7} x}{22 \sqrt {5 x^2+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {3}{11} \int \frac {\sqrt {2 x^2+1}}{\left (5 x^2+3\right )^{3/2} \sqrt {11 x^2+7}}dx+\frac {153 \sqrt {2 x^2+1} \sqrt {\frac {11 x^2+7}{5 x^2+3}} \int \frac {1}{\left (1-\frac {5 x^2}{5 x^2+3}\right ) \sqrt {7-\frac {2 x^2}{5 x^2+3}} \sqrt {\frac {x^2}{5 x^2+3}+1}}d\frac {x}{\sqrt {5 x^2+3}}}{110 \sqrt {\frac {2 x^2+1}{5 x^2+3}} \sqrt {11 x^2+7}}+\frac {6 \sqrt {5 x^2+3} \sqrt {\frac {11 x^2+7}{2 x^2+1}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {3} \sqrt {2 x^2+1}}\right ),\frac {9}{7}\right )}{55 \sqrt {7} \sqrt {\frac {5 x^2+3}{2 x^2+1}} \sqrt {11 x^2+7}}+\frac {5 \sqrt {2 x^2+1} \sqrt {11 x^2+7} x}{22 \sqrt {5 x^2+3}}\)

\(\Big \downarrow \) 412

\(\displaystyle -\frac {3}{11} \int \frac {\sqrt {2 x^2+1}}{\left (5 x^2+3\right )^{3/2} \sqrt {11 x^2+7}}dx+\frac {6 \sqrt {5 x^2+3} \sqrt {\frac {11 x^2+7}{2 x^2+1}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {3} \sqrt {2 x^2+1}}\right ),\frac {9}{7}\right )}{55 \sqrt {7} \sqrt {\frac {5 x^2+3}{2 x^2+1}} \sqrt {11 x^2+7}}+\frac {153 \sqrt {2 x^2+1} \sqrt {\frac {11 x^2+7}{5 x^2+3}} \operatorname {EllipticPi}\left (\frac {35}{2},\arcsin \left (\frac {\sqrt {\frac {2}{7}} x}{\sqrt {5 x^2+3}}\right ),-\frac {7}{2}\right )}{110 \sqrt {2} \sqrt {\frac {2 x^2+1}{5 x^2+3}} \sqrt {11 x^2+7}}+\frac {5 \sqrt {2 x^2+1} \sqrt {11 x^2+7} x}{22 \sqrt {5 x^2+3}}\)

