\(\int \frac {(A+B x^2) \sqrt {a+b x^2+c x^4}}{(d+e x^2)^{7/2}} \, dx\) [229]

Optimal result

Integrand size = 35, antiderivative size = 810 \[ \int \frac {\left (A+B x^2\right ) \sqrt {a+b x^2+c x^4}}{\left (d+e x^2\right )^{7/2}} \, dx=-\frac {(B d-A e) x \sqrt {a+b x^2+c x^4}}{5 d e \left (d+e x^2\right )^{5/2}}+\frac {\left (A e \left (2 c d^2-e (3 b d-4 a e)\right )+B d \left (3 c d^2-e (2 b d-a e)\right )\right ) x \sqrt {a+b x^2+c x^4}}{15 d^2 e \left (c d^2-b d e+a e^2\right ) \left (d+e x^2\right )^{3/2}}-\frac {\left (b^2 d^2 (2 B d+3 A e)-b d \left (5 A c d^2+2 a B d e+13 a A e^2\right )-2 a \left (3 B c d^3-8 A c d^2 e-a B d e^2-4 a A e^3\right )\right ) \sqrt {a+b x^2+c x^4}}{15 d^2 \left (c d^2-b d e+a e^2\right )^2 x \sqrt {d+e x^2}}+\frac {\sqrt {b^2-4 a c} \left (b^2 d^2 (2 B d+3 A e)-b d \left (5 A c d^2+2 a B d e+13 a A e^2\right )-2 a \left (3 B c d^3-8 A c d^2 e-a B d e^2-4 a A e^3\right )\right ) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} x \sqrt {d+e x^2} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{15 \sqrt {2} d^3 \left (c d^2-b d e+a e^2\right )^2 \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} \sqrt {a+b x^2+c x^4}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (b B d^2-10 A c d^2+9 A b d e-2 a B d e-8 a A e^2\right ) \sqrt {-\frac {a \left (c+\frac {a}{x^4}+\frac {b}{x^2}\right )}{b^2-4 a c}} \sqrt {-\frac {a \left (e+\frac {d}{x^2}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) d-2 a e}} x^3 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+\frac {2 a}{x^2}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {b^2-4 a c} d}{b d+\sqrt {b^2-4 a c} d-2 a e}\right )}{15 d^3 \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x^2} \sqrt {a+b x^2+c x^4}} \] Output:

-1/5*(-A*e+B*d)*x*(c*x^4+b*x^2+a)^(1/2)/d/e/(e*x^2+d)^(5/2)+1/15*(A*e*(2*c 
*d^2-e*(-4*a*e+3*b*d))+B*d*(3*c*d^2-e*(-a*e+2*b*d)))*x*(c*x^4+b*x^2+a)^(1/ 
2)/d^2/e/(a*e^2-b*d*e+c*d^2)/(e*x^2+d)^(3/2)-1/15*(b^2*d^2*(3*A*e+2*B*d)-b 
*d*(13*A*a*e^2+5*A*c*d^2+2*B*a*d*e)-2*a*(-4*A*a*e^3-8*A*c*d^2*e-B*a*d*e^2+ 
3*B*c*d^3))*(c*x^4+b*x^2+a)^(1/2)/d^2/(a*e^2-b*d*e+c*d^2)^2/x/(e*x^2+d)^(1 
/2)+1/30*(-4*a*c+b^2)^(1/2)*(b^2*d^2*(3*A*e+2*B*d)-b*d*(13*A*a*e^2+5*A*c*d 
^2+2*B*a*d*e)-2*a*(-4*A*a*e^3-8*A*c*d^2*e-B*a*d*e^2+3*B*c*d^3))*(-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)/d^3/(a*e^2-b*d*e+c*d^2)^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)+1/15 
*2^(1/2)*(-4*a*c+b^2)^(1/2)*(-8*A*a*e^2+9*A*b*d*e-10*A*c*d^2-2*B*a*d*e+B*b 
*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))/d^3/(a*e^2-b*d*e+c*d^2)/(e*x^2+d)^(1/2)/(c*x^4+b*x^2+a)^(1 
/2)
 

Mathematica [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {a+b x^2+c x^4}}{\left (d+e x^2\right )^{7/2}} \, dx=\int \frac {\left (A+B x^2\right ) \sqrt {a+b x^2+c x^4}}{\left (d+e x^2\right )^{7/2}} \, dx \] Input:

Integrate[((A + B*x^2)*Sqrt[a + b*x^2 + c*x^4])/(d + e*x^2)^(7/2),x]
 

Output:

Integrate[((A + B*x^2)*Sqrt[a + b*x^2 + c*x^4])/(d + e*x^2)^(7/2), 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.

