\(\int \frac {(e+f x^2)^2}{(a+b x^2)^{5/2} (c+d x^2)^{5/2}} \, dx\) [91]

Optimal result

Integrand size = 32, antiderivative size = 613 \[ \int \frac {\left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}} \, dx=\frac {(b e-a f)^2 x}{3 a b (b c-a d) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}+\frac {2 (b e-a f) \left (b^2 c e+a^2 d f-a b (4 d e-2 c f)\right ) x}{3 a^2 b (b c-a d)^2 \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}+\frac {d \left (2 b^3 c^2 e^2-3 a^3 c d f^2-a b^2 c e (9 d e-2 c f)-a^2 b \left (d^2 e^2-14 c d e f+5 c^2 f^2\right )\right ) x \sqrt {a+b x^2}}{3 a^2 b c (b c-a d)^3 \left (c+d x^2\right )^{3/2}}+\frac {2 \sqrt {d} \left (b^3 c^3 e^2-a b^2 c^2 e (5 d e-c f)+a^3 d \left (d^2 e^2+c d e f-4 c^2 f^2\right )-a^2 b c \left (5 d^2 e^2-14 c d e f+4 c^2 f^2\right )\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a^2 c^{3/2} (b c-a d)^4 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {\left (b^3 c^2 d e^2-3 a^3 c d^2 f^2+a^2 b d \left (d^2 e^2+16 c d e f-10 c^2 f^2\right )-a b^2 c \left (18 d^2 e^2-16 c d e f+3 c^2 f^2\right )\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a^2 \sqrt {c} \sqrt {d} (b c-a d)^4 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:

1/3*(-a*f+b*e)^2*x/a/b/(-a*d+b*c)/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2)+2/3*(-a* 
f+b*e)*(b^2*c*e+a^2*d*f-a*b*(-2*c*f+4*d*e))*x/a^2/b/(-a*d+b*c)^2/(b*x^2+a) 
^(1/2)/(d*x^2+c)^(3/2)+1/3*d*(2*b^3*c^2*e^2-3*a^3*c*d*f^2-a*b^2*c*e*(-2*c* 
f+9*d*e)-a^2*b*(5*c^2*f^2-14*c*d*e*f+d^2*e^2))*x*(b*x^2+a)^(1/2)/a^2/b/c/( 
-a*d+b*c)^3/(d*x^2+c)^(3/2)+2/3*d^(1/2)*(b^3*c^3*e^2-a*b^2*c^2*e*(-c*f+5*d 
*e)+a^3*d*(-4*c^2*f^2+c*d*e*f+d^2*e^2)-a^2*b*c*(4*c^2*f^2-14*c*d*e*f+5*d^2 
*e^2))*(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))/a^2/c^(3/2)/(-a*d+b*c)^4/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d* 
x^2+c)^(1/2)-1/3*(b^3*c^2*d*e^2-3*a^3*c*d^2*f^2+a^2*b*d*(-10*c^2*f^2+16*c* 
d*e*f+d^2*e^2)-a*b^2*c*(3*c^2*f^2-16*c*d*e*f+18*d^2*e^2))*(b*x^2+a)^(1/2)* 
InverseJacobiAM(arctan(d^(1/2)*x/c^(1/2)),(1-b*c/a/d)^(1/2))/a^2/c^(1/2)/d 
^(1/2)/(-a*d+b*c)^4/(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 = 12.07 (sec) , antiderivative size = 502, normalized size of antiderivative = 0.82 \[ \int \frac {\left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}} \, dx=\frac {-\sqrt {\frac {b}{a}} x \left (a^2 c d (b c-a d) (d e-c f)^2 \left (a+b x^2\right )^2-2 a^2 d (d e-c f) (b c (-5 d e+2 c f)+a d (d e+2 c f)) \left (a+b x^2\right )^2 \left (c+d x^2\right )+a b c^2 (-b c+a d) (b e-a f)^2 \left (c+d x^2\right )^2-2 b c^2 (b e-a f) \left (b^2 c e+2 a^2 d f+a b (-5 d e+2 c f)\right ) \left (a+b x^2\right ) \left (c+d x^2\right )^2\right )-i c \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \left (c+d x^2\right ) \sqrt {1+\frac {d x^2}{c}} \left (-2 b \left (b^3 c^3 e^2+a b^2 c^2 e (-5 d e+c f)+a^3 d \left (d^2 e^2+c d e f-4 c^2 f^2\right )+a^2 b c \left (-5 d^2 e^2+14 c d e f-4 c^2 f^2\right )\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+(b c-a d) \left (2 b^3 c^2 e^2-3 a^3 c d f^2+a b^2 c e (-9 d e+2 c f)-a^2 b \left (d^2 e^2-14 c d e f+5 c^2 f^2\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{3 a^2 \sqrt {\frac {b}{a}} c^2 (b c-a d)^4 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \] Input:

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

Output:

(-(Sqrt[b/a]*x*(a^2*c*d*(b*c - a*d)*(d*e - c*f)^2*(a + b*x^2)^2 - 2*a^2*d* 
(d*e - c*f)*(b*c*(-5*d*e + 2*c*f) + a*d*(d*e + 2*c*f))*(a + b*x^2)^2*(c + 
d*x^2) + a*b*c^2*(-(b*c) + a*d)*(b*e - a*f)^2*(c + d*x^2)^2 - 2*b*c^2*(b*e 
 - a*f)*(b^2*c*e + 2*a^2*d*f + a*b*(-5*d*e + 2*c*f))*(a + b*x^2)*(c + d*x^ 
2)^2)) - I*c*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*(c + d*x^2)*Sqrt[1 + (d*x^2)/ 
c]*(-2*b*(b^3*c^3*e^2 + a*b^2*c^2*e*(-5*d*e + c*f) + a^3*d*(d^2*e^2 + c*d* 
e*f - 4*c^2*f^2) + a^2*b*c*(-5*d^2*e^2 + 14*c*d*e*f - 4*c^2*f^2))*Elliptic 
E[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (b*c - a*d)*(2*b^3*c^2*e^2 - 3*a^ 
3*c*d*f^2 + a*b^2*c*e*(-9*d*e + 2*c*f) - a^2*b*(d^2*e^2 - 14*c*d*e*f + 5*c 
^2*f^2))*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(3*a^2*Sqrt[b/a] 
*c^2*(b*c - a*d)^4*(a + b*x^2)^(3/2)*(c + d*x^2)^(3/2))
 

Rubi [A] (verified)

Time = 1.62 (sec) , antiderivative size = 1170, normalized size of antiderivative = 1.91, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {433, 2009}

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.

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

\(\Big \downarrow \) 433

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \sqrt {d} (b c+a d) \left (b^2 c^2-6 a b d c+a^2 d^2\right ) \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right ) e^2}{3 a^2 c^{3/2} (b c-a d)^4 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {b \sqrt {d} \left (b^2 c^2-18 a b d c+a^2 d^2\right ) \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) e^2}{3 a^2 \sqrt {c} (b c-a d)^4 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {d \left (2 b^2 c^2-9 a b d c-a^2 d^2\right ) x \sqrt {b x^2+a} e^2}{3 a^2 c (b c-a d)^3 \left (d x^2+c\right )^{3/2}}+\frac {2 b (b c-4 a d) x e^2}{3 a^2 (b c-a d)^2 \sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}+\frac {b x e^2}{3 a (b c-a d) \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}+\frac {2 \sqrt {d} \left (b^2 c^2+14 a b d c+a^2 d^2\right ) f \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right ) e}{3 a \sqrt {c} (b c-a d)^4 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {16 b \sqrt {c} \sqrt {d} (b c+a d) f \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right ) e}{3 a (b c-a d)^4 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {2 d (b c+7 a d) f x \sqrt {b x^2+a} e}{3 a (b c-a d)^3 \left (d x^2+c\right )^{3/2}}+\frac {2 (b c+5 a d) f x e}{3 a (b c-a d)^2 \sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}-\frac {2 f x e}{3 (b c-a d) \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}-\frac {8 \sqrt {c} \sqrt {d} (b c+a d) f^2 \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 (b c-a d)^4 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {\sqrt {c} (3 b c+a d) (b c+3 a d) f^2 \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a \sqrt {d} (b c-a d)^4 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {d (5 b c+3 a d) f^2 x \sqrt {b x^2+a}}{3 b (b c-a d)^3 \left (d x^2+c\right )^{3/2}}-\frac {2 (2 b c+a d) f^2 x}{3 b (b c-a d)^2 \sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}+\frac {a f^2 x}{3 b (b c-a d) \left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}\)

Input:

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

Output:

(b*e^2*x)/(3*a*(b*c - a*d)*(a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)) - (2*e*f*x 
)/(3*(b*c - a*d)*(a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)) + (a*f^2*x)/(3*b*(b* 
c - a*d)*(a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)) + (2*b*(b*c - 4*a*d)*e^2*x)/ 
(3*a^2*(b*c - a*d)^2*Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)) + (2*(b*c + 5*a*d) 
*e*f*x)/(3*a*(b*c - a*d)^2*Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)) - (2*(2*b*c 
+ a*d)*f^2*x)/(3*b*(b*c - a*d)^2*Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)) + (d*( 
2*b^2*c^2 - 9*a*b*c*d - a^2*d^2)*e^2*x*Sqrt[a + b*x^2])/(3*a^2*c*(b*c - a* 
d)^3*(c + d*x^2)^(3/2)) + (2*d*(b*c + 7*a*d)*e*f*x*Sqrt[a + b*x^2])/(3*a*( 
b*c - a*d)^3*(c + d*x^2)^(3/2)) - (d*(5*b*c + 3*a*d)*f^2*x*Sqrt[a + b*x^2] 
)/(3*b*(b*c - a*d)^3*(c + d*x^2)^(3/2)) + (2*Sqrt[d]*(b*c + a*d)*(b^2*c^2 
- 6*a*b*c*d + a^2*d^2)*e^2*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sq 
rt[c]], 1 - (b*c)/(a*d)])/(3*a^2*c^(3/2)*(b*c - a*d)^4*Sqrt[(c*(a + b*x^2) 
)/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (2*Sqrt[d]*(b^2*c^2 + 14*a*b*c*d + a 
^2*d^2)*e*f*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b* 
c)/(a*d)])/(3*a*Sqrt[c]*(b*c - a*d)^4*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2)) 
]*Sqrt[c + d*x^2]) - (8*Sqrt[c]*Sqrt[d]*(b*c + a*d)*f^2*Sqrt[a + b*x^2]*El 
lipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*(b*c - a*d)^4*Sq 
rt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (b*Sqrt[d]*(b^2*c^2 
 - 18*a*b*c*d + a^2*d^2)*e^2*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/ 
Sqrt[c]], 1 - (b*c)/(a*d)])/(3*a^2*Sqrt[c]*(b*c - a*d)^4*Sqrt[(c*(a + b...
 

Defintions of rubi rules used

rule 433

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_ 
)^2)^(r_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*x^2)^p*(c + d*x^2) 
^q*(e + f*x^2)^r, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, p, 
 q, r}, x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1266\) vs. \(2(578)=1156\).

Time = 20.42 (sec) , antiderivative size = 1267, normalized size of antiderivative = 2.07

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1267\)
default	\(\text {Expression too large to display}\)	\(5222\)

Input:

int((f*x^2+e)^2/(b*x^2+a)^(5/2)/(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/b^2/d^2* 
(a^2*c*d*f^2+a*b*c^2*f^2-4*a*b*c*d*e*f+a*b*d^2*e^2+b^2*c*d*e^2)/a/c/(a^2*d 
^2-2*a*b*c*d+b^2*c^2)*x^3+1/3/b^2/d^2*(2*a^2*c^2*f^2-2*a^2*c*d*e*f+a^2*d^2 
*e^2-2*a*b*c^2*e*f+b^2*c^2*e^2)/a/c/(a^2*d^2-2*a*b*c*d+b^2*c^2)*x)*(b*d*x^ 
4+a*d*x^2+b*c*x^2+a*c)^(1/2)/(x^4+(a*d+b*c)/d/b*x^2+a*c/d/b)^2-2*b*d*(1/3* 
(4*a^3*c^2*d*f^2-a^3*c*d^2*e*f-a^3*d^3*e^2+4*a^2*b*c^3*f^2-14*a^2*b*c^2*d* 
e*f+5*a^2*b*c*d^2*e^2-a*b^2*c^3*e*f+5*a*b^2*c^2*d*e^2-b^3*c^3*e^2)/a^2/c^2 
/(a^2*d^2-2*a*b*c*d+b^2*c^2)^2*x^3+1/6*(5*a^4*c^2*d^2*f^2-2*a^4*c*d^3*e*f- 
2*a^4*d^4*e^2+6*a^3*b*c^3*d*f^2-14*a^3*b*c^2*d^2*e*f+9*a^3*b*c*d^3*e^2+5*a 
^2*b^2*c^4*f^2-14*a^2*b^2*c^3*d*e*f+2*a^2*b^2*c^2*d^2*e^2-2*a*b^3*c^4*e*f+ 
9*a*b^3*c^3*d*e^2-2*b^4*c^4*e^2)/a^2/c^2/(a^2*d^2-2*a*b*c*d+b^2*c^2)^2/b/d 
*x)/((x^4+(a*d+b*c)/d/b*x^2+a*c/d/b)*b*d)^(1/2)+(-2/3*(a^2*c^2*f^2-a^2*c*d 
*e*f-a^2*d^2*e^2-a*b*c^2*e*f+3*a*b*c*d*e^2-b^2*c^2*e^2)/(a^2*d^2-2*a*b*c*d 
+b^2*c^2)/a^2/c^2+1/3*(5*a^4*c^2*d^2*f^2-2*a^4*c*d^3*e*f-2*a^4*d^4*e^2+6*a 
^3*b*c^3*d*f^2-14*a^3*b*c^2*d^2*e*f+9*a^3*b*c*d^3*e^2+5*a^2*b^2*c^4*f^2-14 
*a^2*b^2*c^3*d*e*f+2*a^2*b^2*c^2*d^2*e^2-2*a*b^3*c^4*e*f+9*a*b^3*c^3*d*e^2 
-2*b^4*c^4*e^2)/a^2/c^2/(a^2*d^2-2*a*b*c*d+b^2*c^2)^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)*Elliptic 
F(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-2/3*b*(4*a^3*c^2*d*f^2-a^3*c*d^ 
2*e*f-a^3*d^3*e^2+4*a^2*b*c^3*f^2-14*a^2*b*c^2*d*e*f+5*a^2*b*c*d^2*e^2-...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3168 vs. \(2 (578) = 1156\).

