\(\int \frac {(a+b x^2)^{3/2} (e+f x^2)}{(c+d x^2)^{3/2}} \, dx\) [11]

Optimal result

Integrand size = 30, antiderivative size = 294 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {(3 b d e-4 b c f+3 a d f) x \sqrt {a+b x^2}}{3 d^2 \sqrt {c+d x^2}}+\frac {f x \left (a+b x^2\right )^{3/2}}{3 d \sqrt {c+d x^2}}+\frac {(a d (3 d e-7 c f)-2 b c (3 d e-4 c f)) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 \sqrt {c} d^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {\sqrt {c} (3 b d e-4 b c f+3 a d f) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 d^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:

1/3*(3*a*d*f-4*b*c*f+3*b*d*e)*x*(b*x^2+a)^(1/2)/d^2/(d*x^2+c)^(1/2)+1/3*f* 
x*(b*x^2+a)^(3/2)/d/(d*x^2+c)^(1/2)+1/3*(a*d*(-7*c*f+3*d*e)-2*b*c*(-4*c*f+ 
3*d*e))*(b*x^2+a)^(1/2)*EllipticE(d^(1/2)*x/c^(1/2)/(1+d*x^2/c)^(1/2),(1-b 
*c/a/d)^(1/2))/c^(1/2)/d^(5/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^( 
1/2)+1/3*c^(1/2)*(3*a*d*f-4*b*c*f+3*b*d*e)*(b*x^2+a)^(1/2)*InverseJacobiAM 
(arctan(d^(1/2)*x/c^(1/2)),(1-b*c/a/d)^(1/2))/d^(5/2)/(c*(b*x^2+a)/a/(d*x^ 
2+c))^(1/2)/(d*x^2+c)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 5.59 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (3 a d (d e-c f)+b c \left (-3 d e+4 c f+d f x^2\right )\right )-i b c \left (a d (-3 d e+7 c f)+b \left (6 c d e-8 c^2 f\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c (-b c+a d) (6 b d e-8 b c f+3 a d f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3 \sqrt {\frac {b}{a}} c d^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

Integrate[((a + b*x^2)^(3/2)*(e + f*x^2))/(c + d*x^2)^(3/2),x]
 

Output:

(Sqrt[b/a]*d*x*(a + b*x^2)*(3*a*d*(d*e - c*f) + b*c*(-3*d*e + 4*c*f + d*f* 
x^2)) - I*b*c*(a*d*(-3*d*e + 7*c*f) + b*(6*c*d*e - 8*c^2*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)*(6*b*d*e - 8*b*c*f + 3*a*d*f)*Sqrt[1 + (b*x^2)/a]*Sqr 
t[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(3*Sqrt[b 
/a]*c*d^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.16, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {401, 25, 403, 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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 401

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 403

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

\(\Big \downarrow \) 406

\(\displaystyle \frac {\frac {a c (3 a d f-4 b c f+3 b d e) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx-b (a d (3 d e-7 c f)-2 b c (3 d e-4 c f)) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}-\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 d e-4 c f)}{3 d}}{c d}+\frac {x \left (a+b x^2\right )^{3/2} (d e-c f)}{c d \sqrt {c+d x^2}}\)

\(\Big \downarrow \) 320

\(\displaystyle \frac {\frac {\frac {c^{3/2} \sqrt {a+b x^2} (3 a d f-4 b c f+3 b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-b (a d (3 d e-7 c f)-2 b c (3 d e-4 c f)) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}-\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 d e-4 c f)}{3 d}}{c d}+\frac {x \left (a+b x^2\right )^{3/2} (d e-c f)}{c d \sqrt {c+d x^2}}\)

