\(\int \frac {(c+d x^2)^{5/2} (e+f x^2)}{(a+b x^2)^{5/2}} \, dx\) [33]

Optimal result

Integrand size = 30, antiderivative size = 393 \[ \int \frac {\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=-\frac {d \left (b^2 c e-4 a b d e-7 a b c f+8 a^2 d f\right ) x \sqrt {c+d x^2}}{3 a b^3 \sqrt {a+b x^2}}-\frac {d (b e-2 a f) x \left (c+d x^2\right )^{3/2}}{3 a b^2 \sqrt {a+b x^2}}+\frac {(b e-a f) x \left (c+d x^2\right )^{5/2}}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {\left (2 b^3 c^2 e+16 a^3 d^2 f+a b^2 c (3 d e+c f)-8 a^2 b d (d e+2 c f)\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 a^{3/2} b^{7/2} \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {d \left (b^2 c e-4 a b d e-7 a b c f+8 a^2 d f\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 \sqrt {a} b^{7/2} \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

-1/3*d*(8*a^2*d*f-7*a*b*c*f-4*a*b*d*e+b^2*c*e)*x*(d*x^2+c)^(1/2)/a/b^3/(b* 
x^2+a)^(1/2)-1/3*d*(-2*a*f+b*e)*x*(d*x^2+c)^(3/2)/a/b^2/(b*x^2+a)^(1/2)+1/ 
3*(-a*f+b*e)*x*(d*x^2+c)^(5/2)/a/b/(b*x^2+a)^(3/2)+1/3*(2*b^3*c^2*e+16*a^3 
*d^2*f+a*b^2*c*(c*f+3*d*e)-8*a^2*b*d*(2*c*f+d*e))*(d*x^2+c)^(1/2)*Elliptic 
E(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2),(1-a*d/b/c)^(1/2))/a^(3/2)/b^(7/2)/( 
b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)-1/3*d*(8*a^2*d*f-7*a*b*c*f- 
4*a*b*d*e+b^2*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)/b^(7/2)/(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 = 7.92 (sec) , antiderivative size = 372, normalized size of antiderivative = 0.95 \[ \int \frac {\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {\left (\frac {b}{a}\right )^{3/2} \left (\sqrt {\frac {b}{a}} x \left (c+d x^2\right ) \left (8 a^4 d^2 f+2 b^4 c^2 e x^2+a b^3 c \left (3 c e+3 d e x^2+c f x^2\right )+a^3 b d \left (-4 d e-7 c f+10 d f x^2\right )+a^2 b^2 d \left (-5 d e x^2+d f x^4+c \left (e-9 f x^2\right )\right )\right )+i c \left (2 b^3 c^2 e+16 a^3 d^2 f+a b^2 c (3 d e+c f)-8 a^2 b d (d e+2 c f)\right ) \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c (-b c+a d) \left (-2 b^2 c e+8 a^2 d f-a b (4 d e+c f)\right ) \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{3 b^5 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \] Input:

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

Output:

((b/a)^(3/2)*(Sqrt[b/a]*x*(c + d*x^2)*(8*a^4*d^2*f + 2*b^4*c^2*e*x^2 + a*b 
^3*c*(3*c*e + 3*d*e*x^2 + c*f*x^2) + a^3*b*d*(-4*d*e - 7*c*f + 10*d*f*x^2) 
 + a^2*b^2*d*(-5*d*e*x^2 + d*f*x^4 + c*(e - 9*f*x^2))) + I*c*(2*b^3*c^2*e 
+ 16*a^3*d^2*f + a*b^2*c*(3*d*e + c*f) - 8*a^2*b*d*(d*e + 2*c*f))*(a + b*x 
^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*(-(b*c) + a*d)*(-2*b^2*c*e + 8*a^2*d*f - a*b*(4*d*e 
 + c*f))*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*A 
rcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(3*b^5*(a + b*x^2)^(3/2)*Sqrt[c + d*x^ 
2])
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 456, 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.

