\(\int \frac {(A+B x^2) \sqrt {a-c x^4}}{x^4 (d+e x^2)^{3/2}} \, dx\) [64]

Optimal result

Integrand size = 34, antiderivative size = 431 \[ \int \frac {\left (A+B x^2\right ) \sqrt {a-c x^4}}{x^4 \left (d+e x^2\right )^{3/2}} \, dx=-\frac {e (B d-A e) x \sqrt {a-c x^4}}{d^3 \sqrt {d+e x^2}}-\frac {A \sqrt {d+e x^2} \sqrt {a-c x^4}}{3 d^2 x^3}+\frac {(B d-A e) \sqrt {d+e x^2} \sqrt {a-c x^4}}{d^3 x}+\frac {2 c (3 B d-4 A e) \left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{3 d^3 \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {2 \sqrt {c} \left (A c d^2+3 a B d e-4 a A e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{3 \sqrt {a} d^3 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

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

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 2251

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 2251

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(sqrt(-c*x^4 + a)*(B*x^2 + A)*sqrt(e*x^2 + d)/(e^2*x^8 + 2*d*e*x^6 
 + d^2*x^4), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 2*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*d*e**2 + 2*sqrt(d + e*x**2)*s 
qrt(a - c*x**4)*a*b*d*e**2*x**2 + 4*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*b* 
e**3*x**4 - sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*c*d**2*e*x**2 - 6*sqrt(d + 
 e*x**2)*sqrt(a - c*x**4)*a*c*d*e**2*x**4 + sqrt(d + e*x**2)*sqrt(a - c*x* 
*4)*b*c*d**3*x**2 + 2*sqrt(d + e*x**2)*sqrt(a - c*x**4)*b*c*d**2*e*x**4 + 
8*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**6)/(a*d**2 + 2*a*d*e*x**2 + a* 
e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x**6 - c*e**2*x**8),x)*a*b*c*d*e**4*x**3 
 + 8*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**6)/(a*d**2 + 2*a*d*e*x**2 + 
 a*e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x**6 - c*e**2*x**8),x)*a*b*c*e**5*x** 
5 - 12*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**6)/(a*d**2 + 2*a*d*e*x**2 
 + a*e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x**6 - c*e**2*x**8),x)*a*c**2*d**2* 
e**3*x**3 - 12*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**6)/(a*d**2 + 2*a* 
d*e*x**2 + a*e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x**6 - c*e**2*x**8),x)*a*c* 
*2*d*e**4*x**5 + 4*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**6)/(a*d**2 + 
2*a*d*e*x**2 + a*e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x**6 - c*e**2*x**8),x)* 
b*c**2*d**3*e**2*x**3 + 4*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**6)/(a* 
d**2 + 2*a*d*e*x**2 + a*e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x**6 - c*e**2*x* 
*8),x)*b*c**2*d**2*e**3*x**5 + 12*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x 
**4)/(a*d**2 + 2*a*d*e*x**2 + a*e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x**6 - c 
*e**2*x**8),x)*a*b*c*d**2*e**3*x**3 + 12*int((sqrt(d + e*x**2)*sqrt(a -...