\(\int \frac {1}{\sqrt {a+b x^2} (c+d x^2)^{5/2} (e+f x^2)} \, dx\) [117]

Optimal result

Integrand size = 32, antiderivative size = 481 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=-\frac {d^2 x \sqrt {a+b x^2}}{3 c (b c-a d) (d e-c f) \left (c+d x^2\right )^{3/2}}-\frac {d^{3/2} (b c (4 d e-7 c f)-a d (2 d e-5 c f)) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 c^{3/2} (b c-a d)^2 (d e-c f)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {\sqrt {d} \left (3 a^2 c d^2 f^2+3 b^2 c \left (d^2 e^2-3 c d e f+3 c^2 f^2\right )-a b d \left (d^2 e^2-5 c d e f+10 c^2 f^2\right )\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a \sqrt {c} (b c-a d)^2 (d e-c f)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {c^{3/2} f^3 \sqrt {a+b x^2} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e (d e-c f)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:

-1/3*d^2*x*(b*x^2+a)^(1/2)/c/(-a*d+b*c)/(-c*f+d*e)/(d*x^2+c)^(3/2)-1/3*d^( 
3/2)*(b*c*(-7*c*f+4*d*e)-a*d*(-5*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)/(-a*d+b*c)^2/( 
-c*f+d*e)^2/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)+1/3*d^(1/2)*(3 
*a^2*c*d^2*f^2+3*b^2*c*(3*c^2*f^2-3*c*d*e*f+d^2*e^2)-a*b*d*(10*c^2*f^2-5*c 
*d*e*f+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/c^(1/2)/(-a*d+b*c)^2/(-c*f+d*e)^3/(c*(b*x^2+a)/a/(d* 
x^2+c))^(1/2)/(d*x^2+c)^(1/2)-c^(3/2)*f^3*(b*x^2+a)^(1/2)*EllipticPi(d^(1/ 
2)*x/c^(1/2)/(1+d*x^2/c)^(1/2),1-c*f/d/e,(1-b*c/a/d)^(1/2))/a/d^(1/2)/e/(- 
c*f+d*e)^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 = 6.39 (sec) , antiderivative size = 1037, normalized size of antiderivative = 2.16 \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx =\text {Too large to display} \] Input:

Integrate[1/(Sqrt[a + b*x^2]*(c + d*x^2)^(5/2)*(e + f*x^2)),x]
 

Output:

(-5*a*b*Sqrt[b/a]*c^2*d^3*e^2*x + 3*a^2*Sqrt[b/a]*c*d^4*e^2*x + 8*a*b*Sqrt 
[b/a]*c^3*d^2*e*f*x - 6*a^2*Sqrt[b/a]*c^2*d^3*e*f*x - 5*a*b*(b/a)^(3/2)*c^ 
2*d^3*e^2*x^3 - a*b*Sqrt[b/a]*c*d^4*e^2*x^3 + 2*a^2*Sqrt[b/a]*d^5*e^2*x^3 
+ 8*a*b*(b/a)^(3/2)*c^3*d^2*e*f*x^3 + a*b*Sqrt[b/a]*c^2*d^3*e*f*x^3 - 5*a^ 
2*Sqrt[b/a]*c*d^4*e*f*x^3 - 4*a*b*(b/a)^(3/2)*c*d^4*e^2*x^5 + 2*a*b*Sqrt[b 
/a]*d^5*e^2*x^5 + 7*a*b*(b/a)^(3/2)*c^2*d^3*e*f*x^5 - 5*a*b*Sqrt[b/a]*c*d^ 
4*e*f*x^5 - I*b*c*d*e*(b*c*(4*d*e - 7*c*f) + a*d*(-2*d*e + 5*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*d*(-(b*c) + a*d)*e*(-(d*e) + c*f)*Sqrt[1 + (b*x^ 
2)/a]*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a 
*d)/(b*c)] - (3*I)*b^2*c^5*f^2*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Ell 
ipticPi[(a*f)/(b*e), I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (6*I)*a*b*c^4* 
d*f^2*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticPi[(a*f)/(b*e), I*Ar 
cSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - (3*I)*a^2*c^3*d^2*f^2*Sqrt[1 + (b*x^2)/ 
a]*Sqrt[1 + (d*x^2)/c]*EllipticPi[(a*f)/(b*e), I*ArcSinh[Sqrt[b/a]*x], (a* 
d)/(b*c)] - (3*I)*b^2*c^4*d*f^2*x^2*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c 
]*EllipticPi[(a*f)/(b*e), I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (6*I)*a*b 
*c^3*d^2*f^2*x^2*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticPi[(a*f)/ 
(b*e), I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - (3*I)*a^2*c^2*d^3*f^2*x^2*Sq 
rt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticPi[(a*f)/(b*e), I*ArcSinh...
 

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 436, normalized size of antiderivative = 0.91, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {421, 25, 402, 25, 400, 313, 320, 413, 413, 412}

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.

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

\(\Big \downarrow \) 421

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 400

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

\(\Big \downarrow \) 313

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

\(\Big \downarrow \) 320

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

\(\Big \downarrow \) 413

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

\(\Big \downarrow \) 413

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

\(\Big \downarrow \) 412

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

Input:

Int[1/(Sqrt[a + b*x^2]*(c + d*x^2)^(5/2)*(e + f*x^2)),x]
 

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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]
 

rule 412

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* 
(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, 
 f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && S 
implerSqrtQ[-f/e, -d/c])
 

rule 413

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[1/((a + 
 b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, 
e, f}, x] &&  !GtQ[c, 0]
 

rule 421

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1817\) vs. \(2(457)=914\).

