\(\int \frac {\sqrt {a+b x^2-c x^4}}{a d+c d x^4} \, dx\) [283]

Optimal result

Integrand size = 30, antiderivative size = 239 \[ \int \frac {\sqrt {a+b x^2-c x^4}}{a d+c d x^4} \, dx=-\frac {\sqrt {b+\sqrt {b^2+4 a c}} \arctan \left (\frac {\sqrt {b+\sqrt {b^2+4 a c}} x \left (b-\sqrt {b^2+4 a c}-2 c x^2\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} \sqrt {a+b x^2-c x^4}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} d}+\frac {\sqrt {-b+\sqrt {b^2+4 a c}} \text {arctanh}\left (\frac {\sqrt {-b+\sqrt {b^2+4 a c}} x \left (b+\sqrt {b^2+4 a c}-2 c x^2\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} \sqrt {a+b x^2-c x^4}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} d} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.84 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.63 \[ \int \frac {\sqrt {a+b x^2-c x^4}}{a d+c d x^4} \, dx=\frac {i \left (\sqrt {-b-2 i \sqrt {a} \sqrt {c}} \arctan \left (\frac {\sqrt {-b-2 i \sqrt {a} \sqrt {c}} x}{\sqrt {a+b x^2-c x^4}}\right )-\sqrt {-b+2 i \sqrt {a} \sqrt {c}} \arctan \left (\frac {\sqrt {-b+2 i \sqrt {a} \sqrt {c}} x}{\sqrt {a+b x^2-c x^4}}\right )\right )}{4 \sqrt {a} \sqrt {c} d} \] Input:

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

Output:

((I/4)*(Sqrt[-b - (2*I)*Sqrt[a]*Sqrt[c]]*ArcTan[(Sqrt[-b - (2*I)*Sqrt[a]*S 
qrt[c]]*x)/Sqrt[a + b*x^2 - c*x^4]] - Sqrt[-b + (2*I)*Sqrt[a]*Sqrt[c]]*Arc 
Tan[(Sqrt[-b + (2*I)*Sqrt[a]*Sqrt[c]]*x)/Sqrt[a + b*x^2 - c*x^4]]))/(Sqrt[ 
a]*Sqrt[c]*d)
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {2518}

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.

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

\(\Big \downarrow \) 2518

\(\displaystyle \frac {\sqrt {\sqrt {4 a c+b^2}-b} \text {arctanh}\left (\frac {x \sqrt {\sqrt {4 a c+b^2}-b} \left (\sqrt {4 a c+b^2}+b-2 c x^2\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} \sqrt {a+b x^2-c x^4}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} d}-\frac {\sqrt {\sqrt {4 a c+b^2}+b} \arctan \left (\frac {x \sqrt {\sqrt {4 a c+b^2}+b} \left (-\sqrt {4 a c+b^2}+b-2 c x^2\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} \sqrt {a+b x^2-c x^4}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {c} d}\)

Input:

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

Output:

-1/2*(Sqrt[b + Sqrt[b^2 + 4*a*c]]*ArcTan[(Sqrt[b + Sqrt[b^2 + 4*a*c]]*x*(b 
 - Sqrt[b^2 + 4*a*c] - 2*c*x^2))/(2*Sqrt[2]*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x^2 
 - c*x^4])])/(Sqrt[2]*Sqrt[a]*Sqrt[c]*d) + (Sqrt[-b + Sqrt[b^2 + 4*a*c]]*A 
rcTanh[(Sqrt[-b + Sqrt[b^2 + 4*a*c]]*x*(b + Sqrt[b^2 + 4*a*c] - 2*c*x^2))/ 
(2*Sqrt[2]*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x^2 - c*x^4])])/(2*Sqrt[2]*Sqrt[a]*S 
qrt[c]*d)
 

Defintions of rubi rules used

rule 2518

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

Maple [A] (verified)

