\(\int \frac {x^2 \sqrt {a+b x^2} (e+f x^2)}{\sqrt {c+d x^2}} \, dx\) [117]

Optimal result

Integrand size = 33, antiderivative size = 388 \[ \int \frac {x^2 \sqrt {a+b x^2} \left (e+f x^2\right )}{\sqrt {c+d x^2}} \, dx=-\frac {\left (2 a^2 d^2 f+2 b^2 c (5 d e-4 c f)-a b d (5 d e-3 c f)\right ) x \sqrt {c+d x^2}}{15 b d^3 \sqrt {a+b x^2}}+\frac {(5 b d e-4 b c f+a d f) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 b d^2}+\frac {f x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 d}+\frac {\sqrt {a} \left (2 a^2 d^2 f+2 b^2 c (5 d e-4 c f)-a b d (5 d e-3 c f)\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{15 b^{3/2} d^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} (5 b d e-4 b c f+a d f) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{15 b^{3/2} d^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

-1/15*(2*a^2*d^2*f+2*b^2*c*(-4*c*f+5*d*e)-a*b*d*(-3*c*f+5*d*e))*x*(d*x^2+c 
)^(1/2)/b/d^3/(b*x^2+a)^(1/2)+1/15*(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/d^2+1/5*f*x^3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/d+1/15* 
a^(1/2)*(2*a^2*d^2*f+2*b^2*c*(-4*c*f+5*d*e)-a*b*d*(-3*c*f+5*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 
^(3/2)/d^3/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)-1/15*a^(3/2)*(a 
*d*f-4*b*c*f+5*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^(3/2)/d^2/(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 = 1.73 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.70 \[ \int \frac {x^2 \sqrt {a+b x^2} \left (e+f x^2\right )}{\sqrt {c+d x^2}} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (a d f+b \left (5 d e-4 c f+3 d f x^2\right )\right )+i c \left (2 a^2 d^2 f+2 b^2 c (5 d e-4 c f)+a b d (-5 d e+3 c f)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c (-b c+a d) (-10 b d e+8 b c f+a d f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{15 b \sqrt {\frac {b}{a}} d^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

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

Output:

(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(a*d*f + b*(5*d*e - 4*c*f + 3*d*f*x 
^2)) + I*c*(2*a^2*d^2*f + 2*b^2*c*(5*d*e - 4*c*f) + a*b*d*(-5*d*e + 3*c*f) 
)*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*(-(b*c) + a*d)*(-10*b*d*e + 8*b*c*f + a*d*f)*Sqrt[1 + 
 (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b 
*c)])/(15*b*Sqrt[b/a]*d^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 356, normalized size of antiderivative = 0.92, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {443, 444, 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.

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

\(\Big \downarrow \) 443

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

\(\Big \downarrow \) 444

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

\(\Big \downarrow \) 406

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

\(\Big \downarrow \) 320

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

\(\Big \downarrow \) 388

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

\(\Big \downarrow \) 313

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

Input:

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

Output:

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

Defintions of rubi rules used

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 443

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

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 = 7.07 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.07