Time = 0.37 (sec) , antiderivative size = 3168, normalized size of antiderivative = 5.17 \[ \int \frac {\left (e+f x^2\right )^2}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - sqrt(c + d*x**2)*sqrt(a + b*x**2)*e*f*x + int((sqrt(c + d*x**2)*sqrt(a 
 + b*x**2)*x**4)/(a**4*c**3*d + 3*a**4*c**2*d**2*x**2 + 3*a**4*c*d**3*x**4 
 + a**4*d**4*x**6 + a**3*b*c**4 + 6*a**3*b*c**3*d*x**2 + 12*a**3*b*c**2*d* 
*2*x**4 + 10*a**3*b*c*d**3*x**6 + 3*a**3*b*d**4*x**8 + 3*a**2*b**2*c**4*x* 
*2 + 12*a**2*b**2*c**3*d*x**4 + 18*a**2*b**2*c**2*d**2*x**6 + 12*a**2*b**2 
*c*d**3*x**8 + 3*a**2*b**2*d**4*x**10 + 3*a*b**3*c**4*x**4 + 10*a*b**3*c** 
3*d*x**6 + 12*a*b**3*c**2*d**2*x**8 + 6*a*b**3*c*d**3*x**10 + a*b**3*d**4* 
x**12 + b**4*c**4*x**6 + 3*b**4*c**3*d*x**8 + 3*b**4*c**2*d**2*x**10 + b** 
4*c*d**3*x**12),x)*a**4*c**2*d**2*f**2 + 2*int((sqrt(c + d*x**2)*sqrt(a + 
b*x**2)*x**4)/(a**4*c**3*d + 3*a**4*c**2*d**2*x**2 + 3*a**4*c*d**3*x**4 + 
a**4*d**4*x**6 + a**3*b*c**4 + 6*a**3*b*c**3*d*x**2 + 12*a**3*b*c**2*d**2* 
x**4 + 10*a**3*b*c*d**3*x**6 + 3*a**3*b*d**4*x**8 + 3*a**2*b**2*c**4*x**2 
+ 12*a**2*b**2*c**3*d*x**4 + 18*a**2*b**2*c**2*d**2*x**6 + 12*a**2*b**2*c* 
d**3*x**8 + 3*a**2*b**2*d**4*x**10 + 3*a*b**3*c**4*x**4 + 10*a*b**3*c**3*d 
*x**6 + 12*a*b**3*c**2*d**2*x**8 + 6*a*b**3*c*d**3*x**10 + a*b**3*d**4*x** 
12 + b**4*c**4*x**6 + 3*b**4*c**3*d*x**8 + 3*b**4*c**2*d**2*x**10 + b**4*c 
*d**3*x**12),x)*a**4*c*d**3*f**2*x**2 + int((sqrt(c + d*x**2)*sqrt(a + b*x 
**2)*x**4)/(a**4*c**3*d + 3*a**4*c**2*d**2*x**2 + 3*a**4*c*d**3*x**4 + a** 
4*d**4*x**6 + a**3*b*c**4 + 6*a**3*b*c**3*d*x**2 + 12*a**3*b*c**2*d**2*x** 
4 + 10*a**3*b*c*d**3*x**6 + 3*a**3*b*d**4*x**8 + 3*a**2*b**2*c**4*x**2 ...