Time = 1.84 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.47

method	result	size
pseudoelliptic	\(\frac {\sqrt {2 \sqrt {4 a c +b^{2}}+2 b}\, \left (b -\sqrt {4 a c +b^{2}}\right ) \left (\ln \left (\frac {c \,x^{4}+\sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2 \sqrt {4 a c +b^{2}}+2 b}\, x -\sqrt {4 a c +b^{2}}\, x^{2}-b \,x^{2}-a}{x^{2}}\right )-\ln \left (\frac {-c \,x^{4}+\sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2 \sqrt {4 a c +b^{2}}+2 b}\, x +\sqrt {4 a c +b^{2}}\, x^{2}+b \,x^{2}+a}{x^{2}}\right )\right ) \sqrt {2 \sqrt {4 a c +b^{2}}-2 b}+16 a c \left (\arctan \left (\frac {\sqrt {2 \sqrt {4 a c +b^{2}}+2 b}\, x -2 \sqrt {-c \,x^{4}+b \,x^{2}+a}}{x \sqrt {2 \sqrt {4 a c +b^{2}}-2 b}}\right )-\arctan \left (\frac {\sqrt {2 \sqrt {4 a c +b^{2}}+2 b}\, x +2 \sqrt {-c \,x^{4}+b \,x^{2}+a}}{x \sqrt {2 \sqrt {4 a c +b^{2}}-2 b}}\right )\right )}{32 \sqrt {2 \sqrt {4 a c +b^{2}}-2 b}\, a c d}\)	\(351\)
elliptic	\(\frac {\left (\frac {\sqrt {b +\sqrt {4 a c +b^{2}}}\, b \ln \left (\frac {\sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}\, \sqrt {b +\sqrt {4 a c +b^{2}}}}{x}-\frac {-c \,x^{4}+b \,x^{2}+a}{x^{2}}-\sqrt {4 a c +b^{2}}\right )}{16 d a c}-\frac {b^{2} \arctan \left (\frac {2 \sqrt {b +\sqrt {4 a c +b^{2}}}-\frac {2 \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}}{x}}{2 \sqrt {-b +\sqrt {4 a c +b^{2}}}}\right )}{8 d a c \sqrt {-b +\sqrt {4 a c +b^{2}}}}-\frac {\sqrt {b +\sqrt {4 a c +b^{2}}}\, \sqrt {4 a c +b^{2}}\, \ln \left (\frac {\sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}\, \sqrt {b +\sqrt {4 a c +b^{2}}}}{x}-\frac {-c \,x^{4}+b \,x^{2}+a}{x^{2}}-\sqrt {4 a c +b^{2}}\right )}{16 d a c}+\frac {\left (4 a c +b^{2}\right ) \arctan \left (\frac {2 \sqrt {b +\sqrt {4 a c +b^{2}}}-\frac {2 \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}}{x}}{2 \sqrt {-b +\sqrt {4 a c +b^{2}}}}\right )}{8 d a c \sqrt {-b +\sqrt {4 a c +b^{2}}}}-\frac {\sqrt {b +\sqrt {4 a c +b^{2}}}\, b \ln \left (\frac {-c \,x^{4}+b \,x^{2}+a}{x^{2}}+\frac {\sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}\, \sqrt {b +\sqrt {4 a c +b^{2}}}}{x}+\sqrt {4 a c +b^{2}}\right )}{16 d a c}+\frac {b^{2} \arctan \left (\frac {\frac {2 \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}}{x}+2 \sqrt {b +\sqrt {4 a c +b^{2}}}}{2 \sqrt {-b +\sqrt {4 a c +b^{2}}}}\right )}{8 d a c \sqrt {-b +\sqrt {4 a c +b^{2}}}}+\frac {\sqrt {b +\sqrt {4 a c +b^{2}}}\, \sqrt {4 a c +b^{2}}\, \ln \left (\frac {-c \,x^{4}+b \,x^{2}+a}{x^{2}}+\frac {\sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}\, \sqrt {b +\sqrt {4 a c +b^{2}}}}{x}+\sqrt {4 a c +b^{2}}\right )}{16 d a c}-\frac {\left (4 a c +b^{2}\right ) \arctan \left (\frac {\frac {2 \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}}{x}+2 \sqrt {b +\sqrt {4 a c +b^{2}}}}{2 \sqrt {-b +\sqrt {4 a c +b^{2}}}}\right )}{8 d a c \sqrt {-b +\sqrt {4 a c +b^{2}}}}\right ) \sqrt {2}}{2}\)	\(761\)
default	\(\frac {\left (\frac {\sqrt {b +\sqrt {4 a c +b^{2}}}\, b \ln \left (\frac {\sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}\, \sqrt {b +\sqrt {4 a c +b^{2}}}}{x}-\frac {-c \,x^{4}+b \,x^{2}+a}{x^{2}}-\sqrt {4 a c +b^{2}}\right )}{16 a c}-\frac {\left (b +\sqrt {4 a c +b^{2}}\right ) b \arctan \left (\frac {2 \sqrt {b +\sqrt {4 a c +b^{2}}}-\frac {2 \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}}{x}}{2 \sqrt {-b +\sqrt {4 a c +b^{2}}}}\right )}{8 a c \sqrt {-b +\sqrt {4 a c +b^{2}}}}-\frac {\sqrt {b +\sqrt {4 a c +b^{2}}}\, \sqrt {4 a c +b^{2}}\, \ln \left (\frac {\sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}\, \sqrt {b +\sqrt {4 a c +b^{2}}}}{x}-\frac {-c \,x^{4}+b \,x^{2}+a}{x^{2}}-\sqrt {4 a c +b^{2}}\right )}{16 a c}+\frac {\left (b +\sqrt {4 a c +b^{2}}\right ) \sqrt {4 a c +b^{2}}\, \arctan \left (\frac {2 \sqrt {b +\sqrt {4 a c +b^{2}}}-\frac {2 \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}}{x}}{2 \sqrt {-b +\sqrt {4 a c +b^{2}}}}\right )}{8 a c \sqrt {-b +\sqrt {4 a c +b^{2}}}}-\frac {\sqrt {b +\sqrt {4 a c +b^{2}}}\, b \ln \left (\frac {-c \,x^{4}+b \,x^{2}+a}{x^{2}}+\frac {\sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}\, \sqrt {b +\sqrt {4 a c +b^{2}}}}{x}+\sqrt {4 a c +b^{2}}\right )}{16 a c}+\frac {\left (b +\sqrt {4 a c +b^{2}}\right ) b \arctan \left (\frac {\frac {2 \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}}{x}+2 \sqrt {b +\sqrt {4 a c +b^{2}}}}{2 \sqrt {-b +\sqrt {4 a c +b^{2}}}}\right )}{8 a c \sqrt {-b +\sqrt {4 a c +b^{2}}}}+\frac {\sqrt {b +\sqrt {4 a c +b^{2}}}\, \sqrt {4 a c +b^{2}}\, \ln \left (\frac {-c \,x^{4}+b \,x^{2}+a}{x^{2}}+\frac {\sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}\, \sqrt {b +\sqrt {4 a c +b^{2}}}}{x}+\sqrt {4 a c +b^{2}}\right )}{16 a c}-\frac {\left (b +\sqrt {4 a c +b^{2}}\right ) \sqrt {4 a c +b^{2}}\, \arctan \left (\frac {\frac {2 \sqrt {-c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}}{x}+2 \sqrt {b +\sqrt {4 a c +b^{2}}}}{2 \sqrt {-b +\sqrt {4 a c +b^{2}}}}\right )}{8 a c \sqrt {-b +\sqrt {4 a c +b^{2}}}}\right ) \sqrt {2}}{2 d}\)	\(788\)