\(\Big \downarrow \) 388

\(\displaystyle \frac {\frac {\frac {c^{3/2} \sqrt {a+b x^2} (3 a d f-4 b c f+3 b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-b (a d (3 d e-7 c f)-2 b c (3 d e-4 c f)) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )}{3 d}-\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 d e-4 c f)}{3 d}}{c d}+\frac {x \left (a+b x^2\right )^{3/2} (d e-c f)}{c d \sqrt {c+d x^2}}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {\frac {\frac {c^{3/2} \sqrt {a+b x^2} (3 a d f-4 b c f+3 b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-b (a d (3 d e-7 c f)-2 b c (3 d e-4 c f)) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 d}-\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2} (3 d e-4 c f)}{3 d}}{c d}+\frac {x \left (a+b x^2\right )^{3/2} (d e-c f)}{c d \sqrt {c+d x^2}}\)

Input:

Int[((a + b*x^2)^(3/2)*(e + f*x^2))/(c + d*x^2)^(3/2),x]
 

Output:

((d*e - c*f)*x*(a + b*x^2)^(3/2))/(c*d*Sqrt[c + d*x^2]) + (-1/3*(b*(3*d*e 
- 4*c*f)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/d + (-(b*(a*d*(3*d*e - 7*c*f) 
- 2*b*c*(3*d*e - 4*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)*(3*b*d*e - 4*b*c*f + 3*a*d*f)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqr 
t[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*d))/(c*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 401

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

rule 403

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

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(542\) vs. \(2(265)=530\).

Time = 11.56 (sec) , antiderivative size = 543, normalized size of antiderivative = 1.85

method	result	size
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {\left (b d \,x^{2}+a d \right ) \left (a c d f -a \,d^{2} e -b \,c^{2} f +b c d e \right ) x}{c \,d^{3} \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (b d \,x^{2}+a d \right )}}+\frac {f b x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 d^{2}}+\frac {\left (\frac {f \,d^{2} a^{2}-2 f d c b a +2 a b \,d^{2} e +f \,c^{2} b^{2}-d \,b^{2} c e}{d^{3}}-\frac {\left (a c d f -a \,d^{2} e -b \,c^{2} f +b c d e \right ) \left (a d -b c \right )}{d^{3} c}+\frac {a \left (a c d f -a \,d^{2} e -b \,c^{2} f +b c d e \right )}{d^{2} c}-\frac {a b c f}{3 d^{2}}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (\frac {b \left (2 a d f -b c f +b d e \right )}{d^{2}}+\frac {\left (a c d f -a \,d^{2} e -b \,c^{2} f +b c d e \right ) b}{d^{2} c}-\frac {f b \left (2 a d +2 b c \right )}{3 d^{2}}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(543\)
risch	\(\frac {f x \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, b}{3 d^{2}}+\frac {\left (\frac {\left (3 f \,d^{2} a^{2}-7 f d c b a +6 a b \,d^{2} e +3 f \,c^{2} b^{2}-3 d \,b^{2} c e \right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{d \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {b \left (4 a d f -5 b c f +3 b d e \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}-\frac {3 \left (a^{2} c f \,d^{2}-a^{2} d^{3} e -2 a b \,c^{2} d f +2 a b c \,d^{2} e +b^{2} c^{3} f -b^{2} c^{2} d e \right ) \left (\frac {\left (b d \,x^{2}+a d \right ) x}{c \left (a d -b c \right ) \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (b d \,x^{2}+a d \right )}}+\frac {\left (\frac {1}{c}-\frac {a d}{\left (a d -b c \right ) c}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {b \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\left (a d -b c \right ) \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{d}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{3 d^{2} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(684\)
default	\(\frac {\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \left (\sqrt {-\frac {b}{a}}\, b^{2} c \,d^{2} f \,x^{5}-2 \sqrt {-\frac {b}{a}}\, a b c \,d^{2} f \,x^{3}+3 \sqrt {-\frac {b}{a}}\, a b \,d^{3} e \,x^{3}+4 \sqrt {-\frac {b}{a}}\, b^{2} c^{2} d f \,x^{3}-3 \sqrt {-\frac {b}{a}}\, b^{2} c \,d^{2} e \,x^{3}+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 -11 \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 +6 \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 -6 \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 +7 \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 -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 \,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 +6 \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 -3 \sqrt {-\frac {b}{a}}\, a^{2} c \,d^{2} f x +3 \sqrt {-\frac {b}{a}}\, a^{2} d^{3} e x +4 \sqrt {-\frac {b}{a}}\, a b \,c^{2} d f x -3 \sqrt {-\frac {b}{a}}\, a b c \,d^{2} e x \right )}{3 \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right ) d^{3} c \sqrt {-\frac {b}{a}}}\)	\(750\)

