\(\int \frac {(A+B x^2) (d+e x^2)^{3/2} \sqrt {a+b x^2+c x^4}}{x^4} \, dx\) [116]

Optimal result

Integrand size = 38, antiderivative size = 874 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^{3/2} \sqrt {a+b x^2+c x^4}}{x^4} \, dx=-\frac {A d \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{3 x^3}+\frac {(5 B c d+b B e+4 A c e) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{8 c x}+\frac {1}{4} B e x \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}-\frac {\sqrt {b^2-4 a c} (8 A b c d+39 a B c d+3 a b B e+44 a A c e) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} x \sqrt {d+e x^2} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{24 \sqrt {2} a c \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt {b^2-4 a c} \left (24 b B c d^2+16 A c^2 d^2+32 A b c d e+3 a B c d e-3 a b B e^2-20 a A c e^2\right ) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} x^3 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{12 \sqrt {2} a c \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}+\frac {\sqrt {b^2-4 a c} \left (4 A c e (3 c d+b e)+B \left (3 c^2 d^2-b^2 e^2+2 c e (3 b d+2 a e)\right )\right ) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} x^3 \operatorname {EllipticPi}\left (\frac {2 \sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}},\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{2 \sqrt {2} c \left (b+\sqrt {b^2-4 a c}\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \] Output:

-1/3*A*d*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/x^3+1/8*(4*A*c*e+B*b*e+5*B* 
c*d)*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/c/x+1/4*B*e*x*(e*x^2+d)^(1/2)*( 
c*x^4+b*x^2+a)^(1/2)-1/48*(-4*a*c+b^2)^(1/2)*(44*A*a*c*e+8*A*b*c*d+3*B*a*b 
*e+39*B*a*c*d)*(-a*(c+a/x^4+b/x^2)/(-4*a*c+b^2))^(1/2)*x*(e*x^2+d)^(1/2)*E 
llipticE(1/2*(1+(b+2*a/x^2)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2)*((-4 
*a*c+b^2)^(1/2)*d/(b*d+(-4*a*c+b^2)^(1/2)*d-2*a*e))^(1/2))*2^(1/2)/a/c/(-a 
*(e+d/x^2)/((b+(-4*a*c+b^2)^(1/2))*d-2*a*e))^(1/2)/(c*x^4+b*x^2+a)^(1/2)-1 
/24*(-4*a*c+b^2)^(1/2)*(-20*A*a*c*e^2+32*A*b*c*d*e+16*A*c^2*d^2-3*B*a*b*e^ 
2+3*B*a*c*d*e+24*B*b*c*d^2)*(-a*(c+a/x^4+b/x^2)/(-4*a*c+b^2))^(1/2)*(-a*(e 
+d/x^2)/((b+(-4*a*c+b^2)^(1/2))*d-2*a*e))^(1/2)*x^3*EllipticF(1/2*(1+(b+2* 
a/x^2)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2^(1/2)*((-4*a*c+b^2)^(1/2)*d/(b* 
d+(-4*a*c+b^2)^(1/2)*d-2*a*e))^(1/2))*2^(1/2)/a/c/(e*x^2+d)^(1/2)/(c*x^4+b 
*x^2+a)^(1/2)+1/4*(-4*a*c+b^2)^(1/2)*(4*A*c*e*(b*e+3*c*d)+B*(3*c^2*d^2-b^2 
*e^2+2*c*e*(2*a*e+3*b*d)))*(-a*(c+a/x^4+b/x^2)/(-4*a*c+b^2))^(1/2)*(-a*(e+ 
d/x^2)/((b+(-4*a*c+b^2)^(1/2))*d-2*a*e))^(1/2)*x^3*EllipticPi(1/2*(1+(b+2* 
a/x^2)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),2*(-4*a*c+b^2)^(1/2)/(b+(-4*a*c+b 
^2)^(1/2)),2^(1/2)*((-4*a*c+b^2)^(1/2)*d/(b*d+(-4*a*c+b^2)^(1/2)*d-2*a*e)) 
^(1/2))*2^(1/2)/c/(b+(-4*a*c+b^2)^(1/2))/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^( 
1/2)
 

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 2250

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 2250

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

integrate((B*x^2+A)*(e*x^2+d)^(3/2)*(c*x^4+b*x^2+a)^(1/2)/x^4,x, algorithm 
="fricas")
 

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((B*x^2+A)*(e*x^2+d)^(3/2)*(c*x^4+b*x^2+a)^(1/2)/x^4,x, algorithm 
="maxima")
 

Output:

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

Giac [F]

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

integrate((B*x^2+A)*(e*x^2+d)^(3/2)*(c*x^4+b*x^2+a)^(1/2)/x^4,x, algorithm 
="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 7*sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*a**2*b*d*e**2 + 6*sqrt(d 
+ e*x**2)*sqrt(a + b*x**2 + c*x**4)*a**2*b*e**3*x**2 - 8*sqrt(d + e*x**2)* 
sqrt(a + b*x**2 + c*x**4)*a**2*c*d**2*e + 8*sqrt(d + e*x**2)*sqrt(a + b*x* 
*2 + c*x**4)*a**2*c*d*e**2*x**2 - 7*sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x 
**4)*a*b**2*d**2*e + 12*sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*a*b**2* 
d*e**2*x**2 + sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*a*b**2*e**3*x**4 
- 6*sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*a*b*c*d**3 + 12*sqrt(d + e* 
x**2)*sqrt(a + b*x**2 + c*x**4)*a*b*c*d**2*e*x**2 + sqrt(d + e*x**2)*sqrt( 
a + b*x**2 + c*x**4)*a*b*c*d*e**2*x**4 + 6*sqrt(d + e*x**2)*sqrt(a + b*x** 
2 + c*x**4)*b**3*d**2*e*x**2 + sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)* 
b**3*d*e**2*x**4 + 4*sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*b**2*c*d** 
3*x**2 + sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*b**2*c*d**2*e*x**4 - 8 
*int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*x**4)/(a**2*b*d*e**2 + a* 
*2*b*e**3*x**2 + a**2*c*d**2*e + a**2*c*d*e**2*x**2 + a*b**2*d**2*e + 2*a* 
b**2*d*e**2*x**2 + a*b**2*e**3*x**4 + a*b*c*d**3 + 2*a*b*c*d**2*e*x**2 + 2 
*a*b*c*d*e**2*x**4 + a*b*c*e**3*x**6 + a*c**2*d**2*e*x**4 + a*c**2*d*e**2* 
x**6 + b**3*d**2*e*x**2 + b**3*d*e**2*x**4 + b**2*c*d**3*x**2 + 2*b**2*c*d 
**2*e*x**4 + b**2*c*d*e**2*x**6 + b*c**2*d**3*x**4 + b*c**2*d**2*e*x**6),x 
)*a**3*b**2*c*e**6*x**3 - 20*int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x** 
4)*x**4)/(a**2*b*d*e**2 + a**2*b*e**3*x**2 + a**2*c*d**2*e + a**2*c*d*e...