\(\int \frac {x^4}{\sqrt {(a+b x^2) (c+d x^2)}} \, dx\) [702]

Optimal result

Integrand size = 23, antiderivative size = 296 \[ \int \frac {x^4}{\sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \, dx=-\frac {2 (b c+a d) x \left (c+d x^2\right )}{3 b d^2 \sqrt {a c+(b c+a d) x^2+b d x^4}}+\frac {x \sqrt {a c+(b c+a d) x^2+b d x^4}}{3 b d}+\frac {2 c (b c+a d) \left (a+b x^2\right ) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 \sqrt {a} b^{3/2} d^2 \sqrt {a c+(b c+a d) x^2+b d x^4}}-\frac {\sqrt {a} c \left (a+b x^2\right ) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 b^{3/2} d \sqrt {a c+(b c+a d) x^2+b d x^4}} \] Output:

-2/3*(a*d+b*c)*x*(d*x^2+c)/b/d^2/(a*c+(a*d+b*c)*x^2+b*d*x^4)^(1/2)+1/3*x*( 
a*c+(a*d+b*c)*x^2+b*d*x^4)^(1/2)/b/d+2/3*c*(a*d+b*c)*(b*x^2+a)*(a*(d*x^2+c 
)/c/(b*x^2+a))^(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^(1/2)/b^(3/2)/d^2/(a*c+(a*d+b*c)*x^2+b*d*x^4)^(1/2)-1/3*a^(1 
/2)*c*(b*x^2+a)*(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)*InverseJacobiAM(arctan(b^( 
1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/b^(3/2)/d/(a*c+(a*d+b*c)*x^2+b*d*x^4)^( 
1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.54 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.67 \[ \int \frac {x^4}{\sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right )+2 i c (b c+a d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c (2 b c+a d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3 b \sqrt {\frac {b}{a}} d^2 \sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \] Input:

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

Output:

(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2) + (2*I)*c*(b*c + a*d)*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*(2*b*c + a*d)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*Ar 
cSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(3*b*Sqrt[b/a]*d^2*Sqrt[(a + b*x^2)*(c + 
 d*x^2)])
 

Rubi [A] (verified)

Time = 1.03 (sec) , antiderivative size = 531, normalized size of antiderivative = 1.79, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2048, 1442, 1511, 27, 1416, 1509}

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.

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

\(\Big \downarrow \) 2048

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

\(\Big \downarrow \) 1442

\(\displaystyle \frac {x \sqrt {x^2 (a d+b c)+a c+b d x^4}}{3 b d}-\frac {\int \frac {2 (b c+a d) x^2+a c}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{3 b d}\)

\(\Big \downarrow \) 1511

\(\displaystyle \frac {x \sqrt {x^2 (a d+b c)+a c+b d x^4}}{3 b d}-\frac {\frac {\sqrt {a} \sqrt {c} \left (\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}+2 a d+2 b c\right ) \int \frac {1}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}-\frac {2 \sqrt {a} \sqrt {c} (a d+b c) \int \frac {\sqrt {a} \sqrt {c}-\sqrt {b} \sqrt {d} x^2}{\sqrt {a} \sqrt {c} \sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}}{3 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x \sqrt {x^2 (a d+b c)+a c+b d x^4}}{3 b d}-\frac {\frac {\sqrt {a} \sqrt {c} \left (\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}+2 a d+2 b c\right ) \int \frac {1}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}-\frac {2 (a d+b c) \int \frac {\sqrt {a} \sqrt {c}-\sqrt {b} \sqrt {d} x^2}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}}{3 b d}\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {x \sqrt {x^2 (a d+b c)+a c+b d x^4}}{3 b d}-\frac {\frac {\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}+2 a d+2 b c\right ) \left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right ) \sqrt {\frac {x^2 (a d+b c)+a c+b d x^4}{\left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{d} x}{\sqrt [4]{a} \sqrt [4]{c}}\right ),\frac {1}{4} \left (2-\frac {b c+a d}{\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}}\right )\right )}{2 b^{3/4} d^{3/4} \sqrt {x^2 (a d+b c)+a c+b d x^4}}-\frac {2 (a d+b c) \int \frac {\sqrt {a} \sqrt {c}-\sqrt {b} \sqrt {d} x^2}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}}{3 b d}\)

\(\Big \downarrow \) 1509

\(\displaystyle \frac {x \sqrt {x^2 (a d+b c)+a c+b d x^4}}{3 b d}-\frac {\frac {\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}+2 a d+2 b c\right ) \left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right ) \sqrt {\frac {x^2 (a d+b c)+a c+b d x^4}{\left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{d} x}{\sqrt [4]{a} \sqrt [4]{c}}\right ),\frac {1}{4} \left (2-\frac {b c+a d}{\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}}\right )\right )}{2 b^{3/4} d^{3/4} \sqrt {x^2 (a d+b c)+a c+b d x^4}}-\frac {2 (a d+b c) \left (\frac {\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right ) \sqrt {\frac {x^2 (a d+b c)+a c+b d x^4}{\left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{d} x}{\sqrt [4]{a} \sqrt [4]{c}}\right )|\frac {1}{4} \left (2-\frac {b c+a d}{\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}}\right )\right )}{\sqrt [4]{b} \sqrt [4]{d} \sqrt {x^2 (a d+b c)+a c+b d x^4}}-\frac {x \sqrt {x^2 (a d+b c)+a c+b d x^4}}{\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2}\right )}{\sqrt {b} \sqrt {d}}}{3 b d}\)

