\(\int \frac {e+f x^2}{x^2 \sqrt {a+b x^2} \sqrt {c-d x^2}} \, dx\) [165]

Optimal result

Integrand size = 34, antiderivative size = 222 \[ \int \frac {e+f x^2}{x^2 \sqrt {a+b x^2} \sqrt {c-d x^2}} \, dx=-\frac {e \sqrt {a+b x^2} \sqrt {c-d x^2}}{a c x}-\frac {\sqrt {d} e \sqrt {a+b x^2} \sqrt {1-\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{a \sqrt {c} \sqrt {1+\frac {b x^2}{a}} \sqrt {c-d x^2}}+\frac {(d e+c f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {a+b x^2} \sqrt {c-d x^2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.35 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.92 \[ \int \frac {e+f x^2}{x^2 \sqrt {a+b x^2} \sqrt {c-d x^2}} \, dx=\frac {\sqrt {\frac {b}{a}} \left (\sqrt {\frac {b}{a}} e \left (a+b x^2\right ) \left (-c+d x^2\right )-i b c e x \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )-i c (-b e+a f) x \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )\right )}{b c x \sqrt {a+b x^2} \sqrt {c-d x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.03, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.265, Rules used = {445, 25, 399, 323, 323, 321, 331, 330, 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.

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

\(\Big \downarrow \) 445

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 399

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 331

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

\(\Big \downarrow \) 330

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

\(\Big \downarrow \) 327

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

Input:

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

Output:

-((e*Sqrt[a + b*x^2]*Sqrt[c - d*x^2])/(a*c*x)) + (-((Sqrt[c]*Sqrt[d]*e*Sqr 
t[a + b*x^2]*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -( 
(b*c)/(a*d))])/(Sqrt[1 + (b*x^2)/a]*Sqrt[c - d*x^2])) + (a*Sqrt[c]*(d*e + 
c*f)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[d]*x)/ 
Sqrt[c]], -((b*c)/(a*d))])/(Sqrt[d]*Sqrt[a + b*x^2]*Sqrt[c - d*x^2]))/(a*c 
)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 323

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + ( 
d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] &&  !GtQ[c, 0]
 

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 330

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2]   Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^ 
2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &&  !GtQ[a, 
0]
 

rule 331

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[Sqrt[a + b*x^2]/Sqrt[1 + (d/c)*x^ 
2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] &&  !GtQ[c, 0]
 

rule 399

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) 
^2]), x_Symbol] :> Simp[f/b   Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + 
 Simp[(b*e - a*f)/b   Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr 
eeQ[{a, b, c, d, e, f}, x] &&  !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && 
(PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
 

rule 445

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

Maple [A] (verified)

Time = 6.32 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.27

method	result	size
default	\(\frac {\left (\sqrt {\frac {d}{c}}\, b d e \,x^{4}+\sqrt {\frac {-x^{2} d +c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) a c f x +\sqrt {\frac {-x^{2} d +c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) a d e x -\sqrt {\frac {-x^{2} d +c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) a d e x +\sqrt {\frac {d}{c}}\, a d e \,x^{2}-\sqrt {\frac {d}{c}}\, b c e \,x^{2}-\sqrt {\frac {d}{c}}\, a c e \right ) \sqrt {-x^{2} d +c}\, \sqrt {b \,x^{2}+a}}{\sqrt {\frac {d}{c}}\, x c a \left (-b d \,x^{4}-a d \,x^{2}+x^{2} b c +a c \right )}\)	\(282\)
risch	\(-\frac {e \sqrt {b \,x^{2}+a}\, \sqrt {-x^{2} d +c}}{a c x}+\frac {\left (\frac {d e a \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+x^{2} b c +a c}}+\frac {a c f \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (-x^{2} d +c \right ) \left (b \,x^{2}+a \right )}}{a c \sqrt {-x^{2} d +c}\, \sqrt {b \,x^{2}+a}}\)	\(295\)
elliptic	\(\frac {\sqrt {\left (-x^{2} d +c \right ) \left (b \,x^{2}+a \right )}\, \left (-\frac {e \sqrt {-b d \,x^{4}-a d \,x^{2}+x^{2} b c +a c}}{a c x}+\frac {f \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+x^{2} b c +a c}}+\frac {d e \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )\right )}{c \sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+x^{2} b c +a c}}\right )}{\sqrt {-x^{2} d +c}\, \sqrt {b \,x^{2}+a}}\)	\(295\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.56 \[ \int \frac {e+f x^2}{x^2 \sqrt {a+b x^2} \sqrt {c-d x^2}} \, dx=-\frac {\sqrt {a c} d^{2} e x \sqrt {\frac {d}{c}} E(\arcsin \left (x \sqrt {\frac {d}{c}}\right )\,|\,-\frac {b c}{a d}) + \sqrt {b x^{2} + a} \sqrt {-d x^{2} + c} c d e - {\left (d^{2} e + c^{2} f\right )} \sqrt {a c} x \sqrt {\frac {d}{c}} F(\arcsin \left (x \sqrt {\frac {d}{c}}\right )\,|\,-\frac {b c}{a d})}{a c^{2} d x} \] Input:

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

Output:

-(sqrt(a*c)*d^2*e*x*sqrt(d/c)*elliptic_e(arcsin(x*sqrt(d/c)), -b*c/(a*d)) 
+ sqrt(b*x^2 + a)*sqrt(-d*x^2 + c)*c*d*e - (d^2*e + c^2*f)*sqrt(a*c)*x*sqr 
t(d/c)*elliptic_f(arcsin(x*sqrt(d/c)), -b*c/(a*d)))/(a*c^2*d*x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {e+f x^2}{x^2 \sqrt {a+b x^2} \sqrt {c-d x^2}} \, dx=\left (\int \frac {\sqrt {-d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{-b d \,x^{6}-a d \,x^{4}+b c \,x^{4}+a c \,x^{2}}d x \right ) e +\left (\int \frac {\sqrt {-d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{-b d \,x^{4}-a d \,x^{2}+b c \,x^{2}+a c}d x \right ) f \] Input:

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

Output:

int((sqrt(c - d*x**2)*sqrt(a + b*x**2))/(a*c*x**2 - a*d*x**4 + b*c*x**4 - 
b*d*x**6),x)*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)*f