\(\int \frac {(A+B x^2) \sqrt {a-c x^4}}{\sqrt {d+e x^2}} \, dx\) [93]

Optimal result

Integrand size = 31, antiderivative size = 538 \[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{\sqrt {d+e x^2}} \, dx=-\frac {(3 B d-4 A e) \sqrt {d+e x^2} \sqrt {a-c x^4}}{8 e^2 x}+\frac {B x \sqrt {d+e x^2} \sqrt {a-c x^4}}{4 e}-\frac {c (3 B d-4 A e) \left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{8 e^2 \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\sqrt {a} \sqrt {c} (B d+4 A e) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{8 e \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {\left (3 B c d^2-4 A c d e-4 a B e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{8 e^2 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

-1/8*(-4*A*e+3*B*d)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/e^2/x+1/4*B*x*(e*x^2+ 
d)^(1/2)*(-c*x^4+a)^(1/2)/e-1/8*c*(-4*A*e+3*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)*Ell 
ipticE(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)+1/8*a^(1/2)*c^(1/2)*( 
4*A*e+B*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/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)-1/8*(-4*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))/e^2/(e*x^2+d)^(1/2)/(-c* 
x^4+a)^(1/2)
 

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 2261

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

integrate((B*x^2+A)*(-c*x^4+a)^(1/2)/(e*x^2+d)^(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. \[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{\sqrt {d+e x^2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,\sqrt {a-c\,x^4}}{\sqrt {e\,x^2+d}} \,d x \] 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]

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

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

Output:

(sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*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 + 3*int((sqr 
t(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*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 + 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**2*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)/(4*e 
)