\(\int \frac {e+f x^2}{(a+b x^2)^{5/2} (c+d x^2)^{5/2}} \, dx\) [38]

Optimal result

Integrand size = 30, antiderivative size = 496 \[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}} \, dx=\frac {(b e-a f) x}{3 a (b c-a d) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}+\frac {\left (2 b^2 c e+5 a^2 d f-a b (8 d e-c f)\right ) x}{3 a^2 (b c-a d)^2 \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}+\frac {d \left (2 b^2 c^2 e-a^2 d (d e-7 c f)-a b c (9 d e-c f)\right ) x \sqrt {a+b x^2}}{3 a^2 c (b c-a d)^3 \left (c+d x^2\right )^{3/2}}+\frac {\sqrt {d} \left (2 b^3 c^3 e-2 a^2 b c d (5 d e-7 c f)-a b^2 c^2 (10 d e-c f)+a^3 d^2 (2 d e+c f)\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a^2 c^{3/2} (b c-a d)^4 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {b \sqrt {d} \left (b^2 c^2 e-2 a b c (9 d e-4 c f)+a^2 d (d e+8 c f)\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a^2 \sqrt {c} (b c-a d)^4 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:

1/3*(-a*f+b*e)*x/a/(-a*d+b*c)/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2)+1/3*(2*b^2*c 
*e+5*a^2*d*f-a*b*(-c*f+8*d*e))*x/a^2/(-a*d+b*c)^2/(b*x^2+a)^(1/2)/(d*x^2+c 
)^(3/2)+1/3*d*(2*b^2*c^2*e-a^2*d*(-7*c*f+d*e)-a*b*c*(-c*f+9*d*e))*x*(b*x^2 
+a)^(1/2)/a^2/c/(-a*d+b*c)^3/(d*x^2+c)^(3/2)+1/3*d^(1/2)*(2*b^3*c^3*e-2*a^ 
2*b*c*d*(-7*c*f+5*d*e)-a*b^2*c^2*(-c*f+10*d*e)+a^3*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)) 
/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*d^(1/2)*(b^2*c^2*e-2*a*b*c*(-4*c*f+9*d*e)+a^2*d*(8*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))/a^2/ 
c^(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 = 11.65 (sec) , antiderivative size = 436, normalized size of antiderivative = 0.88 \[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}} \, dx=\frac {-\sqrt {\frac {b}{a}} x \left (a^2 c d^2 (-b c+a d) (-d e+c f) \left (a+b x^2\right )^2-a^2 d^2 (a d (2 d e+c f)+b c (-10 d e+7 c f)) \left (a+b x^2\right )^2 \left (c+d x^2\right )-a b^2 c^2 (-b c+a d) (-b e+a f) \left (c+d x^2\right )^2-b^2 c^2 \left (2 b^2 c e+7 a^2 d f+a b (-10 d e+c f)\right ) \left (a+b x^2\right ) \left (c+d x^2\right )^2\right )+i b c \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \left (c+d x^2\right ) \sqrt {1+\frac {d x^2}{c}} \left (\left (2 b^3 c^3 e+a b^2 c^2 (-10 d e+c f)+a^3 d^2 (2 d e+c f)+2 a^2 b c d (-5 d e+7 c f)\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-(b c-a d) \left (2 b^2 c^2 e+a b c (-9 d e+c f)+a^2 d (-d e+7 c f)\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{3 a^2 \sqrt {\frac {b}{a}} c^2 (b c-a d)^4 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \] Input:

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

Output:

(-(Sqrt[b/a]*x*(a^2*c*d^2*(-(b*c) + a*d)*(-(d*e) + c*f)*(a + b*x^2)^2 - a^ 
2*d^2*(a*d*(2*d*e + c*f) + b*c*(-10*d*e + 7*c*f))*(a + b*x^2)^2*(c + d*x^2 
) - a*b^2*c^2*(-(b*c) + a*d)*(-(b*e) + a*f)*(c + d*x^2)^2 - b^2*c^2*(2*b^2 
*c*e + 7*a^2*d*f + a*b*(-10*d*e + c*f))*(a + b*x^2)*(c + d*x^2)^2)) + I*b* 
c*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*((2*b^3* 
c^3*e + a*b^2*c^2*(-10*d*e + c*f) + a^3*d^2*(2*d*e + c*f) + 2*a^2*b*c*d*(- 
5*d*e + 7*c*f))*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - (b*c - a* 
d)*(2*b^2*c^2*e + a*b*c*(-9*d*e + c*f) + a^2*d*(-(d*e) + 7*c*f))*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 = 0.70 (sec) , antiderivative size = 531, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {402, 25, 402, 27, 402, 400, 313, 320}

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.

