\(\int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2} (e+f x^2)^3} \, dx\) [201]

Optimal result

Integrand size = 32, antiderivative size = 669 \[ \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx=\frac {f^2 x \sqrt {a+b x^2} \sqrt {c+d x^2}}{4 e (b e-a f) (d e-c f) \left (e+f x^2\right )^2}-\frac {3 f (b e (3 d e-2 c f)-a f (2 d e-c f)) x \sqrt {a+b x^2}}{8 e^2 (b e-a f)^2 (d e-c f) \sqrt {c+d x^2} \left (e+f x^2\right )}+\frac {3 \sqrt {c} \sqrt {d} f (b e (3 d e-2 c f)-a f (2 d e-c f)) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{8 e^2 (b e-a f)^2 (d e-c f)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {d} \left (a f \left (8 d^2 e^2-8 c d e f+3 c^2 f^2\right )-b e \left (8 d^2 e^2-9 c d e f+4 c^2 f^2\right )\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{8 a e^2 (b e-a f) (d e-c f)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {c^{3/2} f \left (a^2 f^2 \left (8 d^2 e^2-8 c d e f+3 c^2 f^2\right )-2 a b e f \left (10 d^2 e^2-11 c d e f+4 c^2 f^2\right )+b^2 e^2 \left (15 d^2 e^2-20 c d e f+8 c^2 f^2\right )\right ) \sqrt {a+b x^2} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{8 a \sqrt {d} e^3 (b e-a f)^2 (d e-c f)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:

1/4*f^2*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/e/(-a*f+b*e)/(-c*f+d*e)/(f*x^2+e 
)^2-3/8*f*(b*e*(-2*c*f+3*d*e)-a*f*(-c*f+2*d*e))*x*(b*x^2+a)^(1/2)/e^2/(-a* 
f+b*e)^2/(-c*f+d*e)/(d*x^2+c)^(1/2)/(f*x^2+e)+3/8*c^(1/2)*d^(1/2)*f*(b*e*( 
-2*c*f+3*d*e)-a*f*(-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))/e^2/(-a*f+b*e)^2/(-c*f+d*e)^2/(c*(b 
*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)-1/8*c^(1/2)*d^(1/2)*(a*f*(3*c^2 
*f^2-8*c*d*e*f+8*d^2*e^2)-b*e*(4*c^2*f^2-9*c*d*e*f+8*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/e^2/(- 
a*f+b*e)/(-c*f+d*e)^3/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)-1/8* 
c^(3/2)*f*(a^2*f^2*(3*c^2*f^2-8*c*d*e*f+8*d^2*e^2)-2*a*b*e*f*(4*c^2*f^2-11 
*c*d*e*f+10*d^2*e^2)+b^2*e^2*(8*c^2*f^2-20*c*d*e*f+15*d^2*e^2))*(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^3/(-a*f+b*e)^2/(-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 = 4.17 (sec) , antiderivative size = 438, normalized size of antiderivative = 0.65 \[ \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx=\frac {\sqrt {\frac {b}{a}} e f^2 x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (2 e (b e-a f) (d e-c f)+3 (b e (3 d e-2 c f)+a f (-2 d e+c f)) \left (e+f x^2\right )\right )-i \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (e+f x^2\right )^2 \left (3 b c e f (a f (2 d e-c f)+b e (-3 d e+2 c f)) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-b e (d e-c f) (b e (7 d e-6 c f)+a f (-4 d e+3 c f)) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+\left (a^2 f^2 \left (8 d^2 e^2-8 c d e f+3 c^2 f^2\right )-2 a b e f \left (10 d^2 e^2-11 c d e f+4 c^2 f^2\right )+b^2 e^2 \left (15 d^2 e^2-20 c d e f+8 c^2 f^2\right )\right ) \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{8 \sqrt {\frac {b}{a}} e^3 (b e-a f)^2 (d e-c f)^2 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2} \] Input:

Integrate[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^3),x]
 

Output:

(Sqrt[b/a]*e*f^2*x*(a + b*x^2)*(c + d*x^2)*(2*e*(b*e - a*f)*(d*e - c*f) + 
3*(b*e*(3*d*e - 2*c*f) + a*f*(-2*d*e + c*f))*(e + f*x^2)) - I*Sqrt[1 + (b* 
x^2)/a]*Sqrt[1 + (d*x^2)/c]*(e + f*x^2)^2*(3*b*c*e*f*(a*f*(2*d*e - c*f) + 
b*e*(-3*d*e + 2*c*f))*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - b*e 
*(d*e - c*f)*(b*e*(7*d*e - 6*c*f) + a*f*(-4*d*e + 3*c*f))*EllipticF[I*ArcS 
inh[Sqrt[b/a]*x], (a*d)/(b*c)] + (a^2*f^2*(8*d^2*e^2 - 8*c*d*e*f + 3*c^2*f 
^2) - 2*a*b*e*f*(10*d^2*e^2 - 11*c*d*e*f + 4*c^2*f^2) + b^2*e^2*(15*d^2*e^ 
2 - 20*c*d*e*f + 8*c^2*f^2))*EllipticPi[(a*f)/(b*e), I*ArcSinh[Sqrt[b/a]*x 
], (a*d)/(b*c)]))/(8*Sqrt[b/a]*e^3*(b*e - a*f)^2*(d*e - c*f)^2*Sqrt[a + b* 
x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^2)
 

Rubi [F]

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.

