\(\int \frac {e+f x^2}{(a+b x^2)^{5/2} (c+d x^2)^{3/2}} \, dx\) [37]

Optimal result

Integrand size = 30, antiderivative size = 383 \[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}} \, dx=\frac {(b e-a f) x}{3 a (b c-a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}+\frac {\left (2 b^2 c e+3 a^2 d f-a b (6 d e-c f)\right ) x}{3 a^2 (b c-a d)^2 \sqrt {a+b x^2} \sqrt {c+d x^2}}+\frac {\sqrt {d} \left (2 b^2 c^2 e-a^2 d (3 d e-7 c f)-a b c (7 d e-c f)\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a^2 \sqrt {c} (b c-a d)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {d} \left (b^2 c e+3 a^2 d f-a b (9 d e-5 c f)\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a^2 (b c-a d)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:

1/3*(-a*f+b*e)*x/a/(-a*d+b*c)/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)+1/3*(2*b^2*c 
*e+3*a^2*d*f-a*b*(-c*f+6*d*e))*x/a^2/(-a*d+b*c)^2/(b*x^2+a)^(1/2)/(d*x^2+c 
)^(1/2)+1/3*d^(1/2)*(2*b^2*c^2*e-a^2*d*(-7*c*f+3*d*e)-a*b*c*(-c*f+7*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^(1/2)/(-a*d+b*c)^3/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^ 
(1/2)-1/3*c^(1/2)*d^(1/2)*(b^2*c*e+3*a^2*d*f-a*b*(-5*c*f+9*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/(- 
a*d+b*c)^3/(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 = 12.18 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.08 \[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}} \, dx=\frac {-\sqrt {\frac {b}{a}} x \left (3 a^4 d^2 (-d e+c f)+2 b^4 c^2 e x^2 \left (c+d x^2\right )+a b^3 c \left (c+d x^2\right ) \left (3 c e-7 d e x^2+c f x^2\right )+a^3 b d \left (5 c^2 f-6 d^2 e x^2+11 c d f x^2\right )+a^2 b^2 d \left (-3 d^2 e x^4+c^2 \left (-8 e+4 f x^2\right )+c d \left (-8 e x^2+7 f x^4\right )\right )\right )-i b c \left (2 b^2 c^2 e+a b c (-7 d e+c f)+a^2 d (-3 d e+7 c f)\right ) \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c (-b c+a d) \left (2 b^2 c e+3 a^2 d f+a b (-6 d e+c f)\right ) \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3 a^2 \sqrt {\frac {b}{a}} c (-b c+a d)^3 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \] Input:

Integrate[(e + f*x^2)/((a + b*x^2)^(5/2)*(c + d*x^2)^(3/2)),x]
 

Output:

(-(Sqrt[b/a]*x*(3*a^4*d^2*(-(d*e) + c*f) + 2*b^4*c^2*e*x^2*(c + d*x^2) + a 
*b^3*c*(c + d*x^2)*(3*c*e - 7*d*e*x^2 + c*f*x^2) + a^3*b*d*(5*c^2*f - 6*d^ 
2*e*x^2 + 11*c*d*f*x^2) + a^2*b^2*d*(-3*d^2*e*x^4 + c^2*(-8*e + 4*f*x^2) + 
 c*d*(-8*e*x^2 + 7*f*x^4)))) - I*b*c*(2*b^2*c^2*e + a*b*c*(-7*d*e + c*f) + 
 a^2*d*(-3*d*e + 7*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 + 3*a^2*d*f + a*b*(-6*d*e + c*f))*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]* 
Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(3*a^2 
*Sqrt[b/a]*c*(-(b*c) + a*d)^3*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.05, 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, 25, 27, 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.

