\(\int \frac {x^4 (A+B x^2+C x^4+D x^6)}{\sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx\) [75]

Optimal result

Integrand size = 44, antiderivative size = 815 \[ \int \frac {x^4 \left (A+B x^2+C x^4+D x^6\right )}{\sqrt {d+e x^2} \sqrt {a-c x^4}} \, dx=\frac {\left (4 a e^2 (25 d D-32 C e)+3 c \left (35 d^3 D-40 C d^2 e+48 B d e^2-64 A e^3\right )\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{384 c^2 e^4 x}-\frac {\left (36 a D e^2+c \left (35 d^2 D-40 C d e+48 B e^2\right )\right ) x \sqrt {d+e x^2} \sqrt {a-c x^4}}{192 c^2 e^3}+\frac {(7 d D-8 C e) x^3 \sqrt {d+e x^2} \sqrt {a-c x^4}}{48 c e^2}-\frac {D x^5 \sqrt {d+e x^2} \sqrt {a-c x^4}}{8 c e}+\frac {\left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (4 a e^2 (25 d D-32 C e)+3 c \left (35 d^3 D-40 C d^2 e+48 B d e^2-64 A e^3\right )\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{384 c e^4 \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {\sqrt {a} \left (4 a e^2 (7 d D-32 C e)+c \left (35 d^3 D-40 C d^2 e+48 B d e^2-192 A e^3\right )\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{384 c^{3/2} e^3 \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\left (48 a^2 D e^4+8 a c e^2 \left (3 d^2 D-4 C d e+8 B e^2\right )+c^2 d \left (35 d^3 D-40 C d^2 e+48 B d e^2-64 A e^3\right )\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{128 c^2 e^4 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

1/384*(4*a*e^2*(-32*C*e+25*D*d)+3*c*(-64*A*e^3+48*B*d*e^2-40*C*d^2*e+35*D* 
d^3))*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/c^2/e^4/x-1/192*(36*a*D*e^2+c*(48*B 
*e^2-40*C*d*e+35*D*d^2))*x*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/c^2/e^3+1/48*( 
-8*C*e+7*D*d)*x^3*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/c/e^2-1/8*D*x^5*(e*x^2+ 
d)^(1/2)*(-c*x^4+a)^(1/2)/c/e+1/384*(d+a^(1/2)*e/c^(1/2))*(4*a*e^2*(-32*C* 
e+25*D*d)+3*c*(-64*A*e^3+48*B*d*e^2-40*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/e^4/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)-1/384*a^(1/2)*(4*a*e^2*(-32*C*e+ 
7*D*d)+c*(-192*A*e^3+48*B*d*e^2-40*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)*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)+1/128*(48*a^2*D*e^4+8*a*c*e^2* 
(8*B*e^2-4*C*d*e+3*D*d^2)+c^2*d*(-64*A*e^3+48*B*d*e^2-40*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^2/e^4/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)
 

Mathematica [F]

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

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

Output:

Integrate[(x^4*(A + B*x^2 + C*x^4 + D*x^6))/(Sqrt[d + e*x^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.

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

\(\Big \downarrow \) 2251

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

Input:

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

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

Input:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Maxima [F]

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

integrate(x^4*(D*x^6+C*x^4+B*x^2+A)/(e*x^2+d)^(1/2)/(-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)*sqrt(e*x^2 + d 
)), x)
 

Giac [F]

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

integrate(x^4*(D*x^6+C*x^4+B*x^2+A)/(e*x^2+d)^(1/2)/(-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)*sqrt(e*x^2 + d 
)), x)
 

Mupad [F(-1)]

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

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

Reduce [F]

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

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

Output:

( - 36*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*d*e**2*x - 48*sqrt(d + e*x**2)* 
sqrt(a - c*x**4)*b*c*e**2*x + 40*sqrt(d + e*x**2)*sqrt(a - c*x**4)*c**2*d* 
e*x - 32*sqrt(d + e*x**2)*sqrt(a - c*x**4)*c**2*e**2*x**3 - 35*sqrt(d + e* 
x**2)*sqrt(a - c*x**4)*c*d**3*x + 28*sqrt(d + e*x**2)*sqrt(a - c*x**4)*c*d 
**2*e*x**3 - 24*sqrt(d + e*x**2)*sqrt(a - c*x**4)*c*d*e**2*x**5 + 320*int( 
(sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**4)/(a*d + a*e*x**2 - c*d*x**4 - c*e* 
x**6),x)*a*c**2*e**3 - 100*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**4)/(a 
*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a*c*d**2*e**2 - 144*int((sqrt(d + 
e*x**2)*sqrt(a - c*x**4)*x**4)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*b 
*c**2*d*e**2 + 120*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**4)/(a*d + a*e 
*x**2 - c*d*x**4 - c*e*x**6),x)*c**3*d**2*e - 105*int((sqrt(d + e*x**2)*sq 
rt(a - c*x**4)*x**4)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*c**2*d**4 + 
 72*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**2)/(a*d + a*e*x**2 - c*d*x** 
4 - c*e*x**6),x)*a**2*d*e**3 + 96*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x 
**2)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a*b*c*e**3 + 16*int((sqrt(d 
 + e*x**2)*sqrt(a - c*x**4)*x**2)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x 
)*a*c**2*d*e**2 - 14*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**2)/(a*d + a 
*e*x**2 - c*d*x**4 - c*e*x**6),x)*a*c*d**3*e + 36*int((sqrt(d + e*x**2)*sq 
rt(a - c*x**4))/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a**2*d**2*e**2 + 
 48*int((sqrt(d + e*x**2)*sqrt(a - c*x**4))/(a*d + a*e*x**2 - c*d*x**4 ...