\(\displaystyle \int \frac {\left (A+B x^2\right ) \sqrt {a+b x^2+c x^4}}{\left (d+e x^2\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 2260

\(\displaystyle \int \frac {\left (A+B x^2\right ) \sqrt {a+b x^2+c x^4}}{\left (d+e x^2\right )^{7/2}}dx\)

Input:

Int[((A + B*x^2)*Sqrt[a + b*x^2 + c*x^4])/(d + e*x^2)^(7/2),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]

\[\int \frac {\left (B \,x^{2}+A \right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{\left (e \,x^{2}+d \right )^{\frac {7}{2}}}d x\]

Input:

int((B*x^2+A)*(c*x^4+b*x^2+a)^(1/2)/(e*x^2+d)^(7/2),x)
 

Output:

int((B*x^2+A)*(c*x^4+b*x^2+a)^(1/2)/(e*x^2+d)^(7/2),x)
 

Fricas [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {a+b x^2+c x^4}}{\left (d+e x^2\right )^{7/2}} \, dx=\int { \frac {\sqrt {c x^{4} + b x^{2} + a} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((B*x^2+A)*(c*x^4+b*x^2+a)^(1/2)/(e*x^2+d)^(7/2),x, algorithm="fr 
icas")
 

Output:

integral(sqrt(c*x^4 + b*x^2 + a)*(B*x^2 + A)*sqrt(e*x^2 + d)/(e^4*x^8 + 4* 
d*e^3*x^6 + 6*d^2*e^2*x^4 + 4*d^3*e*x^2 + d^4), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (A+B x^2\right ) \sqrt {a+b x^2+c x^4}}{\left (d+e x^2\right )^{7/2}} \, dx=\text {Timed out} \] Input:

integrate((B*x**2+A)*(c*x**4+b*x**2+a)**(1/2)/(e*x**2+d)**(7/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {a+b x^2+c x^4}}{\left (d+e x^2\right )^{7/2}} \, dx=\int { \frac {\sqrt {c x^{4} + b x^{2} + a} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((B*x^2+A)*(c*x^4+b*x^2+a)^(1/2)/(e*x^2+d)^(7/2),x, algorithm="ma 
xima")
 

Output:

integrate(sqrt(c*x^4 + b*x^2 + a)*(B*x^2 + A)/(e*x^2 + d)^(7/2), x)
 

Giac [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {a+b x^2+c x^4}}{\left (d+e x^2\right )^{7/2}} \, dx=\int { \frac {\sqrt {c x^{4} + b x^{2} + a} {\left (B x^{2} + A\right )}}{{\left (e x^{2} + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((B*x^2+A)*(c*x^4+b*x^2+a)^(1/2)/(e*x^2+d)^(7/2),x, algorithm="gi 
ac")
 

Output:

integrate(sqrt(c*x^4 + b*x^2 + a)*(B*x^2 + A)/(e*x^2 + d)^(7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (A+B x^2\right ) \sqrt {a+b x^2+c x^4}}{\left (d+e x^2\right )^{7/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,\sqrt {c\,x^4+b\,x^2+a}}{{\left (e\,x^2+d\right )}^{7/2}} \,d x \] Input:

int(((A + B*x^2)*(a + b*x^2 + c*x^4)^(1/2))/(d + e*x^2)^(7/2),x)
 

Output:

int(((A + B*x^2)*(a + b*x^2 + c*x^4)^(1/2))/(d + e*x^2)^(7/2), x)
 

Reduce [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {a+b x^2+c x^4}}{\left (d+e x^2\right )^{7/2}} \, dx=\int \frac {\left (B \,x^{2}+A \right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{\left (e \,x^{2}+d \right )^{\frac {7}{2}}}d x \] Input:

int((B*x^2+A)*(c*x^4+b*x^2+a)^(1/2)/(e*x^2+d)^(7/2),x)
 

Output:

int((B*x^2+A)*(c*x^4+b*x^2+a)^(1/2)/(e*x^2+d)^(7/2),x)