\(\int \frac {\sqrt {a+b x^2} (e+f x^2)}{x^6 \sqrt {c+d x^2}} \, dx\) [121]

Optimal result

Integrand size = 33, antiderivative size = 391 \[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )}{x^6 \sqrt {c+d x^2}} \, dx=\frac {\left (2 b^2 c^2 e+a b c (3 d e-5 c f)-2 a^2 d (4 d e-5 c f)\right ) \sqrt {c+d x^2}}{15 a c^3 x \sqrt {a+b x^2}}-\frac {e \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 c x^5}-\frac {(b c e-4 a d e+5 a c f) \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 a c^2 x^3}+\frac {\sqrt {b} \left (2 b^2 c^2 e+a b c (3 d e-5 c f)-2 a^2 d (4 d e-5 c f)\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{15 a^{3/2} c^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {b} d (b c e-4 a d e+5 a c f) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{15 \sqrt {a} c^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

1/15*(2*b^2*c^2*e+a*b*c*(-5*c*f+3*d*e)-2*a^2*d*(-5*c*f+4*d*e))*(d*x^2+c)^( 
1/2)/a/c^3/x/(b*x^2+a)^(1/2)-1/5*e*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/c/x^5-1 
/15*(5*a*c*f-4*a*d*e+b*c*e)*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a/c^2/x^3+1/15 
*b^(1/2)*(2*b^2*c^2*e+a*b*c*(-5*c*f+3*d*e)-2*a^2*d*(-5*c*f+4*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))/ 
a^(3/2)/c^3/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)-1/15*b^(1/2)*d 
*(5*a*c*f-4*a*d*e+b*c*e)*(d*x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x/ 
a^(1/2)),(1-a*d/b/c)^(1/2))/a^(1/2)/c^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 = 2.28 (sec) , antiderivative size = 342, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )}{x^6 \sqrt {c+d x^2}} \, dx=\frac {\sqrt {\frac {b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right ) \left (2 b^2 c^2 e x^4-a b c x^2 \left (-3 d e x^2+c \left (e+5 f x^2\right )\right )+a^2 \left (-8 d^2 e x^4+2 c d x^2 \left (2 e+5 f x^2\right )-c^2 \left (3 e+5 f x^2\right )\right )\right )+i b c \left (2 b^2 c^2 e+a b c (3 d e-5 c f)+2 a^2 d (-4 d e+5 c f)\right ) x^5 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i b c (-b c+a d) (-2 b c e-4 a d e+5 a c f) x^5 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{15 a^2 \sqrt {\frac {b}{a}} c^3 x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.10, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {442, 445, 445, 25, 27, 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.

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

\(\Big \downarrow \) 442

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

\(\Big \downarrow \) 445

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

\(\Big \downarrow \) 445

\(\displaystyle \frac {-\frac {-\frac {\int -\frac {b d \left (\left (-2 d (4 d e-5 c f) a^2+b c (3 d e-5 c f) a+2 b^2 c^2 e\right ) x^2+a c (b c e-4 a d e+5 a c f)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {2 b^2 c e}{a}-\frac {8 a d^2 e}{c}+10 a d f-5 b c f+3 b d e\right )}{x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (5 a c f-4 a d e+b c e)}{3 a c x^3}}{5 c}-\frac {e \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 c x^5}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {-\frac {\frac {\int \frac {b d \left (\left (-2 d (4 d e-5 c f) a^2+b c (3 d e-5 c f) a+2 b^2 c^2 e\right ) x^2+a c (b c e-4 a d e+5 a c f)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {2 b^2 c e}{a}-\frac {8 a d^2 e}{c}+10 a d f-5 b c f+3 b d e\right )}{x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (5 a c f-4 a d e+b c e)}{3 a c x^3}}{5 c}-\frac {e \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 c x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\frac {b d \int \frac {\left (-2 d (4 d e-5 c f) a^2+b c (3 d e-5 c f) a+2 b^2 c^2 e\right ) x^2+a c (b c e-4 a d e+5 a c f)}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {2 b^2 c e}{a}-\frac {8 a d^2 e}{c}+10 a d f-5 b c f+3 b d e\right )}{x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (5 a c f-4 a d e+b c e)}{3 a c x^3}}{5 c}-\frac {e \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 c x^5}\)

\(\Big \downarrow \) 406

\(\displaystyle \frac {-\frac {\frac {b d \left (\left (-2 a^2 d (4 d e-5 c f)+a b c (3 d e-5 c f)+2 b^2 c^2 e\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+a c (5 a c f-4 a d e+b c e) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {2 b^2 c e}{a}-\frac {8 a d^2 e}{c}+10 a d f-5 b c f+3 b d e\right )}{x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (5 a c f-4 a d e+b c e)}{3 a c x^3}}{5 c}-\frac {e \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 c x^5}\)