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

\(\Big \downarrow \) 401

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 401

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 403

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

\(\Big \downarrow \) 406

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

\(\Big \downarrow \) 320

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

\(\Big \downarrow \) 388

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

\(\Big \downarrow \) 313

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

Input:

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

Output:

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

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. \(844\) vs. \(2(358)=716\).

Time = 16.26 (sec) , antiderivative size = 845, normalized size of antiderivative = 2.15

method	result	size
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {\left (a^{3} d^{2} f -2 a^{2} b c d f -a^{2} b \,d^{2} e +b^{2} c^{2} f a +2 a \,b^{2} c d e -b^{3} c^{2} e \right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 a \,b^{5} \left (x^{2}+\frac {a}{b}\right )^{2}}+\frac {\left (b d \,x^{2}+b c \right ) \left (8 a^{3} d^{2} f -9 a^{2} b c d f -5 a^{2} b \,d^{2} e +b^{2} c^{2} f a +3 a \,b^{2} c d e +2 b^{3} c^{2} e \right ) x}{3 b^{4} a^{2} \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}+\frac {d^{2} f x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b^{3}}+\frac {\left (\frac {d \left (3 f \,d^{2} a^{2}-6 f d c b a -2 a b \,d^{2} e +3 f \,c^{2} b^{2}+3 d \,b^{2} c e \right )}{b^{4}}-\frac {\left (a^{3} d^{2} f -2 a^{2} b c d f -a^{2} b \,d^{2} e +b^{2} c^{2} f a +2 a \,b^{2} c d e -b^{3} c^{2} e \right ) d}{3 b^{4} a}-\frac {\left (8 a^{3} d^{2} f -9 a^{2} b c d f -5 a^{2} b \,d^{2} e +b^{2} c^{2} f a +3 a \,b^{2} c d e +2 b^{3} c^{2} e \right ) \left (a d -b c \right )}{3 b^{4} a^{2}}-\frac {c \left (8 a^{3} d^{2} f -9 a^{2} b c d f -5 a^{2} b \,d^{2} e +b^{2} c^{2} f a +3 a \,b^{2} c d e +2 b^{3} c^{2} e \right )}{3 b^{3} a^{2}}-\frac {a c \,d^{2} f}{3 b^{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 {d^{2} \left (2 a d f -3 b c f -b d e \right )}{b^{3}}-\frac {\left (8 a^{3} d^{2} f -9 a^{2} b c d f -5 a^{2} b \,d^{2} e +b^{2} c^{2} f a +3 a \,b^{2} c d e +2 b^{3} c^{2} e \right ) d}{3 b^{3} a^{2}}-\frac {d^{2} f \left (2 a d +2 b c \right )}{3 b^{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}}\)	\(845\)
risch	\(\text {Expression too large to display}\)	\(1205\)
default	\(\text {Expression too large to display}\)	\(1916\)

Input:

