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

Optimal result

Integrand size = 42, antiderivative size = 471 \[ \int \frac {x^4 \left (A+B x^2\right )}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=-\frac {(2 a B d f+b (3 B d e+2 B c f-3 A d f)) x \sqrt {c+d x^2}}{3 b d^2 f^2 \sqrt {a+b x^2}}+\frac {B x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 b d f}+\frac {\sqrt {a} (2 a B d f+b (3 B d e+2 B c f-3 A d f)) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 b^{3/2} d^2 f^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {a^{3/2} \left (a B c f^2-b e (3 B d e+B c f-3 A d f)\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 b^{3/2} c d f^2 (b e-a f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {a^{3/2} e (B e-A f) \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c f^2 (b e-a f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

-1/3*(2*a*B*d*f+b*(-3*A*d*f+2*B*c*f+3*B*d*e))*x*(d*x^2+c)^(1/2)/b/d^2/f^2/ 
(b*x^2+a)^(1/2)+1/3*B*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b/d/f+1/3*a^(1/2)* 
(2*a*B*d*f+b*(-3*A*d*f+2*B*c*f+3*B*d*e))*(d*x^2+c)^(1/2)*EllipticE(b^(1/2) 
*x/a^(1/2)/(1+b*x^2/a)^(1/2),(1-a*d/b/c)^(1/2))/b^(3/2)/d^2/f^2/(b*x^2+a)^ 
(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)+1/3*a^(3/2)*(a*B*c*f^2-b*e*(-3*A*d*f 
+B*c*f+3*B*d*e))*(d*x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)) 
,(1-a*d/b/c)^(1/2))/b^(3/2)/c/d/f^2/(-a*f+b*e)/(b*x^2+a)^(1/2)/(a*(d*x^2+c 
)/c/(b*x^2+a))^(1/2)+a^(3/2)*e*(-A*f+B*e)*(d*x^2+c)^(1/2)*EllipticPi(b^(1/ 
2)*x/a^(1/2)/(1+b*x^2/a)^(1/2),1-a*f/b/e,(1-a*d/b/c)^(1/2))/b^(1/2)/c/f^2/ 
(-a*f+b*e)/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 11.40 (sec) , antiderivative size = 346, normalized size of antiderivative = 0.73 \[ \int \frac {x^4 \left (A+B x^2\right )}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\frac {i c f (2 a B d f+b (3 B d e+2 B c f-3 A d f)) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i \left (a B c d f^2-3 A b d f (d e+c f)+b B \left (3 d^2 e^2+3 c d e f+2 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+d \left (\sqrt {\frac {b}{a}} B f^2 x \left (a+b x^2\right ) \left (c+d x^2\right )+3 i b d e (B e-A f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{3 b \sqrt {\frac {b}{a}} d^2 f^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

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

Output:

(I*c*f*(2*a*B*d*f + b*(3*B*d*e + 2*B*c*f - 3*A*d*f))*Sqrt[1 + (b*x^2)/a]*S 
qrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*(a*B 
*c*d*f^2 - 3*A*b*d*f*(d*e + c*f) + b*B*(3*d^2*e^2 + 3*c*d*e*f + 2*c^2*f^2) 
)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x] 
, (a*d)/(b*c)] + d*(Sqrt[b/a]*B*f^2*x*(a + b*x^2)*(c + d*x^2) + (3*I)*b*d* 
e*(B*e - A*f)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticPi[(a*f)/(b* 
e), I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(3*b*Sqrt[b/a]*d^2*f^3*Sqrt[a + 
 b*x^2]*Sqrt[c + d*x^2])
 

Rubi [A] (verified)

Time = 2.01 (sec) , antiderivative size = 623, normalized size of antiderivative = 1.32, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {7276, 2009}

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.

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\sqrt {-a} e \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} (B e-A f) \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right )}{\sqrt {b} f^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}+\frac {\sqrt {c} e \sqrt {a+b x^2} (B e-A f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} f^3 \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {\sqrt {c} \sqrt {a+b x^2} (B e-A f) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} f^2 \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {x \sqrt {a+b x^2} (B e-A f)}{b f^2 \sqrt {c+d x^2}}+\frac {2 B \sqrt {c} \sqrt {a+b x^2} (a d+b c) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 b^2 d^{3/2} f \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {B c^{3/2} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 b d^{3/2} f \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {2 B x \sqrt {a+b x^2} (a d+b c)}{3 b^2 d f \sqrt {c+d x^2}}+\frac {B x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 b d f}\)

Input:

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

Output:

