\(\int \frac {x^4 (A+B x^2+C x^4+D x^6)}{(d+e x^2)^{3/2} \sqrt {a-c x^4}} \, dx\) [83]

Optimal result

Integrand size = 44, antiderivative size = 842 \[ \int \frac {x^4 \left (A+B x^2+C x^4+D x^6\right )}{\left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}} \, dx=\frac {d \left (d^3 D-C d^2 e+B d e^2-A e^3\right ) x \sqrt {a-c x^4}}{e^3 \left (c d^2-a e^2\right ) \sqrt {d+e x^2}}+\frac {\left (16 a^2 D e^4+a c e^2 \left (41 d^2 D-42 C d e+24 B e^2\right )-3 c^2 d \left (35 d^3 D-30 C d^2 e+24 B d e^2-16 A e^3\right )\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{48 c^2 e^4 \left (c d^2-a e^2\right ) x}+\frac {(11 d D-6 C e) x \sqrt {d+e x^2} \sqrt {a-c x^4}}{24 c e^3}-\frac {D x^3 \sqrt {d+e x^2} \sqrt {a-c x^4}}{6 c e^2}+\frac {\left (16 a^2 D e^4+a c e^2 \left (41 d^2 D-42 C d e+24 B e^2\right )-3 c^2 d \left (35 d^3 D-30 C d^2 e+24 B d e^2-16 A e^3\right )\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{48 c^{3/2} e^4 \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\sqrt {a} \left (16 a D e^2+c \left (35 d^2 D-30 C d e+24 B e^2\right )\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{48 c^{3/2} e^3 \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {\left (4 a e^2 (3 d D-2 C e)+c \left (35 d^3 D-30 C d^2 e+24 B d e^2-16 A e^3\right )\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{16 c e^4 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

d*(-A*e^3+B*d*e^2-C*d^2*e+D*d^3)*x*(-c*x^4+a)^(1/2)/e^3/(-a*e^2+c*d^2)/(e* 
x^2+d)^(1/2)+1/48*(16*a^2*D*e^4+a*c*e^2*(24*B*e^2-42*C*d*e+41*D*d^2)-3*c^2 
*d*(-16*A*e^3+24*B*d*e^2-30*C*d^2*e+35*D*d^3))*(e*x^2+d)^(1/2)*(-c*x^4+a)^ 
(1/2)/c^2/e^4/(-a*e^2+c*d^2)/x+1/24*(-6*C*e+11*D*d)*x*(e*x^2+d)^(1/2)*(-c* 
x^4+a)^(1/2)/c/e^3-1/6*D*x^3*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/c/e^2+1/48*( 
16*a^2*D*e^4+a*c*e^2*(24*B*e^2-42*C*d*e+41*D*d^2)-3*c^2*d*(-16*A*e^3+24*B* 
d*e^2-30*C*d^2*e+35*D*d^3))*(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))/c^(3/2)/e^4/(c^(1/2)*d-a^(1 
/2)*e)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)+1/48*a^(1/2)*(16*a*D*e^2+c*(24*B*e 
^2-30*C*d*e+35*D*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))/c^(3/2)/e^3/(e*x^2+d)^(1/2)/(-c* 
x^4+a)^(1/2)-1/16*(4*a*e^2*(-2*C*e+3*D*d)+c*(-16*A*e^3+24*B*d*e^2-30*C*d^2 
*e+35*D*d^3))*(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^4/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)
 

Mathematica [F]

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

Integrate[(x^4*(A + B*x^2 + C*x^4 + D*x^6))/((d + e*x^2)^(3/2)*Sqrt[a - c* 
x^4]),x]
 

Output:

Integrate[(x^4*(A + B*x^2 + C*x^4 + D*x^6))/((d + e*x^2)^(3/2)*Sqrt[a - c* 
x^4]), 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.

\(\displaystyle \int \frac {x^4 \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {a-c x^4} \left (d+e x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 2251

\(\displaystyle \int \frac {x^4 \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {a-c x^4} \left (d+e x^2\right )^{3/2}}dx\)

Input:

Int[(x^4*(A + B*x^2 + C*x^4 + D*x^6))/((d + e*x^2)^(3/2)*Sqrt[a - c*x^4]), 
x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 2251

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

Maple [F]

\[\int \frac {x^{4} \left (D x^{6}+C \,x^{4}+B \,x^{2}+A \right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}} \sqrt {-c \,x^{4}+a}}d x\]

Input:

int(x^4*(D*x^6+C*x^4+B*x^2+A)/(e*x^2+d)^(3/2)/(-c*x^4+a)^(1/2),x)
 

Output:

int(x^4*(D*x^6+C*x^4+B*x^2+A)/(e*x^2+d)^(3/2)/(-c*x^4+a)^(1/2),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {x^4 \left (A+B x^2+C x^4+D x^6\right )}{\left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}} \, dx=\text {Timed out} \] Input:

integrate(x^4*(D*x^6+C*x^4+B*x^2+A)/(e*x^2+d)^(3/2)/(-c*x^4+a)^(1/2),x, al 
gorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

integrate(x**4*(D*x**6+C*x**4+B*x**2+A)/(e*x**2+d)**(3/2)/(-c*x**4+a)**(1/ 
2),x)
 