\(\Big \downarrow \) 429

\(\displaystyle -\frac {\sqrt {2 x^2+1} \sqrt {\frac {11 x^2+7}{5 x^2+3}} \int \frac {\sqrt {7} \sqrt {\frac {x^2}{5 x^2+3}+1}}{\sqrt {7-\frac {2 x^2}{5 x^2+3}}}d\frac {x}{\sqrt {5 x^2+3}}}{11 \sqrt {7} \sqrt {\frac {2 x^2+1}{5 x^2+3}} \sqrt {11 x^2+7}}+\frac {6 \sqrt {5 x^2+3} \sqrt {\frac {11 x^2+7}{2 x^2+1}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {3} \sqrt {2 x^2+1}}\right ),\frac {9}{7}\right )}{55 \sqrt {7} \sqrt {\frac {5 x^2+3}{2 x^2+1}} \sqrt {11 x^2+7}}+\frac {153 \sqrt {2 x^2+1} \sqrt {\frac {11 x^2+7}{5 x^2+3}} \operatorname {EllipticPi}\left (\frac {35}{2},\arcsin \left (\frac {\sqrt {\frac {2}{7}} x}{\sqrt {5 x^2+3}}\right ),-\frac {7}{2}\right )}{110 \sqrt {2} \sqrt {\frac {2 x^2+1}{5 x^2+3}} \sqrt {11 x^2+7}}+\frac {5 \sqrt {2 x^2+1} \sqrt {11 x^2+7} x}{22 \sqrt {5 x^2+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\sqrt {2 x^2+1} \sqrt {\frac {11 x^2+7}{5 x^2+3}} \int \frac {\sqrt {\frac {x^2}{5 x^2+3}+1}}{\sqrt {7-\frac {2 x^2}{5 x^2+3}}}d\frac {x}{\sqrt {5 x^2+3}}}{11 \sqrt {\frac {2 x^2+1}{5 x^2+3}} \sqrt {11 x^2+7}}+\frac {6 \sqrt {5 x^2+3} \sqrt {\frac {11 x^2+7}{2 x^2+1}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {3} \sqrt {2 x^2+1}}\right ),\frac {9}{7}\right )}{55 \sqrt {7} \sqrt {\frac {5 x^2+3}{2 x^2+1}} \sqrt {11 x^2+7}}+\frac {153 \sqrt {2 x^2+1} \sqrt {\frac {11 x^2+7}{5 x^2+3}} \operatorname {EllipticPi}\left (\frac {35}{2},\arcsin \left (\frac {\sqrt {\frac {2}{7}} x}{\sqrt {5 x^2+3}}\right ),-\frac {7}{2}\right )}{110 \sqrt {2} \sqrt {\frac {2 x^2+1}{5 x^2+3}} \sqrt {11 x^2+7}}+\frac {5 \sqrt {2 x^2+1} \sqrt {11 x^2+7} x}{22 \sqrt {5 x^2+3}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {6 \sqrt {5 x^2+3} \sqrt {\frac {11 x^2+7}{2 x^2+1}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {3} \sqrt {2 x^2+1}}\right ),\frac {9}{7}\right )}{55 \sqrt {7} \sqrt {\frac {5 x^2+3}{2 x^2+1}} \sqrt {11 x^2+7}}-\frac {\sqrt {2 x^2+1} \sqrt {\frac {11 x^2+7}{5 x^2+3}} E\left (\arcsin \left (\frac {\sqrt {\frac {2}{7}} x}{\sqrt {5 x^2+3}}\right )|-\frac {7}{2}\right )}{11 \sqrt {2} \sqrt {\frac {2 x^2+1}{5 x^2+3}} \sqrt {11 x^2+7}}+\frac {153 \sqrt {2 x^2+1} \sqrt {\frac {11 x^2+7}{5 x^2+3}} \operatorname {EllipticPi}\left (\frac {35}{2},\arcsin \left (\frac {\sqrt {\frac {2}{7}} x}{\sqrt {5 x^2+3}}\right ),-\frac {7}{2}\right )}{110 \sqrt {2} \sqrt {\frac {2 x^2+1}{5 x^2+3}} \sqrt {11 x^2+7}}+\frac {5 \sqrt {2 x^2+1} \sqrt {11 x^2+7} x}{22 \sqrt {5 x^2+3}}\)

Input:

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

Output:

(5*x*Sqrt[1 + 2*x^2]*Sqrt[7 + 11*x^2])/(22*Sqrt[3 + 5*x^2]) - (Sqrt[1 + 2* 
x^2]*Sqrt[(7 + 11*x^2)/(3 + 5*x^2)]*EllipticE[ArcSin[(Sqrt[2/7]*x)/Sqrt[3 
+ 5*x^2]], -7/2])/(11*Sqrt[2]*Sqrt[(1 + 2*x^2)/(3 + 5*x^2)]*Sqrt[7 + 11*x^ 
2]) + (6*Sqrt[3 + 5*x^2]*Sqrt[(7 + 11*x^2)/(1 + 2*x^2)]*EllipticF[ArcSin[x 
/(Sqrt[3]*Sqrt[1 + 2*x^2])], 9/7])/(55*Sqrt[7]*Sqrt[(3 + 5*x^2)/(1 + 2*x^2 
)]*Sqrt[7 + 11*x^2]) + (153*Sqrt[1 + 2*x^2]*Sqrt[(7 + 11*x^2)/(3 + 5*x^2)] 
*EllipticPi[35/2, ArcSin[(Sqrt[2/7]*x)/Sqrt[3 + 5*x^2]], -7/2])/(110*Sqrt[ 
2]*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 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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

int((2*x^2+1)^(1/2)*(5*x^2+3)^(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)