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 402

\(\displaystyle \frac {\frac {x \left (3 a^2 d f-a b (6 d e-c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)}-\frac {\int -\frac {d \left (\left (3 d f a^2-b (6 d e-c f) a+2 b^2 c e\right ) x^2+a (b c e+3 a d e-4 a c f)\right )}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{a (b c-a d)}}{3 a (b c-a d)}+\frac {x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {d \left (\left (3 d f a^2-b (6 d e-c f) a+2 b^2 c e\right ) x^2+a (b c e+3 a d e-4 a c f)\right )}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{a (b c-a d)}+\frac {x \left (3 a^2 d f-a b (6 d e-c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)}}{3 a (b c-a d)}+\frac {x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {d \int \frac {\left (3 d f a^2-b (6 d e-c f) a+2 b^2 c e\right ) x^2+a (b c e+3 a d e-4 a c f)}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{a (b c-a d)}+\frac {x \left (3 a^2 d f-a b (6 d e-c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)}}{3 a (b c-a d)}+\frac {x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d)}\)

\(\Big \downarrow \) 400

\(\displaystyle \frac {\frac {d \left (\frac {\left (a^2 (-d) (3 d e-7 c f)-a b c (7 d e-c f)+2 b^2 c^2 e\right ) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b c-a d}-\frac {a \left (3 a^2 d f-a b (9 d e-5 c f)+b^2 c e\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}\right )}{a (b c-a d)}+\frac {x \left (3 a^2 d f-a b (6 d e-c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)}}{3 a (b c-a d)}+\frac {x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d)}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {\frac {d \left (\frac {\sqrt {a+b x^2} \left (a^2 (-d) (3 d e-7 c f)-a b c (7 d e-c f)+2 b^2 c^2 e\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {a \left (3 a^2 d f-a b (9 d e-5 c f)+b^2 c e\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}\right )}{a (b c-a d)}+\frac {x \left (3 a^2 d f-a b (6 d e-c f)+2 b^2 c e\right )}{a \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)}}{3 a (b c-a d)}+\frac {x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d)}\)

\(\Big \downarrow \) 320

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

Input:

Int[(e + f*x^2)/((a + b*x^2)^(5/2)*(c + d*x^2)^(3/2)),x]
 

Output:

((b*e - a*f)*x)/(3*a*(b*c - a*d)*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2]) + (((2 
*b^2*c*e + 3*a^2*d*f - a*b*(6*d*e - c*f))*x)/(a*(b*c - a*d)*Sqrt[a + b*x^2 
]*Sqrt[c + d*x^2]) + (d*(((2*b^2*c^2*e - a^2*d*(3*d*e - 7*c*f) - a*b*c*(7* 
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]) - (Sqrt[c]*(b^2*c*e + 3*a^2*d*f - a*b*(9*d*e - 5*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 
])))/(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 [A] (verified)

Time = 12.40 (sec) , antiderivative size = 669, normalized size of antiderivative = 1.75

method	result	size
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {x \left (a f -b e \right ) \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b a \left (a d -b c \right )^{2} \left (x^{2}+\frac {a}{b}\right )^{2}}-\frac {\left (b d \,x^{2}+b c \right ) x \left (4 a^{2} d f +a b c f -7 a b d e +2 c e \,b^{2}\right )}{3 a^{2} \left (a d -b c \right )^{3} \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}-\frac {\left (b d \,x^{2}+a d \right ) d x \left (c f -d e \right )}{c \left (a d -b c \right )^{3} \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (b d \,x^{2}+a d \right )}}+\frac {\left (-\frac {\left (a f -b e \right ) d}{3 a \left (a d -b c \right )^{2}}+\frac {4 a^{2} d f +a b c f -7 a b d e +2 c e \,b^{2}}{3 \left (a d -b c \right )^{2} a^{2}}+\frac {b c \left (4 a^{2} d f +a b c f -7 a b d e +2 c e \,b^{2}\right )}{3 a^{2} \left (a d -b c \right )^{3}}-\frac {\left (c f -d e \right ) d}{\left (a d -b c \right )^{2} c}+\frac {a \,d^{2} \left (c f -d e \right )}{c \left (a d -b c \right )^{3}}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (\frac {b d \left (4 a^{2} d f +a b c f -7 a b d e +2 c e \,b^{2}\right )}{3 \left (a d -b c \right )^{3} a^{2}}+\frac {b \,d^{2} \left (c f -d e \right )}{\left (a d -b c \right )^{3} c}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(669\)
default	\(\text {Expression too large to display}\)	\(1744\)

