\(\int \sqrt {d+e x^2} \sqrt {a-c x^4} \, dx\) [437]

Optimal result

Integrand size = 24, antiderivative size = 499 \[ \int \sqrt {d+e x^2} \sqrt {a-c x^4} \, dx=\frac {d \sqrt {d+e x^2} \sqrt {a-c x^4}}{8 e x}+\frac {1}{4} x \sqrt {d+e x^2} \sqrt {a-c x^4}+\frac {c d \left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{8 e \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {5 \sqrt {a} \sqrt {c} d \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{8 \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\left (c d^2+4 a e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{8 e \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

1/8*d*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/e/x+1/4*x*(e*x^2+d)^(1/2)*(-c*x^4+a 
)^(1/2)+1/8*c*d*(d+a^(1/2)*e/c^(1/2))*(1-a/c/x^4)^(1/2)*x^3*(a^(1/2)*(e*x^ 
2+d)/(c^(1/2)*d+a^(1/2)*e)/x^2)^(1/2)*EllipticE(1/2*(1-a^(1/2)/c^(1/2)/x^2 
)^(1/2)*2^(1/2),2^(1/2)*(d/(d+a^(1/2)*e/c^(1/2)))^(1/2))/e/(e*x^2+d)^(1/2) 
/(-c*x^4+a)^(1/2)+5/8*a^(1/2)*c^(1/2)*d*(1-a/c/x^4)^(1/2)*x^3*(a^(1/2)*(e* 
x^2+d)/(c^(1/2)*d+a^(1/2)*e)/x^2)^(1/2)*EllipticF(1/2*(1-a^(1/2)/c^(1/2)/x 
^2)^(1/2)*2^(1/2),2^(1/2)*(d/(d+a^(1/2)*e/c^(1/2)))^(1/2))/(e*x^2+d)^(1/2) 
/(-c*x^4+a)^(1/2)+1/8*(4*a*e^2+c*d^2)*(1-a/c/x^4)^(1/2)*x^3*(a^(1/2)*(e*x^ 
2+d)/(c^(1/2)*d+a^(1/2)*e)/x^2)^(1/2)*EllipticPi(1/2*(1-a^(1/2)/c^(1/2)/x^ 
2)^(1/2)*2^(1/2),2,2^(1/2)*(d/(d+a^(1/2)*e/c^(1/2)))^(1/2))/e/(e*x^2+d)^(1 
/2)/(-c*x^4+a)^(1/2)
 

Mathematica [F]

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

Integrate[Sqrt[d + e*x^2]*Sqrt[a - c*x^4],x]
 

Output:

Integrate[Sqrt[d + e*x^2]*Sqrt[a - c*x^4], x]
 

Rubi [F]

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.

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

\(\Big \downarrow \) 1571

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

Input:

Int[Sqrt[d + e*x^2]*Sqrt[a - c*x^4],x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 1571

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> U 
nintegrable[(d + e*x^2)^q*(a + c*x^4)^p, x] /; FreeQ[{a, c, d, e, p, q}, x]
 

Maple [F]

\[\int \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(sqrt(-c*x^4 + a)*sqrt(e*x^2 + d), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \sqrt {d+e x^2} \sqrt {a-c x^4} \, dx=\frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x}{4}-\frac {\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) c d}{4}+\frac {\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a e}{2}+\frac {3 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a d}{4} \] Input:

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

Output:

(sqrt(d + e*x**2)*sqrt(a - c*x**4)*x - int((sqrt(d + e*x**2)*sqrt(a - c*x* 
*4)*x**4)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*c*d + 2*int((sqrt(d + 
e*x**2)*sqrt(a - c*x**4)*x**2)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a 
*e + 3*int((sqrt(d + e*x**2)*sqrt(a - c*x**4))/(a*d + a*e*x**2 - c*d*x**4 
- c*e*x**6),x)*a*d)/4