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 402

\(\displaystyle \frac {\frac {3 d \left (\frac {\int \frac {b \left (-d (d e-7 c f) a^2-b c (9 d e-c f) a+2 b^2 c^2 e\right ) x^2+a \left (-d (2 d e+c f) a^2+b c (9 d e-7 c f) a+b^2 c^2 e\right )}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c (b c-a d)}+\frac {x \sqrt {a+b x^2} \left (a^2 (-d) (d e-7 c f)-a b c (9 d e-c f)+2 b^2 c^2 e\right )}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{a (b c-a d)}+\frac {x \left (5 a^2 d f-a b (8 d e-c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} (b c-a d)}}{3 a (b c-a d)}+\frac {x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 400

\(\displaystyle \frac {\frac {3 d \left (\frac {\frac {\left (a^3 d^2 (c f+2 d e)-2 a^2 b c d (5 d e-7 c f)-a b^2 c^2 (10 d e-c f)+2 b^3 c^3 e\right ) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b c-a d}-\frac {a b \left (a^2 d (8 c f+d e)-2 a b c (9 d e-4 c f)+b^2 c^2 e\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}}{3 c (b c-a d)}+\frac {x \sqrt {a+b x^2} \left (a^2 (-d) (d e-7 c f)-a b c (9 d e-c f)+2 b^2 c^2 e\right )}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{a (b c-a d)}+\frac {x \left (5 a^2 d f-a b (8 d e-c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} (b c-a d)}}{3 a (b c-a d)}+\frac {x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {\frac {3 d \left (\frac {\frac {\sqrt {a+b x^2} \left (a^3 d^2 (c f+2 d e)-2 a^2 b c d (5 d e-7 c f)-a b^2 c^2 (10 d e-c f)+2 b^3 c^3 e\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {a b \left (a^2 d (8 c f+d e)-2 a b c (9 d e-4 c f)+b^2 c^2 e\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}}{3 c (b c-a d)}+\frac {x \sqrt {a+b x^2} \left (a^2 (-d) (d e-7 c f)-a b c (9 d e-c f)+2 b^2 c^2 e\right )}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{a (b c-a d)}+\frac {x \left (5 a^2 d f-a b (8 d e-c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} (b c-a d)}}{3 a (b c-a d)}+\frac {x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 320

\(\displaystyle \frac {\frac {x \left (5 a^2 d f-a b (8 d e-c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac {3 d \left (\frac {x \sqrt {a+b x^2} \left (a^2 (-d) (d e-7 c f)-a b c (9 d e-c f)+2 b^2 c^2 e\right )}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}+\frac {\frac {\sqrt {a+b x^2} \left (a^3 d^2 (c f+2 d e)-2 a^2 b c d (5 d e-7 c f)-a b^2 c^2 (10 d e-c f)+2 b^3 c^3 e\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {a+b x^2} \left (a^2 d (8 c f+d e)-2 a b c (9 d e-4 c f)+b^2 c^2 e\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c (b c-a d)}\right )}{a (b c-a d)}}{3 a (b c-a d)}+\frac {x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} (b c-a d)}\)

Input:

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

Output:

((b*e - a*f)*x)/(3*a*(b*c - a*d)*(a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)) + (( 
(2*b^2*c*e + 5*a^2*d*f - a*b*(8*d*e - c*f))*x)/(a*(b*c - a*d)*Sqrt[a + b*x 
^2]*(c + d*x^2)^(3/2)) + (3*d*(((2*b^2*c^2*e - a^2*d*(d*e - 7*c*f) - a*b*c 
*(9*d*e - c*f))*x*Sqrt[a + b*x^2])/(3*c*(b*c - a*d)*(c + d*x^2)^(3/2)) + ( 
((2*b^3*c^3*e - 2*a^2*b*c*d*(5*d*e - 7*c*f) - a*b^2*c^2*(10*d*e - c*f) + a 
^3*d^2*(2*d*e + c*f))*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c] 
], 1 - (b*c)/(a*d)])/(Sqrt[c]*Sqrt[d]*(b*c - a*d)*Sqrt[(c*(a + b*x^2))/(a* 
(c + d*x^2))]*Sqrt[c + d*x^2]) - (b*Sqrt[c]*(b^2*c^2*e - 2*a*b*c*(9*d*e - 
4*c*f) + a^2*d*(d*e + 8*c*f))*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x) 
/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*(b*c - a*d)*Sqrt[(c*(a + b*x^2))/(a* 
(c + d*x^2))]*Sqrt[c + d*x^2]))/(3*c*(b*c - a*d))))/(a*(b*c - a*d)))/(3*a* 
(b*c - a*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 400

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)^ 
(3/2)), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d)   Int[1/(Sqrt[a + b*x^2]* 
Sqrt[c + d*x^2]), x], x] - Simp[(d*e - c*f)/(b*c - a*d)   Int[Sqrt[a + b*x^ 
2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[b/a] & 
& PosQ[d/c]
 

rule 402

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 + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) 
 Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) 
*(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b 
, c, d, e, f, q}, x] && LtQ[p, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1012\) vs. \(2(461)=922\).

