\(\int \frac {\sqrt {a+b x^2} (e+f x^2)}{\sqrt {c+d x^2}} \, dx\) [3]

Optimal result

Integrand size = 30, antiderivative size = 284 \[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )}{\sqrt {c+d x^2}} \, dx=\frac {(3 b d e-2 b c f+a d f) x \sqrt {c+d x^2}}{3 d^2 \sqrt {a+b x^2}}+\frac {f x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 d}-\frac {\sqrt {a} (3 b d e-2 b c f+a d f) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 \sqrt {b} d^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {a^{3/2} (3 d e-c f) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 \sqrt {b} c d \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

1/3*(a*d*f-2*b*c*f+3*b*d*e)*x*(d*x^2+c)^(1/2)/d^2/(b*x^2+a)^(1/2)+1/3*f*x* 
(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/d-1/3*a^(1/2)*(a*d*f-2*b*c*f+3*b*d*e)*(d*x 
^2+c)^(1/2)*EllipticE(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2),(1-a*d/b/c)^(1/2 
))/b^(1/2)/d^2/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)+1/3*a^(3/2) 
*(-c*f+3*d*e)*(d*x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)),(1 
-a*d/b/c)^(1/2))/b^(1/2)/c/d/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/ 
2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.93 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )}{\sqrt {c+d x^2}} \, dx=\frac {\sqrt {\frac {b}{a}} d f x \left (a+b x^2\right ) \left (c+d x^2\right )-i c (3 b d e-2 b c f+a d f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i (-b c+a d) (-3 d e+2 c f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3 \sqrt {\frac {b}{a}} d^2 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

Integrate[(Sqrt[a + b*x^2]*(e + f*x^2))/Sqrt[c + d*x^2],x]
 

Output:

(Sqrt[b/a]*d*f*x*(a + b*x^2)*(c + d*x^2) - I*c*(3*b*d*e - 2*b*c*f + a*d*f) 
*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], 
 (a*d)/(b*c)] + I*(-(b*c) + a*d)*(-3*d*e + 2*c*f)*Sqrt[1 + (b*x^2)/a]*Sqrt 
[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(3*Sqrt[b/ 
a]*d^2*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.93, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {403, 406, 320, 388, 313}

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.

\(\displaystyle \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )}{\sqrt {c+d x^2}} \, dx\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {\int \frac {(3 b d e-2 b c f+a d f) x^2+a (3 d e-c f)}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}+\frac {f x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 d}\)

\(\Big \downarrow \) 406

\(\displaystyle \frac {a (3 d e-c f) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+(a d f-2 b c f+3 b d e) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}+\frac {f x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 d}\)

\(\Big \downarrow \) 320

\(\displaystyle \frac {(a d f-2 b c f+3 b d e) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {\sqrt {c} \sqrt {a+b x^2} (3 d e-c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 d}+\frac {f x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 d}\)

\(\Big \downarrow \) 388

\(\displaystyle \frac {(a d f-2 b c f+3 b d e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {\sqrt {c} \sqrt {a+b x^2} (3 d e-c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 d}+\frac {f x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 d}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {\frac {\sqrt {c} \sqrt {a+b x^2} (3 d e-c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+(a d f-2 b c f+3 b d e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 d}+\frac {f x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 d}\)

Input:

Int[(Sqrt[a + b*x^2]*(e + f*x^2))/Sqrt[c + d*x^2],x]
 

Output:

(f*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*d) + ((3*b*d*e - 2*b*c*f + a*d*f) 
*((x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[a + b*x^2]*Ellip 
ticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b*Sqrt[d]*Sqrt[(c*(a 
+ b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])) + (Sqrt[c]*(3*d*e - c*f)*Sqrt 
[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt 
[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]))/(3*d)
 

Defintions of rubi rules used

rule 313

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim 
p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c 
+ d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ 
[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
 

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 388

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] 
 :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b   Int[Sqrt[ 
a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 403

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2), x_Symbol] :> Simp[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + 
 q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1))   Int[(a + b*x^2)^p*(c 
+ d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + 
f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, 
 d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
 

rule 406

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2), x_Symbol] :> Simp[e   Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim 
p[f   Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, 
f, p, q}, x]
 

Maple [A] (verified)

Time = 7.90 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.10

method	result	size
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {f x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 d}+\frac {\left (a e -\frac {a c f}{3 d}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (a f +b e -\frac {f \left (2 a d +2 b c \right )}{3 d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(312\)
risch	\(\frac {f x \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{3 d}-\frac {\left (\frac {\left (a d f -2 b c f +3 b d e \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}+\frac {a c f \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {3 a d e \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{3 d \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(384\)
default	\(-\frac {\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \left (-\sqrt {-\frac {b}{a}}\, b \,d^{2} f \,x^{5}-\sqrt {-\frac {b}{a}}\, a \,d^{2} f \,x^{3}-\sqrt {-\frac {b}{a}}\, b c d f \,x^{3}+2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a c d f -3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,d^{2} e -2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,c^{2} f +3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b c d e -\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a c d f +2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,c^{2} f -3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b c d e -\sqrt {-\frac {b}{a}}\, a c d f x \right )}{3 \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right ) d^{2} \sqrt {-\frac {b}{a}}}\)	\(506\)

