\(\int \frac {x^6 (A+B x^2)}{\sqrt {a+b x^2} \sqrt {c+d x^2} (e+f x^2)} \, dx\) [1]

Optimal result

Integrand size = 42, antiderivative size = 691 \[ \int \frac {x^6 \left (A+B x^2\right )}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\frac {\left (8 a^2 B d f+a b (10 B d e+7 B c f-10 A d f)-b^2 \left (5 A (3 d e+2 c f)-B \left (10 c e+\frac {15 d e^2}{f}+\frac {8 c^2 f}{d}\right )\right )\right ) x \sqrt {c+d x^2}}{15 b^2 d^2 f^2 \sqrt {a+b x^2}}-\frac {(4 a B d f+b (5 B d e+4 B c f-5 A d f)) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 b^2 d^2 f^2}+\frac {B x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b d f}-\frac {\sqrt {a} \left (8 a^2 B d^2 f^2+a b d f (10 B d e+7 B c f-10 A d f)-b^2 \left (5 A d f (3 d e+2 c f)-B \left (15 d^2 e^2+10 c d e f+8 c^2 f^2\right )\right )\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{15 b^{5/2} d^3 f^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} \left (4 a^2 B c d f^3+a b c f^2 (B d e+4 B c f-5 A d f)+b^2 e \left (5 A d f (3 d e+c f)-B \left (15 d^2 e^2+5 c d e f+4 c^2 f^2\right )\right )\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{15 b^{5/2} c d^2 f^3 (b e-a f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} e^2 (B e-A f) \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c f^3 (b e-a f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

1/15*(8*a^2*B*d*f+a*b*(-10*A*d*f+7*B*c*f+10*B*d*e)-b^2*(5*A*(2*c*f+3*d*e)- 
B*(10*c*e+15*d*e^2/f+8*c^2*f/d)))*x*(d*x^2+c)^(1/2)/b^2/d^2/f^2/(b*x^2+a)^ 
(1/2)-1/15*(4*a*B*d*f+b*(-5*A*d*f+4*B*c*f+5*B*d*e))*x*(b*x^2+a)^(1/2)*(d*x 
^2+c)^(1/2)/b^2/d^2/f^2+1/5*B*x^3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b/d/f-1/ 
15*a^(1/2)*(8*a^2*B*d^2*f^2+a*b*d*f*(-10*A*d*f+7*B*c*f+10*B*d*e)-b^2*(5*A* 
d*f*(2*c*f+3*d*e)-B*(8*c^2*f^2+10*c*d*e*f+15*d^2*e^2)))*(d*x^2+c)^(1/2)*El 
lipticE(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2),(1-a*d/b/c)^(1/2))/b^(5/2)/d^3 
/f^3/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)-1/15*a^(3/2)*(4*a^2*B 
*c*d*f^3+a*b*c*f^2*(-5*A*d*f+4*B*c*f+B*d*e)+b^2*e*(5*A*d*f*(c*f+3*d*e)-B*( 
4*c^2*f^2+5*c*d*e*f+15*d^2*e^2)))*(d*x^2+c)^(1/2)*InverseJacobiAM(arctan(b 
^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/b^(5/2)/c/d^2/f^3/(-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^2*(-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^3/(-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 = 13.26 (sec) , antiderivative size = 499, normalized size of antiderivative = 0.72 \[ \int \frac {x^6 \left (A+B x^2\right )}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\frac {-i c f \left (8 a^2 B d^2 f^2+a b d f (10 B d e+7 B c f-10 A d f)+b^2 \left (-5 A d f (3 d e+2 c f)+B \left (15 d^2 e^2+10 c d e f+8 c^2 f^2\right )\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i \left (4 a^2 B c d^2 f^3+a b c d f^2 (5 B d e+3 B c f-5 A d f)+b^2 \left (-5 A d f \left (3 d^2 e^2+3 c d e f+2 c^2 f^2\right )+B \left (15 d^3 e^3+15 c d^2 e^2 f+10 c^2 d e f^2+8 c^3 f^3\right )\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )-d \left (\sqrt {\frac {b}{a}} f^2 x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (-5 A b d f+4 a B d f+b B \left (5 d e+4 c f-3 d f x^2\right )\right )+15 i b^2 d^2 e^2 (B e-A f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{15 a^2 \left (\frac {b}{a}\right )^{5/2} d^3 f^4 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

Integrate[(x^6*(A + B*x^2))/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)), 
x]
 

Output:

((-I)*c*f*(8*a^2*B*d^2*f^2 + a*b*d*f*(10*B*d*e + 7*B*c*f - 10*A*d*f) + b^2 
*(-5*A*d*f*(3*d*e + 2*c*f) + B*(15*d^2*e^2 + 10*c*d*e*f + 8*c^2*f^2)))*Sqr 
t[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a* 
d)/(b*c)] + I*(4*a^2*B*c*d^2*f^3 + a*b*c*d*f^2*(5*B*d*e + 3*B*c*f - 5*A*d* 
f) + b^2*(-5*A*d*f*(3*d^2*e^2 + 3*c*d*e*f + 2*c^2*f^2) + B*(15*d^3*e^3 + 1 
5*c*d^2*e^2*f + 10*c^2*d*e*f^2 + 8*c^3*f^3)))*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]* 
f^2*x*(a + b*x^2)*(c + d*x^2)*(-5*A*b*d*f + 4*a*B*d*f + b*B*(5*d*e + 4*c*f 
 - 3*d*f*x^2)) + (15*I)*b^2*d^2*e^2*(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)] 
))/(15*a^2*(b/a)^(5/2)*d^3*f^4*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
 

Rubi [A] (verified)

Time = 2.56 (sec) , antiderivative size = 1018, normalized size of antiderivative = 1.47, 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.

\(\displaystyle \int \frac {x^6 \left (A+B x^2\right )}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx\)

\(\Big \downarrow \) 7276

\(\displaystyle \int \left (-\frac {e^2 (B e-A f)}{f^4 \sqrt {a+b x^2} \sqrt {c+d x^2}}+\frac {B e^4-A e^3 f}{f^4 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )}+\frac {e x^2 (B e-A f)}{f^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}-\frac {x^4 (B e-A f)}{f^2 \sqrt {a+b x^2} \sqrt {c+d x^2}}+\frac {B x^6}{f \sqrt {a+b x^2} \sqrt {c+d x^2}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {B \sqrt {b x^2+a} \sqrt {d x^2+c} x^3}{5 b d f}-\frac {(B e-A f) \sqrt {b x^2+a} \sqrt {d x^2+c} x}{3 b d f^2}-\frac {4 B (b c+a d) \sqrt {b x^2+a} \sqrt {d x^2+c} x}{15 b^2 d^2 f}+\frac {2 (b c+a d) (B e-A f) \sqrt {b x^2+a} x}{3 b^2 d f^2 \sqrt {d x^2+c}}+\frac {e (B e-A f) \sqrt {b x^2+a} x}{b f^3 \sqrt {d x^2+c}}+\frac {B \left (8 b^2 c^2+7 a b d c+8 a^2 d^2\right ) \sqrt {b x^2+a} x}{15 b^3 d^2 f \sqrt {d x^2+c}}-\frac {2 \sqrt {c} (b c+a d) (B e-A f) \sqrt {b x^2+a} 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^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {c} e (B e-A f) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} f^3 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {B \sqrt {c} \left (8 b^2 c^2+7 a b d c+8 a^2 d^2\right ) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 b^3 d^{5/2} f \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {c^{3/2} (B e-A f) \sqrt {b x^2+a} \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^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {c} e^2 (B e-A f) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} f^4 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {4 B c^{3/2} (b c+a d) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{15 b^2 d^{5/2} f \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {\sqrt {-a} e^2 (B e-A f) \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \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^4 \sqrt {b x^2+a} \sqrt {d x^2+c}}\)

Input:

Int[(x^6*(A + B*x^2))/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)),x]
 

Output:

(B*(8*b^2*c^2 + 7*a*b*c*d + 8*a^2*d^2)*x*Sqrt[a + b*x^2])/(15*b^3*d^2*f*Sq 
rt[c + d*x^2]) + (e*(B*e - A*f)*x*Sqrt[a + b*x^2])/(b*f^3*Sqrt[c + d*x^2]) 
 + (2*(b*c + a*d)*(B*e - A*f)*x*Sqrt[a + b*x^2])/(3*b^2*d*f^2*Sqrt[c + d*x 
^2]) - (4*B*(b*c + a*d)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(15*b^2*d^2*f) 
- ((B*e - A*f)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*b*d*f^2) + (B*x^3*Sqr 
t[a + b*x^2]*Sqrt[c + d*x^2])/(5*b*d*f) - (B*Sqrt[c]*(8*b^2*c^2 + 7*a*b*c* 
d + 8*a^2*d^2)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - 
(b*c)/(a*d)])/(15*b^3*d^(5/2)*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]*EllipticE[ArcTan[(Sq 
rt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b*Sqrt[d]*f^3*Sqrt[(c*(a + b*x^2))/( 
a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (2*Sqrt[c]*(b*c + a*d)*(B*e - A*f)*Sqrt 
[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*b^ 
2*d^(3/2)*f^2*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (4* 
B*c^(3/2)*(b*c + a*d)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c] 
], 1 - (b*c)/(a*d)])/(15*b^2*d^(5/2)*f*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2) 
)]*Sqrt[c + d*x^2]) - (Sqrt[c]*e^2*(B*e - A*f)*Sqrt[a + b*x^2]*EllipticF[A 
rcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[d]*f^4*Sqrt[(c*(a + 
b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (c^(3/2)*(B*e - A*f)*Sqrt[a + 
b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*b*d^(3/ 
2)*f^2*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (Sqrt[-...
 

