\(\int \frac {x^6 (e+f x^2)}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx\) [174]

Optimal result

Integrand size = 33, antiderivative size = 524 \[ \int \frac {x^6 \left (e+f x^2\right )}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=-\frac {\left (48 a^3 d^3 f-a b^2 c d (49 d e-40 c f)-8 b^3 c^2 (7 d e-6 c f)-8 a^2 b d^2 (7 d e-5 c f)\right ) x \sqrt {c+d x^2}}{105 b^3 d^4 \sqrt {a+b x^2}}-\frac {(25 a b c d f+4 (b c+a d) (7 b d e-6 b c f-6 a d f)) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{105 b^3 d^3}+\frac {(7 b d e-6 b c f-6 a d f) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{35 b^2 d^2}+\frac {f x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{7 b d}+\frac {\sqrt {a} \left (48 a^3 d^3 f-a b^2 c d (49 d e-40 c f)-8 b^3 c^2 (7 d e-6 c f)-8 a^2 b d^2 (7 d e-5 c f)\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{105 b^{7/2} d^4 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {a^{3/2} (25 a b c d f+4 (b c+a d) (7 b d e-6 b c f-6 a d f)) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{105 b^{7/2} d^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

-1/105*(48*a^3*d^3*f-a*b^2*c*d*(-40*c*f+49*d*e)-8*b^3*c^2*(-6*c*f+7*d*e)-8 
*a^2*b*d^2*(-5*c*f+7*d*e))*x*(d*x^2+c)^(1/2)/b^3/d^4/(b*x^2+a)^(1/2)-1/105 
*(25*a*b*c*d*f+4*(a*d+b*c)*(-6*a*d*f-6*b*c*f+7*b*d*e))*x*(b*x^2+a)^(1/2)*( 
d*x^2+c)^(1/2)/b^3/d^3+1/35*(-6*a*d*f-6*b*c*f+7*b*d*e)*x^3*(b*x^2+a)^(1/2) 
*(d*x^2+c)^(1/2)/b^2/d^2+1/7*f*x^5*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b/d+1/1 
05*a^(1/2)*(48*a^3*d^3*f-a*b^2*c*d*(-40*c*f+49*d*e)-8*b^3*c^2*(-6*c*f+7*d* 
e)-8*a^2*b*d^2*(-5*c*f+7*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^(7/2)/d^4/(b*x^2+a)^(1/2)/(a*(d*x^ 
2+c)/c/(b*x^2+a))^(1/2)+1/105*a^(3/2)*(25*a*b*c*d*f+4*(a*d+b*c)*(-6*a*d*f- 
6*b*c*f+7*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^(7/2)/d^3/(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 = 7.40 (sec) , antiderivative size = 382, normalized size of antiderivative = 0.73 \[ \int \frac {x^6 \left (e+f x^2\right )}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (24 a^2 d^2 f+a b d \left (23 c f-2 d \left (14 e+9 f x^2\right )\right )+b^2 \left (24 c^2 f+3 d^2 x^2 \left (7 e+5 f x^2\right )-2 c d \left (14 e+9 f x^2\right )\right )\right )+i c \left (48 a^3 d^3 f-8 a^2 b d^2 (7 d e-5 c f)+8 b^3 c^2 (-7 d e+6 c f)+a b^2 c d (-49 d e+40 c f)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c \left (24 a^3 d^3 f+8 b^3 c^2 (-7 d e+6 c f)+a b^2 c d (-21 d e+16 c f)+a^2 b d^2 (-28 d e+17 c f)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{105 b^3 \sqrt {\frac {b}{a}} d^4 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

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

Output:

(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(24*a^2*d^2*f + a*b*d*(23*c*f - 2*d 
*(14*e + 9*f*x^2)) + b^2*(24*c^2*f + 3*d^2*x^2*(7*e + 5*f*x^2) - 2*c*d*(14 
*e + 9*f*x^2))) + I*c*(48*a^3*d^3*f - 8*a^2*b*d^2*(7*d*e - 5*c*f) + 8*b^3* 
c^2*(-7*d*e + 6*c*f) + a*b^2*c*d*(-49*d*e + 40*c*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*c*(2 
4*a^3*d^3*f + 8*b^3*c^2*(-7*d*e + 6*c*f) + a*b^2*c*d*(-21*d*e + 16*c*f) + 
a^2*b*d^2*(-28*d*e + 17*c*f))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Elli 
pticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(105*b^3*Sqrt[b/a]*d^4*Sqrt[a 
+ b*x^2]*Sqrt[c + d*x^2])
 

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 489, normalized size of antiderivative = 0.93, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {444, 444, 25, 444, 25, 406, 320, 388, 313}

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.