Input:

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

Output:

1/32/(2*(4*a*c+b^2)^(1/2)-2*b)^(1/2)*((2*(4*a*c+b^2)^(1/2)+2*b)^(1/2)*(b-( 
4*a*c+b^2)^(1/2))*(ln((c*x^4+(-c*x^4+b*x^2+a)^(1/2)*(2*(4*a*c+b^2)^(1/2)+2 
*b)^(1/2)*x-(4*a*c+b^2)^(1/2)*x^2-b*x^2-a)/x^2)-ln((-c*x^4+(-c*x^4+b*x^2+a 
)^(1/2)*(2*(4*a*c+b^2)^(1/2)+2*b)^(1/2)*x+(4*a*c+b^2)^(1/2)*x^2+b*x^2+a)/x 
^2))*(2*(4*a*c+b^2)^(1/2)-2*b)^(1/2)+16*a*c*(arctan(((2*(4*a*c+b^2)^(1/2)+ 
2*b)^(1/2)*x-2*(-c*x^4+b*x^2+a)^(1/2))/x/(2*(4*a*c+b^2)^(1/2)-2*b)^(1/2))- 
arctan(((2*(4*a*c+b^2)^(1/2)+2*b)^(1/2)*x+2*(-c*x^4+b*x^2+a)^(1/2))/x/(2*( 
4*a*c+b^2)^(1/2)-2*b)^(1/2))))/a/c/d
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 669 vs. \(2 (187) = 374\).

