\(\int \frac {x^6 (A+B x^2)}{\sqrt {a-b x^2} \sqrt {c+d x^2} (e+f x^2)} \, dx\) [8]

Optimal result

Integrand size = 43, antiderivative size = 601 \[ \int \frac {x^6 \left (A+B x^2\right )}{\sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=-\frac {(4 a B d f-b (5 B d e+4 B c f-5 A d f)) x \sqrt {a-b x^2} \sqrt {c+d x^2}}{15 b^2 d^2 f^2}-\frac {B x^3 \sqrt {a-b x^2} \sqrt {c+d x^2}}{5 b d f}+\frac {\sqrt {a} \left (8 a^2 B d^2 f^2-a b d f (10 B d e+7 B c f-10 A d f)-b^2 \left (5 A d f (3 d e+2 c f)-B \left (15 d^2 e^2+10 c d e f+8 c^2 f^2\right )\right )\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{15 b^{5/2} d^3 f^3 \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}}}-\frac {\sqrt {a} \left (4 a^2 B c d^2 f^3-a b c d f^2 (5 B d e+3 B c f-5 A d f)-b^2 \left (5 A d f \left (3 d^2 e^2+3 c d e f+2 c^2 f^2\right )-B \left (15 d^3 e^3+15 c d^2 e^2 f+10 c^2 d e f^2+8 c^3 f^3\right )\right )\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{15 b^{5/2} d^3 f^4 \sqrt {a-b x^2} \sqrt {c+d x^2}}+\frac {\sqrt {a} e^2 (B e-A f) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} f^4 \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Output:

-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)))*(1-b*x^2/a)^(1/2)*(d* 
x^2+c)^(1/2)*EllipticE(b^(1/2)*x/a^(1/2),(-a*d/b/c)^(1/2))/b^(5/2)/d^3/f^3 
/(-b*x^2+a)^(1/2)/(1+d*x^2/c)^(1/2)-1/15*a^(1/2)*(4*a^2*B*c*d^2*f^3-a*b*c* 
d*f^2*(-5*A*d*f+3*B*c*f+5*B*d*e)-b^2*(5*A*d*f*(2*c^2*f^2+3*c*d*e*f+3*d^2*e 
^2)-B*(8*c^3*f^3+10*c^2*d*e*f^2+15*c*d^2*e^2*f+15*d^3*e^3)))*(1-b*x^2/a)^( 
1/2)*(1+d*x^2/c)^(1/2)*EllipticF(b^(1/2)*x/a^(1/2),(-a*d/b/c)^(1/2))/b^(5/ 
2)/d^3/f^4/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)+a^(1/2)*e^2*(-A*f+B*e)*(1-b*x^ 
2/a)^(1/2)*(1+d*x^2/c)^(1/2)*EllipticPi(b^(1/2)*x/a^(1/2),-a*f/b/e,(-a*d/b 
/c)^(1/2))/b^(1/2)/f^4/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 13.63 (sec) , antiderivative size = 511, normalized size of antiderivative = 0.85 \[ \int \frac {x^6 \left (A+B x^2\right )}{\sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\frac {-\sqrt {-\frac {b}{a}} d f^2 x \left (a-b x^2\right ) \left (c+d x^2\right ) \left (5 A b d f+4 a B d f+b B \left (-5 d e-4 c f+3 d f x^2\right )\right )-i c f \left (8 a^2 B d^2 f^2+a b d f (-10 B d e-7 B c f+10 A d f)+b^2 \left (-5 A d f (3 d e+2 c f)+B \left (15 d^2 e^2+10 c d e f+8 c^2 f^2\right )\right )\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )+i \left (4 a^2 B c d^2 f^3+a b c d f^2 (-5 B d e-3 B c f+5 A d f)+b^2 \left (-5 A d f \left (3 d^2 e^2+3 c d e f+2 c^2 f^2\right )+B \left (15 d^3 e^3+15 c d^2 e^2 f+10 c^2 d e f^2+8 c^3 f^3\right )\right )\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )-15 i b^2 d^3 e^2 (B e-A f) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )}{15 b^2 \sqrt {-\frac {b}{a}} d^3 f^4 \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Input:

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

Output:

(-(Sqrt[-(b/a)]*d*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))) - 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)))*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Elli 
pticE[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 + 15*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[Sq 
rt[-(b/a)]*x], -((a*d)/(b*c))] - (15*I)*b^2*d^3*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*b^2*Sqrt[-(b/a)]*d^3*f^4*Sqrt[a - b*x^2]*Sq 
rt[c + d*x^2])
 

Rubi [A] (verified)

