\(\int \frac {x^6 \sqrt {a+b x^2} \sqrt {c+d x^2}}{e+f x^2} \, dx\) [69]

Optimal result

Integrand size = 35, antiderivative size = 787 \[ \int \frac {x^6 \sqrt {a+b x^2} \sqrt {c+d x^2}}{e+f x^2} \, dx=-\frac {(7 b d e+4 b c f-a d f) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{35 b d f^2}+\frac {x^3 \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{7 d f}+\frac {\frac {\left (8 a^3 d^3 f^3+a^2 b d^2 f^2 (14 d e-5 c f)+a b^2 d f \left (35 d^2 e^2-14 c d e f-5 c^2 f^2\right )-b^3 \left (105 d^3 e^3-35 c d^2 e^2 f-14 c^2 d e f^2-8 c^3 f^3\right )\right ) x \sqrt {c+d x^2}}{b d^2 f^2 \sqrt {a+b x^2}}-\left (7 b c e+7 a d e-\frac {35 b d e^2}{f}-2 a c f+\frac {4 b c^2 f}{d}+\frac {4 a^2 d f}{b}\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}-\frac {\sqrt {a} \left (8 a^3 d^3 f^3+a^2 b d^2 f^2 (14 d e-5 c f)+a b^2 d f \left (35 d^2 e^2-14 c d e f-5 c^2 f^2\right )-b^3 \left (105 d^3 e^3-35 c d^2 e^2 f-14 c^2 d e f^2-8 c^3 f^3\right )\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{b^{3/2} d^2 f^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {a^{3/2} \left (4 a^2 c d^2 f^3+a b c d f^2 (7 d e-2 c f)-b^2 \left (105 d^3 e^3-70 c d^2 e^2 f-7 c^2 d e f^2-4 c^3 f^3\right )\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{b^{3/2} c d f^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {105 a^{3/2} \sqrt {b} d e^2 (d e-c f) \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{c f^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}}{105 b d f^2} \] Output:

-1/35*(-a*d*f+4*b*c*f+7*b*d*e)*x^3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b/d/f^2 
+1/7*x^3*(b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/d/f+1/105*((8*a^3*d^3*f^3+a^2*b*d 
^2*f^2*(-5*c*f+14*d*e)+a*b^2*d*f*(-5*c^2*f^2-14*c*d*e*f+35*d^2*e^2)-b^3*(- 
8*c^3*f^3-14*c^2*d*e*f^2-35*c*d^2*e^2*f+105*d^3*e^3))*x*(d*x^2+c)^(1/2)/b/ 
d^2/f^2/(b*x^2+a)^(1/2)-(7*b*c*e+7*a*d*e-35*b*d*e^2/f-2*a*c*f+4*b*c^2*f/d+ 
4*a^2*d*f/b)*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)-a^(1/2)*(8*a^3*d^3*f^3+a^2* 
b*d^2*f^2*(-5*c*f+14*d*e)+a*b^2*d*f*(-5*c^2*f^2-14*c*d*e*f+35*d^2*e^2)-b^3 
*(-8*c^3*f^3-14*c^2*d*e*f^2-35*c*d^2*e^2*f+105*d^3*e^3))*(d*x^2+c)^(1/2)*E 
llipticE(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)+a^(3/2)*(4*a^2*c*d^2 
*f^3+a*b*c*d*f^2*(-2*c*f+7*d*e)-b^2*(-4*c^3*f^3-7*c^2*d*e*f^2-70*c*d^2*e^2 
*f+105*d^3*e^3))*(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/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+ 
a))^(1/2)+105*a^(3/2)*b^(1/2)*d*e^2*(-c*f+d*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))/c/f^2/(b* 
x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2))/b/d/f^2
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 4.64 (sec) , antiderivative size = 596, normalized size of antiderivative = 0.76 \[ \int \frac {x^6 \sqrt {a+b x^2} \sqrt {c+d x^2}}{e+f x^2} \, dx=\frac {-\sqrt {\frac {b}{a}} d f^2 x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (4 a^2 d^2 f^2+a b d f \left (7 d e-2 c f-3 d f x^2\right )+b^2 \left (4 c^2 f^2+c d f \left (7 e-3 f x^2\right )+d^2 \left (-35 e^2+21 e f x^2-15 f^2 x^4\right )\right )\right )-i c f \left (8 a^3 d^3 f^3+a^2 b d^2 f^2 (14 d e-5 c f)+a b^2 d f \left (35 d^2 e^2-14 c d e f-5 c^2 f^2\right )+b^3 \left (-105 d^3 e^3+35 c d^2 e^2 f+14 c^2 d e f^2+8 c^3 f^3\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i \left (4 a^3 c d^3 f^4+a^2 b c d^2 f^3 (7 d e-3 c f)+a b^2 d f \left (105 d^3 e^3-35 c d^2 e^2 f-21 c^2 d e f^2-9 c^3 f^3\right )+b^3 \left (-105 d^4 e^4+35 c^2 d^2 e^2 f^2+14 c^3 d e f^3+8 c^4 f^4\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+105 i b^2 d^3 e^2 (-b e+a f) (-d e+c f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{105 a^2 \left (\frac {b}{a}\right )^{5/2} d^3 f^5 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

