\(\int \frac {e+f x^2}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx\) [179]

Optimal result

Integrand size = 33, antiderivative size = 286 \[ \int \frac {e+f x^2}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {(2 b c e+2 a d e-3 a c f) \sqrt {c+d x^2}}{3 a c^2 x \sqrt {a+b x^2}}-\frac {e \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 a c x^3}+\frac {\sqrt {b} (2 b c e+2 a d e-3 a c f) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 a^{3/2} c^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {b} d e \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 \sqrt {a} c^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

1/3*(-3*a*c*f+2*a*d*e+2*b*c*e)*(d*x^2+c)^(1/2)/a/c^2/x/(b*x^2+a)^(1/2)-1/3 
*e*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a/c/x^3+1/3*b^(1/2)*(-3*a*c*f+2*a*d*e+2 
*b*c*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))/a^(3/2)/c^2/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2) 
-1/3*b^(1/2)*d*e*(d*x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)) 
,(1-a*d/b/c)^(1/2))/a^(1/2)/c^2/(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 = 4.04 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.90 \[ \int \frac {e+f x^2}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {-\sqrt {\frac {b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right ) \left (-2 b c e x^2-2 a d e x^2+a c \left (e+3 f x^2\right )\right )-i b c (-2 b c e-2 a d e+3 a c f) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i b c (-2 b c e-a d e+3 a c f) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3 a^2 \sqrt {\frac {b}{a}} c^2 x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

Integrate[(e + f*x^2)/(x^4*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.14, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {445, 445, 25, 27, 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.

\(\displaystyle \int \frac {e+f x^2}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx\)

\(\Big \downarrow \) 445

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

\(\Big \downarrow \) 445

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 406

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

\(\Big \downarrow \) 320

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

\(\Big \downarrow \) 388

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

\(\Big \downarrow \) 313

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

Input:

Int[(e + f*x^2)/(x^4*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]
 

Output:

-1/3*(e*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(a*c*x^3) - (-(((2*b*c*e + 2*a*d* 
e - 3*a*c*f)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(a*c*x)) + (b*d*((2*b*c*e + 
2*a*d*e - 3*a*c*f)*((x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqr 
t[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b*S 
qrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])) + (c^(3/2)* 
e*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])))/(a*c))/ 
(3*a*c)
 

Defintions of rubi rules used

rule 25

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

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 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 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]
 

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 = 7.23 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.16

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \left (3 a c f \,x^{2}-2 a d e \,x^{2}-2 b c e \,x^{2}+a c e \right )}{3 a^{2} c^{2} x^{3}}-\frac {b d \left (\frac {\left (3 a c f -2 a d e -2 b c e \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}+\frac {a c e \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{3 a^{2} c^{2} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(333\)
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {e \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 a c \,x^{3}}-\frac {\left (3 a c f -2 a d e -2 b c e \right ) \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 a^{2} c^{2} x}-\frac {b d e \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{3 a c \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (3 a c f -2 a d e -2 b c e \right ) b \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{3 a^{2} c \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(363\)
default	\(-\frac {\left (3 \sqrt {-\frac {b}{a}}\, a b c d f \,x^{6}-2 \sqrt {-\frac {b}{a}}\, a b \,d^{2} e \,x^{6}-2 \sqrt {-\frac {b}{a}}\, b^{2} c d e \,x^{6}+3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} f \,x^{3}-\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c d e \,x^{3}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} e \,x^{3}-3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} f \,x^{3}+2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c d e \,x^{3}+2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} e \,x^{3}+3 \sqrt {-\frac {b}{a}}\, a^{2} c d f \,x^{4}-2 \sqrt {-\frac {b}{a}}\, a^{2} d^{2} e \,x^{4}+3 \sqrt {-\frac {b}{a}}\, a b \,c^{2} f \,x^{4}-3 \sqrt {-\frac {b}{a}}\, a b c d e \,x^{4}-2 \sqrt {-\frac {b}{a}}\, b^{2} c^{2} e \,x^{4}+3 \sqrt {-\frac {b}{a}}\, a^{2} c^{2} f \,x^{2}-\sqrt {-\frac {b}{a}}\, a^{2} c d e \,x^{2}-\sqrt {-\frac {b}{a}}\, a b \,c^{2} e \,x^{2}+\sqrt {-\frac {b}{a}}\, a^{2} c^{2} e \right ) \sqrt {x^{2} d +c}\, \sqrt {b \,x^{2}+a}}{3 \sqrt {-\frac {b}{a}}\, x^{3} c^{2} a^{2} \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right )}\)	\(643\)

Input:

int((f*x^2+e)/x^4/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.66 \[ \int \frac {e+f x^2}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {{\left (3 \, a b c f - 2 \, {\left (b^{2} c + a b d\right )} e\right )} \sqrt {a c} x^{3} \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left (3 \, a b c f - {\left (2 \, b^{2} c + {\left (a^{2} + 2 \, a b\right )} d\right )} e\right )} \sqrt {a c} x^{3} \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left (a^{2} c e + {\left (3 \, a^{2} c f - 2 \, {\left (a b c + a^{2} d\right )} e\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{3 \, a^{3} c^{2} x^{3}} \] Input:

integrate((f*x^2+e)/x^4/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="fric 
as")
 

Output:

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

Sympy [F]

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

integrate((f*x**2+e)/x**4/(b*x**2+a)**(1/2)/(d*x**2+c)**(1/2),x)
 

Output:

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

Maxima [F]

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

integrate((f*x^2+e)/x^4/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="maxi 
ma")
 

Output:

integrate((f*x^2 + e)/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*x^4), x)
 

Giac [F]

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

integrate((f*x^2+e)/x^4/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="giac 
")
 

Output:

integrate((f*x^2 + e)/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*x^4), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e+f x^2}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int \frac {f\,x^2+e}{x^4\,\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}} \,d x \] Input:

int((e + f*x^2)/(x^4*(a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)),x)
 

Output:

int((e + f*x^2)/(x^4*(a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {e+f x^2}{x^4 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {-\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, f +\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b d f x +\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{8}+a d \,x^{6}+b c \,x^{6}+a c \,x^{4}}d x \right ) a c e x}{a c x} \] Input:

int((f*x^2+e)/x^4/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x)
 

Output:

( - sqrt(c + d*x**2)*sqrt(a + b*x**2)*f + 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*f*x + int((sqrt 
(c + d*x**2)*sqrt(a + b*x**2))/(a*c*x**4 + a*d*x**6 + b*c*x**6 + b*d*x**8) 
,x)*a*c*e*x)/(a*c*x)