\(\int \frac {\sqrt {a-c x^4}}{(d+e x^2)^{7/2}} \, dx\) [441]

Optimal result

Integrand size = 24, antiderivative size = 506 \[ \int \frac {\sqrt {a-c x^4}}{\left (d+e x^2\right )^{7/2}} \, dx=\frac {x \sqrt {a-c x^4}}{5 d \left (d+e x^2\right )^{5/2}}+\frac {2 \left (c d^2-2 a e^2\right ) x \sqrt {a-c x^4}}{15 d^2 \left (c d^2-a e^2\right ) \left (d+e x^2\right )^{3/2}}+\frac {8 a e \left (2 c d^2-a e^2\right ) \sqrt {a-c x^4}}{15 d^2 \left (c d^2-a e^2\right )^2 x \sqrt {d+e x^2}}+\frac {8 a \sqrt {c} e \left (2 c d^2-a e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{15 d^3 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-a e^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {2 \sqrt {a} \sqrt {c} \left (5 c d^2-4 a e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{15 d^3 \left (c d^2-a e^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

1/5*x*(-c*x^4+a)^(1/2)/d/(e*x^2+d)^(5/2)+2/15*(-2*a*e^2+c*d^2)*x*(-c*x^4+a 
)^(1/2)/d^2/(-a*e^2+c*d^2)/(e*x^2+d)^(3/2)+8/15*a*e*(-a*e^2+2*c*d^2)*(-c*x 
^4+a)^(1/2)/d^2/(-a*e^2+c*d^2)^2/x/(e*x^2+d)^(1/2)+8/15*a*c^(1/2)*e*(-a*e^ 
2+2*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))/d^3/(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)+2/15*a^(1/2)*c^(1/2)*(-4*a*e^2+5*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)*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))/d^3/(-a*e^2+c*d^2)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)
 

Mathematica [F]

\[ \int \frac {\sqrt {a-c x^4}}{\left (d+e x^2\right )^{7/2}} \, dx=\int \frac {\sqrt {a-c x^4}}{\left (d+e x^2\right )^{7/2}} \, dx \] Input:

Integrate[Sqrt[a - c*x^4]/(d + e*x^2)^(7/2),x]
 

Output:

Integrate[Sqrt[a - c*x^4]/(d + e*x^2)^(7/2), 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.

\(\displaystyle \int \frac {\sqrt {a-c x^4}}{\left (d+e x^2\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 1571

\(\displaystyle \int \frac {\sqrt {a-c x^4}}{\left (d+e x^2\right )^{7/2}}dx\)

Input:

Int[Sqrt[a - c*x^4]/(d + e*x^2)^(7/2),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 1571

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

Maple [F]

\[\int \frac {\sqrt {-c \,x^{4}+a}}{\left (e \,x^{2}+d \right )^{\frac {7}{2}}}d x\]

Input:

int((-c*x^4+a)^(1/2)/(e*x^2+d)^(7/2),x)
 

Output:

int((-c*x^4+a)^(1/2)/(e*x^2+d)^(7/2),x)
 

Fricas [F]

\[ \int \frac {\sqrt {a-c x^4}}{\left (d+e x^2\right )^{7/2}} \, dx=\int { \frac {\sqrt {-c x^{4} + a}}{{\left (e x^{2} + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((-c*x^4+a)^(1/2)/(e*x^2+d)^(7/2),x, algorithm="fricas")
 

Output:

integral(sqrt(-c*x^4 + a)*sqrt(e*x^2 + d)/(e^4*x^8 + 4*d*e^3*x^6 + 6*d^2*e 
^2*x^4 + 4*d^3*e*x^2 + d^4), x)
 

Sympy [F]

\[ \int \frac {\sqrt {a-c x^4}}{\left (d+e x^2\right )^{7/2}} \, dx=\int \frac {\sqrt {a - c x^{4}}}{\left (d + e x^{2}\right )^{\frac {7}{2}}}\, dx \] Input:

integrate((-c*x**4+a)**(1/2)/(e*x**2+d)**(7/2),x)
 

Output:

Integral(sqrt(a - c*x**4)/(d + e*x**2)**(7/2), x)
 

