\(\int \frac {x^6}{\sqrt {(a+b x^2) (c+d x^2)}} \, dx\) [701]

Optimal result

Integrand size = 23, antiderivative size = 370 \[ \int \frac {x^6}{\sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \, dx=-\frac {\left (9 a b c d-8 (b c+a d)^2\right ) x \left (c+d x^2\right )}{15 b^2 d^3 \sqrt {a c+(b c+a d) x^2+b d x^4}}-\frac {4 (b c+a d) x \sqrt {a c+(b c+a d) x^2+b d x^4}}{15 b^2 d^2}+\frac {x^3 \sqrt {a c+(b c+a d) x^2+b d x^4}}{5 b d}+\frac {c \left (9 a b c d-8 (b c+a d)^2\right ) \left (a+b x^2\right ) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{15 \sqrt {a} b^{5/2} d^3 \sqrt {a c+(b c+a d) x^2+b d x^4}}+\frac {4 \sqrt {a} c (b c+a d) \left (a+b x^2\right ) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{15 b^{5/2} d^2 \sqrt {a c+(b c+a d) x^2+b d x^4}} \] Output:

-1/15*(9*a*b*c*d-8*(a*d+b*c)^2)*x*(d*x^2+c)/b^2/d^3/(a*c+(a*d+b*c)*x^2+b*d 
*x^4)^(1/2)-4/15*(a*d+b*c)*x*(a*c+(a*d+b*c)*x^2+b*d*x^4)^(1/2)/b^2/d^2+1/5 
*x^3*(a*c+(a*d+b*c)*x^2+b*d*x^4)^(1/2)/b/d+1/15*c*(9*a*b*c*d-8*(a*d+b*c)^2 
)*(b*x^2+a)*(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)*EllipticE(b^(1/2)*x/a^(1/2)/(1 
+b*x^2/a)^(1/2),(1-a*d/b/c)^(1/2))/a^(1/2)/b^(5/2)/d^3/(a*c+(a*d+b*c)*x^2+ 
b*d*x^4)^(1/2)+4/15*a^(1/2)*c*(a*d+b*c)*(b*x^2+a)*(a*(d*x^2+c)/c/(b*x^2+a) 
)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/b^(5/ 
2)/d^2/(a*c+(a*d+b*c)*x^2+b*d*x^4)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.10 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.66 \[ \int \frac {x^6}{\sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \, dx=\frac {-\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (4 b c+4 a d-3 b d x^2\right )-i c \left (8 b^2 c^2+7 a b c d+8 a^2 d^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i c \left (8 b^2 c^2+3 a b c d+4 a^2 d^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{15 a^2 \left (\frac {b}{a}\right )^{5/2} d^3 \sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \] Input:

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

Output:

(-(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(4*b*c + 4*a*d - 3*b*d*x^2)) - I* 
c*(8*b^2*c^2 + 7*a*b*c*d + 8*a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2) 
/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + I*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[I*Arc 
Sinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(15*a^2*(b/a)^(5/2)*d^3*Sqrt[(a + b*x^2)* 
(c + d*x^2)])
 

Rubi [A] (verified)

Time = 1.38 (sec) , antiderivative size = 614, normalized size of antiderivative = 1.66, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2048, 1442, 1602, 1511, 27, 1416, 1509}

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.

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

\(\Big \downarrow \) 2048

\(\displaystyle \int \frac {x^6}{\sqrt {x^2 (a d+b c)+a c+b d x^4}}dx\)

\(\Big \downarrow \) 1442

\(\displaystyle \frac {x^3 \sqrt {x^2 (a d+b c)+a c+b d x^4}}{5 b d}-\frac {\int \frac {x^2 \left (4 (b c+a d) x^2+3 a c\right )}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{5 b d}\)

\(\Big \downarrow \) 1602

\(\displaystyle \frac {x^3 \sqrt {x^2 (a d+b c)+a c+b d x^4}}{5 b d}-\frac {\frac {4 x (a d+b c) \sqrt {x^2 (a d+b c)+a c+b d x^4}}{3 b d}-\frac {\int \frac {4 a c (b c+a d)-\left (9 a b c d-8 (b c+a d)^2\right ) x^2}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{3 b d}}{5 b d}\)

