\(\int \frac {(A+B x^2) \sqrt {a+c x^4}}{\sqrt {d+e x^2}} \, dx\) [119]

Optimal result

Integrand size = 30, antiderivative size = 1110 \[ \int \frac {\left (A+B x^2\right ) \sqrt {a+c x^4}}{\sqrt {d+e x^2}} \, dx =\text {Too large to display} \] Output:

1/8*a^(1/2)*d*(-4*A*e+3*B*d)*(c+a/x^4)*(e+d/x^2)*x^3/e^2/(a*e^2+c*d^2)^(1/ 
2)/(1+a^(1/2)*(e+d/x^2)/(a*e^2+c*d^2)^(1/2))/(e*x^2+d)^(1/2)/(c*x^4+a)^(1/ 
2)-1/8*(-4*A*e+3*B*d)*(e*x^2+d)^(1/2)*(c*x^4+a)^(1/2)/e^2/x+1/4*B*x*(e*x^2 
+d)^(1/2)*(c*x^4+a)^(1/2)/e+1/16*(-4*A*c*d*e+4*B*a*e^2+3*B*c*d^2)*(c+a/x^4 
)^(1/2)*(e+d/x^2)^(1/2)*x^3*arctanh(c^(1/2)*(e+d/x^2)^(1/2)/e^(1/2)/(c+a/x 
^4)^(1/2))/c^(1/2)/e^(5/2)/(e*x^2+d)^(1/2)/(c*x^4+a)^(1/2)-1/8*a^(1/4)*(-4 
*A*e+3*B*d)*(a*e^2+c*d^2)^(3/4)*(1+a^(1/2)*(e+d/x^2)/(a*e^2+c*d^2)^(1/2))* 
(d^2*(c+a/x^4)/(a*e^2+c*d^2)/(1+a^(1/2)*(e+d/x^2)/(a*e^2+c*d^2)^(1/2))^2)^ 
(1/2)*(e+d/x^2)^(1/2)*x^3*EllipticE(sin(2*arctan(a^(1/4)*(e+d/x^2)^(1/2)/( 
a*e^2+c*d^2)^(1/4))),1/2*(2+2*a^(1/2)/(a*e^2+c*d^2)^(1/2)*e)^(1/2))/d/e^2/ 
(e*x^2+d)^(1/2)/(c*x^4+a)^(1/2)-1/8*a^(1/4)*(a*e^2+c*d^2)^(3/4)*(4*A*c*d*e 
-B*(3*c*d^2+2*a*e^2-2*a^(1/2)*e*(a*e^2+c*d^2)^(1/2)))*(1+a^(1/2)*(e+d/x^2) 
/(a*e^2+c*d^2)^(1/2))*((c+a/x^4)/(c+a*e^2/d^2)/(1+a^(1/2)*(e+d/x^2)/(a*e^2 
+c*d^2)^(1/2))^2)^(1/2)*(e+d/x^2)^(1/2)*x^3*InverseJacobiAM(2*arctan(a^(1/ 
4)*(e+d/x^2)^(1/2)/(a*e^2+c*d^2)^(1/4)),1/2*(2+2*a^(1/2)/(a*e^2+c*d^2)^(1/ 
2)*e)^(1/2))/c/d^2/e^2/(e*x^2+d)^(1/2)/(c*x^4+a)^(1/2)+1/32*(a*e^2+c*d^2)^ 
(1/4)*(-4*A*c*d*e+4*B*a*e^2+3*B*c*d^2)*(a^(1/2)*e-(a*e^2+c*d^2)^(1/2))^2*( 
1+a^(1/2)*(e+d/x^2)/(a*e^2+c*d^2)^(1/2))*((c+a/x^4)/(c+a*e^2/d^2)/(1+a^(1/ 
2)*(e+d/x^2)/(a*e^2+c*d^2)^(1/2))^2)^(1/2)*(e+d/x^2)^(1/2)*x^3*EllipticPi( 
sin(2*arctan(a^(1/4)*(e+d/x^2)^(1/2)/(a*e^2+c*d^2)^(1/4))),1/4*(a^(1/2)...
 

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 2261

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 2261

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {a+c x^4}}{\sqrt {d+e x^2}} \, dx=\frac {\sqrt {e \,x^{2}+d}\, \sqrt {c \,x^{4}+a}\, b x +4 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {c \,x^{4}+a}\, x^{4}}{c e \,x^{6}+c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a c e -3 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {c \,x^{4}+a}\, x^{4}}{c e \,x^{6}+c d \,x^{4}+a e \,x^{2}+a d}d x \right ) b c d +2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {c \,x^{4}+a}\, x^{2}}{c e \,x^{6}+c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a b e +4 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {c \,x^{4}+a}}{c e \,x^{6}+c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a^{2} e -\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {c \,x^{4}+a}}{c e \,x^{6}+c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a b d}{4 e} \] Input:

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

Output:

(sqrt(d + e*x**2)*sqrt(a + c*x**4)*b*x + 4*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*e - 3*int((sqr 
t(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*d + 2*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*e + 4*int((sqrt(d + e*x**2)*sqrt(a + c*x** 
4))/(a*d + a*e*x**2 + c*d*x**4 + c*e*x**6),x)*a**2*e - int((sqrt(d + e*x** 
2)*sqrt(a + c*x**4))/(a*d + a*e*x**2 + c*d*x**4 + c*e*x**6),x)*a*b*d)/(4*e 
)