\(\int \frac {(A+B x^2) \sqrt {d+e x^2} \sqrt {a-c x^4}}{x^{10}} \, dx\) [41]

Optimal result

Integrand size = 34, antiderivative size = 583 \[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{x^{10}} \, dx=-\frac {A \sqrt {d+e x^2} \sqrt {a-c x^4}}{9 x^9}-\frac {(9 B d+A e) \sqrt {d+e x^2} \sqrt {a-c x^4}}{63 d x^7}+\frac {\left (14 A c d^2-9 a B d e+6 a A e^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{315 a d^2 x^5}+\frac {2 \left (15 B c d^3+4 A c d^2 e+6 a B d e^2-4 a A e^3\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{315 a d^3 x^3}-\frac {2 c \left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (12 a B d e \left (2 c d^2-a e^2\right )+A \left (21 c^2 d^4-9 a c d^2 e^2+8 a^2 e^4\right )\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{315 a^2 d^4 \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {2 \sqrt {c} \left (c d^2-a e^2\right ) \left (15 B c d^3-3 A c d^2 e-12 a B d e^2+8 a A e^3\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{315 a^{3/2} d^4 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

-1/9*A*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/x^9-1/63*(A*e+9*B*d)*(e*x^2+d)^(1/ 
2)*(-c*x^4+a)^(1/2)/d/x^7+1/315*(6*A*a*e^2+14*A*c*d^2-9*B*a*d*e)*(e*x^2+d) 
^(1/2)*(-c*x^4+a)^(1/2)/a/d^2/x^5+2/315*(-4*A*a*e^3+4*A*c*d^2*e+6*B*a*d*e^ 
2+15*B*c*d^3)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/a/d^3/x^3-2/315*c*(d+a^(1/2 
)*e/c^(1/2))*(12*a*B*d*e*(-a*e^2+2*c*d^2)+A*(8*a^2*e^4-9*a*c*d^2*e^2+21*c^ 
2*d^4))*(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^2/d^4/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)-2/315* 
c^(1/2)*(-a*e^2+c*d^2)*(8*A*a*e^3-3*A*c*d^2*e-12*B*a*d*e^2+15*B*c*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)*Ell 
ipticF(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)/d^4/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)
 

Mathematica [F]

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

Integrate[((A + B*x^2)*Sqrt[d + e*x^2]*Sqrt[a - c*x^4])/x^10,x]
 

Output:

Integrate[((A + B*x^2)*Sqrt[d + e*x^2]*Sqrt[a - c*x^4])/x^10, 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.

\(\displaystyle \int \frac {\sqrt {a-c x^4} \left (A+B x^2\right ) \sqrt {d+e x^2}}{x^{10}} \, dx\)

\(\Big \downarrow \) 2251

\(\displaystyle \int \frac {\sqrt {a-c x^4} \left (A+B x^2\right ) \sqrt {d+e x^2}}{x^{10}}dx\)

Input:

Int[((A + B*x^2)*Sqrt[d + e*x^2]*Sqrt[a - c*x^4])/x^10,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]

\[\int \frac {\left (B \,x^{2}+A \right ) \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{x^{10}}d x\]

Input:

int((B*x^2+A)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/x^10,x)
 

Output:

int((B*x^2+A)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/x^10,x)
 

Fricas [F]

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

integrate((B*x^2+A)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/x^10,x, algorithm="fr 
icas")
 

Output:

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

Sympy [F]

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

integrate((B*x**2+A)*(e*x**2+d)**(1/2)*(-c*x**4+a)**(1/2)/x**10,x)
 

Output:

Integral((A + B*x**2)*sqrt(a - c*x**4)*sqrt(d + e*x**2)/x**10, x)
 

Maxima [F]

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

integrate((B*x^2+A)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/x^10,x, algorithm="ma 
xima")
 

Output:

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

Giac [F]

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

integrate((B*x^2+A)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/x^10,x, algorithm="gi 
ac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{x^{10}} \, dx=\int \frac {\left (B\,x^2+A\right )\,\sqrt {a-c\,x^4}\,\sqrt {e\,x^2+d}}{x^{10}} \,d x \] Input:

int(((A + B*x^2)*(a - c*x^4)^(1/2)*(d + e*x^2)^(1/2))/x^10,x)
 

Output:

int(((A + B*x^2)*(a - c*x^4)^(1/2)*(d + e*x^2)^(1/2))/x^10, x)
 

Reduce [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{x^{10}} \, dx=\text {too large to display} \] Input:

int((B*x^2+A)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/x^10,x)
 

Output:

( - 1440*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**5*d**2*e**6 + 1440*sqrt(d + 
e*x**2)*sqrt(a - c*x**4)*a**5*d*e**7*x**2 - 1728*sqrt(d + e*x**2)*sqrt(a - 
 c*x**4)*a**5*e**8*x**4 - 1440*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**4*b*d* 
*2*e**6*x**2 - 864*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**4*b*d*e**7*x**4 + 
2640*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**4*c*d**3*e**5*x**2 - 2592*sqrt(d 
 + e*x**2)*sqrt(a - c*x**4)*a**4*c*d**2*e**6*x**4 + 288*sqrt(d + e*x**2)*s 
qrt(a - c*x**4)*a**4*c*d*e**7*x**6 + 1620*sqrt(d + e*x**2)*sqrt(a - c*x**4 
)*a**3*b*c*d**4*e**4*x**2 - 1944*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**3*b* 
c*d**3*e**5*x**4 + 2304*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**3*b*c*d**2*e* 
*6*x**6 + 10*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**3*c**2*d**6*e**2 + 2290* 
sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**3*c**2*d**5*e**3*x**2 - 2748*sqrt(d + 
 e*x**2)*sqrt(a - c*x**4)*a**3*c**2*d**4*e**4*x**4 + 1392*sqrt(d + e*x**2) 
*sqrt(a - c*x**4)*a**3*c**2*d**3*e**5*x**6 + 1485*sqrt(d + e*x**2)*sqrt(a 
- c*x**4)*a**2*b*c**2*d**6*e**2*x**2 - 1764*sqrt(d + e*x**2)*sqrt(a - c*x* 
*4)*a**2*b*c**2*d**5*e**3*x**4 + 324*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a** 
2*b*c**2*d**4*e**4*x**6 + 840*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*c**3* 
d**7*e*x**2 - 1012*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*c**3*d**6*e**2*x 
**4 + 458*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a**2*c**3*d**5*e**3*x**6 + 945 
*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*b*c**3*d**8*x**2 - 1134*sqrt(d + e*x* 
*2)*sqrt(a - c*x**4)*a*b*c**3*d**7*e*x**4 + 279*sqrt(d + e*x**2)*sqrt(a...