\(\Big \downarrow \) 1511

\(\displaystyle \frac {x^3 \sqrt {x^2 (a d+b c)+a c+b d x^4}}{5 b d}-\frac {\frac {4 x (a d+b c) \sqrt {x^2 (a d+b c)+a c+b d x^4}}{3 b d}-\frac {\frac {\sqrt {a} \sqrt {c} \left (9 a b c d-8 (a d+b c)^2\right ) \int \frac {\sqrt {a} \sqrt {c}-\sqrt {b} \sqrt {d} x^2}{\sqrt {a} \sqrt {c} \sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}-\frac {\sqrt {a} \sqrt {c} \left (-8 (a d+b c)^2-4 \sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d} (a d+b c)+9 a b c d\right ) \int \frac {1}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}}{3 b d}}{5 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x^3 \sqrt {x^2 (a d+b c)+a c+b d x^4}}{5 b d}-\frac {\frac {4 x (a d+b c) \sqrt {x^2 (a d+b c)+a c+b d x^4}}{3 b d}-\frac {\frac {\left (9 a b c d-8 (a d+b c)^2\right ) \int \frac {\sqrt {a} \sqrt {c}-\sqrt {b} \sqrt {d} x^2}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}-\frac {\sqrt {a} \sqrt {c} \left (-8 (a d+b c)^2-4 \sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d} (a d+b c)+9 a b c d\right ) \int \frac {1}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}}{3 b d}}{5 b d}\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {x^3 \sqrt {x^2 (a d+b c)+a c+b d x^4}}{5 b d}-\frac {\frac {4 x (a d+b c) \sqrt {x^2 (a d+b c)+a c+b d x^4}}{3 b d}-\frac {\frac {\left (9 a b c d-8 (a d+b c)^2\right ) \int \frac {\sqrt {a} \sqrt {c}-\sqrt {b} \sqrt {d} x^2}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}-\frac {\sqrt [4]{a} \sqrt [4]{c} \left (-8 (a d+b c)^2-4 \sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d} (a d+b c)+9 a b c d\right ) \left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right ) \sqrt {\frac {x^2 (a d+b c)+a c+b d x^4}{\left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{d} x}{\sqrt [4]{a} \sqrt [4]{c}}\right ),\frac {1}{4} \left (2-\frac {b c+a d}{\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}}\right )\right )}{2 b^{3/4} d^{3/4} \sqrt {x^2 (a d+b c)+a c+b d x^4}}}{3 b d}}{5 b d}\)

\(\Big \downarrow \) 1509

\(\displaystyle \frac {x^3 \sqrt {x^2 (a d+b c)+a c+b d x^4}}{5 b d}-\frac {\frac {4 x (a d+b c) \sqrt {x^2 (a d+b c)+a c+b d x^4}}{3 b d}-\frac {\frac {\left (9 a b c d-8 (a d+b c)^2\right ) \left (\frac {\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right ) \sqrt {\frac {x^2 (a d+b c)+a c+b d x^4}{\left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{d} x}{\sqrt [4]{a} \sqrt [4]{c}}\right )|\frac {1}{4} \left (2-\frac {b c+a d}{\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}}\right )\right )}{\sqrt [4]{b} \sqrt [4]{d} \sqrt {x^2 (a d+b c)+a c+b d x^4}}-\frac {x \sqrt {x^2 (a d+b c)+a c+b d x^4}}{\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2}\right )}{\sqrt {b} \sqrt {d}}-\frac {\sqrt [4]{a} \sqrt [4]{c} \left (-8 (a d+b c)^2-4 \sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d} (a d+b c)+9 a b c d\right ) \left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right ) \sqrt {\frac {x^2 (a d+b c)+a c+b d x^4}{\left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{d} x}{\sqrt [4]{a} \sqrt [4]{c}}\right ),\frac {1}{4} \left (2-\frac {b c+a d}{\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}}\right )\right )}{2 b^{3/4} d^{3/4} \sqrt {x^2 (a d+b c)+a c+b d x^4}}}{3 b d}}{5 b d}\)