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 = 20.75 (sec) , antiderivative size = 612, normalized size of antiderivative = 0.89

method	result	size
risch	\(\frac {x \left (3 x^{2} B b d f +5 A b d f -4 a B d f -4 B b c f -5 b B d e \right ) \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{15 b^{2} d^{2} f^{2}}-\frac {\left (\frac {\left (5 A a b c d \,f^{3}-15 A \,b^{2} d^{2} e^{2} f -4 a^{2} B c d \,f^{3}-4 B a b \,c^{2} f^{3}-5 B a b c d e \,f^{2}+15 B \,b^{2} d^{2} e^{3}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{f^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (10 a \,d^{2} f^{2} b A +10 b^{2} c d \,f^{2} A +15 b^{2} d^{2} e f A -8 a^{2} B \,d^{2} f^{2}-7 a b c d \,f^{2} B -10 a b \,d^{2} e f B -8 b^{2} c^{2} f^{2} B -10 b^{2} c d e f B -15 b^{2} B \,d^{2} e^{2}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{f \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}+\frac {15 b^{2} d^{2} e^{2} \left (A f -B e \right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right )}{f^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{15 b^{2} d^{2} f^{2} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(612\)
default	\(\text {Expression too large to display}\)	\(1799\)
elliptic	\(\text {Expression too large to display}\)	\(2306\)

Input:

int(x^6*(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:

1/15*x*(3*B*b*d*f*x^2+5*A*b*d*f-4*B*a*d*f-4*B*b*c*f-5*B*b*d*e)*(b*x^2+a)^( 
1/2)*(d*x^2+c)^(1/2)/b^2/d^2/f^2-1/15/b^2/d^2/f^2*((5*A*a*b*c*d*f^3-15*A*b 
^2*d^2*e^2*f-4*B*a^2*c*d*f^3-4*B*a*b*c^2*f^3-5*B*a*b*c*d*e*f^2+15*B*b^2*d^ 
2*e^3)/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))-1 
/f*(10*A*a*b*d^2*f^2+10*A*b^2*c*d*f^2+15*A*b^2*d^2*e*f-8*B*a^2*d^2*f^2-7*B 
*a*b*c*d*f^2-10*B*a*b*d^2*e*f-8*B*b^2*c^2*f^2-10*B*b^2*c*d*e*f-15*B*b^2*d^ 
2*e^2)*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)))+15*b^2*d^2*e^2*(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. \[ \int \frac {x^6 \left (A+B x^2\right )}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:

integrate(x^6*(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]

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

integrate(x**6*(B*x**2+A)/(b*x**2+a)**(1/2)/(d*x**2+c)**(1/2)/(f*x**2+e),x 
)
 

Output:

Integral(x**6*(A + B*x**2)/(sqrt(a + b*x**2)*sqrt(c + d*x**2)*(e + f*x**2) 
), x)
 

Maxima [F]

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

integrate(x^6*(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^6/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)), x 
)
 

Giac [F]

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

integrate(x^6*(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^6/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)), x 
)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

int((x^6*(A + B*x^2))/((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)),x)
 

Output:

int((x^6*(A + B*x^2))/((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)), x 
)
 

Reduce [F]

\[ \int \frac {x^6 \left (A+B x^2\right )}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx =\text {Too large to display} \] Input:

int(x^6*(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)*a*d*f*x - 4*sqrt(c + d*x**2)*sqrt(a + b 
*x**2)*b*c*f*x - 5*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*d*e*x + 3*sqrt(c + 
d*x**2)*sqrt(a + b*x**2)*b*d*f*x**3 - 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)*a**2*d**2*f**2 - 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)*a*b*c*d*f**2 - 5* 
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 
*b*d**2*e*f + 8*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**2*c**2*f**2 + 10*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**2*c*d*e*f + 15*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**2*d**2*e**2 - int((s 
qrt(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**2*c* 
d*f**2 - 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*...