Integrate[(x^6*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)*(4*a^2*d^2*f^2 + a*b*d*f*(7*d 
*e - 2*c*f - 3*d*f*x^2) + b^2*(4*c^2*f^2 + c*d*f*(7*e - 3*f*x^2) + d^2*(-3 
5*e^2 + 21*e*f*x^2 - 15*f^2*x^4)))) - I*c*f*(8*a^3*d^3*f^3 + a^2*b*d^2*f^2 
*(14*d*e - 5*c*f) + a*b^2*d*f*(35*d^2*e^2 - 14*c*d*e*f - 5*c^2*f^2) + b^3* 
(-105*d^3*e^3 + 35*c*d^2*e^2*f + 14*c^2*d*e*f^2 + 8*c^3*f^3))*Sqrt[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^3*c*d^3*f^4 + a^2*b*c*d^2*f^3*(7*d*e - 3*c*f) + a*b^2*d*f*(105*d 
^3*e^3 - 35*c*d^2*e^2*f - 21*c^2*d*e*f^2 - 9*c^3*f^3) + b^3*(-105*d^4*e^4 
+ 35*c^2*d^2*e^2*f^2 + 14*c^3*d*e*f^3 + 8*c^4*f^4))*Sqrt[1 + (b*x^2)/a]*Sq 
rt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (105*I) 
*b^2*d^3*e^2*(-(b*e) + a*f)*(-(d*e) + c*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)])/(10 
5*a^2*(b/a)^(5/2)*d^3*f^5*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
 

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.

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

\(\Big \downarrow \) 450

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 450

Int[((g_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_)^2)^(p_.)*((c_.) + (d_.)*(x_)^2)^ 
(q_.)*((e_.) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Unintegrable[(g*x)^m*(a + 
b*x^2)^p*(c + d*x^2)^q*(e + f*x^2)^r, x] /; FreeQ[{a, b, c, d, e, f, g, m, 
p, q, r}, x]
 

Maple [A] (verified)

Time = 8.97 (sec) , antiderivative size = 785, normalized size of antiderivative = 1.00

method	result	size
risch	\(-\frac {x \left (-15 f^{2} x^{4} b^{2} d^{2}-3 a b \,d^{2} f^{2} x^{2}-3 b^{2} c d \,f^{2} x^{2}+21 b^{2} d^{2} e f \,x^{2}+4 a^{2} d^{2} f^{2}-2 a b c d \,f^{2}+7 a b \,d^{2} e f +4 b^{2} c^{2} f^{2}+7 b^{2} c d e f -35 b^{2} d^{2} e^{2}\right ) \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{105 b^{2} d^{2} f^{3}}+\frac {\left (\frac {\left (4 a^{3} c \,d^{2} f^{4}-2 a^{2} b \,c^{2} d \,f^{4}+7 a^{2} b c \,d^{2} e \,f^{3}+4 a \,b^{2} c^{3} f^{4}+7 a \,b^{2} c^{2} d e \,f^{3}+70 a \,b^{2} c \,d^{2} e^{2} f^{2}-105 a \,b^{2} d^{3} e^{3} f -105 b^{3} c \,d^{2} e^{3} f +105 b^{3} d^{3} e^{4}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{f^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (8 a^{3} d^{3} f^{3}-5 a^{2} b c \,d^{2} f^{3}+14 a^{2} b \,d^{3} e \,f^{2}-5 a \,b^{2} c^{2} d \,f^{3}-14 a \,b^{2} c \,d^{2} e \,f^{2}+35 a \,b^{2} d^{3} e^{2} f +8 b^{3} c^{3} f^{3}+14 b^{3} c^{2} d e \,f^{2}+35 b^{3} c \,d^{2} e^{2} f -105 b^{3} d^{3} e^{3}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{f \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}-\frac {105 e^{2} \left (a c \,f^{2}-a d e f -b c e f +b d \,e^{2}\right ) b^{2} d^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right )}{f^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{105 d^{2} b^{2} f^{3} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(785\)
default	\(\text {Expression too large to display}\)	\(2263\)
elliptic	\(\text {Expression too large to display}\)	\(2716\)

Input:

int(x^6*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/(f*x^2+e),x,method=_RETURNVERBOSE)
 

Output:

-1/105*x*(-15*b^2*d^2*f^2*x^4-3*a*b*d^2*f^2*x^2-3*b^2*c*d*f^2*x^2+21*b^2*d 
^2*e*f*x^2+4*a^2*d^2*f^2-2*a*b*c*d*f^2+7*a*b*d^2*e*f+4*b^2*c^2*f^2+7*b^2*c 
*d*e*f-35*b^2*d^2*e^2)*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b^2/d^2/f^3+1/105/d 
^2/b^2/f^3*((4*a^3*c*d^2*f^4-2*a^2*b*c^2*d*f^4+7*a^2*b*c*d^2*e*f^3+4*a*b^2 
*c^3*f^4+7*a*b^2*c^2*d*e*f^3+70*a*b^2*c*d^2*e^2*f^2-105*a*b^2*d^3*e^3*f-10 
5*b^3*c*d^2*e^3*f+105*b^3*d^3*e^4)/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*(8*a^3*d^3*f^3-5*a^2*b*c*d^2*f^3+14*a^2*b*d 
^3*e*f^2-5*a*b^2*c^2*d*f^3-14*a*b^2*c*d^2*e*f^2+35*a*b^2*d^3*e^2*f+8*b^3*c 
^3*f^3+14*b^3*c^2*d*e*f^2+35*b^3*c*d^2*e^2*f-105*b^3*d^3*e^3)*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)))-105*e^2*(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2) 
*b^2*d^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)*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. \[ \int \frac {x^6 \sqrt {a+b x^2} \sqrt {c+d x^2}}{e+f x^2} \, dx=\text {Timed out} \] Input:

integrate(x^6*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/(f*x^2+e),x, algorithm="fric 
as")
 

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(x^6*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/(f*x^2+e),x, algorithm="maxi 
ma")
 

Output:

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

Giac [F]

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

integrate(x^6*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/(f*x^2+e),x, algorithm="giac 
")
 

Output:

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

Mupad [F(-1)]

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

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

Reduce [F]

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

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

Output:

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