\(\int \frac {1}{(d+e x^2)^{3/2} (a-c x^4)^{3/2}} \, dx\) [463]

Optimal result

Integrand size = 24, antiderivative size = 464 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^{3/2}} \, dx=\frac {c x \left (d-e x^2\right )}{2 a \left (c d^2-a e^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {e \left (c d^2+a e^2\right ) \sqrt {a-c x^4}}{a \left (c d^2-a e^2\right )^2 x \sqrt {d+e x^2}}-\frac {\sqrt {c} e \left (c d^2+a e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{a d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-a e^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\sqrt {c} \left (c d^2-2 a e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 a^{3/2} d \left (c d^2-a e^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

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

Mathematica [F]

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

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

Output:

Integrate[1/((d + e*x^2)^(3/2)*(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.

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

\(\Big \downarrow \) 1571

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

Input:

Int[1/((d + e*x^2)^(3/2)*(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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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