Input:

int((b*x^2+a)^(1/2)*(f*x^2+e)/(d*x^2+c)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(1/3*f/d*x*(b* 
d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+(a*e-1/3*a*c*f/d)/(-b/a)^(1/2)*(1+b*x^2/a 
)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x* 
(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-(a*f+b*e-1/3*f/d*(2*a*d+2*b*c))*c/( 
-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a 
*c)^(1/2)/d*(EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-EllipticE( 
x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )}{\sqrt {c+d x^2}} \, dx=-\frac {{\left (3 \, b c^{2} d e - {\left (2 \, b c^{3} - a c^{2} d\right )} f\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - \sqrt {b d} {\left (3 \, {\left (b c^{2} d + a d^{3}\right )} e - {\left (2 \, b c^{3} - a c^{2} d + a c d^{2}\right )} f\right )} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (b c d^{2} f x^{2} + 3 \, b c d^{2} e - {\left (2 \, b c^{2} d - a c d^{2}\right )} f\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{3 \, b c d^{3} x} \] Input:

integrate((b*x^2+a)^(1/2)*(f*x^2+e)/(d*x^2+c)^(1/2),x, algorithm="fricas")
 

Output:

-1/3*((3*b*c^2*d*e - (2*b*c^3 - a*c^2*d)*f)*sqrt(b*d)*x*sqrt(-c/d)*ellipti 
c_e(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - sqrt(b*d)*(3*(b*c^2*d + a*d^3)*e - 
(2*b*c^3 - a*c^2*d + a*c*d^2)*f)*x*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/d) 
/x), a*d/(b*c)) - (b*c*d^2*f*x^2 + 3*b*c*d^2*e - (2*b*c^2*d - a*c*d^2)*f)* 
sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(b*c*d^3*x)
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )}{\sqrt {c+d x^2}} \, dx=\int \frac {\sqrt {a + b x^{2}} \left (e + f x^{2}\right )}{\sqrt {c + d x^{2}}}\, dx \] Input:

integrate((b*x**2+a)**(1/2)*(f*x**2+e)/(d*x**2+c)**(1/2),x)
 

Output:

Integral(sqrt(a + b*x**2)*(e + f*x**2)/sqrt(c + d*x**2), x)
 

Maxima [F]

\[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )}{\sqrt {c+d x^2}} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (f x^{2} + e\right )}}{\sqrt {d x^{2} + c}} \,d x } \] Input:

integrate((b*x^2+a)^(1/2)*(f*x^2+e)/(d*x^2+c)^(1/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(b*x^2 + a)*(f*x^2 + e)/sqrt(d*x^2 + c), x)
 

Giac [F]

\[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )}{\sqrt {c+d x^2}} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (f x^{2} + e\right )}}{\sqrt {d x^{2} + c}} \,d x } \] Input:

integrate((b*x^2+a)^(1/2)*(f*x^2+e)/(d*x^2+c)^(1/2),x, algorithm="giac")
 

Output:

integrate(sqrt(b*x^2 + a)*(f*x^2 + e)/sqrt(d*x^2 + c), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )}{\sqrt {c+d x^2}} \, dx=\int \frac {\sqrt {b\,x^2+a}\,\left (f\,x^2+e\right )}{\sqrt {d\,x^2+c}} \,d x \] Input:

int(((a + b*x^2)^(1/2)*(e + f*x^2))/(c + d*x^2)^(1/2),x)
 

Output:

int(((a + b*x^2)^(1/2)*(e + f*x^2))/(c + d*x^2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )}{\sqrt {c+d x^2}} \, dx=\frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, f x +\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a d f -2 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b c f +3 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b d e -\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a c f +3 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a d e}{3 d} \] Input:

int((b*x^2+a)^(1/2)*(f*x^2+e)/(d*x^2+c)^(1/2),x)
 

Output:

(sqrt(c + d*x**2)*sqrt(a + b*x**2)*f*x + int((sqrt(c + d*x**2)*sqrt(a + b* 
x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a*d*f - 2*int((sqrt( 
c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4), 
x)*b*c*f + 3*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x**2 
+ b*c*x**2 + b*d*x**4),x)*b*d*e - int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/ 
(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a*c*f + 3*int((sqrt(c + d*x**2)* 
sqrt(a + b*x**2))/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a*d*e)/(3*d)