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

\(\Big \downarrow \) 433

\(\displaystyle \int \left (-\frac {3 f}{8 e^2 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (-e f-f^2 x^2\right )}-\frac {3 f}{16 e^2 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (\sqrt {-e} \sqrt {f}-f x\right )^2}-\frac {3 f}{16 e^2 \sqrt {a+b x^2} \sqrt {c+d x^2} \left (\sqrt {-e} \sqrt {f}+f x\right )^2}-\frac {f^{3/2}}{8 (-e)^{3/2} \sqrt {a+b x^2} \sqrt {c+d x^2} \left (\sqrt {-e} \sqrt {f}-f x\right )^3}-\frac {f^{3/2}}{8 (-e)^{3/2} \sqrt {a+b x^2} \sqrt {c+d x^2} \left (\sqrt {-e} \sqrt {f}+f x\right )^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 f \int \frac {1}{\left (\sqrt {-e} \sqrt {f}-f x\right )^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{16 e^2}-\frac {3 f \int \frac {1}{\left (f x+\sqrt {-e} \sqrt {f}\right )^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{16 e^2}-\frac {f^{3/2} \int \frac {1}{\left (\sqrt {-e} \sqrt {f}-f x\right )^3 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{8 (-e)^{3/2}}-\frac {f^{3/2} \int \frac {1}{\left (f x+\sqrt {-e} \sqrt {f}\right )^3 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{8 (-e)^{3/2}}+\frac {3 \sqrt {-a} \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {d x^2}{c}+1} \operatorname {EllipticPi}\left (\frac {a f}{b e},\arcsin \left (\frac {\sqrt {b} x}{\sqrt {-a}}\right ),\frac {a d}{b c}\right )}{8 \sqrt {b} e^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}\)

Input:

Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^3),x]
 

Output:

$Aborted
 

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. \(2615\) vs. \(2(637)=1274\).

Time = 11.02 (sec) , antiderivative size = 2616, normalized size of antiderivative = 3.91

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(2616\)
default	\(\text {Expression too large to display}\)	\(4650\)

Input:

int(1/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^3,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)*(1/4*f^2/(a*c* 
f^2-a*d*e*f-b*c*e*f+b*d*e^2)/e*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/(f*x^ 
2+e)^2+3/8*f^2*(a*c*f^2-2*a*d*e*f-2*b*c*e*f+3*b*d*e^2)/(a*c*f^2-a*d*e*f-b* 
c*e*f+b*d*e^2)^2/e^2*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/(f*x^2+e)+11/4/ 
(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)^2/e*f^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)*EllipticPi(x*(-b/a)^( 
1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2))*a*b*c*d-3/4*f^2*b^2/(a*c*f^2-a*d 
*e*f-b*c*e*f+b*d*e^2)^2/e*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)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d 
+b*c)/c/b)^(1/2))+3/4*f^2*b^2/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)^2/e*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))+3/8/(a*c*f^2-a 
*d*e*f-b*c*e*f+b*d*e^2)^2/e^3*f^4/(-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)*EllipticPi(x*(-b/a)^(1/2),a*f 
/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2))*a^2*c^2+1/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e 
^2)^2/e*f^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)*EllipticPi(x*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(- 
b/a)^(1/2))*a^2*d^2-7/8*b^2*d^2/(a*c*f^2-a*d*e*f-b*c*e*f+b*d*e^2)^2*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/(a*c*f^2-a*...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate(1/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^3,x, algorithm="fric 
as")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate(1/(b*x**2+a)**(1/2)/(d*x**2+c)**(1/2)/(f*x**2+e)**3,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(1/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^3,x, algorithm="maxi 
ma")
 

Output:

integrate(1/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^3), x)
 

Giac [F]

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

integrate(1/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^3,x, algorithm="giac 
")
 

Output:

integrate(1/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^3), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^3} \, dx=\int \frac {1}{\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}\,{\left (f\,x^2+e\right )}^3} \,d x \] Input:

int(1/((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^3),x)
 

Output:

int(1/((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^3), x)
 

Reduce [F]

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

int(1/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^3,x)
 

Output:

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