\(\int \frac {1}{(a+b x^2)^{5/2} \sqrt {c+d x^2} (e+f x^2)} \, dx\) [121]

Optimal result

Integrand size = 32, antiderivative size = 480 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\frac {b^2 x \sqrt {c+d x^2}}{3 a (b c-a d) (b e-a f) \left (a+b x^2\right )^{3/2}}+\frac {b^{3/2} \left (2 b^2 c e-4 a b d e-5 a b c f+7 a^2 d f\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 a^{3/2} (b c-a d)^2 (b e-a f)^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {b} \left (b^3 c d e^2-9 a^3 d^2 f^2+a^2 b d f (9 d e+10 c f)-a b^2 \left (3 d^2 e^2+5 c d e f+3 c^2 f^2\right )\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 \sqrt {a} c (b c-a d)^2 (b e-a f)^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} f^3 \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e (b e-a f)^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

1/3*b^2*x*(d*x^2+c)^(1/2)/a/(-a*d+b*c)/(-a*f+b*e)/(b*x^2+a)^(3/2)+1/3*b^(3 
/2)*(7*a^2*d*f-5*a*b*c*f-4*a*b*d*e+2*b^2*c*e)*(d*x^2+c)^(1/2)*EllipticE(b^ 
(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2),(1-a*d/b/c)^(1/2))/a^(3/2)/(-a*d+b*c)^2/ 
(-a*f+b*e)^2/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)-1/3*b^(1/2)*( 
b^3*c*d*e^2-9*a^3*d^2*f^2+a^2*b*d*f*(10*c*f+9*d*e)-a*b^2*(3*c^2*f^2+5*c*d* 
e*f+3*d^2*e^2))*(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)/c/(-a*d+b*c)^2/(-a*f+b*e)^3/(b*x^2+a)^(1/2)/(a* 
(d*x^2+c)/c/(b*x^2+a))^(1/2)-a^(3/2)*f^3*(d*x^2+c)^(1/2)*EllipticPi(b^(1/2 
)*x/a^(1/2)/(1+b*x^2/a)^(1/2),1-a*f/b/e,(1-a*d/b/c)^(1/2))/b^(1/2)/c/e/(-a 
*f+b*e)^3/(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 = 8.00 (sec) , antiderivative size = 437, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\frac {a b \left (\frac {b}{a}\right )^{3/2} e x \left (c+d x^2\right ) \left (8 a^3 d f+2 b^3 c e x^2+a b^2 \left (3 c e-4 d e x^2-5 c f x^2\right )+a^2 b \left (-5 d e-6 c f+7 d f x^2\right )\right )+i b^2 c e \left (2 b^2 c e+7 a^2 d f-a b (4 d e+5 c f)\right ) \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i b (-b c+a d) e \left (2 b^2 c e+6 a^2 d f-a b (3 d e+5 c f)\right ) \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )-3 i a^2 (b c-a d)^2 f^2 \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3 a^2 \sqrt {\frac {b}{a}} (b c-a d)^2 e (b e-a f)^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 440, normalized size of antiderivative = 0.92, 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.

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

\(\Big \downarrow \) 421

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 400

\(\displaystyle \frac {b \left (\frac {\frac {b \left (7 a^2 d f-5 a b c f-4 a b d e+2 b^2 c e\right ) \int \frac {\sqrt {d x^2+c}}{\left (b x^2+a\right )^{3/2}}dx}{b c-a d}-\frac {d \left (6 a^2 d f-a b (4 c f+3 d e)+b^2 c e\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}}{3 a (b c-a d)}+\frac {b x \sqrt {c+d x^2} (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} (b c-a d)}\right )}{(b e-a f)^2}+\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(b e-a f)^2}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {b \left (\frac {\frac {\sqrt {b} \sqrt {c+d x^2} \left (7 a^2 d f-5 a b c f-4 a b d e+2 b^2 c e\right ) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {a+b x^2} (b c-a d) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {d \left (6 a^2 d f-a b (4 c f+3 d e)+b^2 c e\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}}{3 a (b c-a d)}+\frac {b x \sqrt {c+d x^2} (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} (b c-a d)}\right )}{(b e-a f)^2}+\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(b e-a f)^2}\)

\(\Big \downarrow \) 320

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

\(\Big \downarrow \) 413

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

\(\Big \downarrow \) 413

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

\(\Big \downarrow \) 412

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

Input:

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

Output:

(b*((b*(b*e - a*f)*x*Sqrt[c + d*x^2])/(3*a*(b*c - a*d)*(a + b*x^2)^(3/2)) 
+ ((Sqrt[b]*(2*b^2*c*e - 4*a*b*d*e - 5*a*b*c*f + 7*a^2*d*f)*Sqrt[c + d*x^2 
]*EllipticE[ArcTan[(Sqrt[b]*x)/Sqrt[a]], 1 - (a*d)/(b*c)])/(Sqrt[a]*(b*c - 
 a*d)*Sqrt[a + b*x^2]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]) - (Sqrt[c]*Sq 
rt[d]*(b^2*c*e + 6*a^2*d*f - a*b*(3*d*e + 4*c*f))*Sqrt[a + b*x^2]*Elliptic 
F[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*(b*c - a*d)*Sqrt[(c*(a 
 + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]))/(3*a*(b*c - a*d))))/(b*e - a 
*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*(b*e - 
a*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. \(1324\) vs. \(2(456)=912\).

Time = 10.93 (sec) , antiderivative size = 1325, normalized size of antiderivative = 2.76

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1325\)
default	\(\text {Expression too large to display}\)	\(2062\)

Input:

int(1/(b*x^2+a)^(5/2)/(d*x^2+c)^(1/2)/(f*x^2+e),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/(a*d-b* 
c)*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)*b/a^2/(a*d-b*c)^2*x*(7*a^2*d*f-5*a*b*c*f-4*a*b*d*e+2*b^2*c*e)/(a*f- 
b*e)^2/((x^2+a/b)*(b*d*x^2+b*c))^(1/2)+1/3/(-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))*d*b/(a*d-b*c)/a/(a*f-b*e)-7/3/(-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)*El 
lipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))/(a*d-b*c)*b/(a*f-b*e)^2*d 
*f+5/3/(-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))/(a*d- 
b*c)*b^2/a/(a*f-b*e)^2*c*f+4/3/(-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))/(a*d-b*c)*b^2/a/(a*f-b*e)^2*d*e-2/3/(-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))/(a*d-b*c)*b^3/a^2/(a*f-b*e)^2*c 
*e-7/3*b^2/(a*d-b*c)^2/(a*f-b*e)^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)*EllipticE(x*(-b/a)^(1/2),( 
-1+(a*d+b*c)/c/b)^(1/2))*d*f+5/3*b^3/a/(a*d-b*c)^2/(a*f-b*e)^2*c^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)*EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))*f+4/3*b^3/a/(a*d...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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