\(\int \frac {(a+b x^2)^{5/2} (e+f x^2)}{(c+d x^2)^{5/2}} \, dx\) [18]

Optimal result

Integrand size = 30, antiderivative size = 387 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {(d e-c f) x \left (a+b x^2\right )^{5/2}}{3 c d \left (c+d x^2\right )^{3/2}}-\frac {b (a d (d e-7 c f)-4 b c (d e-2 c f)) x \sqrt {a+b x^2}}{3 c d^3 \sqrt {c+d x^2}}-\frac {b (d e-2 c f) x \left (a+b x^2\right )^{3/2}}{3 c d^2 \sqrt {c+d x^2}}+\frac {\left (a b c d (3 d e-16 c f)-8 b^2 c^2 (d e-2 c f)+a^2 d^2 (2 d e+c f)\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 c^{3/2} d^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {b (a d (d e-7 c f)-4 b c (d e-2 c f)) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 \sqrt {c} d^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:

1/3*(-c*f+d*e)*x*(b*x^2+a)^(5/2)/c/d/(d*x^2+c)^(3/2)-1/3*b*(a*d*(-7*c*f+d* 
e)-4*b*c*(-2*c*f+d*e))*x*(b*x^2+a)^(1/2)/c/d^3/(d*x^2+c)^(1/2)-1/3*b*(-2*c 
*f+d*e)*x*(b*x^2+a)^(3/2)/c/d^2/(d*x^2+c)^(1/2)+1/3*(a*b*c*d*(-16*c*f+3*d* 
e)-8*b^2*c^2*(-2*c*f+d*e)+a^2*d^2*(c*f+2*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^(3/2)/d^(7/2)/(c*( 
b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)-1/3*b*(a*d*(-7*c*f+d*e)-4*b*c* 
(-2*c*f+d*e))*(b*x^2+a)^(1/2)*InverseJacobiAM(arctan(d^(1/2)*x/c^(1/2)),(1 
-b*c/a/d)^(1/2))/c^(1/2)/d^(7/2)/(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 = 7.37 (sec) , antiderivative size = 375, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (a^2 d^3 \left (3 c e+2 d e x^2+c f x^2\right )+a b c d \left (-7 c^2 f+3 d^2 e x^2+c d \left (e-9 f x^2\right )\right )+b^2 c^2 \left (8 c^2 f+d^2 x^2 \left (-5 e+f x^2\right )+c d \left (-4 e+10 f x^2\right )\right )\right )+i b c \left (a b c d (3 d e-16 c f)+a^2 d^2 (2 d e+c f)+8 b^2 c^2 (-d e+2 c f)\right ) \sqrt {1+\frac {b x^2}{a}} \left (c+d x^2\right ) \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i b c (-b c+a d) (8 b c (d e-2 c f)+a d (d e+8 c f)) \sqrt {1+\frac {b x^2}{a}} \left (c+d x^2\right ) \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3 \sqrt {\frac {b}{a}} c^2 d^4 \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}} \] Input:

Integrate[((a + b*x^2)^(5/2)*(e + f*x^2))/(c + d*x^2)^(5/2),x]
 

Output:

(Sqrt[b/a]*d*x*(a + b*x^2)*(a^2*d^3*(3*c*e + 2*d*e*x^2 + c*f*x^2) + a*b*c* 
d*(-7*c^2*f + 3*d^2*e*x^2 + c*d*(e - 9*f*x^2)) + b^2*c^2*(8*c^2*f + d^2*x^ 
2*(-5*e + f*x^2) + c*d*(-4*e + 10*f*x^2))) + I*b*c*(a*b*c*d*(3*d*e - 16*c* 
f) + a^2*d^2*(2*d*e + c*f) + 8*b^2*c^2*(-(d*e) + 2*c*f))*Sqrt[1 + (b*x^2)/ 
a]*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d) 
/(b*c)] - I*b*c*(-(b*c) + a*d)*(8*b*c*(d*e - 2*c*f) + a*d*(d*e + 8*c*f))*S 
qrt[1 + (b*x^2)/a]*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqr 
t[b/a]*x], (a*d)/(b*c)])/(3*Sqrt[b/a]*c^2*d^4*Sqrt[a + b*x^2]*(c + d*x^2)^ 
(3/2))
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 450, normalized size of antiderivative = 1.16, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {401, 25, 401, 27, 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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 401