Time = 14.50 (sec) , antiderivative size = 1013, normalized size of antiderivative = 2.04

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1013\)
default	\(\text {Expression too large to display}\)	\(3687\)

Input:

int((f*x^2+e)/(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/d*(2* 
a*c*f-a*d*e-b*c*e)/a/c/(a^2*d^2-2*a*b*c*d+b^2*c^2)*x^3-1/3*(a^2*c*d*f-a^2* 
d^2*e+a*b*c^2*f-b^2*c^2*e)/a/c/(a^2*d^2-2*a*b*c*d+b^2*c^2)/b^2/d^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*(- 
1/6*(a^3*c*d^2*f+2*a^3*d^3*e+14*a^2*b*c^2*d*f-10*a^2*b*c*d^2*e+a*b^2*c^3*f 
-10*a*b^2*c^2*d*e+2*b^3*c^3*e)/a^2/c^2/(a^2*d^2-2*a*b*c*d+b^2*c^2)^2*x^3-1 
/6*(a^4*c*d^3*f+2*a^4*d^4*e+7*a^3*b*c^2*d^2*f-9*a^3*b*c*d^3*e+7*a^2*b^2*c^ 
3*d*f-2*a^2*b^2*c^2*d^2*e+a*b^3*c^4*f-9*a*b^3*c^3*d*e+2*b^4*c^4*e)/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)+(1/3/(a^2*d^2-2*a*b*c*d+b^2*c^2)*(a^2*c*d*f+2*a^2*d^2*e+a*b*c^2*f- 
6*a*b*c*d*e+2*b^2*c^2*e)/a^2/c^2-1/3*(a^4*c*d^3*f+2*a^4*d^4*e+7*a^3*b*c^2* 
d^2*f-9*a^3*b*c*d^3*e+7*a^2*b^2*c^3*d*f-2*a^2*b^2*c^2*d^2*e+a*b^3*c^4*f-9* 
a*b^3*c^3*d*e+2*b^4*c^4*e)/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)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))+1/3*b*(a^3*c*d^2*f+2 
*a^3*d^3*e+14*a^2*b*c^2*d*f-10*a^2*b*c*d^2*e+a*b^2*c^3*f-10*a*b^2*c^2*d*e+ 
2*b^3*c^3*e)/(a^2*d^2-2*a*b*c*d+b^2*c^2)^2/a^2/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))-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. 2280 vs. \(2 (461) = 922\).

