\(\int \frac {A+B x^2}{(d+e x^2)^{5/2} \sqrt {a-c x^4}} \, dx\) [111]

Optimal result

Integrand size = 31, antiderivative size = 510 \[ \int \frac {A+B x^2}{\left (d+e x^2\right )^{5/2} \sqrt {a-c x^4}} \, dx=\frac {e (B d-A e) x \sqrt {a-c x^4}}{3 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )^{3/2}}-\frac {\left (3 B c d^3-6 A c d^2 e+a B d e^2+2 a A e^3\right ) \sqrt {a-c x^4}}{3 d \left (c d^2-a e^2\right )^2 x \sqrt {d+e x^2}}-\frac {\sqrt {c} \left (3 B c d^3-6 A c d^2 e+a B d e^2+2 a A e^3\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{3 d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-a e^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\sqrt {c} \left (3 A c d^2-a B d e-2 a A e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{3 \sqrt {a} d^2 \left (c d^2-a e^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

1/3*e*(-A*e+B*d)*x*(-c*x^4+a)^(1/2)/d/(-a*e^2+c*d^2)/(e*x^2+d)^(3/2)-1/3*( 
2*A*a*e^3-6*A*c*d^2*e+B*a*d*e^2+3*B*c*d^3)*(-c*x^4+a)^(1/2)/d/(-a*e^2+c*d^ 
2)^2/x/(e*x^2+d)^(1/2)-1/3*c^(1/2)*(2*A*a*e^3-6*A*c*d^2*e+B*a*d*e^2+3*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)*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))/d^2/(c^(1/2)*d-a^(1/2)*e)/(-a*e^2+c*d^2)/(e*x^2+d) 
^(1/2)/(-c*x^4+a)^(1/2)+1/3*c^(1/2)*(-2*A*a*e^2+3*A*c*d^2-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)*Ellipti 
cF(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)/d^2/(-a*e^2+c*d^2)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)
 

Mathematica [F]

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

Integrate[(A + B*x^2)/((d + e*x^2)^(5/2)*Sqrt[a - c*x^4]),x]
 

Output:

Integrate[(A + B*x^2)/((d + e*x^2)^(5/2)*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.

\(\displaystyle \int \frac {A+B x^2}{\sqrt {a-c x^4} \left (d+e x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 2261

\(\displaystyle \int \frac {A+B x^2}{\sqrt {a-c x^4} \left (d+e x^2\right )^{5/2}}dx\)

Input:

Int[(A + B*x^2)/((d + e*x^2)^(5/2)*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]

\[\int \frac {B \,x^{2}+A}{\left (e \,x^{2}+d \right )^{\frac {5}{2}} \sqrt {-c \,x^{4}+a}}d x\]

Input:

int((B*x^2+A)/(e*x^2+d)^(5/2)/(-c*x^4+a)^(1/2),x)
 

Output:

int((B*x^2+A)/(e*x^2+d)^(5/2)/(-c*x^4+a)^(1/2),x)
 

Fricas [F]

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

integrate((B*x^2+A)/(e*x^2+d)^(5/2)/(-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)/(c*e^3*x^10 + 3*c*d 
*e^2*x^8 + (3*c*d^2*e - a*e^3)*x^6 - 3*a*d^2*e*x^2 + (c*d^3 - 3*a*d*e^2)*x 
^4 - a*d^3), x)
 

Sympy [F]

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

integrate((B*x**2+A)/(e*x**2+d)**(5/2)/(-c*x**4+a)**(1/2),x)
 

Output:

Integral((A + B*x**2)/(sqrt(a - c*x**4)*(d + e*x**2)**(5/2)), x)
 

Maxima [F]

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

integrate((B*x^2+A)/(e*x^2+d)^(5/2)/(-c*x^4+a)^(1/2),x, algorithm="maxima" 
)
 

