\(\int \frac {A+B x^2}{\sqrt {d+e x^2} (a-c x^4)^{3/2}} \, dx\) [115]

Optimal result

Integrand size = 31, antiderivative size = 413 \[ \int \frac {A+B x^2}{\sqrt {d+e x^2} \left (a-c x^4\right )^{3/2}} \, dx=\frac {(B d-A e) \sqrt {d+e x^2}}{2 \left (c d^2-a e^2\right ) x \sqrt {a-c x^4}}+\frac {(A c d-a B e) x \sqrt {d+e x^2}}{2 a \left (c d^2-a e^2\right ) \sqrt {a-c x^4}}+\frac {\sqrt {c} (B d-A e) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 a \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {A \sqrt {c} \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 a^{3/2} \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

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

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

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

Output:

Integrate[(A + B*x^2)/(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.

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

\(\Big \downarrow \) 2261

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 2261

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol 
] :> Unintegrable[Px*(d + e*x^2)^q*(a + c*x^4)^p, x] /; FreeQ[{a, c, d, e, 
p, q}, x] && PolyQ[Px, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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