\(\int \frac {1}{(d+e x^2)^{5/2} (a-c x^4)^{3/2}} \, dx\) [464]

Optimal result

Integrand size = 24, antiderivative size = 584 \[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \left (a-c x^4\right )^{3/2}} \, dx=\frac {c x \left (d-e x^2\right )}{2 a \left (c d^2-a e^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}}+\frac {e^2 \left (3 c d^2+a e^2\right ) x \sqrt {a-c x^4}}{3 a d \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )^{3/2}}-\frac {e \left (9 c^2 d^4+27 a c d^2 e^2-4 a^2 e^4\right ) \sqrt {a-c x^4}}{6 a d \left (c d^2-a e^2\right )^3 x \sqrt {d+e x^2}}-\frac {\sqrt {c} e \left (9 c^2 d^4+27 a c d^2 e^2-4 a^2 e^4\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{6 a d^2 \left (\sqrt {c} d-\sqrt {a} e\right )^3 \left (\sqrt {c} d+\sqrt {a} e\right )^2 \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\sqrt {c} \left (3 c^2 d^4-15 a c d^2 e^2+4 a^2 e^4\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{6 a^{3/2} d^2 \left (c d^2-a e^2\right )^2 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

1/2*c*x*(-e*x^2+d)/a/(-a*e^2+c*d^2)/(e*x^2+d)^(3/2)/(-c*x^4+a)^(1/2)+1/3*e 
^2*(a*e^2+3*c*d^2)*x*(-c*x^4+a)^(1/2)/a/d/(-a*e^2+c*d^2)^2/(e*x^2+d)^(3/2) 
-1/6*e*(-4*a^2*e^4+27*a*c*d^2*e^2+9*c^2*d^4)*(-c*x^4+a)^(1/2)/a/d/(-a*e^2+ 
c*d^2)^3/x/(e*x^2+d)^(1/2)-1/6*c^(1/2)*e*(-4*a^2*e^4+27*a*c*d^2*e^2+9*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))/a/d^2/(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)+1/6*c^(1/2)*(4*a^2*e^4-15*a*c*d^2*e^2 
+3*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^(3/2)/d^2/(-a*e^2+c*d^2)^2/(e*x^2+d)^(1/2 
)/(-c*x^4+a)^(1/2)
 

Mathematica [F]

\[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \left (a-c x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (d+e x^2\right )^{5/2} \left (a-c x^4\right )^{3/2}} \, dx \] Input:

Integrate[1/((d + e*x^2)^(5/2)*(a - c*x^4)^(3/2)),x]
 

Output:

Integrate[1/((d + e*x^2)^(5/2)*(a - c*x^4)^(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.

\(\displaystyle \int \frac {1}{\left (a-c x^4\right )^{3/2} \left (d+e x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 1571

\(\displaystyle \int \frac {1}{\left (a-c x^4\right )^{3/2} \left (d+e x^2\right )^{5/2}}dx\)

Input:

Int[1/((d + e*x^2)^(5/2)*(a - c*x^4)^(3/2)),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]

\[\int \frac {1}{\left (e \,x^{2}+d \right )^{\frac {5}{2}} \left (-c \,x^{4}+a \right )^{\frac {3}{2}}}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(e*x^2+d)^(5/2)/(-c*x^4+a)^(3/2),x, algorithm="fricas")
 

Output:

integral(sqrt(-c*x^4 + a)*sqrt(e*x^2 + d)/(c^2*e^3*x^14 + 3*c^2*d*e^2*x^12 
 + (3*c^2*d^2*e - 2*a*c*e^3)*x^10 + (c^2*d^3 - 6*a*c*d*e^2)*x^8 + 3*a^2*d^ 
2*e*x^2 - (6*a*c*d^2*e - a^2*e^3)*x^6 + a^2*d^3 - (2*a*c*d^3 - 3*a^2*d*e^2 
)*x^4), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \left (a-c x^4\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x**2+d)**(5/2)/(-c*x**4+a)**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(e*x^2+d)^(5/2)/(-c*x^4+a)^(3/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(e*x^2+d)^(5/2)/(-c*x^4+a)^(3/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \left (a-c x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (a-c\,x^4\right )}^{3/2}\,{\left (e\,x^2+d\right )}^{5/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \left (a-c x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{c^{2} e^{3} x^{14}+3 c^{2} d \,e^{2} x^{12}-2 a c \,e^{3} x^{10}+3 c^{2} d^{2} e \,x^{10}-6 a c d \,e^{2} x^{8}+c^{2} d^{3} x^{8}+a^{2} e^{3} x^{6}-6 a c \,d^{2} e \,x^{6}+3 a^{2} d \,e^{2} x^{4}-2 a c \,d^{3} x^{4}+3 a^{2} d^{2} e \,x^{2}+a^{2} d^{3}}d x \] Input:

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

Output:

int((sqrt(d + e*x**2)*sqrt(a - c*x**4))/(a**2*d**3 + 3*a**2*d**2*e*x**2 + 
3*a**2*d*e**2*x**4 + a**2*e**3*x**6 - 2*a*c*d**3*x**4 - 6*a*c*d**2*e*x**6 
- 6*a*c*d*e**2*x**8 - 2*a*c*e**3*x**10 + c**2*d**3*x**8 + 3*c**2*d**2*e*x* 
*10 + 3*c**2*d*e**2*x**12 + c**2*e**3*x**14),x)