Output:

integrate((B*x^2 + A)/(sqrt(-c*x^4 + a)*(e*x^2 + d)^(5/2)), x)
 

Giac [F]

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

integrate((B*x^2+A)/(e*x^2+d)^(5/2)/(-c*x^4+a)^(1/2),x, algorithm="giac")
 

Output:

integrate((B*x^2 + A)/(sqrt(-c*x^4 + a)*(e*x^2 + d)^(5/2)), x)
 

Mupad [F(-1)]

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

int((A + B*x^2)/((a - c*x^4)^(1/2)*(d + e*x^2)^(5/2)),x)
 

Output:

int((A + B*x^2)/((a - c*x^4)^(1/2)*(d + e*x^2)^(5/2)), x)
 

Reduce [F]

\[ \int \frac {A+B x^2}{\left (d+e x^2\right )^{5/2} \sqrt {a-c x^4}} \, dx =\text {Too large to display} \] Input:

int((B*x^2+A)/(e*x^2+d)^(5/2)/(-c*x^4+a)^(1/2),x)
 

Output:

(sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*x**3 + 2*int((sqrt(d + e*x**2)*sqrt(a 
 - c*x**4)*x**8)/(a*d**3 + 3*a*d**2*e*x**2 + 3*a*d*e**2*x**4 + a*e**3*x**6 
 - c*d**3*x**4 - 3*c*d**2*e*x**6 - 3*c*d*e**2*x**8 - c*e**3*x**10),x)*b*c* 
d**2*e + 4*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**8)/(a*d**3 + 3*a*d**2 
*e*x**2 + 3*a*d*e**2*x**4 + a*e**3*x**6 - c*d**3*x**4 - 3*c*d**2*e*x**6 - 
3*c*d*e**2*x**8 - c*e**3*x**10),x)*b*c*d*e**2*x**2 + 2*int((sqrt(d + e*x** 
2)*sqrt(a - c*x**4)*x**8)/(a*d**3 + 3*a*d**2*e*x**2 + 3*a*d*e**2*x**4 + a* 
e**3*x**6 - c*d**3*x**4 - 3*c*d**2*e*x**6 - 3*c*d*e**2*x**8 - c*e**3*x**10 
),x)*b*c*e**3*x**4 + 5*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**6)/(a*d** 
3 + 3*a*d**2*e*x**2 + 3*a*d*e**2*x**4 + a*e**3*x**6 - c*d**3*x**4 - 3*c*d* 
*2*e*x**6 - 3*c*d*e**2*x**8 - c*e**3*x**10),x)*b*c*d**3 + 10*int((sqrt(d + 
 e*x**2)*sqrt(a - c*x**4)*x**6)/(a*d**3 + 3*a*d**2*e*x**2 + 3*a*d*e**2*x** 
4 + a*e**3*x**6 - c*d**3*x**4 - 3*c*d**2*e*x**6 - 3*c*d*e**2*x**8 - c*e**3 
*x**10),x)*b*c*d**2*e*x**2 + 5*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**6 
)/(a*d**3 + 3*a*d**2*e*x**2 + 3*a*d*e**2*x**4 + a*e**3*x**6 - c*d**3*x**4 
- 3*c*d**2*e*x**6 - 3*c*d*e**2*x**8 - c*e**3*x**10),x)*b*c*d*e**2*x**4 + 3 
*int((sqrt(d + e*x**2)*sqrt(a - c*x**4))/(a*d**3 + 3*a*d**2*e*x**2 + 3*a*d 
*e**2*x**4 + a*e**3*x**6 - c*d**3*x**4 - 3*c*d**2*e*x**6 - 3*c*d*e**2*x**8 
 - c*e**3*x**10),x)*a**2*d**3 + 6*int((sqrt(d + e*x**2)*sqrt(a - c*x**4))/ 
(a*d**3 + 3*a*d**2*e*x**2 + 3*a*d*e**2*x**4 + a*e**3*x**6 - c*d**3*x**4...