Integrand size = 32, antiderivative size = 342 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\frac {b^{3/2} \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} (b c-a d) (b e-a f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {a} \sqrt {b} (b d e+b c f-2 a d f) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{c (b c-a d) (b e-a f)^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {a^{3/2} f^2 \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e (b e-a f)^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
b^(3/2)*(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^(1/2)/(-a*d+b*c)/(-a*f+b*e)/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/ c/(b*x^2+a))^(1/2)-a^(1/2)*b^(1/2)*(-2*a*d*f+b*c*f+b*d*e)*(d*x^2+c)^(1/2)* InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/c/(-a*d+b*c)/ (-a*f+b*e)^2/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)+a^(3/2)*f^2*( 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)^2/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/( b*x^2+a))^(1/2)
Result contains complex when optimal does not.
Time = 4.83 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\frac {\sqrt {\frac {b}{a}} \left (a b \left (\frac {b}{a}\right )^{3/2} c e x+a b \left (\frac {b}{a}\right )^{3/2} d e x^3+i b^2 c e \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i b (-b c+a d) e \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+i a b c f \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )-i a^2 d f \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{b (b c-a d) e (b e-a f) \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[1/((a + b*x^2)^(3/2)*Sqrt[c + d*x^2]*(e + f*x^2)),x]
Output:
(Sqrt[b/a]*(a*b*(b/a)^(3/2)*c*e*x + a*b*(b/a)^(3/2)*d*e*x^3 + I*b^2*c*e*Sq rt[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*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]* EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + I*a*b*c*f*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)] - I*a^2*d*f*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)]))/(b*(b*c - a*d)*e*(b*e - a*f)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
Time = 0.45 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {421, 25, 400, 313, 320, 414}
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.
| \(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {f^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(b e-a f)^2}-\frac {b \int -\frac {-b f x^2+b e-2 a f}{\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}dx}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {f^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(b e-a f)^2}+\frac {b \int \frac {-b f x^2+b e-2 a f}{\left (b x^2+a\right )^{3/2} \sqrt {d x^2+c}}dx}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {f^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(b e-a f)^2}+\frac {b \left (\frac {b (b e-a f) \int \frac {\sqrt {d x^2+c}}{\left (b x^2+a\right )^{3/2}}dx}{b c-a d}-\frac {(-2 a d f+b c f+b d e) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}\right )}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \left (\frac {\sqrt {b} \sqrt {c+d x^2} (b e-a f) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {a+b x^2} (b c-a d) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {(-2 a d f+b c f+b d e) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}\right )}{(b e-a f)^2}+\frac {f^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {f^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{(b e-a f)^2}+\frac {b \left (\frac {\sqrt {b} \sqrt {c+d x^2} (b e-a f) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {a+b x^2} (b c-a d) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {c} \sqrt {a+b x^2} (-2 a d f+b c f+b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 414 |
| \(\displaystyle \frac {a^{3/2} f^2 \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e \sqrt {a+b x^2} (b e-a f)^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {b \left (\frac {\sqrt {b} \sqrt {c+d x^2} (b e-a f) E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {a+b x^2} (b c-a d) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {c} \sqrt {a+b x^2} (-2 a d f+b c f+b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{(b e-a f)^2}\) |
Input:
Int[1/((a + b*x^2)^(3/2)*Sqrt[c + d*x^2]*(e + f*x^2)),x]
Output:
(b*((Sqrt[b]*(b*e - a*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]*(b*d*e + b*c*f - 2*a*d*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])))/(b*e - a*f)^2 + (a^(3/2)*f^2*Sqrt[c + d*x^2]*EllipticPi[1 - (a*f)/(b*e), ArcTan[ (Sqrt[b]*x)/Sqrt[a]], 1 - (a*d)/(b*c)])/(Sqrt[b]*c*e*(b*e - a*f)^2*Sqrt[a + b*x^2]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))])
