\(\int \sqrt {a c+(b c+a d) x^2+b d x^4} \, dx\) [14]

Optimal result

Integrand size = 25, antiderivative size = 282 \[ \int \sqrt {a c+(b c+a d) x^2+b d x^4} \, dx=\frac {(b c+a d) x \left (c+d x^2\right )}{3 d \sqrt {a c+(b c+a d) x^2+b d x^4}}+\frac {1}{3} x \sqrt {a c+(b c+a d) x^2+b d x^4}-\frac {\sqrt {a} (b c+a d) \left (c+d x^2\right ) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 \sqrt {b} d \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {a c+(b c+a d) x^2+b d x^4}}+\frac {2 a^{3/2} \left (c+d x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 \sqrt {b} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {a c+(b c+a d) x^2+b d x^4}} \] Output:

1/3*(a*d+b*c)*x*(d*x^2+c)/d/(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)-1/3*a^(1/2)*(a*d+b*c)*(d*x^2+c)*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/(a*(d*x^2+c)/ 
c/(b*x^2+a))^(1/2)/(a*c+(a*d+b*c)*x^2+b*d*x^4)^(1/2)+2/3*a^(3/2)*(d*x^2+c) 
*InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/b^(1/2)/(a*( 
d*x^2+c)/c/(b*x^2+a))^(1/2)/(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 = 2.39 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.69 \[ \int \sqrt {a c+(b c+a d) x^2+b d x^4} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right )-i c (b c+a d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c (-b c+a d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3 \sqrt {\frac {b}{a}} d \sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 514, normalized size of antiderivative = 1.82, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1404, 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.

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

\(\Big \downarrow \) 1404

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

\(\Big \downarrow \) 1511

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1416

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

\(\Big \downarrow \) 1509

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

Input:

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

Output:

(x*Sqrt[a*c + (b*c + a*d)*x^2 + b*d*x^4])/3 + (-(((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]*Ellipti 
cE[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)*(Sqrt[b]*Sqrt[c] + 
 Sqrt[a]*Sqrt[d])^2*(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]*Ellipt 
icF[2*ArcTan[(b^(1/4)*d^(1/4)*x)/(a^(1/4)*c^(1/4))], (2 - (b*c + a*d)/(Sqr 
t[a]*Sqrt[b]*Sqrt[c]*Sqrt[d]))/4])/(2*b^(3/4)*d^(3/4)*Sqrt[a*c + (b*c + a* 
d)*x^2 + b*d*x^4]))/3
 

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 1404

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

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 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]
 

Maple [A] (verified)

Time = 1.90 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.89

method	result	size
default	\(\frac {x \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}{3}+\frac {2 a c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{3 \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}-\frac {\left (\frac {a d}{3}+\frac {b c}{3}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}\, d}\)	\(250\)
elliptic	\(\frac {x \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}{3}+\frac {2 a c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{3 \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}-\frac {\left (\frac {a d}{3}+\frac {b c}{3}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}\, d}\)	\(250\)
risch	\(\frac {x \left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}{3 \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}-\frac {\left (a d +b c \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{3 \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}\, d}+\frac {2 a c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{3 \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}\)	\(255\)

Input:

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

Output:

1/3*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+2/3*a*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))-(1/3*a*d+1/3*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*(El 
lipticF(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 = 0.11 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.56 \[ \int \sqrt {a c+(b c+a d) x^2+b d x^4} \, dx=-\frac {{\left (b c^{2} + a c d\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (b c^{2} + a c d + 2 \, a d^{2}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - \sqrt {b d x^{4} + {\left (b c + a d\right )} x^{2} + a c} {\left (b d^{2} x^{2} + b c d + a d^{2}\right )}}{3 \, b d^{2} x} \] Input:

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

Output:

-1/3*((b*c^2 + a*c*d)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/ 
x), a*d/(b*c)) - (b*c^2 + a*c*d + 2*a*d^2)*sqrt(b*d)*x*sqrt(-c/d)*elliptic 
_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 + b*c*d + a*d^2))/(b*d^2*x)
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(c + d*x**2)*sqrt(a + b*x**2)*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 + 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 
 + 2*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