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

\(\Big \downarrow \) 444

\(\displaystyle \frac {f x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{7 b d}-\frac {\int \frac {x^4 \left (5 a c f-(7 b d e-6 b c f-6 a d f) x^2\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{7 b d}\)

\(\Big \downarrow \) 444

\(\displaystyle \frac {f x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{7 b d}-\frac {-\frac {\int -\frac {x^2 \left (3 a c (7 b d e-6 b c f-6 a d f)-\left (-4 c (7 d e-6 c f) b^2-a d (28 d e-23 c f) b+24 a^2 d^2 f\right ) x^2\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 b d}-\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (-6 a d f-6 b c f+7 b d e)}{5 b d}}{7 b d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {f x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{7 b d}-\frac {\frac {\int \frac {x^2 \left (3 a c (7 b d e-6 b c f-6 a d f)-\left (-4 c (7 d e-6 c f) b^2-a d (28 d e-23 c f) b+24 a^2 d^2 f\right ) x^2\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 b d}-\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (-6 a d f-6 b c f+7 b d e)}{5 b d}}{7 b d}\)

\(\Big \downarrow \) 444

\(\displaystyle \frac {f x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{7 b d}-\frac {\frac {-\frac {\int -\frac {\left (-8 c^2 (7 d e-6 c f) b^3-a c d (49 d e-40 c f) b^2-8 a^2 d^2 (7 d e-5 c f) b+48 a^3 d^3 f\right ) x^2+a c \left (-4 c (7 d e-6 c f) b^2-a d (28 d e-23 c f) b+24 a^2 d^2 f\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}-\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {24 a^2 d f}{b}-a (28 d e-23 c f)-\frac {4 b c (7 d e-6 c f)}{d}\right )}{5 b d}-\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (-6 a d f-6 b c f+7 b d e)}{5 b d}}{7 b d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {f x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{7 b d}-\frac {\frac {\frac {\int \frac {\left (-8 c^2 (7 d e-6 c f) b^3-a c d (49 d e-40 c f) b^2-8 a^2 d^2 (7 d e-5 c f) b+48 a^3 d^3 f\right ) x^2+a c \left (-4 c (7 d e-6 c f) b^2-a d (28 d e-23 c f) b+24 a^2 d^2 f\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}-\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {24 a^2 d f}{b}-a (28 d e-23 c f)-\frac {4 b c (7 d e-6 c f)}{d}\right )}{5 b d}-\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (-6 a d f-6 b c f+7 b d e)}{5 b d}}{7 b d}\)

\(\Big \downarrow \) 406

\(\displaystyle \frac {f x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{7 b d}-\frac {\frac {\frac {a c \left (24 a^2 d^2 f-a b d (28 d e-23 c f)-4 b^2 c (7 d e-6 c f)\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\left (48 a^3 d^3 f-8 a^2 b d^2 (7 d e-5 c f)-a b^2 c d (49 d e-40 c f)-8 b^3 c^2 (7 d e-6 c f)\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}-\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {24 a^2 d f}{b}-a (28 d e-23 c f)-\frac {4 b c (7 d e-6 c f)}{d}\right )}{5 b d}-\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (-6 a d f-6 b c f+7 b d e)}{5 b d}}{7 b d}\)

\(\Big \downarrow \) 320

\(\displaystyle \frac {f x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{7 b d}-\frac {\frac {\frac {\left (48 a^3 d^3 f-8 a^2 b d^2 (7 d e-5 c f)-a b^2 c d (49 d e-40 c f)-8 b^3 c^2 (7 d e-6 c f)\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} \left (24 a^2 d^2 f-a b d (28 d e-23 c f)-4 b^2 c (7 d e-6 c f)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 b d}-\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {24 a^2 d f}{b}-a (28 d e-23 c f)-\frac {4 b c (7 d e-6 c f)}{d}\right )}{5 b d}-\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (-6 a d f-6 b c f+7 b d e)}{5 b d}}{7 b d}\)