Time = 1.43 (sec) , antiderivative size = 669, normalized size of antiderivative = 2.80 \[ \int \frac {\sqrt {a+b x^2-c x^4}}{a d+c d x^4} \, dx=-\frac {1}{8} \, \sqrt {\frac {2 \, a c d^{2} \sqrt {-\frac {1}{a c d^{4}}} - b}{a c d^{2}}} \log \left (-\frac {\sqrt {-c x^{4} + b x^{2} + a} a d^{2} \sqrt {-\frac {1}{a c d^{4}}} + \sqrt {-c x^{4} + b x^{2} + a} x^{2} + {\left (a c d^{3} x^{3} \sqrt {-\frac {1}{a c d^{4}}} - a d x\right )} \sqrt {\frac {2 \, a c d^{2} \sqrt {-\frac {1}{a c d^{4}}} - b}{a c d^{2}}}}{c x^{4} + a}\right ) + \frac {1}{8} \, \sqrt {\frac {2 \, a c d^{2} \sqrt {-\frac {1}{a c d^{4}}} - b}{a c d^{2}}} \log \left (-\frac {\sqrt {-c x^{4} + b x^{2} + a} a d^{2} \sqrt {-\frac {1}{a c d^{4}}} + \sqrt {-c x^{4} + b x^{2} + a} x^{2} - {\left (a c d^{3} x^{3} \sqrt {-\frac {1}{a c d^{4}}} - a d x\right )} \sqrt {\frac {2 \, a c d^{2} \sqrt {-\frac {1}{a c d^{4}}} - b}{a c d^{2}}}}{c x^{4} + a}\right ) - \frac {1}{8} \, \sqrt {-\frac {2 \, a c d^{2} \sqrt {-\frac {1}{a c d^{4}}} + b}{a c d^{2}}} \log \left (\frac {\sqrt {-c x^{4} + b x^{2} + a} a d^{2} \sqrt {-\frac {1}{a c d^{4}}} - \sqrt {-c x^{4} + b x^{2} + a} x^{2} + {\left (a c d^{3} x^{3} \sqrt {-\frac {1}{a c d^{4}}} + a d x\right )} \sqrt {-\frac {2 \, a c d^{2} \sqrt {-\frac {1}{a c d^{4}}} + b}{a c d^{2}}}}{c x^{4} + a}\right ) + \frac {1}{8} \, \sqrt {-\frac {2 \, a c d^{2} \sqrt {-\frac {1}{a c d^{4}}} + b}{a c d^{2}}} \log \left (\frac {\sqrt {-c x^{4} + b x^{2} + a} a d^{2} \sqrt {-\frac {1}{a c d^{4}}} - \sqrt {-c x^{4} + b x^{2} + a} x^{2} - {\left (a c d^{3} x^{3} \sqrt {-\frac {1}{a c d^{4}}} + a d x\right )} \sqrt {-\frac {2 \, a c d^{2} \sqrt {-\frac {1}{a c d^{4}}} + b}{a c d^{2}}}}{c x^{4} + a}\right ) \] Input:

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

Output:

-1/8*sqrt((2*a*c*d^2*sqrt(-1/(a*c*d^4)) - b)/(a*c*d^2))*log(-(sqrt(-c*x^4 
+ b*x^2 + a)*a*d^2*sqrt(-1/(a*c*d^4)) + sqrt(-c*x^4 + b*x^2 + a)*x^2 + (a* 
c*d^3*x^3*sqrt(-1/(a*c*d^4)) - a*d*x)*sqrt((2*a*c*d^2*sqrt(-1/(a*c*d^4)) - 
 b)/(a*c*d^2)))/(c*x^4 + a)) + 1/8*sqrt((2*a*c*d^2*sqrt(-1/(a*c*d^4)) - b) 
/(a*c*d^2))*log(-(sqrt(-c*x^4 + b*x^2 + a)*a*d^2*sqrt(-1/(a*c*d^4)) + sqrt 
(-c*x^4 + b*x^2 + a)*x^2 - (a*c*d^3*x^3*sqrt(-1/(a*c*d^4)) - a*d*x)*sqrt(( 
2*a*c*d^2*sqrt(-1/(a*c*d^4)) - b)/(a*c*d^2)))/(c*x^4 + a)) - 1/8*sqrt(-(2* 
a*c*d^2*sqrt(-1/(a*c*d^4)) + b)/(a*c*d^2))*log((sqrt(-c*x^4 + b*x^2 + a)*a 
*d^2*sqrt(-1/(a*c*d^4)) - sqrt(-c*x^4 + b*x^2 + a)*x^2 + (a*c*d^3*x^3*sqrt 
(-1/(a*c*d^4)) + a*d*x)*sqrt(-(2*a*c*d^2*sqrt(-1/(a*c*d^4)) + b)/(a*c*d^2) 
))/(c*x^4 + a)) + 1/8*sqrt(-(2*a*c*d^2*sqrt(-1/(a*c*d^4)) + b)/(a*c*d^2))* 
log((sqrt(-c*x^4 + b*x^2 + a)*a*d^2*sqrt(-1/(a*c*d^4)) - sqrt(-c*x^4 + b*x 
^2 + a)*x^2 - (a*c*d^3*x^3*sqrt(-1/(a*c*d^4)) + a*d*x)*sqrt(-(2*a*c*d^2*sq 
rt(-1/(a*c*d^4)) + b)/(a*c*d^2)))/(c*x^4 + a))
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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