Input:

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

Output:

(x^3*Sqrt[a*c + (b*c + a*d)*x^2 + b*d*x^4])/(5*b*d) - ((4*(b*c + a*d)*x*Sq 
rt[a*c + (b*c + a*d)*x^2 + b*d*x^4])/(3*b*d) - (((9*a*b*c*d - 8*(b*c + a*d 
)^2)*(-((x*Sqrt[a*c + (b*c + a*d)*x^2 + b*d*x^4])/(Sqrt[a]*Sqrt[c] + Sqrt[ 
b]*Sqrt[d]*x^2)) + (a^(1/4)*c^(1/4)*(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[d]*x^2 
)*Sqrt[(a*c + (b*c + a*d)*x^2 + b*d*x^4)/(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[d 
]*x^2)^2]*EllipticE[2*ArcTan[(b^(1/4)*d^(1/4)*x)/(a^(1/4)*c^(1/4))], (2 - 
(b*c + a*d)/(Sqrt[a]*Sqrt[b]*Sqrt[c]*Sqrt[d]))/4])/(b^(1/4)*d^(1/4)*Sqrt[a 
*c + (b*c + a*d)*x^2 + b*d*x^4])))/(Sqrt[b]*Sqrt[d]) - (a^(1/4)*c^(1/4)*(9 
*a*b*c*d - 4*Sqrt[a]*Sqrt[b]*Sqrt[c]*Sqrt[d]*(b*c + a*d) - 8*(b*c + a*d)^2 
)*(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[d]*x^2)*Sqrt[(a*c + (b*c + a*d)*x^2 + b* 
d*x^4)/(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[d]*x^2)^2]*EllipticF[2*ArcTan[(b^(1 
/4)*d^(1/4)*x)/(a^(1/4)*c^(1/4))], (2 - (b*c + a*d)/(Sqrt[a]*Sqrt[b]*Sqrt[ 
c]*Sqrt[d]))/4])/(2*b^(3/4)*d^(3/4)*Sqrt[a*c + (b*c + a*d)*x^2 + b*d*x^4]) 
)/(3*b*d))/(5*b*d)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 1416

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c 
/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

rule 1442

Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] 
:> Simp[d^3*(d*x)^(m - 3)*((a + b*x^2 + c*x^4)^(p + 1)/(c*(m + 4*p + 1))), 
x] - Simp[d^4/(c*(m + 4*p + 1))   Int[(d*x)^(m - 4)*Simp[a*(m - 3) + b*(m + 
 2*p - 1)*x^2, x]*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x 
] && NeQ[b^2 - 4*a*c, 0] && GtQ[m, 3] && NeQ[m + 4*p + 1, 0] && IntegerQ[2* 
p] && (IntegerQ[p] || IntegerQ[m])
 

rule 1509

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q 
^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* 
x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 
/(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 
- 4*a*c, 0] && PosQ[c/a]
 

rule 1511

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q   Int[1/Sqrt[a + b*x^2 + c*x^ 
4], x], x] - Simp[e/q   Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; 
NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos 
Q[c/a]
 

rule 1602

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*( 
x_)^4)^(p_), x_Symbol] :> Simp[e*f*(f*x)^(m - 1)*((a + b*x^2 + c*x^4)^(p + 
1)/(c*(m + 4*p + 3))), x] - Simp[f^2/(c*(m + 4*p + 3))   Int[(f*x)^(m - 2)* 
(a + b*x^2 + c*x^4)^p*Simp[a*e*(m - 1) + (b*e*(m + 2*p + 1) - c*d*(m + 4*p 
+ 3))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c 
, 0] && GtQ[m, 1] && NeQ[m + 4*p + 3, 0] && IntegerQ[2*p] && (IntegerQ[p] | 
| IntegerQ[m])
 

rule 2048

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))*((c_) + (d_.)*(x_)^(n_.)))^(p_) 
, x_Symbol] :> Int[u*(a*c*e + (b*c + a*d)*e*x^n + b*d*e*x^(2*n))^p, x] /; F 
reeQ[{a, b, c, d, e, n, p}, x]
 

