\(\int \frac {(A+B x^2) (d+e x^2)^{5/2}}{(a-c x^4)^{3/2}} \, dx\) [112]

Optimal result

Integrand size = 31, antiderivative size = 669 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{5/2}}{\left (a-c x^4\right )^{3/2}} \, dx=\frac {x \left (A+B x^2\right ) \left (d+e x^2\right )^{5/2}}{2 a \sqrt {a-c x^4}}+\frac {\left (B c d^2+2 A c d e+2 a B e^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{2 a c^2 x}+\frac {e (2 B d+A e) x \sqrt {d+e x^2} \sqrt {a-c x^4}}{2 a c}+\frac {B e^2 x^3 \sqrt {d+e x^2} \sqrt {a-c x^4}}{2 a c}+\frac {\left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (B c d^2+2 A c d e+2 a B e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 a c \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\left (A c d \left (c d^2-3 a e^2\right )-a B e \left (3 c d^2+2 a e^2\right )\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 a^{3/2} c^{3/2} \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {e^2 (5 B d+2 A e) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{2 c \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

1/2*x*(B*x^2+A)*(e*x^2+d)^(5/2)/a/(-c*x^4+a)^(1/2)+1/2*(2*A*c*d*e+2*B*a*e^ 
2+B*c*d^2)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/a/c^2/x+1/2*e*(A*e+2*B*d)*x*(e 
*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/a/c+1/2*B*e^2*x^3*(e*x^2+d)^(1/2)*(-c*x^4+a 
)^(1/2)/a/c+1/2*(d+a^(1/2)*e/c^(1/2))*(2*A*c*d*e+2*B*a*e^2+B*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)*Ellipt 
icE(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/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)+1/2*(A*c*d*(-3*a*e^2+c*d 
^2)-a*B*e*(2*a*e^2+3*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)/c^(3/2)/(e*x^2+d)^ 
(1/2)/(-c*x^4+a)^(1/2)-1/2*e^2*(2*A*e+5*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)*EllipticPi(1/2*(1-a^(1/2)/c^( 
1/2)/x^2)^(1/2)*2^(1/2),2,2^(1/2)*(d/(d+a^(1/2)*e/c^(1/2)))^(1/2))/c/(e*x^ 
2+d)^(1/2)/(-c*x^4+a)^(1/2)
 

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 2261

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

Input:

Int[((A + B*x^2)*(d + e*x^2)^(5/2))/(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]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

Integral((A + B*x**2)*(d + e*x**2)**(5/2)/(a - c*x**4)**(3/2), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{5/2}}{\left (a-c x^4\right )^{3/2}} \, dx=\text {too large to display} \] Input:

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

Output:

(4*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*e**4*x + 12*sqrt(d + e*x**2)*sqr 
t(a - c*x**4)*a*b*d*e**3*x + 3*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*c*d**2* 
e**2*x - sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*c*d*e**3*x**3 + 3*sqrt(d + e* 
x**2)*sqrt(a - c*x**4)*b*c*d**3*e*x - 3*sqrt(d + e*x**2)*sqrt(a - c*x**4)* 
b*c*d**2*e**2*x**3 - 2*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**8)/(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**2*c**2*d*e**4 - 5*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**8)/(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*b*c**2*d**2*e**3 + 2*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**8 
)/(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*c**3*d*e**4*x**4 + 5*int((sqrt(d + e*x**2)*sqrt(a - c*x**4) 
*x**8)/(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*c**3*d**2*e**3*x**4 - 8*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)*a**4*e**5 - 24*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)*a**3*b*d*e**4 - 3*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)*a**3*c*d**2*e**3 + 8*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*...