Output:

Integral(x**4*(A + B*x**2 + C*x**4 + D*x**6)/(sqrt(a - c*x**4)*(d + e*x**2 
)**(3/2)), x)
 

Maxima [F]

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

integrate(x^4*(D*x^6+C*x^4+B*x^2+A)/(e*x^2+d)^(3/2)/(-c*x^4+a)^(1/2),x, al 
gorithm="maxima")
 

Output:

integrate((D*x^6 + C*x^4 + B*x^2 + A)*x^4/(sqrt(-c*x^4 + a)*(e*x^2 + d)^(3 
/2)), x)
 

Giac [F]

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

integrate(x^4*(D*x^6+C*x^4+B*x^2+A)/(e*x^2+d)^(3/2)/(-c*x^4+a)^(1/2),x, al 
gorithm="giac")
 

Output:

integrate((D*x^6 + C*x^4 + B*x^2 + A)*x^4/(sqrt(-c*x^4 + a)*(e*x^2 + d)^(3 
/2)), x)
 

Mupad [F(-1)]

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

int((x^4*(A + B*x^2 + C*x^4 + x^6*D))/((a - c*x^4)^(1/2)*(d + e*x^2)^(3/2) 
),x)
 

Output:

int((x^4*(A + B*x^2 + C*x^4 + x^6*D))/((a - c*x^4)^(1/2)*(d + e*x^2)^(3/2) 
), x)
 

Reduce [F]

\[ \int \frac {x^4 \left (A+B x^2+C x^4+D x^6\right )}{\left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}} \, dx =\text {Too large to display} \] Input:

int(x^4*(D*x^6+C*x^4+B*x^2+A)/(e*x^2+d)^(3/2)/(-c*x^4+a)^(1/2),x)
 

Output:

( - 12*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*c*e**2*x - 2*sqrt(d + e*x**2)*s 
qrt(a - c*x**4)*a*d**2*e*x - 6*sqrt(d + e*x**2)*sqrt(a - c*x**4)*c**2*d*e* 
x**3 + 7*sqrt(d + e*x**2)*sqrt(a - c*x**4)*c*d**3*x**3 - 4*sqrt(d + e*x**2 
)*sqrt(a - c*x**4)*c*d**2*e*x**5 - 24*int((sqrt(d + e*x**2)*sqrt(a - c*x** 
4)*x**6)/(a*d**2 + 2*a*d*e*x**2 + a*e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x**6 
 - c*e**2*x**8),x)*a*c**2*d*e**3 - 24*int((sqrt(d + e*x**2)*sqrt(a - c*x** 
4)*x**6)/(a*d**2 + 2*a*d*e*x**2 + a*e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x**6 
 - c*e**2*x**8),x)*a*c**2*e**4*x**2 + 12*int((sqrt(d + e*x**2)*sqrt(a - c* 
x**4)*x**6)/(a*d**2 + 2*a*d*e*x**2 + a*e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x 
**6 - c*e**2*x**8),x)*a*c*d**3*e**2 + 12*int((sqrt(d + e*x**2)*sqrt(a - c* 
x**4)*x**6)/(a*d**2 + 2*a*d*e*x**2 + a*e**2*x**4 - c*d**2*x**4 - 2*c*d*e*x 
**6 - c*e**2*x**8),x)*a*c*d**2*e**3*x**2 + 24*int((sqrt(d + e*x**2)*sqrt(a 
 - c*x**4)*x**6)/(a*d**2 + 2*a*d*e*x**2 + a*e**2*x**4 - c*d**2*x**4 - 2*c* 
d*e*x**6 - c*e**2*x**8),x)*b*c**2*d**2*e**2 + 24*int((sqrt(d + e*x**2)*sqr 
t(a - c*x**4)*x**6)/(a*d**2 + 2*a*d*e*x**2 + a*e**2*x**4 - c*d**2*x**4 - 2 
*c*d*e*x**6 - c*e**2*x**8),x)*b*c**2*d*e**3*x**2 - 30*int((sqrt(d + e*x**2 
)*sqrt(a - c*x**4)*x**6)/(a*d**2 + 2*a*d*e*x**2 + a*e**2*x**4 - c*d**2*x** 
4 - 2*c*d*e*x**6 - c*e**2*x**8),x)*c**3*d**3*e - 30*int((sqrt(d + e*x**2)* 
sqrt(a - c*x**4)*x**6)/(a*d**2 + 2*a*d*e*x**2 + a*e**2*x**4 - c*d**2*x**4 
- 2*c*d*e*x**6 - c*e**2*x**8),x)*c**3*d**2*e**2*x**2 + 35*int((sqrt(d +...