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 401

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 403

\(\displaystyle \frac {\frac {x \left (a+b x^2\right )^{3/2} (a d (c f+2 d e)+3 b c (d e-2 c f))}{c d \sqrt {c+d x^2}}-\frac {3 b \left (\frac {\int \frac {\left (-8 b^2 (d e-2 c f) c^2+a b d (3 d e-16 c f) c+a^2 d^2 (2 d e+c f)\right ) x^2+a c (a d (d e-7 c f)-4 b c (d e-2 c f))}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (a d (c f+2 d e)+4 b c (d e-2 c f))}{3 d}\right )}{c d}}{3 c d}+\frac {x \left (a+b x^2\right )^{5/2} (d e-c f)}{3 c d \left (c+d x^2\right )^{3/2}}\)

\(\Big \downarrow \) 406

\(\displaystyle \frac {\frac {x \left (a+b x^2\right )^{3/2} (a d (c f+2 d e)+3 b c (d e-2 c f))}{c d \sqrt {c+d x^2}}-\frac {3 b \left (\frac {\left (a^2 d^2 (c f+2 d e)+a b c d (3 d e-16 c f)-8 b^2 c^2 (d e-2 c f)\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+a c (a d (d e-7 c f)-4 b c (d e-2 c f)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (a d (c f+2 d e)+4 b c (d e-2 c f))}{3 d}\right )}{c d}}{3 c d}+\frac {x \left (a+b x^2\right )^{5/2} (d e-c f)}{3 c d \left (c+d x^2\right )^{3/2}}\)

\(\Big \downarrow \) 320

\(\displaystyle \frac {\frac {x \left (a+b x^2\right )^{3/2} (a d (c f+2 d e)+3 b c (d e-2 c f))}{c d \sqrt {c+d x^2}}-\frac {3 b \left (\frac {\left (a^2 d^2 (c f+2 d e)+a b c d (3 d e-16 c f)-8 b^2 c^2 (d e-2 c f)\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} (a d (d e-7 c f)-4 b c (d e-2 c f)) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 d}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (a d (c f+2 d e)+4 b c (d e-2 c f))}{3 d}\right )}{c d}}{3 c d}+\frac {x \left (a+b x^2\right )^{5/2} (d e-c f)}{3 c d \left (c+d x^2\right )^{3/2}}\)

\(\Big \downarrow \) 388

\(\displaystyle \frac {\frac {x \left (a+b x^2\right )^{3/2} (a d (c f+2 d e)+3 b c (d e-2 c f))}{c d \sqrt {c+d x^2}}-\frac {3 b \left (\frac {\left (a^2 d^2 (c f+2 d e)+a b c d (3 d e-16 c f)-8 b^2 c^2 (d e-2 c f)\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (a d (d e-7 c f)-4 b c (d e-2 c f)) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 d}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (a d (c f+2 d e)+4 b c (d e-2 c f))}{3 d}\right )}{c d}}{3 c d}+\frac {x \left (a+b x^2\right )^{5/2} (d e-c f)}{3 c d \left (c+d x^2\right )^{3/2}}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {\frac {x \left (a+b x^2\right )^{3/2} (a d (c f+2 d e)+3 b c (d e-2 c f))}{c d \sqrt {c+d x^2}}-\frac {3 b \left (\frac {\left (a^2 d^2 (c f+2 d e)+a b c d (3 d e-16 c f)-8 b^2 c^2 (d e-2 c f)\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (a d (d e-7 c f)-4 b c (d e-2 c f)) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 d}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (a d (c f+2 d e)+4 b c (d e-2 c f))}{3 d}\right )}{c d}}{3 c d}+\frac {x \left (a+b x^2\right )^{5/2} (d e-c f)}{3 c d \left (c+d x^2\right )^{3/2}}\)

Input:

Int[((a + b*x^2)^(5/2)*(e + f*x^2))/(c + d*x^2)^(5/2),x]
 

Output:

((d*e - c*f)*x*(a + b*x^2)^(5/2))/(3*c*d*(c + d*x^2)^(3/2)) + (((3*b*c*(d* 
e - 2*c*f) + a*d*(2*d*e + c*f))*x*(a + b*x^2)^(3/2))/(c*d*Sqrt[c + d*x^2]) 
 - (3*b*(((4*b*c*(d*e - 2*c*f) + a*d*(2*d*e + c*f))*x*Sqrt[a + b*x^2]*Sqrt 
[c + d*x^2])/(3*d) + ((a*b*c*d*(3*d*e - 16*c*f) - 8*b^2*c^2*(d*e - 2*c*f) 
+ a^2*d^2*(2*d*e + 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)*(a*d*(d*e - 7*c*f) - 4*b*c*(d*e - 2*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]))/(3*d)))/(c*d))/(3*c*d)
 

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 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. \(842\) vs. \(2(352)=704\).

Time = 16.16 (sec) , antiderivative size = 843, normalized size of antiderivative = 2.18

method	result	size
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {\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 ) x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 c \,d^{5} \left (x^{2}+\frac {c}{d}\right )^{2}}+\frac {\left (b d \,x^{2}+a d \right ) \left (a^{2} c f \,d^{2}+2 a^{2} d^{3} e -9 a b \,c^{2} d f +3 a b c \,d^{2} e +8 b^{2} c^{3} f -5 b^{2} c^{2} d e \right ) x}{3 c^{2} d^{4} \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (b d \,x^{2}+a d \right )}}+\frac {b^{2} f x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 d^{3}}+\frac {\left (\frac {b \left (3 f \,d^{2} a^{2}-6 f d c b a +3 a b \,d^{2} e +3 f \,c^{2} b^{2}-2 d \,b^{2} c e \right )}{d^{4}}-\frac {\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 ) b}{3 d^{4} c}+\frac {\left (a^{2} c f \,d^{2}+2 a^{2} d^{3} e -9 a b \,c^{2} d f +3 a b c \,d^{2} e +8 b^{2} c^{3} f -5 b^{2} c^{2} d e \right ) \left (a d -b c \right )}{3 d^{4} c^{2}}-\frac {a \left (a^{2} c f \,d^{2}+2 a^{2} d^{3} e -9 a b \,c^{2} d f +3 a b c \,d^{2} e +8 b^{2} c^{3} f -5 b^{2} c^{2} d e \right )}{3 d^{3} c^{2}}-\frac {a \,b^{2} c f}{3 d^{3}}\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^{2} \left (3 a d f -2 b c f +b d e \right )}{d^{3}}-\frac {\left (a^{2} c f \,d^{2}+2 a^{2} d^{3} e -9 a b \,c^{2} d f +3 a b c \,d^{2} e +8 b^{2} c^{3} f -5 b^{2} c^{2} d e \right ) b}{3 d^{3} c^{2}}-\frac {b^{2} f \left (2 a d +2 b c \right )}{3 d^{3}}\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}}\)	\(843\)
risch	\(\text {Expression too large to display}\)	\(1198\)
default	\(\text {Expression too large to display}\)	\(1942\)

Input:

int((b*x^2+a)^(5/2)*(f*x^2+e)/(d*x^2+c)^(5/2),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)*(-1/3*(a^2*c*d 
^2*f-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)/c/d^5*x* 
(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/(x^2+c/d)^2+1/3*(b*d*x^2+a*d)*(a^2*c*d 
^2*f+2*a^2*d^3*e-9*a*b*c^2*d*f+3*a*b*c*d^2*e+8*b^2*c^3*f-5*b^2*c^2*d*e)/c^ 
2/d^4*x/((x^2+c/d)*(b*d*x^2+a*d))^(1/2)+1/3*b^2*f/d^3*x*(b*d*x^4+a*d*x^2+b 
*c*x^2+a*c)^(1/2)+(b*(3*a^2*d^2*f-6*a*b*c*d*f+3*a*b*d^2*e+3*b^2*c^2*f-2*b^ 
2*c*d*e)/d^4-1/3*(a^2*c*d^2*f-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)/d^4*b/c+1/3*(a^2*c*d^2*f+2*a^2*d^3*e-9*a*b*c^2*d*f+3*a*b* 
c*d^2*e+8*b^2*c^3*f-5*b^2*c^2*d*e)/d^4*(a*d-b*c)/c^2-1/3*a/d^3*(a^2*c*d^2* 
f+2*a^2*d^3*e-9*a*b*c^2*d*f+3*a*b*c*d^2*e+8*b^2*c^3*f-5*b^2*c^2*d*e)/c^2-1 
/3*a*b^2*c/d^3*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))-(1/d^3*b^2*(3*a*d*f-2*b*c*f+b*d*e)-1/3*(a^2*c*d^2*f+2*a^2*d^3*e-9*a* 
b*c^2*d*f+3*a*b*c*d^2*e+8*b^2*c^3*f-5*b^2*c^2*d*e)/d^3*b/c^2-1/3*b^2*f/d^3 
*(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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 846 vs. \(2 (356) = 712\).

Time = 0.11 (sec) , antiderivative size = 846, normalized size of antiderivative = 2.19 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

integrate((b*x^2+a)^(5/2)*(f*x^2+e)/(d*x^2+c)^(5/2),x, algorithm="fricas")
 

Output:

-1/3*((((8*b^2*c^3*d^3 - 3*a*b*c^2*d^4 - 2*a^2*c*d^5)*e - (16*b^2*c^4*d^2 
- 16*a*b*c^3*d^3 + a^2*c^2*d^4)*f)*x^5 + 2*((8*b^2*c^4*d^2 - 3*a*b*c^3*d^3 
 - 2*a^2*c^2*d^4)*e - (16*b^2*c^5*d - 16*a*b*c^4*d^2 + a^2*c^3*d^3)*f)*x^3 
 + ((8*b^2*c^5*d - 3*a*b*c^4*d^2 - 2*a^2*c^3*d^3)*e - (16*b^2*c^6 - 16*a*b 
*c^5*d + a^2*c^4*d^2)*f)*x)*sqrt(b*d)*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c 
/d)/x), a*d/(b*c)) - (((8*b^2*c^3*d^3 - 3*a*b*c^2*d^4 - a^2*d^6 - 2*(a^2 - 
 2*a*b)*c*d^5)*e - (16*b^2*c^4*d^2 - 16*a*b*c^3*d^3 - 7*a^2*c*d^5 + (a^2 + 
 8*a*b)*c^2*d^4)*f)*x^5 + 2*((8*b^2*c^4*d^2 - 3*a*b*c^3*d^3 - a^2*c*d^5 - 
2*(a^2 - 2*a*b)*c^2*d^4)*e - (16*b^2*c^5*d - 16*a*b*c^4*d^2 - 7*a^2*c^2*d^ 
4 + (a^2 + 8*a*b)*c^3*d^3)*f)*x^3 + ((8*b^2*c^5*d - 3*a*b*c^4*d^2 - a^2*c^ 
2*d^4 - 2*(a^2 - 2*a*b)*c^3*d^3)*e - (16*b^2*c^6 - 16*a*b*c^5*d - 7*a^2*c^ 
3*d^3 + (a^2 + 8*a*b)*c^4*d^2)*f)*x)*sqrt(b*d)*sqrt(-c/d)*elliptic_f(arcsi 
n(sqrt(-c/d)/x), a*d/(b*c)) - (b^2*c^2*d^4*f*x^6 + (3*b^2*c^2*d^4*e - (6*b 
^2*c^3*d^3 - 7*a*b*c^2*d^4)*f)*x^4 + ((12*b^2*c^3*d^3 - 5*a*b*c^2*d^4 - a^ 
2*c*d^5)*e - (24*b^2*c^4*d^2 - 25*a*b*c^3*d^3 + 2*a^2*c^2*d^4)*f)*x^2 + (8 
*b^2*c^4*d^2 - 3*a*b*c^3*d^3 - 2*a^2*c^2*d^4)*e - (16*b^2*c^5*d - 16*a*b*c 
^4*d^2 + a^2*c^3*d^3)*f)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(c^2*d^7*x^5 + 2 
*c^3*d^6*x^3 + c^4*d^5*x)
 