method	result	size
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {f \,x^{3} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{5 d}+\frac {\left (a f +b e -\frac {f \left (4 a d +4 b c \right )}{5 d}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b d}-\frac {\left (a f +b e -\frac {f \left (4 a d +4 b c \right )}{5 d}\right ) a c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{3 b d \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (a e -\frac {3 a c f}{5 d}-\frac {\left (a f +b e -\frac {f \left (4 a d +4 b c \right )}{5 d}\right ) \left (2 a d +2 b c \right )}{3 b d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(417\)
risch	\(\frac {x \left (3 b d f \,x^{2}+a d f -4 b c f +5 b d e \right ) \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{15 b \,d^{2}}-\frac {\left (-\frac {\left (2 f \,d^{2} a^{2}+3 f d c b a -5 a b \,d^{2} e -8 f \,c^{2} b^{2}+10 d \,b^{2} c e \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}+\frac {a^{2} c d f \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {4 a b \,c^{2} f \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {5 a c d e b \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{15 b \,d^{2} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(538\)
default	\(\frac {\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \left (3 \sqrt {-\frac {b}{a}}\, b^{2} d^{3} f \,x^{7}+4 \sqrt {-\frac {b}{a}}\, a b \,d^{3} f \,x^{5}-\sqrt {-\frac {b}{a}}\, b^{2} c \,d^{2} f \,x^{5}+5 \sqrt {-\frac {b}{a}}\, b^{2} d^{3} e \,x^{5}+\sqrt {-\frac {b}{a}}\, a^{2} d^{3} f \,x^{3}+5 \sqrt {-\frac {b}{a}}\, a b \,d^{3} e \,x^{3}-4 \sqrt {-\frac {b}{a}}\, b^{2} c^{2} d f \,x^{3}+5 \sqrt {-\frac {b}{a}}\, b^{2} c \,d^{2} e \,x^{3}+\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c \,d^{2} f +7 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} d f -10 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c \,d^{2} e -8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{3} f +10 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} d e -2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c \,d^{2} f -3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} d f +5 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c \,d^{2} e +8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{3} f -10 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} d e +\sqrt {-\frac {b}{a}}\, a^{2} c \,d^{2} f x -4 \sqrt {-\frac {b}{a}}\, a b \,c^{2} d f x +5 \sqrt {-\frac {b}{a}}\, a b c \,d^{2} e x \right )}{15 d^{3} \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right ) b \sqrt {-\frac {b}{a}}}\)	\(847\)

Input:

int(x^2*(b*x^2+a)^(1/2)*(f*x^2+e)/(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*f/d*x^3*( 
b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+1/3*(a*f+b*e-1/5*f/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*(a*f+b*e-1/5*f/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))-(a* 
e-3/5*a*c*f/d-1/3*(a*f+b*e-1/5*f/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*(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 = 324, normalized size of antiderivative = 0.84 \[ \int \frac {x^2 \sqrt {a+b x^2} \left (e+f x^2\right )}{\sqrt {c+d x^2}} \, dx=\frac {\sqrt {b d} {\left (5 \, {\left (2 \, b^{2} c^{2} d - a b c d^{2}\right )} e - {\left (8 \, b^{2} c^{3} - 3 \, a b c^{2} d - 2 \, a^{2} c d^{2}\right )} f\right )} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - \sqrt {b d} {\left (5 \, {\left (2 \, b^{2} c^{2} d - a b c d^{2} + a b d^{3}\right )} e - {\left (8 \, b^{2} c^{3} - 3 \, a b c^{2} d - a^{2} d^{3} - 2 \, {\left (a^{2} - 2 \, a b\right )} c d^{2}\right )} f\right )} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) + {\left (3 \, b^{2} d^{3} f x^{4} + {\left (5 \, b^{2} d^{3} e - {\left (4 \, b^{2} c d^{2} - a b d^{3}\right )} f\right )} x^{2} - 5 \, {\left (2 \, b^{2} c d^{2} - a b d^{3}\right )} e + {\left (8 \, b^{2} c^{2} d - 3 \, a b c d^{2} - 2 \, a^{2} d^{3}\right )} f\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{15 \, b^{2} d^{4} x} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^2 \sqrt {a+b x^2} \left (e+f x^2\right )}{\sqrt {c+d x^2}} \, dx=\frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a d f x -4 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b c f x +5 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b d e x +3 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b d f \,x^{3}-2 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{2} d^{2} f -3 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a b c d f +5 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a b \,d^{2} e +8 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b^{2} c^{2} f -10 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b^{2} c d e -\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{2} c d f +4 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a b \,c^{2} f -5 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a b c d e}{15 b \,d^{2}} \] Input:

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

Output:

(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 - 2*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*d**2*f - 3*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*f + 5*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*d**2*e + 8*int((sqrt(c + d*x**2)*sq 
rt(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*f 
 - 10*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*d*e - 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*c*d*f + 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*f 
- 5*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*d*e)/(15*b*d**2)