Input:

int((b*x^2+a)^(3/2)*(f*x^2+e)/(d*x^2+c)^(3/2),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)*(-(b*d*x^2+a*d 
)*(a*c*d*f-a*d^2*e-b*c^2*f+b*c*d*e)/c/d^3*x/((x^2+c/d)*(b*d*x^2+a*d))^(1/2 
)+1/3*f*b/d^2*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+((a^2*d^2*f-2*a*b*c*d* 
f+2*a*b*d^2*e+b^2*c^2*f-b^2*c*d*e)/d^3-(a*c*d*f-a*d^2*e-b*c^2*f+b*c*d*e)/d 
^3*(a*d-b*c)/c+a/d^2*(a*c*d*f-a*d^2*e-b*c^2*f+b*c*d*e)/c-1/3*a*b*c/d^2*f)/ 
(-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))-(b/d^2*(2*a* 
d*f-b*c*f+b*d*e)+(a*c*d*f-a*d^2*e-b*c^2*f+b*c*d*e)/d^2*b/c-1/3*f*b/d^2*(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 = 459, normalized size of antiderivative = 1.56 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{3/2}} \, dx=-\frac {{\left ({\left (3 \, {\left (2 \, b^{2} c^{2} d^{2} - a b c d^{3}\right )} e - {\left (8 \, b^{2} c^{3} d - 7 \, a b c^{2} d^{2}\right )} f\right )} x^{3} + {\left (3 \, {\left (2 \, b^{2} c^{3} d - a b c^{2} d^{2}\right )} e - {\left (8 \, b^{2} c^{4} - 7 \, a b c^{3} d\right )} f\right )} x\right )} \sqrt {b d} \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left ({\left (3 \, {\left (2 \, b^{2} c^{2} d^{2} - a b c d^{3} + a b d^{4}\right )} e - {\left (8 \, b^{2} c^{3} d - 7 \, a b c^{2} d^{2} + 4 \, a b c d^{3} - 3 \, a^{2} d^{4}\right )} f\right )} x^{3} + {\left (3 \, {\left (2 \, b^{2} c^{3} d - a b c^{2} d^{2} + a b c d^{3}\right )} e - {\left (8 \, b^{2} c^{4} - 7 \, a b c^{3} d + 4 \, a b c^{2} d^{2} - 3 \, a^{2} c d^{3}\right )} f\right )} x\right )} \sqrt {b d} \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (b^{2} c d^{3} f x^{4} + {\left (3 \, b^{2} c d^{3} e - 4 \, {\left (b^{2} c^{2} d^{2} - a b c d^{3}\right )} f\right )} x^{2} + 3 \, {\left (2 \, b^{2} c^{2} d^{2} - a b c d^{3}\right )} e - {\left (8 \, b^{2} c^{3} d - 7 \, a b c^{2} d^{2}\right )} f\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{3 \, {\left (b c d^{5} x^{3} + b c^{2} d^{4} x\right )}} \] Input:

integrate((b*x^2+a)^(3/2)*(f*x^2+e)/(d*x^2+c)^(3/2),x, algorithm="fricas")
 

Output:

-1/3*(((3*(2*b^2*c^2*d^2 - a*b*c*d^3)*e - (8*b^2*c^3*d - 7*a*b*c^2*d^2)*f) 
*x^3 + (3*(2*b^2*c^3*d - a*b*c^2*d^2)*e - (8*b^2*c^4 - 7*a*b*c^3*d)*f)*x)* 
sqrt(b*d)*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - ((3*(2* 
b^2*c^2*d^2 - a*b*c*d^3 + a*b*d^4)*e - (8*b^2*c^3*d - 7*a*b*c^2*d^2 + 4*a* 
b*c*d^3 - 3*a^2*d^4)*f)*x^3 + (3*(2*b^2*c^3*d - a*b*c^2*d^2 + a*b*c*d^3)*e 
 - (8*b^2*c^4 - 7*a*b*c^3*d + 4*a*b*c^2*d^2 - 3*a^2*c*d^3)*f)*x)*sqrt(b*d) 
*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - (b^2*c*d^3*f*x^4 
 + (3*b^2*c*d^3*e - 4*(b^2*c^2*d^2 - a*b*c*d^3)*f)*x^2 + 3*(2*b^2*c^2*d^2 
- a*b*c*d^3)*e - (8*b^2*c^3*d - 7*a*b*c^2*d^2)*f)*sqrt(b*x^2 + a)*sqrt(d*x 
^2 + c))/(b*c*d^5*x^3 + b*c^2*d^4*x)
 

Sympy [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {3}{2}} \left (e + f x^{2}\right )}{\left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((b*x**2+a)**(3/2)*(f*x**2+e)/(d*x**2+c)**(3/2),x)
 

Output:

Integral((a + b*x**2)**(3/2)*(e + f*x**2)/(c + d*x**2)**(3/2), x)
 

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(3/2)*(f*x^2+e)/(d*x^2+c)^(3/2),x, algorithm="maxima")
 

Output:

integrate((b*x^2 + a)^(3/2)*(f*x^2 + e)/(d*x^2 + c)^(3/2), x)
 

Giac [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(3/2)*(f*x^2+e)/(d*x^2+c)^(3/2),x, algorithm="giac")
 

Output:

integrate((b*x^2 + a)^(3/2)*(f*x^2 + e)/(d*x^2 + c)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{3/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,\left (f\,x^2+e\right )}{{\left (d\,x^2+c\right )}^{3/2}} \,d x \] Input:

int(((a + b*x^2)^(3/2)*(e + f*x^2))/(c + d*x^2)^(3/2),x)
 

Output:

int(((a + b*x^2)^(3/2)*(e + f*x^2))/(c + d*x^2)^(3/2), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

int((b*x^2+a)^(3/2)*(f*x^2+e)/(d*x^2+c)^(3/2),x)
 

Output:

(3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*d*f*x - 3*sqrt(c + d*x**2)*sqrt( 
a + b*x**2)*a*b*c*f*x + 6*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*d*e*x + 2* 
sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*c*f*x**3 - 3*int((sqrt(c + d*x**2)* 
sqrt(a + b*x**2)*x**4)/(a*c**2 + 2*a*c*d*x**2 + a*d**2*x**4 + b*c**2*x**2 
+ 2*b*c*d*x**4 + b*d**2*x**6),x)*a**2*b*c*d**2*f - 3*int((sqrt(c + d*x**2) 
*sqrt(a + b*x**2)*x**4)/(a*c**2 + 2*a*c*d*x**2 + a*d**2*x**4 + b*c**2*x**2 
 + 2*b*c*d*x**4 + b*d**2*x**6),x)*a**2*b*d**3*f*x**2 + 11*int((sqrt(c + d* 
x**2)*sqrt(a + b*x**2)*x**4)/(a*c**2 + 2*a*c*d*x**2 + a*d**2*x**4 + b*c**2 
*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*a*b**2*c**2*d*f - 6*int((sqrt(c + d 
*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**2 + 2*a*c*d*x**2 + a*d**2*x**4 + b*c** 
2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*a*b**2*c*d**2*e + 11*int((sqrt(c + 
 d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**2 + 2*a*c*d*x**2 + a*d**2*x**4 + b*c 
**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*a*b**2*c*d**2*f*x**2 - 6*int((sq 
rt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**2 + 2*a*c*d*x**2 + a*d**2*x**4 
 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*a*b**2*d**3*e*x**2 - 8*int 
((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**2 + 2*a*c*d*x**2 + a*d**2* 
x**4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*b**3*c**3*f + 6*int((s 
qrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**2 + 2*a*c*d*x**2 + a*d**2*x** 
4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*b**3*c**2*d*e - 8*int((sq 
rt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**2 + 2*a*c*d*x**2 + a*d**2*x...