\(\int \frac {1}{(d+e x^2)^{7/2} \sqrt {a-c x^4}} \, dx\) [457]

Optimal result

Integrand size = 24, antiderivative size = 570 \[ \int \frac {1}{\left (d+e x^2\right )^{7/2} \sqrt {a-c x^4}} \, dx=-\frac {e^2 x \sqrt {a-c x^4}}{5 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )^{5/2}}-\frac {4 e^2 \left (3 c d^2-a e^2\right ) x \sqrt {a-c x^4}}{15 d^2 \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )^{3/2}}+\frac {e \left (45 c^2 d^4-21 a c d^2 e^2+8 a^2 e^4\right ) \sqrt {a-c x^4}}{15 d^2 \left (c d^2-a e^2\right )^3 x \sqrt {d+e x^2}}+\frac {\sqrt {c} e \left (45 c^2 d^4-21 a c d^2 e^2+8 a^2 e^4\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{15 d^3 \left (\sqrt {c} d-\sqrt {a} e\right )^3 \left (\sqrt {c} d+\sqrt {a} e\right )^2 \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\sqrt {c} \left (15 c^2 d^4-15 a c d^2 e^2+8 a^2 e^4\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{15 \sqrt {a} d^3 \left (c d^2-a e^2\right )^2 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

-1/5*e^2*x*(-c*x^4+a)^(1/2)/d/(-a*e^2+c*d^2)/(e*x^2+d)^(5/2)-4/15*e^2*(-a* 
e^2+3*c*d^2)*x*(-c*x^4+a)^(1/2)/d^2/(-a*e^2+c*d^2)^2/(e*x^2+d)^(3/2)+1/15* 
e*(8*a^2*e^4-21*a*c*d^2*e^2+45*c^2*d^4)*(-c*x^4+a)^(1/2)/d^2/(-a*e^2+c*d^2 
)^3/x/(e*x^2+d)^(1/2)+1/15*c^(1/2)*e*(8*a^2*e^4-21*a*c*d^2*e^2+45*c^2*d^4) 
*(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)^3/(c^(1/2)*d+a^(1/2)*e)^2/(e 
*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)+1/15*c^(1/2)*(8*a^2*e^4-15*a*c*d^2*e^2+15*c 
^2*d^4)*(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)/d^3/(-a*e^2+c*d^2)^2/(e*x^2+d)^(1/2)/(- 
c*x^4+a)^(1/2)
 

Mathematica [F]

\[ \int \frac {1}{\left (d+e x^2\right )^{7/2} \sqrt {a-c x^4}} \, dx=\int \frac {1}{\left (d+e x^2\right )^{7/2} \sqrt {a-c x^4}} \, dx \] Input:

Integrate[1/((d + e*x^2)^(7/2)*Sqrt[a - c*x^4]),x]
 

Output:

Integrate[1/((d + e*x^2)^(7/2)*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.

\(\displaystyle \int \frac {1}{\sqrt {a-c x^4} \left (d+e x^2\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 1571

\(\displaystyle \int \frac {1}{\sqrt {a-c x^4} \left (d+e x^2\right )^{7/2}}dx\)

Input:

Int[1/((d + e*x^2)^(7/2)*Sqrt[a - c*x^4]),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]

\[\int \frac {1}{\left (e \,x^{2}+d \right )^{\frac {7}{2}} \sqrt {-c \,x^{4}+a}}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {1}{\left (d+e x^2\right )^{7/2} \sqrt {a-c x^4}} \, dx=\int { \frac {1}{\sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

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

Output:

integral(-sqrt(-c*x^4 + a)*sqrt(e*x^2 + d)/(c*e^4*x^12 + 4*c*d*e^3*x^10 + 
(6*c*d^2*e^2 - a*e^4)*x^8 - 4*a*d^3*e*x^2 + 4*(c*d^3*e - a*d*e^3)*x^6 - a* 
d^4 + (c*d^4 - 6*a*d^2*e^2)*x^4), x)
 

Sympy [F]

\[ \int \frac {1}{\left (d+e x^2\right )^{7/2} \sqrt {a-c x^4}} \, dx=\int \frac {1}{\sqrt {a - c x^{4}} \left (d + e x^{2}\right )^{\frac {7}{2}}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {1}{\left (d+e x^2\right )^{7/2} \sqrt {a-c x^4}} \, dx=\int { \frac {1}{\sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {1}{\left (d+e x^2\right )^{7/2} \sqrt {a-c x^4}} \, dx=\int { \frac {1}{\sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^{7/2} \sqrt {a-c x^4}} \, dx=\int \frac {1}{\sqrt {a-c\,x^4}\,{\left (e\,x^2+d\right )}^{7/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\left (d+e x^2\right )^{7/2} \sqrt {a-c x^4}} \, dx=\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c \,e^{4} x^{12}-4 c d \,e^{3} x^{10}+a \,e^{4} x^{8}-6 c \,d^{2} e^{2} x^{8}+4 a d \,e^{3} x^{6}-4 c \,d^{3} e \,x^{6}+6 a \,d^{2} e^{2} x^{4}-c \,d^{4} x^{4}+4 a \,d^{3} e \,x^{2}+a \,d^{4}}d x \] Input:

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

Output:

int((sqrt(d + e*x**2)*sqrt(a - c*x**4))/(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)