Time = 0.36 (sec) , antiderivative size = 2280, normalized size of antiderivative = 4.60 \[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/3*(((2*(b^6*c^3*d^2 - 5*a*b^5*c^2*d^3 - 5*a^2*b^4*c*d^4 + a^3*b^3*d^5)* 
e + (a*b^5*c^3*d^2 + 14*a^2*b^4*c^2*d^3 + a^3*b^3*c*d^4)*f)*x^8 + 2*(2*(b^ 
6*c^4*d - 4*a*b^5*c^3*d^2 - 10*a^2*b^4*c^2*d^3 - 4*a^3*b^3*c*d^4 + a^4*b^2 
*d^5)*e + (a*b^5*c^4*d + 15*a^2*b^4*c^3*d^2 + 15*a^3*b^3*c^2*d^3 + a^4*b^2 
*c*d^4)*f)*x^6 + (2*(b^6*c^5 - a*b^5*c^4*d - 24*a^2*b^4*c^3*d^2 - 24*a^3*b 
^3*c^2*d^3 - a^4*b^2*c*d^4 + a^5*b*d^5)*e + (a*b^5*c^5 + 18*a^2*b^4*c^4*d 
+ 58*a^3*b^3*c^3*d^2 + 18*a^4*b^2*c^2*d^3 + a^5*b*c*d^4)*f)*x^4 + 2*(2*(a* 
b^5*c^5 - 4*a^2*b^4*c^4*d - 10*a^3*b^3*c^3*d^2 - 4*a^4*b^2*c^2*d^3 + a^5*b 
*c*d^4)*e + (a^2*b^4*c^5 + 15*a^3*b^3*c^4*d + 15*a^4*b^2*c^3*d^2 + a^5*b*c 
^2*d^3)*f)*x^2 + 2*(a^2*b^4*c^5 - 5*a^3*b^3*c^4*d - 5*a^4*b^2*c^3*d^2 + a^ 
5*b*c^2*d^3)*e + (a^3*b^3*c^5 + 14*a^4*b^2*c^4*d + a^5*b*c^3*d^2)*f)*sqrt( 
a*c)*sqrt(-b/a)*elliptic_e(arcsin(x*sqrt(-b/a)), a*d/(b*c)) - (((2*b^6*c^3 
*d^2 + (a^2*b^4 - 10*a*b^5)*c^2*d^3 - 2*(9*a^3*b^3 + 5*a^2*b^4)*c*d^4 + (a 
^4*b^2 + 2*a^3*b^3)*d^5)*e + (a*b^5*c^3*d^2 + 2*(4*a^3*b^3 + 7*a^2*b^4)*c^ 
2*d^3 + (8*a^4*b^2 + a^3*b^3)*c*d^4)*f)*x^8 + 2*((2*b^6*c^4*d + (a^2*b^4 - 
 8*a*b^5)*c^3*d^2 - (17*a^3*b^3 + 20*a^2*b^4)*c^2*d^3 - (17*a^4*b^2 + 8*a^ 
3*b^3)*c*d^4 + (a^5*b + 2*a^4*b^2)*d^5)*e + (a*b^5*c^4*d + (8*a^3*b^3 + 15 
*a^2*b^4)*c^3*d^2 + (16*a^4*b^2 + 15*a^3*b^3)*c^2*d^3 + (8*a^5*b + a^4*b^2 
)*c*d^4)*f)*x^6 + ((2*b^6*c^5 + (a^2*b^4 - 2*a*b^5)*c^4*d - 2*(7*a^3*b^3 + 
 24*a^2*b^4)*c^3*d^2 - 2*(35*a^4*b^2 + 24*a^3*b^3)*c^2*d^3 - 2*(7*a^5*b...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(c + d*x**2)*sqrt(a + b*x**2)*f*x**3 + 3*int((sqrt(c + d*x**2)*sqrt(a 
 + b*x**2)*x**6)/(a**3*c**3 + 3*a**3*c**2*d*x**2 + 3*a**3*c*d**2*x**4 + a* 
*3*d**3*x**6 + 3*a**2*b*c**3*x**2 + 9*a**2*b*c**2*d*x**4 + 9*a**2*b*c*d**2 
*x**6 + 3*a**2*b*d**3*x**8 + 3*a*b**2*c**3*x**4 + 9*a*b**2*c**2*d*x**6 + 9 
*a*b**2*c*d**2*x**8 + 3*a*b**2*d**3*x**10 + b**3*c**3*x**6 + 3*b**3*c**2*d 
*x**8 + 3*b**3*c*d**2*x**10 + b**3*d**3*x**12),x)*a**2*b*c**2*d*f + 6*int( 
(sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**6)/(a**3*c**3 + 3*a**3*c**2*d*x**2 + 
 3*a**3*c*d**2*x**4 + a**3*d**3*x**6 + 3*a**2*b*c**3*x**2 + 9*a**2*b*c**2* 
d*x**4 + 9*a**2*b*c*d**2*x**6 + 3*a**2*b*d**3*x**8 + 3*a*b**2*c**3*x**4 + 
9*a*b**2*c**2*d*x**6 + 9*a*b**2*c*d**2*x**8 + 3*a*b**2*d**3*x**10 + b**3*c 
**3*x**6 + 3*b**3*c**2*d*x**8 + 3*b**3*c*d**2*x**10 + b**3*d**3*x**12),x)* 
a**2*b*c*d**2*f*x**2 + 3*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**6)/(a** 
3*c**3 + 3*a**3*c**2*d*x**2 + 3*a**3*c*d**2*x**4 + a**3*d**3*x**6 + 3*a**2 
*b*c**3*x**2 + 9*a**2*b*c**2*d*x**4 + 9*a**2*b*c*d**2*x**6 + 3*a**2*b*d**3 
*x**8 + 3*a*b**2*c**3*x**4 + 9*a*b**2*c**2*d*x**6 + 9*a*b**2*c*d**2*x**8 + 
 3*a*b**2*d**3*x**10 + b**3*c**3*x**6 + 3*b**3*c**2*d*x**8 + 3*b**3*c*d**2 
*x**10 + b**3*d**3*x**12),x)*a**2*b*d**3*f*x**4 + 6*int((sqrt(c + d*x**2)* 
sqrt(a + b*x**2)*x**6)/(a**3*c**3 + 3*a**3*c**2*d*x**2 + 3*a**3*c*d**2*x** 
4 + a**3*d**3*x**6 + 3*a**2*b*c**3*x**2 + 9*a**2*b*c**2*d*x**4 + 9*a**2*b* 
c*d**2*x**6 + 3*a**2*b*d**3*x**8 + 3*a*b**2*c**3*x**4 + 9*a*b**2*c**2*d...