\(\int \frac {(A+B x^2) (d+e x^2)^{3/2}}{\sqrt {a-c x^4}} \, dx\) [107]

Optimal result

Integrand size = 31, antiderivative size = 545 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{\sqrt {a-c x^4}} \, dx=-\frac {(5 B d+4 A e) \sqrt {d+e x^2} \sqrt {a-c x^4}}{8 c x}-\frac {B e x \sqrt {d+e x^2} \sqrt {a-c x^4}}{4 c}-\frac {(5 B d+4 A e) \left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{8 \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\left (8 A c d^2+7 a B d e+4 a A e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{8 \sqrt {a} \sqrt {c} \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\left (3 B c d^2+12 A c d e+4 a B e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{8 c \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

-1/8*(4*A*e+5*B*d)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/c/x-1/4*B*e*x*(e*x^2+d 
)^(1/2)*(-c*x^4+a)^(1/2)/c-1/8*(4*A*e+5*B*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)*Ellipti 
cE(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*a*e^2+8*A*c*d^2+7*B* 
a*d*e)*(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))/a^(1/2)/c^(1/2)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2) 
+1/8*(12*A*c*d*e+4*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)*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))/c/(e*x^2+d)^(1 
/2)/(-c*x^4+a)^(1/2)
 

Mathematica [F]

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

Integrate[((A + B*x^2)*(d + e*x^2)^(3/2))/Sqrt[a - c*x^4],x]
 

Output:

Integrate[((A + B*x^2)*(d + e*x^2)^(3/2))/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.

\(\displaystyle \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{\sqrt {a-c x^4}} \, dx\)

\(\Big \downarrow \) 2261

\(\displaystyle \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{\sqrt {a-c x^4}}dx\)

Input:

Int[((A + B*x^2)*(d + e*x^2)^(3/2))/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]

\[\int \frac {\left (B \,x^{2}+A \right ) \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{\sqrt {-c \,x^{4}+a}}d x\]

Input:

int((B*x^2+A)*(e*x^2+d)^(3/2)/(-c*x^4+a)^(1/2),x)
 

Output:

int((B*x^2+A)*(e*x^2+d)^(3/2)/(-c*x^4+a)^(1/2),x)
 

Fricas [F]

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

integrate((B*x^2+A)*(e*x^2+d)^(3/2)/(-c*x^4+a)^(1/2),x, algorithm="fricas" 
)
 

Output:

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

Sympy [F]

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

integrate((B*x**2+A)*(e*x**2+d)**(3/2)/(-c*x**4+a)**(1/2),x)
 

Output:

Integral((A + B*x**2)*(d + e*x**2)**(3/2)/sqrt(a - c*x**4), x)
 

Maxima [F]

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

integrate((B*x^2+A)*(e*x^2+d)^(3/2)/(-c*x^4+a)^(1/2),x, algorithm="maxima" 
)
 

Output:

integrate((B*x^2 + A)*(e*x^2 + d)^(3/2)/sqrt(-c*x^4 + a), x)
 

Giac [F]

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

integrate((B*x^2+A)*(e*x^2+d)^(3/2)/(-c*x^4+a)^(1/2),x, algorithm="giac")
 

Output:

integrate((B*x^2 + A)*(e*x^2 + d)^(3/2)/sqrt(-c*x^4 + a), x)
 

Mupad [F(-1)]

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

int(((A + B*x^2)*(d + e*x^2)^(3/2))/(a - c*x^4)^(1/2),x)
 

Output:

int(((A + B*x^2)*(d + e*x^2)^(3/2))/(a - c*x^4)^(1/2), x)
 

Reduce [F]

\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{\sqrt {a-c x^4}} \, dx=\frac {-\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, b e x +4 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a c \,e^{2}+5 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) b c d e +2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a b \,e^{2}+8 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a c d e +4 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) b c \,d^{2}+\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a b d e +4 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a c \,d^{2}}{4 c} \] Input:

int((B*x^2+A)*(e*x^2+d)^(3/2)/(-c*x^4+a)^(1/2),x)
 

Output:

( - sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*e*x + 4*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*e**2 + 5* 
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*e + 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*b*e**2 + 8*int((sqrt(d + e*x**2)*s 
qrt(a - c*x**4)*x**2)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a*c*d*e + 
4*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)*b*c*d**2 + 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*e + 4*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*c*d**2)/(4*c)