Input:

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

Output:

(x*Sqrt[a*c + (b*c + a*d)*x^2 + b*d*x^4])/(3*b*d) - ((-2*(b*c + a*d)*(-((x 
*Sqrt[a*c + (b*c + a*d)*x^2 + b*d*x^4])/(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[d] 
*x^2)) + (a^(1/4)*c^(1/4)*(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[d]*x^2)*Sqrt[(a* 
c + (b*c + a*d)*x^2 + b*d*x^4)/(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[d]*x^2)^2]* 
EllipticE[2*ArcTan[(b^(1/4)*d^(1/4)*x)/(a^(1/4)*c^(1/4))], (2 - (b*c + a*d 
)/(Sqrt[a]*Sqrt[b]*Sqrt[c]*Sqrt[d]))/4])/(b^(1/4)*d^(1/4)*Sqrt[a*c + (b*c 
+ a*d)*x^2 + b*d*x^4])))/(Sqrt[b]*Sqrt[d]) + (a^(1/4)*c^(1/4)*(2*b*c + Sqr 
t[a]*Sqrt[b]*Sqrt[c]*Sqrt[d] + 2*a*d)*(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[d]*x 
^2)*Sqrt[(a*c + (b*c + a*d)*x^2 + b*d*x^4)/(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt 
[d]*x^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*d^(1/4)*x)/(a^(1/4)*c^(1/4))], (2 
- (b*c + a*d)/(Sqrt[a]*Sqrt[b]*Sqrt[c]*Sqrt[d]))/4])/(2*b^(3/4)*d^(3/4)*Sq 
rt[a*c + (b*c + a*d)*x^2 + b*d*x^4]))/(3*b*d)
 

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 1416

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c 
/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

rule 1442

Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] 
:> Simp[d^3*(d*x)^(m - 3)*((a + b*x^2 + c*x^4)^(p + 1)/(c*(m + 4*p + 1))), 
x] - Simp[d^4/(c*(m + 4*p + 1))   Int[(d*x)^(m - 4)*Simp[a*(m - 3) + b*(m + 
 2*p - 1)*x^2, x]*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x 
] && NeQ[b^2 - 4*a*c, 0] && GtQ[m, 3] && NeQ[m + 4*p + 1, 0] && IntegerQ[2* 
p] && (IntegerQ[p] || IntegerQ[m])
 

rule 1509

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q 
^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* 
x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 
/(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 
- 4*a*c, 0] && PosQ[c/a]
 

rule 1511

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q   Int[1/Sqrt[a + b*x^2 + c*x^ 
4], x], x] - Simp[e/q   Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; 
NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos 
Q[c/a]
 

rule 2048

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))*((c_) + (d_.)*(x_)^(n_.)))^(p_) 
, x_Symbol] :> Int[u*(a*c*e + (b*c + a*d)*e*x^n + b*d*e*x^(2*n))^p, x] /; F 
reeQ[{a, b, c, d, e, n, p}, x]
 

Maple [A] (verified)

Time = 3.93 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.90

method	result	size
default	\(\frac {x \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}{3 b d}-\frac {a c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{3 b d \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}+\frac {\left (2 a d +2 b c \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{3 b \,d^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}\)	\(265\)
elliptic	\(\frac {x \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}{3 b d}-\frac {a c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{3 b d \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}+\frac {\left (2 a d +2 b c \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{3 b \,d^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}\)	\(265\)
risch	\(\frac {x \left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}{3 b d \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}-\frac {\frac {a c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}-\frac {\left (2 a d +2 b c \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}\, d}}{3 b d}\)	\(271\)

Input:

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

Output:

1/3/b/d*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)-1/3*a*c/b/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)*Ellip 
ticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))+1/3/b/d^2*(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)*(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 = 0.07 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.54 \[ \int \frac {x^4}{\sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \, dx=\frac {2 \, {\left (b c^{2} + a c d\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (2 \, b c^{2} + 2 \, a c d + a d^{2}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) + \sqrt {b d x^{4} + {\left (b c + a d\right )} x^{2} + a c} {\left (b d^{2} x^{2} - 2 \, b c d - 2 \, a d^{2}\right )}}{3 \, b^{2} d^{3} x} \] Input:

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

Output:

1/3*(2*(b*c^2 + a*c*d)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d) 
/x), a*d/(b*c)) - (2*b*c^2 + 2*a*c*d + a*d^2)*sqrt(b*d)*x*sqrt(-c/d)*ellip 
tic_f(arcsin(sqrt(-c/d)/x), a*d/(b*c)) + sqrt(b*d*x^4 + (b*c + a*d)*x^2 + 
a*c)*(b*d^2*x^2 - 2*b*c*d - 2*a*d^2))/(b^2*d^3*x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(c + d*x**2)*sqrt(a + b*x**2)*x - 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)*a*d - 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 - 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)/(3*b*d)