\(\Big \downarrow \) 320

\(\displaystyle \frac {-\frac {\frac {b d \left (\left (-2 a^2 d (4 d e-5 c f)+a b c (3 d e-5 c f)+2 b^2 c^2 e\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} (5 a c f-4 a d e+b c e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {2 b^2 c e}{a}-\frac {8 a d^2 e}{c}+10 a d f-5 b c f+3 b d e\right )}{x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (5 a c f-4 a d e+b c e)}{3 a c x^3}}{5 c}-\frac {e \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 c x^5}\)

\(\Big \downarrow \) 388

\(\displaystyle \frac {-\frac {\frac {b d \left (\left (-2 a^2 d (4 d e-5 c f)+a b c (3 d e-5 c f)+2 b^2 c^2 e\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (5 a c f-4 a d e+b c e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {2 b^2 c e}{a}-\frac {8 a d^2 e}{c}+10 a d f-5 b c f+3 b d e\right )}{x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (5 a c f-4 a d e+b c e)}{3 a c x^3}}{5 c}-\frac {e \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 c x^5}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {-\frac {\frac {b d \left (\left (-2 a^2 d (4 d e-5 c f)+a b c (3 d e-5 c f)+2 b^2 c^2 e\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (5 a c f-4 a d e+b c e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {2 b^2 c e}{a}-\frac {8 a d^2 e}{c}+10 a d f-5 b c f+3 b d e\right )}{x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (5 a c f-4 a d e+b c e)}{3 a c x^3}}{5 c}-\frac {e \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 c x^5}\)

Input:

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

Output:

-1/5*(e*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(c*x^5) + (-1/3*((b*c*e - 4*a*d*e 
 + 5*a*c*f)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(a*c*x^3) - (-((((2*b^2*c*e)/ 
a + 3*b*d*e - (8*a*d^2*e)/c - 5*b*c*f + 10*a*d*f)*Sqrt[a + b*x^2]*Sqrt[c + 
 d*x^2])/x) + (b*d*((2*b^2*c^2*e + a*b*c*(3*d*e - 5*c*f) - 2*a^2*d*(4*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]*Sq 
rt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])) + (c^(3/2)*(b*c*e - 
4*a*d*e + 5*a*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))]*Sqrt[c + 
d*x^2])))/(a*c))/(3*a*c))/(5*c)
 

Defintions of rubi rules used

rule 25

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

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 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 442

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ 
.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p 
+ 1)*((c + d*x^2)^q/(a*g*(m + 1))), x] - Simp[1/(a*g^2*(m + 1))   Int[(g*x) 
^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f)*(m + 1) + e*2 
*(b*c*(p + 1) + a*d*q) + d*((b*e - a*f)*(m + 1) + b*e*2*(p + q + 1))*x^2, x 
], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && GtQ[q, 0] && LtQ[m, -1] 
&&  !(EqQ[q, 1] && SimplerQ[e + f*x^2, c + d*x^2])
 

rule 445

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

Maple [A] (verified)

Time = 8.32 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.22

method	result	size
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {e \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{5 c \,x^{5}}-\frac {\left (5 a c f -4 a d e +b c e \right ) \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{15 a \,c^{2} x^{3}}+\frac {\left (10 a^{2} c f d -8 a^{2} d^{2} e -5 a b \,c^{2} f +3 a b c d e +2 b^{2} c^{2} e \right ) \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{15 a^{2} c^{3} x}-\frac {\left (5 a c f -4 a d e +b c e \right ) b d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{15 a \,c^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {b \left (10 a^{2} c f d -8 a^{2} d^{2} e -5 a b \,c^{2} f +3 a b c d e +2 b^{2} c^{2} e \right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{15 a^{2} c^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(476\)
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \left (-10 a^{2} c d f \,x^{4}+8 a^{2} d^{2} e \,x^{4}+5 a b \,c^{2} f \,x^{4}-3 a b c d e \,x^{4}-2 b^{2} c^{2} e \,x^{4}+5 a^{2} c^{2} f \,x^{2}-4 a^{2} c d e \,x^{2}+a b \,c^{2} e \,x^{2}+3 a^{2} c^{2} e \right )}{15 c^{3} x^{5} a^{2}}-\frac {b d \left (-\frac {\left (10 a^{2} c f d -8 a^{2} d^{2} e -5 a b \,c^{2} f +3 a b c d e +2 b^{2} c^{2} e \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}+\frac {a b \,c^{2} e \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {5 a^{2} c^{2} f \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {4 a^{2} c d e \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{15 a^{2} c^{3} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(620\)
default	\(\text {Expression too large to display}\)	\(1094\)

