\(\int \frac {\sqrt {a-c x^4}}{(d+e x^2)^{5/2}} \, dx\) [440]

Optimal result

Integrand size = 24, antiderivative size = 382 \[ \int \frac {\sqrt {a-c x^4}}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {x \sqrt {a-c x^4}}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 a e \sqrt {a-c x^4}}{3 d \left (c d^2-a e^2\right ) x \sqrt {d+e x^2}}+\frac {2 a \sqrt {c} e \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{3 d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{3 d^2 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

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

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 1571

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral(sqrt(a - c*x**4)/(d + e*x**2)**(5/2), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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