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]
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]
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]
Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)*Sqrt[(e_) + (f_.)*(x_) ^2]), x_Symbol] :> Simp[c*(Sqrt[e + f*x^2]/(a*e*Rt[d/c, 2]*Sqrt[c + d*x^2]* Sqrt[c*((e + f*x^2)/(e*(c + d*x^2)))]))*EllipticPi[1 - b*(c/(a*d)), ArcTan[ Rt[d/c, 2]*x], 1 - c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ [d/c]
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]
Time = 8.44 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.20
| method | result | size |
| default | \(\frac {\left (\sqrt {-\frac {b}{a}}\, b^{2} d e \,x^{3}-\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b d e +\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c e -\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c e +\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right ) a^{2} d f -\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right ) a b c f +\sqrt {-\frac {b}{a}}\, b^{2} c e x \right ) \sqrt {x^{2} d +c}\, \sqrt {b \,x^{2}+a}}{e \sqrt {-\frac {b}{a}}\, a \left (a d -b c \right ) \left (a f -b e \right ) \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right )}\) | \(412\) |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {\left (b d \,x^{2}+b c \right ) b x}{a \left (a d -b c \right ) \left (a f -b e \right ) \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}-\frac {\sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right ) b}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, a \left (a f -b e \right )}-\frac {b^{2} c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{a \left (a d -b c \right ) \left (a f -b e \right ) \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {f \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right )}{\left (a f -b e \right ) e \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(419\) |
Input:
int(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x,method=_RETURNVERBOSE)
Output:
((-b/a)^(1/2)*b^2*d*e*x^3-((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*Elliptic F(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a*b*d*e+((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*e-((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*e+ ((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*d*f-((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*f+ (-b/a)^(1/2)*b^2*c*e*x)*(d*x^2+c)^(1/2)*(b*x^2+a)^(1/2)/e/(-b/a)^(1/2)/a/( a*d-b*c)/(a*f-b*e)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:
integrate(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algorithm="fricas ")
Output:
Timed out
\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int \frac {1}{\left (a + b x^{2}\right )^{\frac {3}{2}} \sqrt {c + d x^{2}} \left (e + f x^{2}\right )}\, dx \] Input:
integrate(1/(b*x**2+a)**(3/2)/(d*x**2+c)**(1/2)/(f*x**2+e),x)
Output:
Integral(1/((a + b*x**2)**(3/2)*sqrt(c + d*x**2)*(e + f*x**2)), x)
\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}} \,d x } \] Input:
integrate(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algorithm="maxima ")
Output:
integrate(1/((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*(f*x^2 + e)), x)
\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}} \,d x } \] Input:
integrate(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x, algorithm="giac")
Output:
integrate(1/((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*(f*x^2 + e)), x)
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{3/2}\,\sqrt {d\,x^2+c}\,\left (f\,x^2+e\right )} \,d x \] Input:
int(1/((a + b*x^2)^(3/2)*(c + d*x^2)^(1/2)*(e + f*x^2)),x)
Output:
int(1/((a + b*x^2)^(3/2)*(c + d*x^2)^(1/2)*(e + f*x^2)), x)
\[ \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} \left (e+f x^2\right )} \, dx=\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b^{2} d f \,x^{8}+2 a b d f \,x^{6}+b^{2} c f \,x^{6}+b^{2} d e \,x^{6}+a^{2} d f \,x^{4}+2 a b c f \,x^{4}+2 a b d e \,x^{4}+b^{2} c e \,x^{4}+a^{2} c f \,x^{2}+a^{2} d e \,x^{2}+2 a b c e \,x^{2}+a^{2} c e}d x \] Input:
int(1/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)/(f*x^2+e),x)
Output:
int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a**2*c*e + a**2*c*f*x**2 + a**2*d *e*x**2 + a**2*d*f*x**4 + 2*a*b*c*e*x**2 + 2*a*b*c*f*x**4 + 2*a*b*d*e*x**4 + 2*a*b*d*f*x**6 + b**2*c*e*x**4 + b**2*c*f*x**6 + b**2*d*e*x**6 + b**2*d *f*x**8),x)