\(\int \frac {(A+B x^2) (d+e x^2)^{3/2} \sqrt {a+b x^2+c x^4}}{x^2} \, dx\) [115]

Optimal result

Integrand size = 38, antiderivative size = 993 \[ \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^2} \, dx=\frac {\left (6 A c e (5 c d+b e)+B \left (3 c^2 d^2-3 b^2 e^2+8 c e (b d+a e)\right )\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{48 c^2 e x}+\frac {(7 B c d+b B e+6 A c e) x \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{24 c}+\frac {1}{6} B e x^3 \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}-\frac {\sqrt {b^2-4 a c} \left (6 A c e (13 c d+b e)+B \left (3 c^2 d^2-3 b^2 e^2+8 c e (b d+a e)\right )\right ) \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 )}{48 \sqrt {2} c^2 e \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 (6 A c \left (8 b c d^2+a c d e-a b e^2\right )+a B \left (31 c^2 d^2+3 b^2 e^2-2 c e (5 b d+4 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 {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 )}{24 \sqrt {2} a c^2 \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}+\frac {\sqrt {b^2-4 a c} \left (2 A c e \left (3 c^2 d^2-b^2 e^2+2 c e (3 b d+2 a e)\right )-B \left (c^3 d^3-b^3 e^3-3 c^2 d e (b d+4 a e)+b c e^2 (3 b d+4 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 )}{4 \sqrt {2} c^2 \left (b+\sqrt {b^2-4 a c}\right ) e \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \] Output:

1/48*(6*A*c*e*(b*e+5*c*d)+B*(3*c^2*d^2-3*b^2*e^2+8*c*e*(a*e+b*d)))*(e*x^2+ 
d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/c^2/e/x+1/24*(6*A*c*e+B*b*e+7*B*c*d)*x*(e*x 
^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/c+1/6*B*e*x^3*(e*x^2+d)^(1/2)*(c*x^4+b*x 
^2+a)^(1/2)-1/96*(-4*a*c+b^2)^(1/2)*(6*A*c*e*(b*e+13*c*d)+B*(3*c^2*d^2-3*b 
^2*e^2+8*c*e*(a*e+b*d)))*(-a*(c+a/x^4+b/x^2)/(-4*a*c+b^2))^(1/2)*x*(e*x^2+ 
d)^(1/2)*EllipticE(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)/c^2/e/(-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)-1/48*(-4*a*c+b^2)^(1/2)*(6*A*c*(-a*b*e^2+a*c*d*e+8*b*c*d^2)+a*B 
*(31*c^2*d^2+3*b^2*e^2-2*c*e*(4*a*e+5*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*Ellip 
ticF(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^2/(e*x^ 
2+d)^(1/2)/(c*x^4+b*x^2+a)^(1/2)+1/8*(-4*a*c+b^2)^(1/2)*(2*A*c*e*(3*c^2*d^ 
2-b^2*e^2+2*c*e*(2*a*e+3*b*d))-B*(c^3*d^3-b^3*e^3-3*c^2*d*e*(4*a*e+b*d)+b* 
c*e^2*(4*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^2/(b+(-4*a*c+b^2)^(1/2))/e/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)...
 

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 2250

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(12*sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*a**2*c*e**2 - 2*sqrt(d + e* 
x**2)*sqrt(a + b*x**2 + c*x**4)*a*b**2*e**2 + 58*sqrt(d + e*x**2)*sqrt(a + 
 b*x**2 + c*x**4)*a*b*c*d*e + 6*sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4) 
*a*b*c*e**2*x**2 + 24*sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*a*c**2*d* 
*2 + 6*sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*a*c**2*d*e*x**2 - 2*sqrt 
(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*b**3*d*e + sqrt(d + e*x**2)*sqrt(a 
+ b*x**2 + c*x**4)*b**3*e**2*x**2 + 10*sqrt(d + e*x**2)*sqrt(a + b*x**2 + 
c*x**4)*b**2*c*d**2 + 8*sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*b**2*c* 
d*e*x**2 + 4*sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*b**2*c*e**2*x**4 + 
 7*sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*b*c**2*d**2*x**2 + 4*sqrt(d 
+ e*x**2)*sqrt(a + b*x**2 + c*x**4)*b*c**2*d*e*x**4 - 24*int((sqrt(d + e*x 
**2)*sqrt(a + b*x**2 + c*x**4)*x**4)/(a*b*d*e + a*b*e**2*x**2 + a*c*d**2 + 
 a*c*d*e*x**2 + b**2*d*e*x**2 + b**2*e**2*x**4 + b*c*d**2*x**2 + 2*b*c*d*e 
*x**4 + b*c*e**2*x**6 + c**2*d**2*x**4 + c**2*d*e*x**6),x)*a**2*b*c**2*e** 
4*x - 24*int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*x**4)/(a*b*d*e + 
a*b*e**2*x**2 + a*c*d**2 + a*c*d*e*x**2 + b**2*d*e*x**2 + b**2*e**2*x**4 + 
 b*c*d**2*x**2 + 2*b*c*d*e*x**4 + b*c*e**2*x**6 + c**2*d**2*x**4 + c**2*d* 
e*x**6),x)*a**2*c**3*d*e**3*x + 18*int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + 
 c*x**4)*x**4)/(a*b*d*e + a*b*e**2*x**2 + a*c*d**2 + a*c*d*e*x**2 + b**2*d 
*e*x**2 + b**2*e**2*x**4 + b*c*d**2*x**2 + 2*b*c*d*e*x**4 + b*c*e**2*x*...