Time = 3.09 (sec) , antiderivative size = 1018, normalized size of antiderivative = 1.69, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, 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 {a-b x^2} \sqrt {d x^2+c} x^3}{5 b d f}+\frac {(B e-A f) \sqrt {a-b x^2} \sqrt {d x^2+c} x}{3 b d f^2}+\frac {4 B (b c-a d) \sqrt {a-b x^2} \sqrt {d x^2+c} x}{15 b^2 d^2 f}+\frac {2 \sqrt {a} (b c-a d) (B e-A f) \sqrt {1-\frac {b x^2}{a}} \sqrt {d x^2+c} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{3 b^{3/2} d^2 f^2 \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}+\frac {\sqrt {a} e (B e-A f) \sqrt {1-\frac {b x^2}{a}} \sqrt {d x^2+c} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d f^3 \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}+\frac {\sqrt {a} B \left (8 b^2 c^2-7 a b d c+8 a^2 d^2\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {d x^2+c} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{15 b^{5/2} d^3 f \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}-\frac {\sqrt {a} c (2 b c-a d) (B e-A f) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{3 b^{3/2} d^2 f^2 \sqrt {a-b x^2} \sqrt {d x^2+c}}-\frac {\sqrt {a} c e (B e-A f) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d f^3 \sqrt {a-b x^2} \sqrt {d x^2+c}}-\frac {\sqrt {a} e^2 (B e-A f) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} f^4 \sqrt {a-b x^2} \sqrt {d x^2+c}}-\frac {\sqrt {a} B c \left (8 b^2 c^2-3 a b d c+4 a^2 d^2\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{15 b^{5/2} d^3 f \sqrt {a-b x^2} \sqrt {d x^2+c}}+\frac {\sqrt {a} e^2 (B e-A f) \sqrt {1-\frac {b x^2}{a}} \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 {a-b x^2} \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:

(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*Sqrt[a - b 
*x^2]*Sqrt[c + d*x^2])/(5*b*d*f) + (Sqrt[a]*B*(8*b^2*c^2 - 7*a*b*c*d + 8*a 
^2*d^2)*Sqrt[1 - (b*x^2)/a]*Sqrt[c + d*x^2]*EllipticE[ArcSin[(Sqrt[b]*x)/S 
qrt[a]], -((a*d)/(b*c))])/(15*b^(5/2)*d^3*f*Sqrt[a - b*x^2]*Sqrt[1 + (d*x^ 
2)/c]) + (Sqrt[a]*e*(B*e - A*f)*Sqrt[1 - (b*x^2)/a]*Sqrt[c + d*x^2]*Ellipt 
icE[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(Sqrt[b]*d*f^3*Sqrt[a - 
b*x^2]*Sqrt[1 + (d*x^2)/c]) + (2*Sqrt[a]*(b*c - a*d)*(B*e - A*f)*Sqrt[1 - 
(b*x^2)/a]*Sqrt[c + d*x^2]*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/ 
(b*c))])/(3*b^(3/2)*d^2*f^2*Sqrt[a - b*x^2]*Sqrt[1 + (d*x^2)/c]) - (Sqrt[a 
]*B*c*(8*b^2*c^2 - 3*a*b*c*d + 4*a^2*d^2)*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d* 
x^2)/c]*EllipticF[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(15*b^(5/2 
)*d^3*f*Sqrt[a - b*x^2]*Sqrt[c + d*x^2]) - (Sqrt[a]*e^2*(B*e - A*f)*Sqrt[1 
 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[b]*x)/Sqrt[a]], - 
((a*d)/(b*c))])/(Sqrt[b]*f^4*Sqrt[a - b*x^2]*Sqrt[c + d*x^2]) - (Sqrt[a]*c 
*e*(B*e - A*f)*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[ArcSin[(S 
qrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(Sqrt[b]*d*f^3*Sqrt[a - b*x^2]*Sqrt[c 
 + d*x^2]) - (Sqrt[a]*c*(2*b*c - a*d)*(B*e - A*f)*Sqrt[1 - (b*x^2)/a]*Sqrt 
[1 + (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(3 
*b^(3/2)*d^2*f^2*Sqrt[a - b*x^2]*Sqrt[c + d*x^2]) + (Sqrt[a]*e^2*(B*e -...
 

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 = 21.18 (sec) , antiderivative size = 623, normalized size of antiderivative = 1.04

method	result	size
risch	\(-\frac {x \left (3 x^{2} B b d f +5 A b d f +4 a B d f -4 B b c f -5 b B d e \right ) \sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{15 b^{2} d^{2} f^{2}}+\frac {\left (\frac {\left (5 A a b c d \,f^{3}+15 A \,b^{2} d^{2} e^{2} f +4 a^{2} B c d \,f^{3}-4 B a b \,c^{2} f^{3}-5 B a b c d e \,f^{2}-15 B \,b^{2} d^{2} e^{3}\right ) \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{f^{2} \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}-\frac {\left (10 a \,d^{2} f^{2} b A -10 b^{2} c d \,f^{2} A -15 b^{2} d^{2} e f A +8 a^{2} B \,d^{2} f^{2}-7 a b c d \,f^{2} B -10 a b \,d^{2} e f B +8 b^{2} c^{2} f^{2} B +10 b^{2} c d e f B +15 b^{2} B \,d^{2} e^{2}\right ) c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )\right )}{f \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}\, d}-\frac {15 b^{2} d^{2} e^{2} \left (A f -B e \right ) \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {b}{a}}, -\frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {\frac {b}{a}}}\right )}{f^{2} \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}\right ) \sqrt {\left (-b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{15 b^{2} d^{2} f^{2} \sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(623\)
default	\(\text {Expression too large to display}\)	\(1804\)
elliptic	\(\text {Expression too large to display}\)	\(2374\)

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=_RET 
URNVERBOSE)
 

Output:

-1/15/b^2/d^2*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)/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, algo 
rithm="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, algo 
rithm="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, algo 
rithm="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. \[ \int \frac {x^6 \left (A+B x^2\right )}{\sqrt {a-b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int \frac {x^6\,\left (B\,x^2+A\right )}{\sqrt {a-b\,x^2}\,\sqrt {d\,x^2+c}\,\left (f\,x^2+e\right )} \,d x \] 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:

( - 9*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 + 18*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 - 17*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 - 25*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 
+ 9*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**2*c*d*f**2 + 18*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*...