Sympy [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{5/2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {5}{2}} \left (e + f x^{2}\right )}{\left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate((b*x**2+a)**(5/2)*(f*x**2+e)/(d*x**2+c)**(5/2),x)
 

Output:

Integral((a + b*x**2)**(5/2)*(e + f*x**2)/(c + d*x**2)**(5/2), x)
 

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(5/2)*(f*x^2+e)/(d*x^2+c)^(5/2),x, algorithm="maxima")
 

Output:

integrate((b*x^2 + a)^(5/2)*(f*x^2 + e)/(d*x^2 + c)^(5/2), x)
 

Giac [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(5/2)*(f*x^2+e)/(d*x^2+c)^(5/2),x, algorithm="giac")
 

Output:

integrate((b*x^2 + a)^(5/2)*(f*x^2 + e)/(d*x^2 + c)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{5/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/2}\,\left (f\,x^2+e\right )}{{\left (d\,x^2+c\right )}^{5/2}} \,d x \] Input:

int(((a + b*x^2)^(5/2)*(e + f*x^2))/(c + d*x^2)^(5/2),x)
 

Output:

int(((a + b*x^2)^(5/2)*(e + f*x^2))/(c + d*x^2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{5/2}} \, dx=\text {too large to display} \] Input:

int((b*x^2+a)^(5/2)*(f*x^2+e)/(d*x^2+c)^(5/2),x)
 

Output:

( - 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*d*f*x + 18*sqrt(c + d*x**2)*s 
qrt(a + b*x**2)*a*b*c*f*x - 9*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*d*e*x 
+ 14*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*d*f*x**3 - 12*sqrt(c + d*x**2)* 
sqrt(a + b*x**2)*b**2*c*f*x**3 + 6*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2* 
d*e*x**3 + 2*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*d*f*x**5 + 15*int((sqr 
t(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**3 + 3*a*c**2*d*x**2 + 3*a*c*d** 
2*x**4 + a*d**3*x**6 + b*c**3*x**2 + 3*b*c**2*d*x**4 + 3*b*c*d**2*x**6 + b 
*d**3*x**8),x)*a**2*b*c**2*d**2*f + 30*int((sqrt(c + d*x**2)*sqrt(a + b*x* 
*2)*x**4)/(a*c**3 + 3*a*c**2*d*x**2 + 3*a*c*d**2*x**4 + a*d**3*x**6 + b*c* 
*3*x**2 + 3*b*c**2*d*x**4 + 3*b*c*d**2*x**6 + b*d**3*x**8),x)*a**2*b*c*d** 
3*f*x**2 + 15*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**3 + 3*a*c 
**2*d*x**2 + 3*a*c*d**2*x**4 + a*d**3*x**6 + b*c**3*x**2 + 3*b*c**2*d*x**4 
 + 3*b*c*d**2*x**6 + b*d**3*x**8),x)*a**2*b*d**4*f*x**4 - 48*int((sqrt(c + 
 d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**3 + 3*a*c**2*d*x**2 + 3*a*c*d**2*x** 
4 + a*d**3*x**6 + b*c**3*x**2 + 3*b*c**2*d*x**4 + 3*b*c*d**2*x**6 + b*d**3 
*x**8),x)*a*b**2*c**3*d*f + 9*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4) 
/(a*c**3 + 3*a*c**2*d*x**2 + 3*a*c*d**2*x**4 + a*d**3*x**6 + b*c**3*x**2 + 
 3*b*c**2*d*x**4 + 3*b*c*d**2*x**6 + b*d**3*x**8),x)*a*b**2*c**2*d**2*e - 
96*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**3 + 3*a*c**2*d*x**2 
+ 3*a*c*d**2*x**4 + a*d**3*x**6 + b*c**3*x**2 + 3*b*c**2*d*x**4 + 3*b*c...