int((d*x^2+c)^(5/2)*(f*x^2+e)/(b*x^2+a)^(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^3*d^2 
*f-2*a^2*b*c*d*f-a^2*b*d^2*e+a*b^2*c^2*f+2*a*b^2*c*d*e-b^3*c^2*e)/a/b^5*x* 
(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/(x^2+a/b)^2+1/3*(b*d*x^2+b*c)*(8*a^3*d 
^2*f-9*a^2*b*c*d*f-5*a^2*b*d^2*e+a*b^2*c^2*f+3*a*b^2*c*d*e+2*b^3*c^2*e)/b^ 
4/a^2*x/((x^2+a/b)*(b*d*x^2+b*c))^(1/2)+1/3*d^2*f/b^3*x*(b*d*x^4+a*d*x^2+b 
*c*x^2+a*c)^(1/2)+(d*(3*a^2*d^2*f-6*a*b*c*d*f-2*a*b*d^2*e+3*b^2*c^2*f+3*b^ 
2*c*d*e)/b^4-1/3*(a^3*d^2*f-2*a^2*b*c*d*f-a^2*b*d^2*e+a*b^2*c^2*f+2*a*b^2* 
c*d*e-b^3*c^2*e)/b^4*d/a-1/3*(8*a^3*d^2*f-9*a^2*b*c*d*f-5*a^2*b*d^2*e+a*b^ 
2*c^2*f+3*a*b^2*c*d*e+2*b^3*c^2*e)/b^4*(a*d-b*c)/a^2-1/3/b^3*c*(8*a^3*d^2* 
f-9*a^2*b*c*d*f-5*a^2*b*d^2*e+a*b^2*c^2*f+3*a*b^2*c*d*e+2*b^3*c^2*e)/a^2-1 
/3*a/b^3*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))-(-1/b^3*d^2*(2*a*d*f-3*b*c*f-b*d*e)-1/3*(8*a^3*d^2*f-9*a^2*b*c*d*f-5 
*a^2*b*d^2*e+a*b^2*c^2*f+3*a*b^2*c*d*e+2*b^3*c^2*e)/b^3*d/a^2-1/3*d^2*f/b^ 
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. 899 vs. \(2 (358) = 716\).

Time = 0.14 (sec) , antiderivative size = 899, normalized size of antiderivative = 2.29 \[ \int \frac {\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(18*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*c*d*f*x - 12*sqrt(c + d*x**2)*sqrt 
(a + b*x**2)*a*d**2*f*x**3 - 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*c**2*f* 
x - 9*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*c*d*e*x + 14*sqrt(c + d*x**2)*sq 
rt(a + b*x**2)*b*c*d*f*x**3 + 6*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*d**2*e 
*x**3 + 2*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*d**2*f*x**5 + 48*int((sqrt(c 
 + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**3*c + a**3*d*x**2 + 3*a**2*b*c*x**2 
+ 3*a**2*b*d*x**4 + 3*a*b**2*c*x**4 + 3*a*b**2*d*x**6 + b**3*c*x**6 + b**3 
*d*x**8),x)*a**4*d**3*f - 48*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/ 
(a**3*c + a**3*d*x**2 + 3*a**2*b*c*x**2 + 3*a**2*b*d*x**4 + 3*a*b**2*c*x** 
4 + 3*a*b**2*d*x**6 + b**3*c*x**6 + b**3*d*x**8),x)*a**3*b*c*d**2*f - 24*i 
nt((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**3*c + a**3*d*x**2 + 3*a**2 
*b*c*x**2 + 3*a**2*b*d*x**4 + 3*a*b**2*c*x**4 + 3*a*b**2*d*x**6 + b**3*c*x 
**6 + b**3*d*x**8),x)*a**3*b*d**3*e + 96*int((sqrt(c + d*x**2)*sqrt(a + b* 
x**2)*x**4)/(a**3*c + a**3*d*x**2 + 3*a**2*b*c*x**2 + 3*a**2*b*d*x**4 + 3* 
a*b**2*c*x**4 + 3*a*b**2*d*x**6 + b**3*c*x**6 + b**3*d*x**8),x)*a**3*b*d** 
3*f*x**2 + 15*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**3*c + a**3* 
d*x**2 + 3*a**2*b*c*x**2 + 3*a**2*b*d*x**4 + 3*a*b**2*c*x**4 + 3*a*b**2*d* 
x**6 + b**3*c*x**6 + b**3*d*x**8),x)*a**2*b**2*c**2*d*f + 9*int((sqrt(c + 
d*x**2)*sqrt(a + b*x**2)*x**4)/(a**3*c + a**3*d*x**2 + 3*a**2*b*c*x**2 + 3 
*a**2*b*d*x**4 + 3*a*b**2*c*x**4 + 3*a*b**2*d*x**6 + b**3*c*x**6 + b**3...