Maxima [F]

\[ \int \frac {\sqrt {a-c x^4}}{\left (d+e x^2\right )^{7/2}} \, dx=\int { \frac {\sqrt {-c x^{4} + a}}{{\left (e x^{2} + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((-c*x^4+a)^(1/2)/(e*x^2+d)^(7/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(-c*x^4 + a)/(e*x^2 + d)^(7/2), x)
 

Giac [F]

\[ \int \frac {\sqrt {a-c x^4}}{\left (d+e x^2\right )^{7/2}} \, dx=\int { \frac {\sqrt {-c x^{4} + a}}{{\left (e x^{2} + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((-c*x^4+a)^(1/2)/(e*x^2+d)^(7/2),x, algorithm="giac")
 

Output:

integrate(sqrt(-c*x^4 + a)/(e*x^2 + d)^(7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a-c x^4}}{\left (d+e x^2\right )^{7/2}} \, dx=\int \frac {\sqrt {a-c\,x^4}}{{\left (e\,x^2+d\right )}^{7/2}} \,d x \] Input:

int((a - c*x^4)^(1/2)/(d + e*x^2)^(7/2),x)
 

Output:

int((a - c*x^4)^(1/2)/(d + e*x^2)^(7/2), x)
 

Reduce [F]

\[ \int \frac {\sqrt {a-c x^4}}{\left (d+e x^2\right )^{7/2}} \, dx =\text {Too large to display} \] Input:

int((-c*x^4+a)^(1/2)/(e*x^2+d)^(7/2),x)
 

Output:

( - sqrt(d + e*x**2)*sqrt(a - c*x**4)*c*x**5 - 2*int((sqrt(d + e*x**2)*sqr 
t(a - c*x**4)*x**10)/(a*d**4 + 4*a*d**3*e*x**2 + 6*a*d**2*e**2*x**4 + 4*a* 
d*e**3*x**6 + a*e**4*x**8 - c*d**4*x**4 - 4*c*d**3*e*x**6 - 6*c*d**2*e**2* 
x**8 - 4*c*d*e**3*x**10 - c*e**4*x**12),x)*c**2*d**3*e - 6*int((sqrt(d + e 
*x**2)*sqrt(a - c*x**4)*x**10)/(a*d**4 + 4*a*d**3*e*x**2 + 6*a*d**2*e**2*x 
**4 + 4*a*d*e**3*x**6 + a*e**4*x**8 - c*d**4*x**4 - 4*c*d**3*e*x**6 - 6*c* 
d**2*e**2*x**8 - 4*c*d*e**3*x**10 - c*e**4*x**12),x)*c**2*d**2*e**2*x**2 - 
 6*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**10)/(a*d**4 + 4*a*d**3*e*x**2 
 + 6*a*d**2*e**2*x**4 + 4*a*d*e**3*x**6 + a*e**4*x**8 - c*d**4*x**4 - 4*c* 
d**3*e*x**6 - 6*c*d**2*e**2*x**8 - 4*c*d*e**3*x**10 - c*e**4*x**12),x)*c** 
2*d*e**3*x**4 - 2*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**10)/(a*d**4 + 
4*a*d**3*e*x**2 + 6*a*d**2*e**2*x**4 + 4*a*d*e**3*x**6 + a*e**4*x**8 - c*d 
**4*x**4 - 4*c*d**3*e*x**6 - 6*c*d**2*e**2*x**8 - 4*c*d*e**3*x**10 - c*e** 
4*x**12),x)*c**2*e**4*x**6 - 7*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**8 
)/(a*d**4 + 4*a*d**3*e*x**2 + 6*a*d**2*e**2*x**4 + 4*a*d*e**3*x**6 + a*e** 
4*x**8 - c*d**4*x**4 - 4*c*d**3*e*x**6 - 6*c*d**2*e**2*x**8 - 4*c*d*e**3*x 
**10 - c*e**4*x**12),x)*c**2*d**4 - 21*int((sqrt(d + e*x**2)*sqrt(a - c*x* 
*4)*x**8)/(a*d**4 + 4*a*d**3*e*x**2 + 6*a*d**2*e**2*x**4 + 4*a*d*e**3*x**6 
 + a*e**4*x**8 - c*d**4*x**4 - 4*c*d**3*e*x**6 - 6*c*d**2*e**2*x**8 - 4*c* 
d*e**3*x**10 - c*e**4*x**12),x)*c**2*d**3*e*x**2 - 21*int((sqrt(d + e*x...