Input:

int((f*x^2+e)/(b*x^2+a)^(5/2)/(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)*(-1/3/b/a/(a*d 
-b*c)^2*x*(a*f-b*e)*(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)/a^2/(a*d-b*c)^3*x*(4*a^2*d*f+a*b*c*f-7*a*b*d*e+2*b^2*c*e)/((x^ 
2+a/b)*(b*d*x^2+b*c))^(1/2)-(b*d*x^2+a*d)/c*d/(a*d-b*c)^3*x*(c*f-d*e)/((x^ 
2+c/d)*(b*d*x^2+a*d))^(1/2)+(-1/3*(a*f-b*e)*d/a/(a*d-b*c)^2+1/3/(a*d-b*c)^ 
2*(4*a^2*d*f+a*b*c*f-7*a*b*d*e+2*b^2*c*e)/a^2+1/3*b*c/a^2/(a*d-b*c)^3*(4*a 
^2*d*f+a*b*c*f-7*a*b*d*e+2*b^2*c*e)-(c*f-d*e)*d/(a*d-b*c)^2/c+a*d^2/c/(a*d 
-b*c)^3*(c*f-d*e))/(-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*d*(4*a^2*d*f+a*b*c*f-7*a*b*d*e+2*b^2*c*e)/(a*d-b*c)^3/a^2+b* 
d^2*(c*f-d*e)/(a*d-b*c)^3/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. 1342 vs. \(2 (354) = 708\).

Time = 0.18 (sec) , antiderivative size = 1342, normalized size of antiderivative = 3.50 \[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate((f*x^2+e)/(b*x^2+a)^(5/2)/(d*x^2+c)^(3/2),x, algorithm="fricas")
 

Output:

-1/3*((((2*b^6*c^2*d - 7*a*b^5*c*d^2 - 3*a^2*b^4*d^3)*e + (a*b^5*c^2*d + 7 
*a^2*b^4*c*d^2)*f)*x^6 + ((2*b^6*c^3 - 3*a*b^5*c^2*d - 17*a^2*b^4*c*d^2 - 
6*a^3*b^3*d^3)*e + (a*b^5*c^3 + 9*a^2*b^4*c^2*d + 14*a^3*b^3*c*d^2)*f)*x^4 
 + ((4*a*b^5*c^3 - 12*a^2*b^4*c^2*d - 13*a^3*b^3*c*d^2 - 3*a^4*b^2*d^3)*e 
+ (2*a^2*b^4*c^3 + 15*a^3*b^3*c^2*d + 7*a^4*b^2*c*d^2)*f)*x^2 + (2*a^2*b^4 
*c^3 - 7*a^3*b^3*c^2*d - 3*a^4*b^2*c*d^2)*e + (a^3*b^3*c^3 + 7*a^4*b^2*c^2 
*d)*f)*sqrt(a*c)*sqrt(-b/a)*elliptic_e(arcsin(x*sqrt(-b/a)), a*d/(b*c)) - 
(((2*b^6*c^2*d + (a^2*b^4 - 7*a*b^5)*c*d^2 - 3*(3*a^3*b^3 + a^2*b^4)*d^3)* 
e + (a*b^5*c^2*d + 3*a^4*b^2*d^3 + (5*a^3*b^3 + 7*a^2*b^4)*c*d^2)*f)*x^6 + 
 ((2*b^6*c^3 + (a^2*b^4 - 3*a*b^5)*c^2*d - (7*a^3*b^3 + 17*a^2*b^4)*c*d^2 
- 6*(3*a^4*b^2 + a^3*b^3)*d^3)*e + (a*b^5*c^3 + 6*a^5*b*d^3 + (5*a^3*b^3 + 
 9*a^2*b^4)*c^2*d + (13*a^4*b^2 + 14*a^3*b^3)*c*d^2)*f)*x^4 + ((4*a*b^5*c^ 
3 + 2*(a^3*b^3 - 6*a^2*b^4)*c^2*d - (17*a^4*b^2 + 13*a^3*b^3)*c*d^2 - 3*(3 
*a^5*b + a^4*b^2)*d^3)*e + (2*a^2*b^4*c^3 + 3*a^6*d^3 + 5*(2*a^4*b^2 + 3*a 
^3*b^3)*c^2*d + (11*a^5*b + 7*a^4*b^2)*c*d^2)*f)*x^2 + (2*a^2*b^4*c^3 + (a 
^4*b^2 - 7*a^3*b^3)*c^2*d - 3*(3*a^5*b + a^4*b^2)*c*d^2)*e + (a^3*b^3*c^3 
+ 3*a^6*c*d^2 + (5*a^5*b + 7*a^4*b^2)*c^2*d)*f)*sqrt(a*c)*sqrt(-b/a)*ellip 
tic_f(arcsin(x*sqrt(-b/a)), a*d/(b*c)) - (((2*a*b^5*c^2*d - 7*a^2*b^4*c*d^ 
2 - 3*a^3*b^3*d^3)*e + (a^2*b^4*c^2*d + 7*a^3*b^3*c*d^2)*f)*x^5 + (2*(a*b^ 
5*c^3 - 2*a^2*b^4*c^2*d - 4*a^3*b^3*c*d^2 - 3*a^4*b^2*d^3)*e + (a^2*b^4...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate((f*x**2+e)/(b*x**2+a)**(5/2)/(d*x**2+c)**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {f x^{2} + e}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((f*x^2+e)/(b*x^2+a)^(5/2)/(d*x^2+c)^(3/2),x, algorithm="maxima")
 