\(\Big \downarrow \) 388

\(\displaystyle \frac {f x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{7 b d}-\frac {\frac {\frac {\left (48 a^3 d^3 f-8 a^2 b d^2 (7 d e-5 c f)-a b^2 c d (49 d e-40 c f)-8 b^3 c^2 (7 d e-6 c f)\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} \left (24 a^2 d^2 f-a b d (28 d e-23 c f)-4 b^2 c (7 d e-6 c f)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 b d}-\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {24 a^2 d f}{b}-a (28 d e-23 c f)-\frac {4 b c (7 d e-6 c f)}{d}\right )}{5 b d}-\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (-6 a d f-6 b c f+7 b d e)}{5 b d}}{7 b d}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {f x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}}{7 b d}-\frac {\frac {\frac {\frac {c^{3/2} \sqrt {a+b x^2} \left (24 a^2 d^2 f-a b d (28 d e-23 c f)-4 b^2 c (7 d e-6 c f)\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\left (48 a^3 d^3 f-8 a^2 b d^2 (7 d e-5 c f)-a b^2 c d (49 d e-40 c f)-8 b^3 c^2 (7 d e-6 c f)\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 b d}-\frac {1}{3} x \sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {24 a^2 d f}{b}-a (28 d e-23 c f)-\frac {4 b c (7 d e-6 c f)}{d}\right )}{5 b d}-\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2} (-6 a d f-6 b c f+7 b d e)}{5 b d}}{7 b d}\)

Input:

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

Output:

(f*x^5*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(7*b*d) - (-1/5*((7*b*d*e - 6*b*c* 
f - 6*a*d*f)*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(b*d) + (-1/3*(((24*a^2* 
d*f)/b - a*(28*d*e - 23*c*f) - (4*b*c*(7*d*e - 6*c*f))/d)*x*Sqrt[a + b*x^2 
]*Sqrt[c + d*x^2]) + ((48*a^3*d^3*f - a*b^2*c*d*(49*d*e - 40*c*f) - 8*b^3* 
c^2*(7*d*e - 6*c*f) - 8*a^2*b*d^2*(7*d*e - 5*c*f))*((x*Sqrt[a + b*x^2])/(b 
*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/ 
Sqrt[c]], 1 - (b*c)/(a*d)])/(b*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2) 
)]*Sqrt[c + d*x^2])) + (c^(3/2)*(24*a^2*d^2*f - a*b*d*(28*d*e - 23*c*f) - 
4*b^2*c*(7*d*e - 6*c*f))*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt 
[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqr 
t[c + d*x^2]))/(3*b*d))/(5*b*d))/(7*b*d)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 313

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim 
p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c 
+ d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ 
[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
 

rule 320

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( 
c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre 
eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 388

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] 
 :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b   Int[Sqrt[ 
a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 406

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

rule 444

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q 
_.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[f*g*(g*x)^(m - 1)*(a + b*x^2)^ 
(p + 1)*((c + d*x^2)^(q + 1)/(b*d*(m + 2*(p + q + 1) + 1))), x] - Simp[g^2/ 
(b*d*(m + 2*(p + q + 1) + 1))   Int[(g*x)^(m - 2)*(a + b*x^2)^p*(c + d*x^2) 
^q*Simp[a*f*c*(m - 1) + (a*f*d*(m + 2*q + 1) + b*(f*c*(m + 2*p + 1) - e*d*( 
m + 2*(p + q + 1) + 1)))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, 
q}, x] && GtQ[m, 1]
 

Maple [A] (verified)

