\(\int \frac {(A+B x^2) \sqrt {a+b x^2+c x^4}}{(d+e x^2)^{3/2}} \, dx\) [227]

Optimal result

Integrand size = 35, antiderivative size = 754 \[ \int \frac {\left (A+B x^2\right ) \sqrt {a+b x^2+c x^4}}{\left (d+e x^2\right )^{3/2}} \, dx=-\frac {(B d-A e) x \sqrt {a+b x^2+c x^4}}{d e \sqrt {d+e x^2}}+\frac {(3 B d-2 A e) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}{2 d e^2 x}-\frac {\sqrt {b^2-4 a c} (3 B d-2 A e) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} x \sqrt {d+e x^2} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{2 \sqrt {2} d e^2 \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} \sqrt {a+b x^2+c x^4}}+\frac {\sqrt {b^2-4 a c} (B d-2 A e) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} x^3 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{\sqrt {2} d e \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} (3 B c d-b B e-2 A c e) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} x^3 \operatorname {EllipticPi}\left (\frac {2 \sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}},\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) e^2 \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \] Output:

-(-A*e+B*d)*x*(c*x^4+b*x^2+a)^(1/2)/d/e/(e*x^2+d)^(1/2)+1/2*(-2*A*e+3*B*d) 
*(e*x^2+d)^(1/2)*(c*x^4+b*x^2+a)^(1/2)/d/e^2/x-1/4*(-4*a*c+b^2)^(1/2)*(-2* 
A*e+3*B*d)*(-a*(c+a/x^4+b/x^2)/(-4*a*c+b^2))^(1/2)*x*(e*x^2+d)^(1/2)*Ellip 
ticE(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)/d/e^2/(-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)+1/2 
*(-4*a*c+b^2)^(1/2)*(-2*A*e+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*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)/d/e/(e*x^2+d)^(1/2)/(c* 
x^4+b*x^2+a)^(1/2)-2^(1/2)*(-4*a*c+b^2)^(1/2)*(-2*A*c*e-B*b*e+3*B*c*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))/(b+(-4*a*c+b^2)^(1/2)) 
/e^2/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(1/2)
 

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 2260

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 2260

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(d + e*x**2)*sqrt(a + b*x**2 + c*x**4)*a*c*x + sqrt(d + e*x**2)*sqrt( 
a + b*x**2 + c*x**4)*b**2*x - 2*int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + c* 
x**4)*x**6)/(a*b*d**2*e + 2*a*b*d*e**2*x**2 + a*b*e**3*x**4 + 3*a*c*d**3 + 
 6*a*c*d**2*e*x**2 + 3*a*c*d*e**2*x**4 + b**2*d**2*e*x**2 + 2*b**2*d*e**2* 
x**4 + b**2*e**3*x**6 + 3*b*c*d**3*x**2 + 7*b*c*d**2*e*x**4 + 5*b*c*d*e**2 
*x**6 + b*c*e**3*x**8 + 3*c**2*d**3*x**4 + 6*c**2*d**2*e*x**6 + 3*c**2*d*e 
**2*x**8),x)*a*b*c**2*d*e**2 - 2*int((sqrt(d + e*x**2)*sqrt(a + b*x**2 + c 
*x**4)*x**6)/(a*b*d**2*e + 2*a*b*d*e**2*x**2 + a*b*e**3*x**4 + 3*a*c*d**3 
+ 6*a*c*d**2*e*x**2 + 3*a*c*d*e**2*x**4 + b**2*d**2*e*x**2 + 2*b**2*d*e**2 
*x**4 + b**2*e**3*x**6 + 3*b*c*d**3*x**2 + 7*b*c*d**2*e*x**4 + 5*b*c*d*e** 
2*x**6 + b*c*e**3*x**8 + 3*c**2*d**3*x**4 + 6*c**2*d**2*e*x**6 + 3*c**2*d* 
e**2*x**8),x)*a*b*c**2*e**3*x**2 - 6*int((sqrt(d + e*x**2)*sqrt(a + b*x**2 
 + c*x**4)*x**6)/(a*b*d**2*e + 2*a*b*d*e**2*x**2 + a*b*e**3*x**4 + 3*a*c*d 
**3 + 6*a*c*d**2*e*x**2 + 3*a*c*d*e**2*x**4 + b**2*d**2*e*x**2 + 2*b**2*d* 
e**2*x**4 + b**2*e**3*x**6 + 3*b*c*d**3*x**2 + 7*b*c*d**2*e*x**4 + 5*b*c*d 
*e**2*x**6 + b*c*e**3*x**8 + 3*c**2*d**3*x**4 + 6*c**2*d**2*e*x**6 + 3*c** 
2*d*e**2*x**8),x)*a*c**3*d**2*e - 6*int((sqrt(d + e*x**2)*sqrt(a + b*x**2 
+ c*x**4)*x**6)/(a*b*d**2*e + 2*a*b*d*e**2*x**2 + a*b*e**3*x**4 + 3*a*c*d* 
*3 + 6*a*c*d**2*e*x**2 + 3*a*c*d*e**2*x**4 + b**2*d**2*e*x**2 + 2*b**2*d*e 
**2*x**4 + b**2*e**3*x**6 + 3*b*c*d**3*x**2 + 7*b*c*d**2*e*x**4 + 5*b*c...