\(\int \frac {1}{(a+b x^2)^{3/2} (c+d x^2)^{3/2}} \, dx\) [163]

Optimal result

Integrand size = 23, antiderivative size = 242 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx=\frac {b x}{a (b c-a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}+\frac {\sqrt {d} (b c+a d) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{a \sqrt {c} (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {2 b \sqrt {c} \sqrt {d} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:

b*x/a/(-a*d+b*c)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)+d^(1/2)*(a*d+b*c)*(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/c^(1/2)/(-a*d+b*c)^2/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)-2*b 
*c^(1/2)*d^(1/2)*(b*x^2+a)^(1/2)*InverseJacobiAM(arctan(d^(1/2)*x/c^(1/2)) 
,(1-b*c/a/d)^(1/2))/a/(-a*d+b*c)^2/(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 = 4.01 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx=\frac {\sqrt {\frac {b}{a}} \left (\sqrt {\frac {b}{a}} x \left (a^2 d^2+a b d^2 x^2+b^2 c \left (c+d x^2\right )\right )+i b c (b c+a d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i b c (-b c+a d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{b c (b c-a d)^2 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

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

Output:

(Sqrt[b/a]*(Sqrt[b/a]*x*(a^2*d^2 + a*b*d^2*x^2 + b^2*c*(c + d*x^2)) + I*b* 
c*(b*c + a*d)*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*c*(-(b*c) + a*d)*Sqrt[1 + (b*x^2)/a]*Sqrt 
[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(b*c*(b*c 
 - a*d)^2*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {316, 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.

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

\(\Big \downarrow \) 316

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 400

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

\(\Big \downarrow \) 313

\(\displaystyle \frac {b x}{a \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)}-\frac {d \left (\frac {2 a b \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}-\frac {\sqrt {a+b x^2} (a d+b c) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a (b c-a d)}\)

\(\Big \downarrow \) 320

\(\displaystyle \frac {b x}{a \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)}-\frac {d \left (\frac {2 b \sqrt {c} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\sqrt {a+b x^2} (a d+b c) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a (b c-a d)}\)

Input:

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

Output:

(b*x)/(a*(b*c - a*d)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]) - (d*(-(((b*c + a*d) 
*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])) + (2*b*Sqrt[c]*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqr 
t[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))
 

Defintions of rubi rules used

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 316

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) 
), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d))   Int[(a + b*x^2)^(p + 1)*(c + d*x 
^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x 
], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  ! 
( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 2, 
 p, q, x]
 

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]
 

Maple [A] (verified)

Time = 8.20 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.46

method	result	size
default	\(\frac {\left (\sqrt {-\frac {b}{a}}\, a b \,d^{2} x^{3}+\sqrt {-\frac {b}{a}}\, b^{2} c d \,x^{3}-\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c d +\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2}-\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c d -\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2}+\sqrt {-\frac {b}{a}}\, a^{2} d^{2} x +\sqrt {-\frac {b}{a}}\, b^{2} c^{2} x \right ) \sqrt {x^{2} d +c}\, \sqrt {b \,x^{2}+a}}{c \sqrt {-\frac {b}{a}}\, a \left (a d -b c \right )^{2} \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right )}\)	\(354\)
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {2 b d \left (-\frac {\left (a d +b c \right ) x^{3}}{2 a c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {\left (a^{2} d^{2}+b^{2} c^{2}\right ) x}{2 a c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b d}\right )}{\sqrt {\left (x^{4}+\frac {\left (a d +b c \right ) x^{2}}{d b}+\frac {a c}{d b}\right ) b d}}+\frac {\left (\frac {1}{a c}-\frac {a^{2} d^{2}+b^{2} c^{2}}{a c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {b \left (a d +b c \right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{a \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(464\)

Input:

int(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

((-b/a)^(1/2)*a*b*d^2*x^3+(-b/a)^(1/2)*b^2*c*d*x^3-((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+((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^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-((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^2+(-b/a)^(1/2)*a^2*d^2*x+(-b/a)^(1/ 
2)*b^2*c^2*x)*(d*x^2+c)^(1/2)*(b*x^2+a)^(1/2)/c/(-b/a)^(1/2)/a/(a*d-b*c)^2 
/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.68 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx=-\frac {{\left (a b^{2} c^{2} + a^{2} b c d + {\left (b^{3} c d + a b^{2} d^{2}\right )} x^{4} + {\left (b^{3} c^{2} + 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{2}\right )} \sqrt {a c} \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left (a b^{2} c^{2} + {\left (b^{3} c d + {\left (2 \, a^{2} b + a b^{2}\right )} d^{2}\right )} x^{4} + {\left (2 \, a^{3} + a^{2} b\right )} c d + {\left (b^{3} c^{2} + 2 \, {\left (a^{2} b + a b^{2}\right )} c d + {\left (2 \, a^{3} + a^{2} b\right )} d^{2}\right )} x^{2}\right )} \sqrt {a c} \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left ({\left (a b^{2} c d + a^{2} b d^{2}\right )} x^{3} + {\left (a b^{2} c^{2} + a^{3} d^{2}\right )} x\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{a^{3} b^{2} c^{4} - 2 \, a^{4} b c^{3} d + a^{5} c^{2} d^{2} + {\left (a^{2} b^{3} c^{3} d - 2 \, a^{3} b^{2} c^{2} d^{2} + a^{4} b c d^{3}\right )} x^{4} + {\left (a^{2} b^{3} c^{4} - a^{3} b^{2} c^{3} d - a^{4} b c^{2} d^{2} + a^{5} c d^{3}\right )} x^{2}} \] Input:

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

Output:

-((a*b^2*c^2 + a^2*b*c*d + (b^3*c*d + a*b^2*d^2)*x^4 + (b^3*c^2 + 2*a*b^2* 
c*d + a^2*b*d^2)*x^2)*sqrt(a*c)*sqrt(-b/a)*elliptic_e(arcsin(x*sqrt(-b/a)) 
, a*d/(b*c)) - (a*b^2*c^2 + (b^3*c*d + (2*a^2*b + a*b^2)*d^2)*x^4 + (2*a^3 
 + a^2*b)*c*d + (b^3*c^2 + 2*(a^2*b + a*b^2)*c*d + (2*a^3 + a^2*b)*d^2)*x^ 
2)*sqrt(a*c)*sqrt(-b/a)*elliptic_f(arcsin(x*sqrt(-b/a)), a*d/(b*c)) - ((a* 
b^2*c*d + a^2*b*d^2)*x^3 + (a*b^2*c^2 + a^3*d^2)*x)*sqrt(b*x^2 + a)*sqrt(d 
*x^2 + c))/(a^3*b^2*c^4 - 2*a^4*b*c^3*d + a^5*c^2*d^2 + (a^2*b^3*c^3*d - 2 
*a^3*b^2*c^2*d^2 + a^4*b*c*d^3)*x^4 + (a^2*b^3*c^4 - a^3*b^2*c^3*d - a^4*b 
*c^2*d^2 + a^5*c*d^3)*x^2)
 

Sympy [F]

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

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

Output:

Integral(1/((a + b*x**2)**(3/2)*(c + d*x**2)**(3/2)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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