Time = 10.67 (sec) , antiderivative size = 575, normalized size of antiderivative = 1.10

method	result	size
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {f \,x^{5} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{7 b d}+\frac {\left (e -\frac {f \left (6 a d +6 b c \right )}{7 b d}\right ) x^{3} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{5 b d}+\frac {\left (-\frac {5 a c f}{7 b d}-\frac {\left (e -\frac {f \left (6 a d +6 b c \right )}{7 b d}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b d}-\frac {\left (-\frac {5 a c f}{7 b d}-\frac {\left (e -\frac {f \left (6 a d +6 b c \right )}{7 b d}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) a c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{3 b d \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (-\frac {3 \left (e -\frac {f \left (6 a d +6 b c \right )}{7 b d}\right ) a c}{5 b d}-\frac {\left (-\frac {5 a c f}{7 b d}-\frac {\left (e -\frac {f \left (6 a d +6 b c \right )}{7 b d}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) \left (2 a d +2 b c \right )}{3 b d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(575\)
risch	\(\frac {x \left (15 f \,x^{4} b^{2} d^{2}-18 a b \,d^{2} f \,x^{2}-18 b^{2} c f \,x^{2} d +21 b^{2} d^{2} e \,x^{2}+24 f \,d^{2} a^{2}+23 f d c b a -28 a b \,d^{2} e +24 f \,c^{2} b^{2}-28 d \,b^{2} c e \right ) \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{105 b^{3} d^{3}}-\frac {\left (-\frac {\left (48 f \,d^{3} a^{3}+40 a^{2} b c \,d^{2} f -56 a^{2} b \,d^{3} e +40 a \,b^{2} c^{2} d f -49 a \,b^{2} c \,d^{2} e +48 b^{3} c^{3} f -56 b^{3} c^{2} 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 )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}+\frac {24 a \,b^{2} c^{3} f \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 )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {24 a^{3} c \,d^{2} f \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 )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {28 a \,c^{2} e d \,b^{2} \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 )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {28 a^{2} b c \,d^{2} e \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 )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {23 a^{2} b \,c^{2} d f \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 )}{\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 b^{3} d^{3} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(834\)
default	\(\text {Expression too large to display}\)	\(1332\)

Input:

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

Output:

((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(1/7/b/d*f*x^5 
*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+1/5*(e-1/7/b/d*f*(6*a*d+6*b*c))/b/d*x 
^3*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+1/3*(-5/7*a/b*c/d*f-1/5*(e-1/7/b/d* 
f*(6*a*d+6*b*c))/b/d*(4*a*d+4*b*c))/b/d*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1 
/2)-1/3*(-5/7*a/b*c/d*f-1/5*(e-1/7/b/d*f*(6*a*d+6*b*c))/b/d*(4*a*d+4*b*c)) 
/b/d*a*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)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-(-3 
/5*(e-1/7/b/d*f*(6*a*d+6*b*c))/b/d*a*c-1/3*(-5/7*a/b*c/d*f-1/5*(e-1/7/b/d* 
f*(6*a*d+6*b*c))/b/d*(4*a*d+4*b*c))/b/d*(2*a*d+2*b*c))*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*(Ell 
ipticF(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))))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 494, normalized size of antiderivative = 0.94 \[ \int \frac {x^6 \left (e+f x^2\right )}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=-\frac {\sqrt {b d} {\left (7 \, {\left (8 \, b^{3} c^{3} d + 7 \, a b^{2} c^{2} d^{2} + 8 \, a^{2} b c d^{3}\right )} e - 8 \, {\left (6 \, b^{3} c^{4} + 5 \, a b^{2} c^{3} d + 5 \, a^{2} b c^{2} d^{2} + 6 \, a^{3} c d^{3}\right )} f\right )} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - \sqrt {b d} {\left (7 \, {\left (8 \, b^{3} c^{3} d + 7 \, a b^{2} c^{2} d^{2} + 4 \, a^{2} b d^{4} + 4 \, {\left (2 \, a^{2} b + a b^{2}\right )} c d^{3}\right )} e - {\left (48 \, b^{3} c^{4} + 40 \, a b^{2} c^{3} d + 24 \, a^{3} d^{4} + 8 \, {\left (5 \, a^{2} b + 3 \, a b^{2}\right )} c^{2} d^{2} + {\left (48 \, a^{3} + 23 \, a^{2} b\right )} c d^{3}\right )} f\right )} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (15 \, b^{3} d^{4} f x^{6} + 3 \, {\left (7 \, b^{3} d^{4} e - 6 \, {\left (b^{3} c d^{3} + a b^{2} d^{4}\right )} f\right )} x^{4} - {\left (28 \, {\left (b^{3} c d^{3} + a b^{2} d^{4}\right )} e - {\left (24 \, b^{3} c^{2} d^{2} + 23 \, a b^{2} c d^{3} + 24 \, a^{2} b d^{4}\right )} f\right )} x^{2} + 7 \, {\left (8 \, b^{3} c^{2} d^{2} + 7 \, a b^{2} c d^{3} + 8 \, a^{2} b d^{4}\right )} e - 8 \, {\left (6 \, b^{3} c^{3} d + 5 \, a b^{2} c^{2} d^{2} + 5 \, a^{2} b c d^{3} + 6 \, a^{3} d^{4}\right )} f\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{105 \, b^{4} d^{5} x} \] Input:

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

Output:

-1/105*(sqrt(b*d)*(7*(8*b^3*c^3*d + 7*a*b^2*c^2*d^2 + 8*a^2*b*c*d^3)*e - 8 
*(6*b^3*c^4 + 5*a*b^2*c^3*d + 5*a^2*b*c^2*d^2 + 6*a^3*c*d^3)*f)*x*sqrt(-c/ 
d)*elliptic_e(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - sqrt(b*d)*(7*(8*b^3*c^3*d 
 + 7*a*b^2*c^2*d^2 + 4*a^2*b*d^4 + 4*(2*a^2*b + a*b^2)*c*d^3)*e - (48*b^3* 
c^4 + 40*a*b^2*c^3*d + 24*a^3*d^4 + 8*(5*a^2*b + 3*a*b^2)*c^2*d^2 + (48*a^ 
3 + 23*a^2*b)*c*d^3)*f)*x*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/d)/x), a*d/ 
(b*c)) - (15*b^3*d^4*f*x^6 + 3*(7*b^3*d^4*e - 6*(b^3*c*d^3 + a*b^2*d^4)*f) 
*x^4 - (28*(b^3*c*d^3 + a*b^2*d^4)*e - (24*b^3*c^2*d^2 + 23*a*b^2*c*d^3 + 
24*a^2*b*d^4)*f)*x^2 + 7*(8*b^3*c^2*d^2 + 7*a*b^2*c*d^3 + 8*a^2*b*d^4)*e - 
 8*(6*b^3*c^3*d + 5*a*b^2*c^2*d^2 + 5*a^2*b*c*d^3 + 6*a^3*d^4)*f)*sqrt(b*x 
^2 + a)*sqrt(d*x^2 + c))/(b^4*d^5*x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(24*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*d**2*f*x + 23*sqrt(c + d*x**2)* 
sqrt(a + b*x**2)*a*b*c*d*f*x - 28*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*d* 
*2*e*x - 18*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*d**2*f*x**3 + 24*sqrt(c 
+ d*x**2)*sqrt(a + b*x**2)*b**2*c**2*f*x - 28*sqrt(c + d*x**2)*sqrt(a + b* 
x**2)*b**2*c*d*e*x - 18*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*c*d*f*x**3 
+ 21*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*d**2*e*x**3 + 15*sqrt(c + d*x* 
*2)*sqrt(a + b*x**2)*b**2*d**2*f*x**5 - 48*int((sqrt(c + d*x**2)*sqrt(a + 
b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a**3*d**3*f - 40*i 
nt((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b 
*d*x**4),x)*a**2*b*c*d**2*f + 56*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x* 
*2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a**2*b*d**3*e - 40*int((sqrt 
(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4) 
,x)*a*b**2*c**2*d*f + 49*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c 
 + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a*b**2*c*d**2*e - 48*int((sqrt(c + d 
*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*b* 
*3*c**3*f + 56*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x** 
2 + b*c*x**2 + b*d*x**4),x)*b**3*c**2*d*e - 24*int((sqrt(c + d*x**2)*sqrt( 
a + b*x**2))/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a**3*c*d**2*f - 23* 
int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a*c + a*d*x**2 + b*c*x**2 + b*d*x 
**4),x)*a**2*b*c**2*d*f + 28*int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a...