\(\int \frac {(A+B x^2) (d+e x^2)^{3/2} \sqrt {a-c x^4}}{x^{14}} \, dx\) [51]

Optimal result

Integrand size = 34, antiderivative size = 825 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}}{x^{14}} \, dx=-\frac {A d \sqrt {d+e x^2} \sqrt {a-c x^4}}{13 x^{13}}-\frac {(13 B d+14 A e) \sqrt {d+e x^2} \sqrt {a-c x^4}}{143 x^{11}}+\frac {\left (22 A c d^2-156 a B d e-3 a A e^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{1287 a d x^9}+\frac {\left (234 B c d^3+274 A c d^2 e-39 a B d e^2+24 a A e^3\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{9009 a d^2 x^7}+\frac {2 \left (39 a B d e \left (31 c d^2+3 a e^2\right )+A \left (539 c^2 d^4+81 a c d^2 e^2-72 a^2 e^4\right )\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{45045 a^2 d^3 x^5}+\frac {2 \left (A e \left (1193 c^2 d^4-113 a c d^2 e^2+96 a^2 e^4\right )+39 B \left (25 c^2 d^5+7 a c d^3 e^2-4 a^2 d e^4\right )\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{45045 a^2 d^4 x^3}-\frac {2 c \left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (39 a B d e \left (103 c^2 d^4-15 a c d^2 e^2+8 a^2 e^4\right )+A \left (1617 c^3 d^6+597 a c^2 d^4 e^2+250 a^2 c d^2 e^4-192 a^3 e^6\right )\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{45045 a^3 d^5 \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {2 \sqrt {c} \left (c d^2-a e^2\right ) \left (2 A e \left (327 c^2 d^4+53 a c d^2 e^2-96 a^2 e^4\right )+39 B \left (25 c^2 d^5-9 a c d^3 e^2+8 a^2 d e^4\right )\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{45045 a^{5/2} d^5 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

-1/13*A*d*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/x^13-1/143*(14*A*e+13*B*d)*(e*x 
^2+d)^(1/2)*(-c*x^4+a)^(1/2)/x^11+1/1287*(-3*A*a*e^2+22*A*c*d^2-156*B*a*d* 
e)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/a/d/x^9+1/9009*(24*A*a*e^3+274*A*c*d^2 
*e-39*B*a*d*e^2+234*B*c*d^3)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/a/d^2/x^7+2/ 
45045*(39*a*B*d*e*(3*a*e^2+31*c*d^2)+A*(-72*a^2*e^4+81*a*c*d^2*e^2+539*c^2 
*d^4))*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/a^2/d^3/x^5+2/45045*(A*e*(96*a^2*e 
^4-113*a*c*d^2*e^2+1193*c^2*d^4)+39*B*(-4*a^2*d*e^4+7*a*c*d^3*e^2+25*c^2*d 
^5))*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/a^2/d^4/x^3-2/45045*c*(d+a^(1/2)*e/c 
^(1/2))*(39*a*B*d*e*(8*a^2*e^4-15*a*c*d^2*e^2+103*c^2*d^4)+A*(-192*a^3*e^6 
+250*a^2*c*d^2*e^4+597*a*c^2*d^4*e^2+1617*c^3*d^6))*(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^3 
/d^5/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)-2/45045*c^(1/2)*(-a*e^2+c*d^2)*(2*A* 
e*(-96*a^2*e^4+53*a*c*d^2*e^2+327*c^2*d^4)+39*B*(8*a^2*d*e^4-9*a*c*d^3*e^2 
+25*c^2*d^5))*(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^(5/2)/d^5/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1 
/2)
 

Mathematica [F]

\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}}{x^{14}} \, dx=\int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}}{x^{14}} \, dx \] Input:

Integrate[((A + B*x^2)*(d + e*x^2)^(3/2)*Sqrt[a - c*x^4])/x^14,x]
 

Output:

Integrate[((A + B*x^2)*(d + e*x^2)^(3/2)*Sqrt[a - c*x^4])/x^14, 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.

\(\displaystyle \int \frac {\sqrt {a-c x^4} \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{x^{14}} \, dx\)

\(\Big \downarrow \) 2251

\(\displaystyle \int \frac {\sqrt {a-c x^4} \left (A+B x^2\right ) \left (d+e x^2\right )^{3/2}}{x^{14}}dx\)

Input:

Int[((A + B*x^2)*(d + e*x^2)^(3/2)*Sqrt[a - c*x^4])/x^14,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]

\[\int \frac {\left (B \,x^{2}+A \right ) \left (e \,x^{2}+d \right )^{\frac {3}{2}} \sqrt {-c \,x^{4}+a}}{x^{14}}d x\]

Input:

int((B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(1/2)/x^14,x)
 

Output:

int((B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(1/2)/x^14,x)
 

Fricas [F]

\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}}{x^{14}} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}}}{x^{14}} \,d x } \] Input:

integrate((B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(1/2)/x^14,x, algorithm="fr 
icas")
 

Output:

integral((B*e*x^4 + (B*d + A*e)*x^2 + A*d)*sqrt(-c*x^4 + a)*sqrt(e*x^2 + d 
)/x^14, x)
 

Sympy [F]

\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}}{x^{14}} \, dx=\int \frac {\left (A + B x^{2}\right ) \sqrt {a - c x^{4}} \left (d + e x^{2}\right )^{\frac {3}{2}}}{x^{14}}\, dx \] Input:

integrate((B*x**2+A)*(e*x**2+d)**(3/2)*(-c*x**4+a)**(1/2)/x**14,x)
 

Output:

Integral((A + B*x**2)*sqrt(a - c*x**4)*(d + e*x**2)**(3/2)/x**14, x)
 

Maxima [F]

\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}}{x^{14}} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}}}{x^{14}} \,d x } \] Input:

integrate((B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(1/2)/x^14,x, algorithm="ma 
xima")
 

Output:

integrate(sqrt(-c*x^4 + a)*(B*x^2 + A)*(e*x^2 + d)^(3/2)/x^14, x)
 

Giac [F]

\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}}{x^{14}} \, dx=\int { \frac {\sqrt {-c x^{4} + a} {\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}}}{x^{14}} \,d x } \] Input:

integrate((B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(1/2)/x^14,x, algorithm="gi 
ac")
 

Output:

integrate(sqrt(-c*x^4 + a)*(B*x^2 + A)*(e*x^2 + d)^(3/2)/x^14, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}}{x^{14}} \, dx=\int \frac {\left (B\,x^2+A\right )\,\sqrt {a-c\,x^4}\,{\left (e\,x^2+d\right )}^{3/2}}{x^{14}} \,d x \] Input:

int(((A + B*x^2)*(a - c*x^4)^(1/2)*(d + e*x^2)^(3/2))/x^14,x)
 

Output:

int(((A + B*x^2)*(a - c*x^4)^(1/2)*(d + e*x^2)^(3/2))/x^14, x)
 

Reduce [F]

\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4}}{x^{14}} \, dx=\int \frac {\left (B \,x^{2}+A \right ) \left (e \,x^{2}+d \right )^{\frac {3}{2}} \sqrt {-c \,x^{4}+a}}{x^{14}}d x \] Input:

int((B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(1/2)/x^14,x)
 

Output:

int((B*x^2+A)*(e*x^2+d)^(3/2)*(-c*x^4+a)^(1/2)/x^14,x)