(-2*B*(b*c + a*d)*x*Sqrt[a + b*x^2])/(3*b^2*d*f*Sqrt[c + d*x^2]) - ((B*e - 
 A*f)*x*Sqrt[a + b*x^2])/(b*f^2*Sqrt[c + d*x^2]) + (B*x*Sqrt[a + b*x^2]*Sq 
rt[c + d*x^2])/(3*b*d*f) + (2*B*Sqrt[c]*(b*c + a*d)*Sqrt[a + b*x^2]*Ellipt 
icE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*b^2*d^(3/2)*f*Sqrt[( 
c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (Sqrt[c]*(B*e - A*f)*Sq 
rt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b* 
Sqrt[d]*f^2*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (B*c^ 
(3/2)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a* 
d)])/(3*b*d^(3/2)*f*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) 
 + (Sqrt[c]*e*(B*e - A*f)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqr 
t[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[d]*f^3*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2 
))]*Sqrt[c + d*x^2]) - (Sqrt[-a]*e*(B*e - A*f)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 
+ (d*x^2)/c]*EllipticPi[(a*f)/(b*e), ArcSin[(Sqrt[b]*x)/Sqrt[-a]], (a*d)/( 
b*c)])/(Sqrt[b]*f^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

Maple [A] (verified)

Time = 10.17 (sec) , antiderivative size = 453, normalized size of antiderivative = 0.96

method	result	size
risch	\(\frac {B x \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{3 b d f}+\frac {\left (-\frac {\left (3 A b d f -2 a B d f -2 B b c f -3 b B d e \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{f \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}-\frac {\left (3 A b d e f +B a c \,f^{2}-3 b B d \,e^{2}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{f^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {3 b d e \left (A f -B e \right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right )}{f^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{3 b d f \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(453\)
default	\(-\frac {\left (-B \sqrt {-\frac {b}{a}}\, b \,d^{2} f^{2} x^{5}-B \sqrt {-\frac {b}{a}}\, a \,d^{2} f^{2} x^{3}-B \sqrt {-\frac {b}{a}}\, b c d \,f^{2} x^{3}+3 A \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b c d \,f^{2}+3 A \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,d^{2} e f -3 A \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b c d \,f^{2}-3 A \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right ) b \,d^{2} e f -B \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a c d \,f^{2}-2 B \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,c^{2} f^{2}-3 B \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b c d e f -3 B \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,d^{2} e^{2}+2 B \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a c d \,f^{2}+2 B \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,c^{2} f^{2}+3 B \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b c d e f +3 B \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right ) b \,d^{2} e^{2}-B \sqrt {-\frac {b}{a}}\, a c d \,f^{2} x \right ) \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{3 \sqrt {-\frac {b}{a}}\, f^{3} d^{2} b \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right )}\)	\(859\)
elliptic	\(\text {Expression too large to display}\)	\(1233\)

Input:

int(x^4*(B*x^2+A)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x,method=_RETU 
RNVERBOSE)
 

Output:

1/3*B*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b/d/f+1/3/b/d/f*(-1/f*(3*A*b*d*f-2 
*B*a*d*f-2*B*b*c*f-3*B*b*d*e)*c/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c) 
^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/d*(EllipticF(x*(-b/a)^(1/2),(-1 
+(a*d+b*c)/c/b)^(1/2))-EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))) 
-(3*A*b*d*e*f+B*a*c*f^2-3*B*b*d*e^2)/f^2/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1 
+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*(-b/a)^(1/ 
2),(-1+(a*d+b*c)/c/b)^(1/2))+3*b*d*e*(A*f-B*e)/f^2/(-b/a)^(1/2)*(1+b*x^2/a 
)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticPi(x 
*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2)))*((b*x^2+a)*(d*x^2+c))^ 
(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(c + d*x**2)*sqrt(a + b*x**2)*x + int((sqrt(c + d*x**2)*sqrt(a + b*x* 
*2)*x**4)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x**2 + b*c 
*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*a*d*f - 2*int((sqrt(c + d*x**2)*sqrt 
(a + b*x**2)*x**4)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x 
**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*b*c*f - 3*int((sqrt(c + d*x 
**2)*sqrt(a + b*x**2)*x**4)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 
+ b*c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*b*d*e - int((sqrt( 
c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d* 
f*x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*a*c*f - 2*i 
nt((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c*e + a*c*f*x**2 + a*d*e*x* 
*2 + a*d*f*x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*a* 
d*e - 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c*e + a*c*f*x**2 + 
 a*d*e*x**2 + a*d*f*x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x* 
*6),x)*b*c*e - int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a*c*e + a*c*f*x**2 
 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f* 
x**6),x)*a*c*e)/(3*d*f)