Input:

int((b*x^2+a)^(1/2)*(f*x^2+e)/x^6/(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)*(-1/5/c*e*(b*d 
*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/x^5-1/15/a/c^2*(5*a*c*f-4*a*d*e+b*c*e)*(b* 
d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/x^3+1/15/a^2/c^3*(10*a^2*c*d*f-8*a^2*d^2* 
e-5*a*b*c^2*f+3*a*b*c*d*e+2*b^2*c^2*e)*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2) 
/x-1/15*(5*a*c*f-4*a*d*e+b*c*e)*b*d/a/c^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))+1/15*b*(10*a^2*c*d*f-8*a^2*d^2*e-5*a*b*c^2*f 
+3*a*b*c*d*e+2*b^2*c^2*e)/a^2/c^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))-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 = 320, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )}{x^6 \sqrt {c+d x^2}} \, dx=-\frac {\sqrt {a c} {\left ({\left (2 \, b^{3} c^{2} + 3 \, a b^{2} c d - 8 \, a^{2} b d^{2}\right )} e - 5 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b c d\right )} f\right )} x^{5} \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - \sqrt {a c} {\left ({\left (2 \, b^{3} c^{2} + {\left (a^{2} b + 3 \, a b^{2}\right )} c d - 4 \, {\left (a^{3} + 2 \, a^{2} b\right )} d^{2}\right )} e - 5 \, {\left (a b^{2} c^{2} - {\left (a^{3} + 2 \, a^{2} b\right )} c d\right )} f\right )} x^{5} \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) + {\left (3 \, a^{3} c^{2} e - {\left ({\left (2 \, a b^{2} c^{2} + 3 \, a^{2} b c d - 8 \, a^{3} d^{2}\right )} e - 5 \, {\left (a^{2} b c^{2} - 2 \, a^{3} c d\right )} f\right )} x^{4} + {\left (5 \, a^{3} c^{2} f + {\left (a^{2} b c^{2} - 4 \, a^{3} c d\right )} e\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{15 \, a^{3} c^{3} x^{5}} \] Input:

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

Output:

-1/15*(sqrt(a*c)*((2*b^3*c^2 + 3*a*b^2*c*d - 8*a^2*b*d^2)*e - 5*(a*b^2*c^2 
 - 2*a^2*b*c*d)*f)*x^5*sqrt(-b/a)*elliptic_e(arcsin(x*sqrt(-b/a)), a*d/(b* 
c)) - sqrt(a*c)*((2*b^3*c^2 + (a^2*b + 3*a*b^2)*c*d - 4*(a^3 + 2*a^2*b)*d^ 
2)*e - 5*(a*b^2*c^2 - (a^3 + 2*a^2*b)*c*d)*f)*x^5*sqrt(-b/a)*elliptic_f(ar 
csin(x*sqrt(-b/a)), a*d/(b*c)) + (3*a^3*c^2*e - ((2*a*b^2*c^2 + 3*a^2*b*c* 
d - 8*a^3*d^2)*e - 5*(a^2*b*c^2 - 2*a^3*c*d)*f)*x^4 + (5*a^3*c^2*f + (a^2* 
b*c^2 - 4*a^3*c*d)*e)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(a^3*c^3*x^5)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*c*e - 5*sqrt(c + d*x**2)*sqrt(a + 
b*x**2)*b*c*f*x**4 + 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*d*e*x**4 + 5*in 
t((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a**2*c*d + a**2*d**2*x**2 + a* 
b*c**2 + 2*a*b*c*d*x**2 + a*b*d**2*x**4 + b**2*c**2*x**2 + b**2*c*d*x**4), 
x)*a*b**2*c*d**2*f*x**5 - 3*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/( 
a**2*c*d + a**2*d**2*x**2 + a*b*c**2 + 2*a*b*c*d*x**2 + a*b*d**2*x**4 + b* 
*2*c**2*x**2 + b**2*c*d*x**4),x)*a*b**2*d**3*e*x**5 + 5*int((sqrt(c + d*x* 
*2)*sqrt(a + b*x**2)*x**2)/(a**2*c*d + a**2*d**2*x**2 + a*b*c**2 + 2*a*b*c 
*d*x**2 + a*b*d**2*x**4 + b**2*c**2*x**2 + b**2*c*d*x**4),x)*b**3*c**2*d*f 
*x**5 - 3*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a**2*c*d + a**2*d* 
*2*x**2 + a*b*c**2 + 2*a*b*c*d*x**2 + a*b*d**2*x**4 + b**2*c**2*x**2 + b** 
2*c*d*x**4),x)*b**3*c*d**2*e*x**5 + 5*int((sqrt(c + d*x**2)*sqrt(a + b*x** 
2))/(a**2*c*d*x**4 + a**2*d**2*x**6 + a*b*c**2*x**4 + 2*a*b*c*d*x**6 + a*b 
*d**2*x**8 + b**2*c**2*x**6 + b**2*c*d*x**8),x)*a**3*c**2*d*f*x**5 - 4*int 
((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a**2*c*d*x**4 + a**2*d**2*x**6 + a*b 
*c**2*x**4 + 2*a*b*c*d*x**6 + a*b*d**2*x**8 + b**2*c**2*x**6 + b**2*c*d*x* 
*8),x)*a**3*c*d**2*e*x**5 + 5*int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a** 
2*c*d*x**4 + a**2*d**2*x**6 + a*b*c**2*x**4 + 2*a*b*c*d*x**6 + a*b*d**2*x* 
*8 + b**2*c**2*x**6 + b**2*c*d*x**8),x)*a**2*b*c**3*f*x**5 - 3*int((sqrt(c 
 + d*x**2)*sqrt(a + b*x**2))/(a**2*c*d*x**4 + a**2*d**2*x**6 + a*b*c**2...