Maple [A] (verified)

Time = 5.26 (sec) , antiderivative size = 343, normalized size of antiderivative = 0.93

method	result	size
default	\(\frac {x^{3} \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}{5 b d}-\frac {\left (4 a d +4 b c \right ) x \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}{15 b^{2} d^{2}}+\frac {\left (4 a d +4 b c \right ) a c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{15 b^{2} d^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}-\frac {\left (-\frac {3 a c}{5 b d}+\frac {\left (4 a d +4 b c \right ) \left (2 a d +2 b c \right )}{15 b^{2} d^{2}}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}\, d}\)	\(343\)
elliptic	\(\frac {x^{3} \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}{5 b d}-\frac {\left (4 a d +4 b c \right ) x \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}{15 b^{2} d^{2}}+\frac {\left (4 a d +4 b c \right ) a c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{15 b^{2} d^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}-\frac {\left (-\frac {3 a c}{5 b d}+\frac {\left (4 a d +4 b c \right ) \left (2 a d +2 b c \right )}{15 b^{2} d^{2}}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}\, d}\)	\(343\)
risch	\(-\frac {x \left (-3 b d \,x^{2}+4 a d +4 b c \right ) \left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}{15 b^{2} d^{2} \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}+\frac {-\frac {\left (8 a^{2} d^{2}+7 a b c d +8 b^{2} c^{2}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}\, d}+\frac {4 a b \,c^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}+\frac {4 a^{2} c d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}}{15 b^{2} d^{2}}\)	\(397\)

Input:

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

Output:

1/5/b/d*x^3*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)-1/15/b^2/d^2*(4*a*d+4*b*c) 
*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+1/15/b^2/d^2*(4*a*d+4*b*c)*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*a*c/b/d+1/ 
15/b^2/d^2*(4*a*d+4*b*c)*(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*(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)))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.64 \[ \int \frac {x^6}{\sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \, dx=-\frac {{\left (8 \, b^{2} c^{3} + 7 \, a b c^{2} d + 8 \, a^{2} c d^{2}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (8 \, b^{2} c^{3} + 7 \, a b c^{2} d + 4 \, a^{2} d^{3} + 4 \, {\left (2 \, a^{2} + a b\right )} c d^{2}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (3 \, b^{2} d^{3} x^{4} + 8 \, b^{2} c^{2} d + 7 \, a b c d^{2} + 8 \, a^{2} d^{3} - 4 \, {\left (b^{2} c d^{2} + a b d^{3}\right )} x^{2}\right )} \sqrt {b d x^{4} + {\left (b c + a d\right )} x^{2} + a c}}{15 \, b^{3} d^{4} x} \] Input:

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

Output:

-1/15*((8*b^2*c^3 + 7*a*b*c^2*d + 8*a^2*c*d^2)*sqrt(b*d)*x*sqrt(-c/d)*elli 
ptic_e(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - (8*b^2*c^3 + 7*a*b*c^2*d + 4*a^2 
*d^3 + 4*(2*a^2 + a*b)*c*d^2)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_f(arcsin(sqr 
t(-c/d)/x), a*d/(b*c)) - (3*b^2*d^3*x^4 + 8*b^2*c^2*d + 7*a*b*c*d^2 + 8*a^ 
2*d^3 - 4*(b^2*c*d^2 + a*b*d^3)*x^2)*sqrt(b*d*x^4 + (b*c + a*d)*x^2 + a*c) 
)/(b^3*d^4*x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^6}{\sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \, dx=\frac {-4 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a d x -4 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b c x +3 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b d \,x^{3}+8 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{2} d^{2}+7 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a b c d +8 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b^{2} c^{2}+4 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{2} c d +4 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a b \,c^{2}}{15 b^{2} d^{2}} \] Input:

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

Output:

( - 4*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*d*x - 4*sqrt(c + d*x**2)*sqrt(a 
+ b*x**2)*b*c*x + 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*d*x**3 + 8*int((sq 
rt(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*d**2 + 7*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*c*d + 8*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**2*c**2 + 4*in 
t((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*c*d + 4*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*b*c**2)/(15*b**2*d**2)