Time = 17.82 (sec) , antiderivative size = 1818, normalized size of antiderivative = 3.78

method	result	size
default	\(\text {Expression too large to display}\)	\(1818\)
elliptic	\(\text {Expression too large to display}\)	\(2008\)

Input:

int(1/(b*x^2+a)^(1/2)/(d*x^2+c)^(5/2)/(f*x^2+e),x,method=_RETURNVERBOSE)
 

Output:

-1/3*(-3*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticPi(x*(-b/a)^(1/2) 
,a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2))*a^2*c^3*d^2*f^2+6*((b*x^2+a)/a)^(1/2 
)*((d*x^2+c)/c)^(1/2)*EllipticPi(x*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b 
/a)^(1/2))*a*b*c^3*d^2*f^2*x^2-3*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*E 
llipticPi(x*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2))*a^2*c^2*d^3* 
f^2*x^2-3*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticPi(x*(-b/a)^(1/2 
),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2))*b^2*c^4*d*f^2*x^2+6*(-b/a)^(1/2)*a^ 
2*c^2*d^3*e*f*x+6*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticPi(x*(-b 
/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2))*a*b*c^4*d*f^2-3*((b*x^2+a)/ 
a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticPi(x*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1 
/2)/(-b/a)^(1/2))*b^2*f^2*c^5+((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*Elli 
pticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*b^2*c^3*d^2*e^2-4*((b*x^2+a)/a)^(1/2 
)*((d*x^2+c)/c)^(1/2)*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*b^2*c^3*d^ 
2*e^2+((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(a* 
d/b/c)^(1/2))*a*b*c^2*d^3*e*f*x^2-5*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2 
)*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a*b*c^2*d^3*e*f*x^2-((b*x^2+a) 
/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a* 
b*c*d^4*e^2*x^2-((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a) 
^(1/2),(a*d/b/c)^(1/2))*b^2*c^3*d^2*e*f*x^2+2*((b*x^2+a)/a)^(1/2)*((d*x^2+ 
c)/c)^(1/2)*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a*b*c*d^4*e^2*x^2...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:

integrate(1/(b*x^2+a)^(1/2)/(d*x^2+c)^(5/2)/(f*x^2+e),x, algorithm="fricas 
")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\int \frac {1}{\sqrt {a + b x^{2}} \left (c + d x^{2}\right )^{\frac {5}{2}} \left (e + f x^{2}\right )}\, dx \] Input:

integrate(1/(b*x**2+a)**(1/2)/(d*x**2+c)**(5/2)/(f*x**2+e),x)
 

Output:

Integral(1/(sqrt(a + b*x**2)*(c + d*x**2)**(5/2)*(e + f*x**2)), x)
 

Maxima [F]

\[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\int { \frac {1}{\sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}} \,d x } \] Input:

integrate(1/(b*x^2+a)^(1/2)/(d*x^2+c)^(5/2)/(f*x^2+e),x, algorithm="maxima 
")
 

Output:

integrate(1/(sqrt(b*x^2 + a)*(d*x^2 + c)^(5/2)*(f*x^2 + e)), x)
 

Giac [F]

\[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\int { \frac {1}{\sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}} \,d x } \] Input:

integrate(1/(b*x^2+a)^(1/2)/(d*x^2+c)^(5/2)/(f*x^2+e),x, algorithm="giac")
 

Output:

integrate(1/(sqrt(b*x^2 + a)*(d*x^2 + c)^(5/2)*(f*x^2 + e)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\int \frac {1}{\sqrt {b\,x^2+a}\,{\left (d\,x^2+c\right )}^{5/2}\,\left (f\,x^2+e\right )} \,d x \] Input:

int(1/((a + b*x^2)^(1/2)*(c + d*x^2)^(5/2)*(e + f*x^2)),x)
 

Output:

int(1/((a + b*x^2)^(1/2)*(c + d*x^2)^(5/2)*(e + f*x^2)), x)
 

Reduce [F]

\[ \int \frac {1}{\sqrt {a+b x^2} \left (c+d x^2\right )^{5/2} \left (e+f x^2\right )} \, dx=\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b \,d^{3} f \,x^{10}+a \,d^{3} f \,x^{8}+3 b c \,d^{2} f \,x^{8}+b \,d^{3} e \,x^{8}+3 a c \,d^{2} f \,x^{6}+a \,d^{3} e \,x^{6}+3 b \,c^{2} d f \,x^{6}+3 b c \,d^{2} e \,x^{6}+3 a \,c^{2} d f \,x^{4}+3 a c \,d^{2} e \,x^{4}+b \,c^{3} f \,x^{4}+3 b \,c^{2} d e \,x^{4}+a \,c^{3} f \,x^{2}+3 a \,c^{2} d e \,x^{2}+b \,c^{3} e \,x^{2}+a \,c^{3} e}d x \] Input:

int(1/(b*x^2+a)^(1/2)/(d*x^2+c)^(5/2)/(f*x^2+e),x)
 

Output:

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