\(\int \frac {\sqrt {d+e x^2}}{(a-c x^4)^{3/2}} \, dx\) [461]

Optimal result

Integrand size = 24, antiderivative size = 173 \[ \int \frac {\sqrt {d+e x^2}}{\left (a-c x^4\right )^{3/2}} \, dx=\frac {x \sqrt {d+e x^2}}{2 a \sqrt {a-c x^4}}+\frac {\sqrt {c} d \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 a^{3/2} \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

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

Mathematica [F]

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

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

Output:

Integrate[Sqrt[d + e*x^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.

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

\(\Big \downarrow \) 1571

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

Input:

Int[Sqrt[d + e*x^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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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