\(\int (A+B x^2) \sqrt {d+e x^2} \sqrt {a-c x^4} \, dx\) [36]

Optimal result

Integrand size = 31, antiderivative size = 623 \[ \int \left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4} \, dx=-\frac {\left (3 B c d^2-6 A c d e+8 a B e^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{48 c e^2 x}+\frac {(B d+6 A e) x \sqrt {d+e x^2} \sqrt {a-c x^4}}{24 e}+\frac {1}{6} B x^3 \sqrt {d+e x^2} \sqrt {a-c x^4}-\frac {\left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (3 B c d^2-6 A c d e+8 a B e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{48 e^2 \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\sqrt {a} \left (B c d^2+30 A c d e+8 a B e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{48 \sqrt {c} e \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {\left (B c d^3-2 A c d^2 e-4 a B d e^2-8 a A e^3\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{16 e^2 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

-1/48*(-6*A*c*d*e+8*B*a*e^2+3*B*c*d^2)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/c/ 
e^2/x+1/24*(6*A*e+B*d)*x*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/e+1/6*B*x^3*(e*x 
^2+d)^(1/2)*(-c*x^4+a)^(1/2)-1/48*(d+a^(1/2)*e/c^(1/2))*(-6*A*c*d*e+8*B*a* 
e^2+3*B*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)*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^2/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)+1/4 
8*a^(1/2)*(30*A*c*d*e+8*B*a*e^2+B*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)*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))/c^(1/2)/e/(e*x 
^2+d)^(1/2)/(-c*x^4+a)^(1/2)-1/16*(-8*A*a*e^3-2*A*c*d^2*e-4*B*a*d*e^2+B*c* 
d^3)*(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^2/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)
 

Mathematica [F]

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

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

Output:

Integrate[(A + B*x^2)*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.

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

\(\Big \downarrow \) 2261

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 2261

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

Maple [F]

\[\int \left (B \,x^{2}+A \right ) \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4} \, dx=\frac {6 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, a e x +\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, b d x +4 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, b e \,x^{3}+8 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a b \,e^{2}-6 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a c d e +3 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) b c \,d^{2}+12 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a^{2} e^{2}+10 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a b d e +18 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a^{2} d e -\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a b \,d^{2}}{24 e} \] Input:

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

Output:

(6*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*e*x + sqrt(d + e*x**2)*sqrt(a - c*x 
**4)*b*d*x + 4*sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*e*x**3 + 8*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) 
*a*b*e**2 - 6*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)*a*c*d*e + 3*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)*b*c*d**2 + 12*int((sqr 
t(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**2*e**2 + 10*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*b*d*e + 18*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**2*d*e - 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 
*b*d**2)/(24*e)