Output:

integrate((f*x^2 + e)/((b*x^2 + a)^(5/2)*(d*x^2 + c)^(3/2)), x)
 

Giac [F]

\[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {f x^{2} + e}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((f*x^2+e)/(b*x^2+a)^(5/2)/(d*x^2+c)^(3/2),x, algorithm="giac")
 

Output:

integrate((f*x^2 + e)/((b*x^2 + a)^(5/2)*(d*x^2 + c)^(3/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {f\,x^2+e}{{\left (b\,x^2+a\right )}^{5/2}\,{\left (d\,x^2+c\right )}^{3/2}} \,d x \] Input:

int((e + f*x^2)/((a + b*x^2)^(5/2)*(c + d*x^2)^(3/2)),x)
 

Output:

int((e + f*x^2)/((a + b*x^2)^(5/2)*(c + d*x^2)^(3/2)), x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}} \, dx=\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b^{3} d^{2} x^{10}+3 a \,b^{2} d^{2} x^{8}+2 b^{3} c d \,x^{8}+3 a^{2} b \,d^{2} x^{6}+6 a \,b^{2} c d \,x^{6}+b^{3} c^{2} x^{6}+a^{3} d^{2} x^{4}+6 a^{2} b c d \,x^{4}+3 a \,b^{2} c^{2} x^{4}+2 a^{3} c d \,x^{2}+3 a^{2} b \,c^{2} x^{2}+a^{3} c^{2}}d x \right ) f +\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b^{3} d^{2} x^{10}+3 a \,b^{2} d^{2} x^{8}+2 b^{3} c d \,x^{8}+3 a^{2} b \,d^{2} x^{6}+6 a \,b^{2} c d \,x^{6}+b^{3} c^{2} x^{6}+a^{3} d^{2} x^{4}+6 a^{2} b c d \,x^{4}+3 a \,b^{2} c^{2} x^{4}+2 a^{3} c d \,x^{2}+3 a^{2} b \,c^{2} x^{2}+a^{3} c^{2}}d x \right ) e \] Input:

int((f*x^2+e)/(b*x^2+a)^(5/2)/(d*x^2+c)^(3/2),x)
 

Output:

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